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[**Weighted $2$-Motzkin Paths**]{}\
William Y.C. Chen$^1$, Sherry H.F. Yan$^2$ and Laura L.M. Yang$^3$\
Center for Combinatorics, LPMC\
Nankai University, Tianjin 300071, P. R. China\
$^1$chen@nankai.edu.cn, $^2$huifangyan@eyou.com, $^3$yanglm@hotmail.com
[**Abstract.** ]{} This paper is motivated by two problems recently proposed by Coker on combinatorial identities related to the Narayana polynomials and the Catalan numbers. We find that a bijection of Chen, Deutsch and Elizalde can be used to provide combinatorial interpretations of the identities of Coker when it is applied to weighted plane trees. For the sake of presentation of our combinatorial correspondences, we provide a description of the bijection of Chen, Deutsch and Elizalde in a slightly different manner in the form of a direct construction from plane trees to $2$-Motzkin paths without the intermediate step involving the Dyck paths.
[**AMS Classification:**]{} 05A15, 05A19
[**Keywords:**]{} Plane tree, Narayana number, Catalan number, $2$-Motzkin path, weighted $2$-Motzkin path, multiple Dyck path, bijection.
[**Suggested Running Title:**]{} Weighted $2$-Motzkin Paths
[**Corresponding Author:**]{} William Y. C. Chen, Email: chen@nankai.edu.cn
Introduction
============
The structure of $2$-Motzkin paths, introduced by Barcucci, Lungo, Pergola and Pinzani [@blpp], has proved to be highly efficient in the study of plane trees, Dyck paths, Motzkin paths, noncrossing partitions, RNA secondary structures, Devenport-Schinzel sequences, and combinatorial identities, see [@ds; @Klazar; @SW]. While it is most natural to establish a correspondence between Dyck paths of length $2n$ and $2$-Motzkin paths of length $n-1$, Deutsch and Shapiro came to the realization that direct correspondences between plane trees and $2$-Motzkin paths can have many applications. Recently, Chen, Deutsch and Elizalde [@cde] found bijections between plane trees and $2$-Motzkin paths for the enumeration of plane trees by the numbers of old and young leaves [@cde]. The main result of this paper is to show that the bijection of Chen, Deutsch and Elizalde, presented in a slightly different manner, can be applied to weighted plane trees in order to give combinatorial interpretations of two identities involving the Narayana numbers and Catalan numbers due to Coker [@coker]. This leads to the solutions of the two open problems left in the paper [@coker].
Recall that a [*$2$-Motzkin path*]{} is a lattice path starting at $(0,0)$ and ending at $(n,0)$ but never going below the $x$-axis, with possible steps $(1,1)$, $(1,0)$ and $(1,-1)$, where the level steps $(1,0)$ can be either of two kinds: straight and wavy. The [*length*]{} of the path is defined to the number of its steps. Deutsch and Shapiro [@ds] presented a bijection between plane trees with $n$ edges and $2$-Motzkin paths of length $n-1$. So the number of $2$-Motzkin paths of length $n-1$ equals the Catalan number $$C_n = {1 \over n+1} \, {2n \choose n}.$$
Recently, Coker [@coker] established very interesting combinatorial identities ((\[q1\]) and (\[q2\]) below) by using generating functions and the Lagrange inversion formula based the study of multiple Dyck paths. A multiple Dyck path is a lattice path starting at $(0,0)$ and ending at $(2n, 0)$ with big steps that can be regarded as segments of consecutive up steps or consecutive down steps in an ordinary Dyck path. Note that the notion of multiple Dyck path is formulated by Coker in different coordinates. The main ingredients in Coker’s identities are the Catalan number and the Narayana numbers: $$N(n,k)=\frac{1}{n}{n \choose k}{n \choose k-1},$$ which counts the number of all plane trees with $n$ edges and $k$ leaves [@SW; @stanley2]. It is sequence $A001263$ in [@sloane]. Coker [@coker] left the following two open problems:
[**Problem 1.**]{} Find a bijective proof of the following identity $$\label{q1}
\sum_{k=1}^n\frac{1}{n}{n \choose k}{n \choose
k-1}4^{n-k}=\sum_{k=0}^{\lfloor{(n-1)}/{2}\rfloor}C_k{n-1 \choose
2k}4^k5^{n-2k-1}.$$ [**Problem 2.**]{} Find a combinatorial explanation for the following identity $$\label{q2}
\sum_{k=1}^n\frac{1}{n}{n \choose k}{n \choose
k-1}x^{2k}(1+x)^{2n-2k}=x^2\sum_{k=0}^{n-1}C_{k+1}{n-1 \choose
k}x^k(1+x)^{k},$$ which is equation (6.2) in [@coker]. The above identity (\[q1\]) is a special case of the following identity: $$\label{q1x}
\sum_{k=0}^n \, {1\over n} \, {n\choose k} {n \choose k-1} t^{n-k}
= \sum_{k=0}^{\lfloor (n-1)/2\rfloor}\, C_k\, {n-1\choose 2k}\,t^k
(1+t)^{n-2k-1},$$ where the left hand side of (\[q1x\]) is the Narayana polynomial, as denoted by ${\cal N}_n(t)$ in [@coker]. The identity (\[q1x\]) is the relation (4.4) in [@coker], which can be derived as an identity on the Narayana numbers and the Catalan numbers due to Simion and Ullman [@SU], see also [@CDD]. Remarkably, (\[q1x\]) has many consequences as pointed by Coker [@coker]. For example, it implies the classical identity of Touchard [@T], and the formula on the little Schröder numbers in terms of the Catalan numbers [@G]. The reason for the evaluation of ${\cal N}_n(t)$ at $t=4$ lies in the fact that ${\cal N}_n(4)$ equals the number $d_n$ of multiple Dyck paths of length $2n$. The first few values of $d_n$ for $n=0, 1,2,3,4, 5,6,7$ are $$1,\;1,\; 5,\; 29,\;
185,\; 1257,\; 8925,\; 65445,$$ which is the sequence $A059231$ in [@sloane]. From the interpretation of Narayana numbers in terms of Dyck paths of length $2n$ and of $k$ peaks, it is not difficult to show that $d_n={\cal N}_n(4)$. However, the right hand side of (\[q1\]) does not seems to be obvious, which is obtained by establishing a functional equation and by using the Lagrange inversion formula. The natural question as raised by Coker [@coker] is to find a combinatorial interpretation of (\[q1\]). Note that the enumeration of multiple Dyck paths has also been studied independently by Sulanke [@Sulanke] and Woan [@Woan].
The relation (\[q2\]) was established from the enumeration of multiple Dyck paths of length $2n$ with a given number of steps. Let $\lambda_{n,j}$ be the number of multiple Dyck paths of length $2n$ and $j$ steps, ${\cal P}_n(x)$ be the polynomial $${\cal P}_n(x)= \sum_{j=2}^{2n} \, \lambda_{n,j}\, x^j.$$ It was shown that $${\cal P}_n(x) = \sum_{k=1}^n \, {1 \over n} {n\choose k} {n
\choose k-1} x^{2k} (1+x)^{2n-2k},$$ which can be restated as $${\cal P}_n(x) = x^{2n} {\cal N}_n((1+x^{-1})^2).$$ Coker [@coker] discovered the connection between ${\cal
P}_n(x)$ and the polynomial ${\cal R}_n(x)$ introduced by Denise and Simion [@denise-s] in their study of the number of exterior pairs of Dyck paths of length $2n$. The polynomials ${\cal R}_n(x)$ have the following expansion: $${\cal R}_n(x) = \sum_{k=0}^{n-1} \, (-1)^k C_{k+1}{n-1 \choose
k}\, x^k (1-x)^k.$$ It now becomes clear that the identity (\[q2\]) can be rewritten as $$\label{PR}
{\cal P}_n(x) =x^2 {\cal R}_n(-x).$$
To give combinatorial interpretations of both (\[q1\]) and (\[q2\]), we apply the bijection of Chen, Deutsch and Elizalde [@cde] to weighted plane trees to get weighted $2$-Motzkin paths. Then we use weight-preserving operations on $2$-Motzkin paths to derive the desired combinatorial identities. More precisely, these weight-preserving operations are essentially the reductions from weighted $2$-Motzkin paths to Dyck paths and $2$-Motzkin paths. It would be interesting to find a direct correspondence on Dyck paths which leads to a combinatorial interpretation of (\[PR\]).
Weighted $2$-Motzkin Paths
==========================
Let us review a bijection between plane trees and $2$-Motzkin paths due to Chen, Deutsch and Elizalde [@cde], which is devised for the enumeration of plane trees with $n$ edges and a fixed number of old leaves and a fixed number of young leaves. Such a consideration of old and young leaves reflects to the four types of steps of $2$-Motzkin paths. For the purpose of this paper, we present a slightly modified version of the nonrecursive bijection in [@cde]. Our terminology is also somewhat different.
For a plane tree $T$, a vertex of $v$ is called a leaf if it does not have any children. An internal vertex is a vertex that has at least one child. An edge is denoted as a pair $(u, v)$ of vertices such that $v$ is a child of $u$. Let $u$ be an internal vertex, and $v_1$, $v_2$, $\ldots$, $v_k$ be the children of $u$ listed from left to right. Then we call $v_k$ an [*exterior vertex*]{} and $(u, v_k)$ an [*exterior edge*]{}. If $k>1$, then the edges $(u, v_1)$, $(u, v_2)$ $\ldots$, and $(u,
v_{k-1})$ are called [*interior edges*]{} and $v_1, v_2, \ldots,
v_{k-1}$ are called [*interior vertices*]{}. An edge containing a leaf vertex is called a [*terminal edge*]{}. Let $u$ be the root of $T$, $(u, u_1)$ be the exterior edge of $u$, $(u_1, u_2)$ be the exterior edge of $u_1$, and so on, finally $(u_{k-1}, u_{k})$ be the exterior edge of $u_{k-1}$ such that $u_k$ is a leaf. The exterior edge $(u_{k-1}, u_k)$ is called the [*critical edge*]{} of $T$. To summarize, the edges of a plane tree $T$ are classified into five categories.
- Non-terminal interior edges.
- Non-terminal exterior edges.
- Terminal interior edges.
- Terminal exterior edges (which do not include the critical edge).
- The critical edge.
Note that the [*critical edge*]{} of $T$ is the last encountered edge when we traverse the edges of $T$ in preorder. From the above classification on the edges of a plane tree, it is easy to describe the Chen-Deutsch-Elizalde bijection between plane trees and $2$-Motzkin paths by the preorder traversal of the edges of $T$. To be precise, let $u$ be the root of $T$, $v_1, v_2,
\ldots, v_k$ be the children of $u$, and $T_1, T_2, \ldots, T_k$ be the subtrees of $T$ rooted at $v_1, v_2, \ldots, v_k$, respectively. Then the preorder traversal of the edges of $T$, denoted by $P(T)$, is a linear order of the edges of $T$ recursively defined by $$(u, v_1) \, P(T_1) \, (u, v_2) \, P(T_2) \, \cdots \, (u, v_k)
\, P(T_k).$$
[**The Bijection of Chen, Deutsch and Elizalde [@cde]:**]{} Let $T$ be any nonempty plane tree. At each step of the traversal of the edges of $T$ in preorder,
\(i) draw an up step for a non-terminal interior edge;
\(ii) draw a straight level step for a non-terminal exterior edge;
\(iii) draw a wavy level step for a terminal interior edge;
\(iv) draw a down step for a terminal exterior edge;
\(v) do nothing for the critical edge.
(400,60) (5,15)[(1,1)[10]{}]{} (15,25) (15,25)[(-1,-2)[5]{}]{} (5,15) (10,15) (5,10)[(0,1)[5]{}]{} (5,10)[(1,-2)[2.5]{}]{} (5,10)[(-1,-2)[2.5]{}]{} (5,10)[(0,-1)[5]{}]{} (5,10) (7.5,5) (2.5,0) (5,5) (5,5)[(-1,-2)[2.5]{}]{} (5,5)[(1,-2)[2.5]{}]{} (2.5,5) (7.5,0) (15,25)[(0,-1)[10]{}]{} (15,15)[(1,-2)[2.5]{}]{} (15,15)[(-1,-2)[2.5]{}]{} (15,15) (17.5,10) (12.5,10)
(17.5,10)[(-1,-2)[2.5]{}]{} (17.5,10)[(1,-2)[2.5]{}]{} (20,5)[(0,-1)[5]{}]{} (15,5) (20,5) (20,0) (15,25) (15,25)[(1,-1)[10]{}]{} (25,15) (25,15)[(2,-1)[10]{}]{} (35,10) (25,15)[(1,-1)[5]{}]{} (30,10) (25,15)[(0,-1)[5]{}]{} (25,10) (35,10)[(0,-1)[5]{}]{} (35,5) (30,10)[(0,-1)[5]{}]{} (30,5) (45,10)[(1,0)[6]{}]{} (51,9)[(-1,0)[6]{}]{} (60,5) (60,5)[(1,1)[4]{}]{}
(64,9) (64,9)[(1,0)[4]{}]{}
(68,9) (68,9)(1,0)[4]{}[(1,1)[0.5]{}]{} (68.5,9.5)(1,0)[4]{}[(1,-1)[0.5]{}]{}
(72,9) (72,9)[(1,1)[4]{}]{}
(76,13) (76,13)(1,0)[4]{}[(1,1)[0.5]{}]{} (76.5,13.5)(1,0)[4]{}[(1,-1)[0.5]{}]{}
(80,13) (80,13)[(1,-1)[4]{}]{}
(84,9) (84,9)[(1,-1)[4]{}]{}
(88,5) (88,5)(1,0)[4]{}[(1,1)[0.5]{}]{} (88.5,5.5)(1,0)[4]{}[(1,-1)[0.5]{}]{}
(92,5) (92,5)[(1,1)[4]{}]{}
(96,9) (96,9)(1,0)[4]{}[(1,1)[0.5]{}]{} (96.5,9.5)(1,0)[4]{}[(1,-1)[0.5]{}]{}
(100,9) (100,9)[(1,0)[4]{}]{}
(104,9) (104,9)(1,0)[4]{}[(1,1)[0.5]{}]{} (104.5,9.5)(1,0)[4]{}[(1,-1)[0.5]{}]{}
(108,9) (108,9)[(1,0)[4]{}]{}
(112,9) (112,9)[(1,-1)[4]{}]{}
(116,5) (116,5)[(1,0)[4]{}]{}
(120,5) (120,5)(1,0)[4]{}[(1,1)[0.5]{}]{} (120.5,5.5)(1,0)[4]{}[(1,-1)[0.5]{}]{}
(124,5) (124,5)[(1,1)[4]{}]{}
(128,9) (128,9)[(1,-1)[4]{}]{}
(132,5) (132,5)[(1,0)[4]{}]{} (136,5)
It is easy to see that we have obtained a $2$-Moztkin path. More precisely, a plane tree with $n$ edges corresponds to a $2$-Motzkin path of length $n-1$. The above bijection is denoted by $\Phi$. As a hint to why the above bijection works, one may check that for any plane tree $T$,
$\#$ non-terminal interior edges $=$ $\#$ terminal exterior edges.
We are now ready to assign weights to the edges of a plane tree $T$ in order to obtain combinatorial interpretations of the identities (\[q1\]) and (\[q2\]). The weights of the edges of a plane tree will translate into weights of steps of the corresponding $2$-Motzkin path. In fact, we will take slightly different formulations of (\[q1x\]) and (\[q2\]).
For $n \geq 1$, we have $$\label{identity1}
\sum_{k=1}^n\frac{1}{n}{n \choose k}{n \choose
k-1}x^{k-1}=\sum_{k=0}^{\lfloor{(n-1)}/{2}\rfloor}C_k{n-1 \choose
2k}x^k(1+x)^{n-2k-1}.$$
Let $T$ be a plane tree with $n$ edges. We assign the weights to the edges of $T$ by the following rule: All the terminal edges except the critical edge are given the weight $x$ and all other edges are given the weight $1$. The weight of $T$ is the product of the weights of its edges. Then the left hand side of (\[identity1\]) is the sum of the weights of all plane trees with $n$ edges.
By the above bijection $\Phi$, the set of weighted plane trees with $n$ edges is mapped to the set of $2$-Motzkin paths of length $n-1$ in which all the down steps and wavy level steps are given the weight $x$, and other steps are given the weight $1$. Consider the weighted $2$-Motzkin paths of length $n-1$ that have $k$ up steps and $k$ down steps. These $k$ up steps and $k$ down steps form a Dyck path of length $2k$. The binomial coefficient ${n-1
\choose 2k}$ comes from the choices of the $2k$ positions for these $k$ up steps and $k$ down steps. The remaining $n-2k-1$ steps are either wavy level steps or straight level steps. Since only a wavy level step carries the weight $x$, the total contributions of the weights of $n-2k-1$ level steps amount to $(1+x)^{n-2k-1}$. The $k$ up steps would contribute $x^k$. Therefore, the right hand side of (\[identity1\]) equals the total contributions of all the weighted $2$-Motzkin paths of length $n-1$, as desired.
For $n \geq 1$, we have $$\label{identity2}
\sum_{k=1}^n\frac{1}{n}{n \choose k}{n \choose
k-1}x^{2(k-1)}(1+x)^{2(n-k)}=\sum_{k=0}^{n-1}C_{k+1}{n-1 \choose
k}x^k(1+x)^{k}.$$
Given a plane tree $T$ with $n$ edges, we assign the weights to the edges of $T$ by the following rule: All the terminal edges except the critical edge are assigned the weight $x^2$, all the non-terminal edges are given the weight $(1+x)^2$, and the critical edge is assigned the weight $1$. Then the bijection $\Phi$ transform $T$ into a $2$-Motzkin path in which all the down steps and wavy level steps have the weight $x^2$ and all the up steps and straight level steps have the weight $(1+x)^2$.
By the above weight assignment, the left hand side of (\[identity2\]) is then the sum of the weights of all plane trees with $n$ edges. We now proceed to show that the right hand side of (\[identity2\]) is the sum of weights of all $2$-Motzkin paths of length $n-1$. Consider a $2$-Motzkin paths that has $k$ up steps and $k$ down steps. Since the up steps have weight $x^2$ and the down steps have weight $(1+x)^2$, it makes no difference with respect to the sum of weights if one changes the weights of both up steps and down steps to $x(1+x)$. In other words, such an operation on the change of weights is a weight-preserving bijection on the set of $2$-Motzkin paths.
Note that for any weight assignment, we may transform the sum of weights of all $2$-Motzkin paths of length $n-1$ to the sum of weights of all Motzkin paths of the same length by the following weight assignment: the up steps and down steps carry the same weight, and the horizontal steps in the Motzkin paths carry the weight as the sum of the weights of a straight level step and a wavy level step in the $2$-Motzkin path. Therefore, for our weight assignment the sum of weights of $2$-Motzkin paths of length $n-1$ equals the sum of Motzkin paths of length $n-1$ given the following weight assignment: up steps and down steps are given the weight $x(1+x)$, and the horizontal steps are given the weight $$x^2+(1+x)^2= 1 + x(1+x) + x(1+x).$$ So we have transformed the sum of weights of $2$-Motzkin paths of length $n-1$ to the sum of weights of all Motzkin paths of length $n-1$ in which all the up steps, down steps have weights $x(1+x)$, and the horizontal steps can be regarded as either a straight level step with weight $x(1+x)$, or a wavy level step with weight $x(1+x)$ or a special dotted step with weight $1$.
We now get the desired sum as on the right hand side of (\[identity2\]) by considering the distribution of the special dotted steps, because the remaining steps (up, down, straight level, wavy level) all have the weight $x(1+x)$ and they form a $2$-Motzkin path.
Setting $x={1/4}$ in (\[identity1\]) we obtain (\[q1\]).
[**Acknowledgments.**]{} This work was done under the auspices of the National Science Foundation, the Ministry of Education, and the Ministry of Science and Technology of China.
[99]{} E. Barcucci, A.D. Lungo, E. Pergola and R. Pinzani, A construction for enumerating $k$-coloured Motzkin paths, Lecture Notes in Computer Science, Vol. 959, Springer, Berlin, 1995, 254–263.
W.Y.C. Chen, E. Deutsch and S. Elizalde, Old and young leaves on plane trees and $2$-Motzkin paths, preprint, 2004.
W.Y.C. Chen, E.Y.P. Deng and R.R.X. Du, Reduction of $m$-regular noncrossing partitions, Europ. J. Combin., to appear.
C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003) 13–28.
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137 (1995) 155–176.
E. Deutsch and L.W. Shapiro, A bijection between ordered trees and $2$-Motzkin paths and its many consequences, Discrete Math., 256 (2002) 655–670.
D. Gouyou-Beauchamps and B. Vauquelin, Deux propriétés combinatoires des nombres de Schröder, Theor. Inform. Appl., 22 (1988), 361-388.
M. Klazar, On numbers of Davenport-Schinzel sequences, Discrete Math., 185 (1998) 77–87.
R. Simion and D. Ullman, On the structure of the lattice of noncrossing partitions, Discrete Math., 98 (1991) 193–206.
N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/$\thicksim$njas/sequences.
R. Sulanke, Counting lattice paths by Narayana polynomials, Electron. J. Combin., 7 (2000) \#R40.
W.R. Schmitt and M.S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51 (1994) 317–323.
R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, UK, 1999.
J. Touchard, Sur certaines équations fonctionnelles, In: Proc. International Congress on Mathematics, Uninversity of Toronto Press, Toronto, 1928, 465–472.
W.J. Woan, Diagonal lattice paths, Proc. 32nd Southeastern International Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, LA, 2001, Congr. Numer., 151 (2001) 173-178.
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abstract: 'We update and compare the capabilities of the purely leptonic mode $B\to\tau\nu$ and the semileptonic mode $B\to D\tau\nu$ in the search for a charged Higgs boson.'
author:
- Stéphanie Trine
title: 'Charged-Higgs effects in $B\rightarrow(D)\tau\nu$ decays'
---
INTRODUCTION
============
Supersymmetric extensions of the standard model (SM) – or more generally extensions that require the existence of at least one additional Higgs doublet – generate new flavour-changing interactions already at tree-level via the exchange of a charged Higgs boson. The coupling of $H^{+}$ to fermions grows with the fermion mass. It is thus natural to look at (semi)leptonic $B$ decays with a $\tau$ in the final state to try to uncover this type of effects. In a two-Higgs-doublet-model (2HDM) of type II, where up-type quarks get their mass from one of the two Higgs doublets and down-type quarks from the other one, $H^{+}$ effects are entirely parametrized by the $H^{+}$ mass, $M_H$, and the ratio of the two Higgs vacuum expectation values, $\tan\beta=v_{u}/v_{d}$. They can compete with the exchange of a $W^+$ boson for large values of $\tan\beta$ [@Hou93]. In the minimal supersymmetric extension of the SM (MSSM), the tree-level type-II structure is spoilt by radiative corrections involving supersymmetry-breaking terms. The effective scalar coupling $g_{S}$ then exhibits an additional dependence on sparticle mass parameters when $\tan\beta$ is large $(q=u,c)$ [@AkeroydR03; @Pheno]: $$H_{eff}^{H^{+}}=-2\sqrt{2}G_{F}V_{qb}\frac{m_{b}m_{\tau}}{M_{B}
^{2}}g_{S}\left[ \overline{q}_{L}b_{R}\right] \left[ \overline{\tau}_{R}
\nu_{L}\right] +h.c.,\qquad g_{S}=\frac{M_{B}^{2}\tan^{2}\beta}{M_{H}^{2}
}\frac{1}{(1+\varepsilon_{0}\tan\beta)(1+\varepsilon_{\tau}\tan\beta)},
\label{Eq1}$$ where $\varepsilon_{0,\tau}$ denote sparticle loop factors. The correction induced can be of order one. However, the access to the Higgs sector remains exceptionally clean. In Eq.(\[Eq1\]), $g_{S}$ has been normalized such that it gives the fraction of effects in the $B\rightarrow\tau\nu$ amplitude, which is very sensitive to $H^{+}$ exchange: $\mathcal{B}(B\rightarrow\tau\nu)/\mathcal{B}(B\rightarrow\tau\nu)^{SM}=|1-g_{S}|^{2}$. The $B\rightarrow D\tau\nu$ channel is less sensitive (though better in this respect than other modes such as $B\to D^*\tau\nu$) but, as we will see, exhibits a number of features that make it, too, play an important part in the hunt for the charged Higgs boson.
$\mathcal{B}(B\rightarrow D\tau\nu)$ VERSUS $\mathcal{B}(B\rightarrow\tau\nu)$
==============================================================================
The current capabilities of $\mathcal{B}(B\rightarrow D\tau\nu)$ and $\mathcal{B}(B\rightarrow\tau\nu)$ to constrain $H^{+}$ effects are compared in Fig.\[Fig1\] for $g_S\geq0$ (as is typically the case in the MSSM or the 2HDM-II). The lower sensitivity of the $B\rightarrow D\tau\nu$ mode comes from the different momentum dependence of the Higgs contribution with respect to the longitudinal $W^+$ one[^1]: $(d\Gamma(B\to D\tau\nu)/dq^2)^{W_{\parallel}^{+}+H^{+}} \propto |1-g_S (q^2/M_B^2)/(1-m_c/m_b)|^2$ with $q\equiv p_B-p_D$. On the other hand, the theory prediction for $\mathcal{B}(B\rightarrow\tau\nu)$ suffers from large parametric uncertainties from the CKM matrix element $V_{ub}$ and the $B$ decay constant $f_B$. In contrast, $V_{cb}$ is known with better than $2\%$ accuracy from inclusive $B\to X_c\ell\nu\ (\ell=e,\mu)$ decays, $|V_{cb}|=(41.6\pm0.6)\times10^{-3}$ [@PDG08], and the form factors $f_+(q^2)$ and $f_0(q^2)$ describing the $B \to D$ transition are very well under control, as we now discuss in more detail.
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![Left: $\mathcal{B}(B\rightarrow\tau\nu)$ as a function of $g_{S}$. Light gray band: $\mathcal{B}^\textrm{exp}=(1.51\pm0.33)\times10^{-4}$ [@HFAG08]. Gray (blue) band: $\mathcal{B}^\textrm{th}$ for $|V_{ub}|=(3.95\pm0.35)\times10^{-3}$ [@PDG08] and $f_B=216\pm38{\,\mbox{MeV}}$ [@HPQCD05] (in this last case, we add errors linearly to stay on the conservative side). Right: $R\equiv\mathcal{B}({B \to D \tau \nu})/\mathcal{B}({B \to D \ell \nu})$ as a function of $g_S$. Light gray band: $R^\textrm{exp}=(41.6\pm11.7\pm5.2)\%$ [@BABAR08_BDTauNu]. Dark gray (dark blue) band: $R^\textrm{th}$ with the HFAG vector form factor before ICHEP08 [@HFAG07]. Gray (blue) band: $R^\textrm{th}$ with the HFAG vector form factor after ICHEP08 [@HFAG08]. The dashed lines indicate the $2$ and $3$-sigma limits. The ratio $m_c/m_b$ in the $\overline{\textrm{MS}}$ scheme has recently been determined with very high accuracy: $m_c/m_b=0.2211\pm0.0044$ [@mcmb]. We inflated the error on this number and set $m_c/m_b=0.22\pm0.01$ to reduce the discrepancy with the HFAG estimation [@HFAG08].[]{data-label="Fig1"}](PlotBTauNu2.eps "fig:"){width="0.43\linewidth"} ![Left: $\mathcal{B}(B\rightarrow\tau\nu)$ as a function of $g_{S}$. Light gray band: $\mathcal{B}^\textrm{exp}=(1.51\pm0.33)\times10^{-4}$ [@HFAG08]. Gray (blue) band: $\mathcal{B}^\textrm{th}$ for $|V_{ub}|=(3.95\pm0.35)\times10^{-3}$ [@PDG08] and $f_B=216\pm38{\,\mbox{MeV}}$ [@HPQCD05] (in this last case, we add errors linearly to stay on the conservative side). Right: $R\equiv\mathcal{B}({B \to D \tau \nu})/\mathcal{B}({B \to D \ell \nu})$ as a function of $g_S$. Light gray band: $R^\textrm{exp}=(41.6\pm11.7\pm5.2)\%$ [@BABAR08_BDTauNu]. Dark gray (dark blue) band: $R^\textrm{th}$ with the HFAG vector form factor before ICHEP08 [@HFAG07]. Gray (blue) band: $R^\textrm{th}$ with the HFAG vector form factor after ICHEP08 [@HFAG08]. The dashed lines indicate the $2$ and $3$-sigma limits. The ratio $m_c/m_b$ in the $\overline{\textrm{MS}}$ scheme has recently been determined with very high accuracy: $m_c/m_b=0.2211\pm0.0044$ [@mcmb]. We inflated the error on this number and set $m_c/m_b=0.22\pm0.01$ to reduce the discrepancy with the HFAG estimation [@HFAG08].[]{data-label="Fig1"}](PlotBDTauNu.eps "fig:"){width="0.43\linewidth"}
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To this end, we introduce the following conformal transformation: $$q^{2}\to z(q^{2},t_{0})\equiv
\frac{\sqrt{(M_{B}+M_{D})^{2}-q^{2}}-\sqrt{(M_{B}+M_{D})^{2}-t_{0}}}{\sqrt{(M_{B}+M_{D})^{2}-q^{2}}+\sqrt{(M_{B}+M_{D})^{2}-t_{0}}},
\label{Eq2}$$ which maps the complex $q^2$ plane, cut along $q^2\geq(M_B+M_D)^2$, onto the disk $|z|<1$. The form factors $f_+$ and $f_0$ are analytic in $z$ in this domain, up to a few subthreshold poles, and can thus be written as a power series in $z$ after these poles are factored out ($i=+,0$) [@zpara]: $$f_{i}(q^{2})=\frac{1}{P_{i}(q^{2})\phi_{i}(q^{2},t_{0})}\left[ a_{0}^{i}(t_{0})+a_{1}^{i}(t_{0})z(q^{2},t_{0})+...\right],
\label{Eq3}$$ where the function $P_{i}$ gathers the pole singularities and an arbitrary analytic function $\phi_{i}$ can be factored out as well.
This parametrization has been used in Ref.[@CLN98] with the choice $t_0=q^2_\textrm{max}=(M_{B}-M_{D})^{2}$, together with heavy-quark spin symmetry inputs, to derive the following ansatz for the vector form factor: $$\begin{aligned}
f_{+}(q^2)\equiv\frac{M_{B}+M_{D}}{2\sqrt{M_{D}M_{B}}}V_{1}(q^2),\quad
V_{1}(q^2)=\mathcal{G}(1)\left[ 1-8\rho^{2}z(q^2,t_0)+(51\rho^{2}-10)z(q^2,t_0)^{2}-(252\rho^{2}-84)z(q^2,t_0)^{3}\right],
\label{Eq4}\end{aligned}$$ where $V_1$ is defined such that it reduces to the Isgur-Wise function in the heavy-quark limit and $\mathcal{G}(1)\equiv V_{1}(q^2_\textrm{max})$. The parameters $|V_{cb}|\mathcal{G}(1)$ and $\rho^2$ can be determined from ${B \to D \ell \nu}$ experimental data. Before this summer, the HFAG averages [@HFAG07] based on BELLE, CLEO, and ALEPH data read: $|V_{cb}|\mathcal{G}(1)=(42.3\pm4.5)\times10^{-3}$ and $\rho^2=1.17\pm0.18$ (with a $|V_{cb}|\mathcal{G}(1)$-$\rho^2$ correlation of $0.93$). The recent BABAR results [@BABAR08_BDLNuT] and [@BABAR08_BDLNuU] have now been included, leading to a substantial improvement [@HFAG08]: $|V_{cb}|\mathcal{G}(1)=(42.4\pm0.7\pm1.4)\times10^{-3}$ and $\rho^2=1.19\pm0.04\pm0.04$ (with $|V_{cb}|\mathcal{G}(1)$-$\rho^2$ correlation $0.88$). The old and new vector form factors are compared in Fig.\[Fig2\] (left), where we have defined as usual $w=(M_{B}^{2}+M_{D}^{2}-q^2)/(2M_{B}M_{D})$.
For the scalar form factor, we adopt the ansatz of Ref.[@Hill06]: $$\begin{aligned}
f_{0}(q^2)\equiv\frac{(w+1)\sqrt{M_{D}M_{B}}}{M_{B}+M_{D}}S_{1}(q^2)
=\frac{1}{z(q^2,M_{1}^{2})z(q^2,M_{2}^{2})\phi_{0}(q^2,t_0)}\left[ a_{0}^{0}(t_0)+a_{1}^{0}(t_0)\,z(q^2,t_0)\right],
\label{Eq5}\end{aligned}$$ where $t_0=(M_B+M_D)^2\left( 1-\sqrt{1-(M_B-M_D)^2/(M_B+M_D)^2}\,\right)$ such that $|z|_\textrm{max}$ is minimized, $M_{1}=6.700{\,\mbox{GeV}}$ and $M_{2}=7.108{\,\mbox{GeV}}$ [@EichtenQ94] are the subthreshold poles, and $\phi_0$ is obtained from Eq.(10) of Ref.[@Hill06] setting $Q^{2}=0$ and $\eta=2$: $$\phi_0(q^2,t_0)=\sqrt{\frac{2(M_{B}^2-M_{D}^2)^{2}}{16\pi}}
\frac{\sqrt{(M_{B}+M_{D})^{2}-q^2}}{((M_{B}+M_{D})^{2}-t_0)^{1/4}}
\frac{z(q^2,0)^2}{(q^2)^2}
\left(\frac{z(q^2,t_0)}{t_0-q^2}\right)^{-1/2}
\left(\frac{z(q^2,(M_{B}-M_{D})^{2})}{(M_{B}-M_{D})^{2}-q^2}\right)^{-1/4}.$$ Following [@NTW08], we truncate the series (\[Eq3\]) after the first two terms. This is motivated by the fact that $|z|_\textrm{max}=0.032$ and that a similar parametrization for $f_+$, when fitted to experimental data, produces the same result as Eq.(\[Eq4\]) in very good approximation [@NTW08]. Then, $|V_{cb}|a^0_0(t_0)$ and $|V_{cb}|a^0_1(t_0)$ are determined imposing the conditions (i) $|V_{cb}|S_1(0)=|V_{cb}|V_1(0)$ and (ii) $|V_{cb}|S_1(q^2_\textrm{max})=(4.24\pm0.27)\%$ (corresponding to $|V_{cb}|=(41.6\pm0.6)\times10^{-3}$ from $B\to X_c\ell\nu$ [@PDG08] and $S_{1}(q^2_\textrm{max})=1.02\pm0.05$ from HQET [@NTW08]). The scalar form factors obtained in this way from the old and new $|V_{cb}|V_1$ are not very different, as one can see on Fig.\[Fig2\] (right).
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![Vector (left) and scalar (right) form factors corresponding to the $|V_{cb}|\mathcal{G}(1)$ and $\rho^2$ determinations of HFAG before (dark gray/dark blue)[@HFAG07] and after (gray/blue)[@HFAG08] ICHEP08. With the new determination, the errors on $|V_{cb}|V_1$ and $|V_{cb}|S_1$ are smaller than $4\%$ and $7\%$, respectively.[]{data-label="Fig2"}](PlotV1.eps "fig:"){width="0.43\linewidth"} ![Vector (left) and scalar (right) form factors corresponding to the $|V_{cb}|\mathcal{G}(1)$ and $\rho^2$ determinations of HFAG before (dark gray/dark blue)[@HFAG07] and after (gray/blue)[@HFAG08] ICHEP08. With the new determination, the errors on $|V_{cb}|V_1$ and $|V_{cb}|S_1$ are smaller than $4\%$ and $7\%$, respectively.[]{data-label="Fig2"}](PlotS1.eps "fig:"){width="0.43\linewidth"}
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The recent progress on $|V_{cb}|V_1$ allows to reduce the errors on the SM predictions for the two ${B \to D \tau \nu}$ branching fractions: $\mathcal{B}(B^-\rightarrow D^0\tau^-\bar\nu)^{SM}=\left(0.70^{+0.06}_{-0.05}\right)\%$, $\mathcal{B}(\bar B^0\rightarrow D^+\tau^-\bar\nu)^{SM}=\left(0.65^{+0.06}_{-0.05}\right)\%$ (differing essentially due to $\tau_{B^0}\not=\tau_{B^+}$). The errors from $|V_{cb}|V_1$, however, already cancel to a large extent in the ratio $R\equiv\mathcal{B}({B \to D \tau \nu})/\mathcal{B}({B \to D \ell \nu})$, which is why the nice improvement in Fig.\[Fig2\] has little impact on Fig.\[Fig1\] (right), already dominated by the error on $S_{1}(q^2_\textrm{max})$: $R^{SM}=0.31\pm0.02$. This estimation is compatible with the one obtained from lattice methods: $R^{SM}_{latt}=0.28\pm0.02$ [@KamenikM08]. Note that replacing condition (ii) by a constraint on $S_1(q^2_\textrm{max})/V_1(q^2_\textrm{max})$ from HQET would lead to a similar error on $R$. Still, an interesting $95\%$ C.L. bound on $g_S$ can already be obtained from $R$: $g_S<1.79$, complementary to the bounds from $\mathcal{B}(B\rightarrow\tau\nu)$: $g_S<0.36\ \cup\ 1.64<g_S<2.73$. The corresponding exclusion zones in the $(M_{H},\tan\beta)$ plane are depicted in Fig.\[Fig3\]. The error assigned to $S_{1}(q^2_\textrm{max})$ is quite conservative, so the above constraints are robust. At the three-sigma level, it is not possible to extract any interesting bound from $R$ yet, but its experimental knowledge is expected to improve in the near future. Its role to constrain $H^+$ effects will then of course depend on the new central value. For the moment, a $15\%$ measurement with the same central value would exclude $g_S>0.29$ at the $95\%$ C.L..
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![$95\%$ C.L. exclusion zones in the $(M_H,\tan\beta)$ plane from $\mathcal{B}(B\rightarrow\tau\nu)$ (dark gray/dark blue) and $R$ (gray/blue) in a 2HDM (left) and in the MSSM with $\varepsilon_{0}=0.01$ and $\varepsilon_{\tau}\simeq0$ (right). The exclusion limits are directly read from the gray (blue) bands in Fig.\[Fig1\]. They differ from those usually found in the literature in that experimental and theory errors are not simply added in quadrature and the dependence of the errors on the $H^+$ contribution is taken into account.[]{data-label="Fig3"}](MHtanB2HDM.eps "fig:"){width="0.43\linewidth"} ![$95\%$ C.L. exclusion zones in the $(M_H,\tan\beta)$ plane from $\mathcal{B}(B\rightarrow\tau\nu)$ (dark gray/dark blue) and $R$ (gray/blue) in a 2HDM (left) and in the MSSM with $\varepsilon_{0}=0.01$ and $\varepsilon_{\tau}\simeq0$ (right). The exclusion limits are directly read from the gray (blue) bands in Fig.\[Fig1\]. They differ from those usually found in the literature in that experimental and theory errors are not simply added in quadrature and the dependence of the errors on the $H^+$ contribution is taken into account.[]{data-label="Fig3"}](MHtanBMSSM.eps "fig:"){width="0.43\linewidth"}
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$B\rightarrow D\tau\nu$ DIFFERENTIAL DISTRIBUTIONS
==================================================
If a hint for a charged Higgs boson is seen at the branching fraction level, ${B \to D \tau \nu}$ has a great advantage over $B\to\tau\nu$: it allows to analyze the same data points on a differential basis, better suited to discriminate between effective scalar-type interactions and other effects. The $d\Gamma({B \to D \tau \nu})/dq^2$ distribution, in particular, has already been studied in great detail [@distr]. The polarization of the $\tau$ is also known as a $H^{+}$ analyzer [@pol], yet it requires the knowledge of the $\tau$ momentum, which cannot be accessed at $B$ factories as the $\tau$ does not travel far enough for a displaced vertex and decays into at least one more neutrino.
A straightforward way to nevertheless exploit the sensitivity of the $\tau$ polarization to $H^+$ effects and at the same time retain the information from the $q^2$ spectrum is to look at the subsequent decay of the $\tau$ into a pion and a neutrino [@NTW08]. The direction of the pion is indeed directly correlated with the polarization of the $\tau$. Integrating over the neutrino momenta, we end up with a triple differential decay distribution $d\Gamma(B\to D\nu\tau[\to\pi\nu])/dq^2 dE_\pi d\cos\theta_{D\pi}$. An explicit formula is given in Eqs.(9-11) of Ref.[@NTW08] (with $F_V\equiv f_+$ and $F_S\equiv f_0$). Its sensitivity to $g_S$ is illustrated in Fig.\[Fig4\] for $E_\pi=1.8{\,\mbox{GeV}}$ and $\cos\theta_{D\pi}=-1$. For comparison, we also display the $q^2$ spectra corresponding to the same $g_S$ values. Of course, in practice, one should not fix $E_\pi$ or $\theta_{D\pi}$, but rather perform a (unbinned) maximum likelihood fit of the triple differential decay distribution to the available data points. The information from the $q^2$ spectrum in the dominant $\tau\to\ell\nu\bar\nu$ decay channel should also be included in the fit to make the most out of experimental data.
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![$d\Gamma({B \to D \tau \nu})/dw$ (left) and $d\Gamma(B\to D\nu\tau[\to\pi\nu])/dE_\pi d\cos\theta_{D\pi} dw$ with $E_\pi=1.8{\,\mbox{GeV}}$ and $\cos\theta_{D\pi}=-1$ (right) for $g_S=0$ (gray/red), $g_S=0.35$ (light gray/light blue), and $g_S=1.75$ (dark gray/dark blue). These values are still allowed by $\mathcal{B}(B\rightarrow D\tau\nu)$ and $\mathcal{B}(B\rightarrow\tau\nu)$ at the $95\%$ C.L.. The various curves have been obtained using the more recent HFAG vector form factor [@HFAG08]. The lighter bands take all errors into account, while the darker bands only take into account the error on $S_1(q^2_\textrm{max})$. One could of course also normalize the above differential distributions to $d\Gamma({B \to D \ell \nu})/dw$ to reduce the impact of the errors on $f_+$.[]{data-label="Fig4"}](DiffDistr.eps "fig:"){width="0.43\linewidth"} ![$d\Gamma({B \to D \tau \nu})/dw$ (left) and $d\Gamma(B\to D\nu\tau[\to\pi\nu])/dE_\pi d\cos\theta_{D\pi} dw$ with $E_\pi=1.8{\,\mbox{GeV}}$ and $\cos\theta_{D\pi}=-1$ (right) for $g_S=0$ (gray/red), $g_S=0.35$ (light gray/light blue), and $g_S=1.75$ (dark gray/dark blue). These values are still allowed by $\mathcal{B}(B\rightarrow D\tau\nu)$ and $\mathcal{B}(B\rightarrow\tau\nu)$ at the $95\%$ C.L.. The various curves have been obtained using the more recent HFAG vector form factor [@HFAG08]. The lighter bands take all errors into account, while the darker bands only take into account the error on $S_1(q^2_\textrm{max})$. One could of course also normalize the above differential distributions to $d\Gamma({B \to D \ell \nu})/dw$ to reduce the impact of the errors on $f_+$.[]{data-label="Fig4"}](TrDiffDistr2.eps "fig:"){width="0.43\linewidth"}
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CONCLUSION
==========
The form factors $f_+(q^2)$ and $f_0(q^2)$ in the $B\rightarrow D\tau\nu$ transition are under good control. As a result, the ratio $R\equiv\mathcal{B}({B \to D \tau \nu})/\mathcal{B}({B \to D \ell \nu})$ can be predicted with $7\%$ accuracy in the SM: $R^{SM}=0.31\pm0.02$, where the $5\%$ uncertainty on the scalar form factor at zero recoil $S_1(q^2_\textrm{max})$ is the main error source. This allows to derive useful constraints on the effective $H^+$ coupling $g_S$. Together with the constraints from $\mathcal{B}(B\rightarrow\tau\nu)$, we obtain: $g_S<0.36\ \cup\ 1.64<g_S<1.79$, i.e., the window around $g_S=2$ left over by $\mathcal{B}(B\rightarrow\tau\nu)$ is now nearly completely excluded by $R$ alone. These bounds should be strengthened soon thanks to the current considerable experimental efforts on both modes. In this respect, one should pay particular attention to the $B\rightarrow D\tau\nu$ differential distributions as these are especially well-suited to discriminate between effective scalar interactions and other types of effects and, if the former are seen, to extract the coupling $g_S$ with good precision.
It’s a pleasure to thank my collaborators Ulrich Nierste and Susanne Westhoff. Discussions with Matthias Steinhauser about the ratio $m_c/m_b$ and with Christoph Schwanda and Laurenz Widhalm about experimental issues are also warmly acknowledged. Work supported by the DFG grant No. NI 1105/1-1, by the DFG-SFB/TR9, and by the EU contract No. MRTN-CT-2006-035482 (FLAVIAnet).
[99]{}
W. Hou, Phys. Rev. **D48**, 2342 (1993).
M. S. Carena, D. Garcia, U. Nierste and C. E. M. Wagner, Nucl. Phys. [**B577**]{}, 88 (2000); A. J. Buras, P. H. Chankowski, J. Rosiek and L. Slawianowska, Nucl. Phys. [**B659**]{}, 3 (2003); A. Akeroyd and S. Recksiegel, J. Phys. G **29**, 2311 (2003); H. Itoh, S. Komine and Y. Okada, Prog. Theor. Phys. [**114**]{}, 179 (2005).
For a recent global phenomenological analysis, see for example D. Eriksson, F. Mahmoudi and O. Stal, arXiv:0808.3551 \[hep-ph\].
C. Amsler *et al.* (Particle Data Group), Phys. Lett. **B667**, 1 (2008).
Heavy Flavor Averaging Group, update after ICHEP08, http://www.slac.stanford.edu/xorg/hfag/.
A. Gray *et al.* (HPQCD Collaboration), Phys. Rev. Lett. **95**, 212001 (2005).
B. Aubert *et al.* (BABAR Collaboration), Phys. Rev. Lett. **100**, 021801 (2008).
Heavy Flavor Averaging Group, arXiv:0808.1297 \[hep-ex\].
K. Chetyrkin, J. Kühn and M. Steinhauser, Comput. Phys. Commun. **133**, 43 (2000). Input values: J. Kühn, M. Steinhauser and C. Sturm, Nucl. Phys. **B778**, 192 (2007).
See for example the discussion by F. J. Yndurain, arXiv:hep-ph/0212282.
I. Caprini, L. Lellouch and M. Neubert, Nucl. Phys. **B530**, 153 (1998). See also the earlier related works: C. Bourrely, B. Machet and E. de Rafael, Nucl. Phys. [**B189**]{}, 157 (1981); C.G. Boyd, B. Grinstein and R.F. Lebed, Phys. Rev. **D56**, 6895 (1997).
B. Aubert *et al.* (BABAR Collaboration), these proceedings, arXiv:0807.4978 \[hep-ex\]. B. Aubert *et al.* (BABAR Collaboration), arXiv:0809.0828 \[hep-ex\].
R. Hill, arXiv:hep-ph/0606023.
E. Eichten and C. Quigg, Phys. Rev. **D49**, 5845 (1994).
U. Nierste, S. Trine and S. Westhoff, Phys. Rev. **D78**, 015006 (2008), arXiv:0801.4938 \[hep-ph\].
J.F. Kamenik and F. Mescia, Phys. Rev. **D78**, 014003 (2008), arXiv:0802.3790\[hep-ph\].
B. Grzadkowski and W. Hou, Phys. Lett. **B283**, 427 (1992); K. Kiers and A. Soni, Phys. Rev. **D56**, 5786 (1997).
J. Kalinowski, Phys. Lett. **B245**, 201 (1990); D. Atwood, G. Eilam and A. Soni, Phys. Rev. Lett. [**71**]{}, 492 (1993); Y. Grossman and Z. Ligeti,Phys. Lett. [**B332**]{}, 373 (1994); M. Tanaka, Z. Phys. **C67**, 321 (1995); R. Garisto, Phys. Rev. [**D51**]{}, 1107 (1995); G. H. Wu, K. Kiers and J. N. Ng, Phys. Rev. [**D56**]{}, 5413 (1997).
[^1]: Note that the latter, though helicity-suppressed, is still (slightly) larger than the transverse $W^+$ contribution for all $q^2$ values.
|
---
abstract: 'Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the *alternating cone*. The integral (respectively, $\{0,1\}$) vectors in the alternating cone are sums of characteristic vectors of closed alternating walks (respectively, trails). We study the basic properties of the alternating cone, determine its dimension and extreme rays, and relate its dimension to the majorization order on degree sequences. We consider whether the alternating cone has integral vectors in a given box, and use residual graph techniques to reduce this problem to the one of searching for an alternating trail connecting two given vertices. The latter problem, called *alternating reachability*, is solved in a companion paper along with related results.'
address:
- |
Bhattacharya: Department of Mathematics, Statistics, and Computer Science\
University of Illinois at Chicago\
Chicago, Illinois 60607-7045, USA\
Phone: (312) 413 2163\
Fax: (312) 996 1491
- |
Peled: Department of Mathematics, Statistics, and Computer Science\
University of Illinois at Chicago\
Chicago, Illinois 60607-7045, USA\
Phone: 312 413 2156 Fax: (312) 996 1491
- |
Srinivasan: Department of Mathematics\
Indian Institute of Technology, Bombay\
Powai, Mumbai 400076, INDIA\
Phone: 91-22-2576 7484\
Fax: 91-22-2572 3480
author:
- Amitava Bhattacharya
- 'Uri N. Peled'
- 'Murali K. Srinivasan\*'
date: 'October 22, 2005; Revised October 12, 2006; minor typo January 9, 2007. To appear in Linear Algebra and Its Applications'
title: Cones of closed alternating walks and trails
---
[^1]
Introduction and Summary {#sec1}
========================
Consider a directed graph. Assign a nonnegative real weight to every arc so that at every vertex, the total weight of the incoming arcs is equal to the total weight of the outgoing arcs. The set of all such assignments forms a convex polyhedral cone in the arc space, called the cone of circulations, and is a basic object of study in network flow theory. For instance, placing integral upper and lower bounds on every arc and asking whether there is an integral vector in the cone of circulations meeting these bounds leads to Hoffman’s circulation theorem (see the book [@ff]). Now consider an undirected analog of the situation above. Take an undirected graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the total weight of the incident red edges equals the total weight of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the alternating cone. In this paper and the companion paper [@BPS2], we study the basic theory of the alternating cone. Here we consider its extreme rays, integral vectors, and dimension. We also relate it to threshold graphs and majorization order on degree sequences. We reduce the problem of finding an integral vector in the alternating cone whose components satisfy given upper and lower bounds to the problem of searching for an alternating trail connecting two given vertices in a 2-colored graph (recall that in the directed case, the corresponding problem is reduced to the problem of searching for a directed path from one given vertex to another in a suitable residual directed graph). This latter problem, called alternating reachability, generalizes the problem of searching for an augmenting path with respect to a matching in a non-bipartite graph and is solved in [@BPS2] by generalizing the blossom forest algorithm of Edmonds. We now give precise definitions and an outline of our results.
Let $G=(V,E)$ be an undirected graph (we allow parallel edges but not loops). Assume that the edges of $G$ are colored red or blue, the coloring being given by ${\mathcal C}: E \rar \{R,B\}$. We say that $(G,\mathcal{C})$ is a *2-colored graph*. Consider the real vector space ${\mathbb R}^E$, with coordinates indexed by the set of edges of $G$. We write an element $x \in {\mathbb R}^E$ as $x= (x(e) : e \in E)$. For a subset $F \subseteq E$ and $v \in V$, $F(v)$ denotes the set of all edges in $F$ incident with $v$. For a subset $F \subseteq E$, $F_R$ (respectively, $F_B$) denotes the set of red (respectively, blue) edges in $F$. For an edge $e \in E$, the characteristic vector $\chi(e) \in {\mathbb R}^E$ is defined by $\chi(e)(f) = \left\{ \ba{cc} 1, & \mbox{if }f=e \\
0, & \mbox{if }f \neq e
\ea \right.$. The *red degree* $r(v)$ (respectively, *blue degree* $b(v)$) of a vertex $v \in V$ is the number of red (respectively, blue) edges incident with $v$.
The *cone of closed alternating walks*, or simply the *alternating cone*, ${\mathcal A}(G,{\mathcal C})$ of a 2-colored graph $(G,\mathcal{C})$ (denoted simply by ${\mathcal
A}(G)$ when the coloring ${\mathcal C}$ is understood) is defined to be the set of all vectors $x = (x(e) : e \in E )$ in ${\mathbb
R}^E$ satisfying the following system of homogeneous linear inequalities: \[3\] \_[e E\_R(v)]{} x(e) - \_[e E\_B(v)]{} x(e) & = & 0, v V,\
\[4\] x(e) & & 0,e E. We refer to (\[3\]) as the *balance condition* at vertex $v$. Figure \[fig1\_1\] illustrates a 2-colored graph together with an integral vector in its alternating cone.
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If $G=(V,E)$ is a simple graph, we think of the elements of $E$ as 2-element subsets of $V$. In this case the 2-colored simple graph *associated* to $G$ is the complete graph ${\widehat G} = \left(V,
{\binom{V}{2}}\right)$, where $e=\{i,j\} \in{\binom{V}{2}}$ is colored red if $e \in E$ and colored blue if $e\not \in E$.
Let $G=(V,E)$ be a graph. A *walk* in $G$ is a sequence \[w\] W &=& (v\_0,e\_1,v\_1,e\_2,v\_2, …, e\_m,v\_m),m 0, where $v_i \in V$ for all $i$, $e_j \in E$ for all $j$, and $e_j$ has endpoints $v_{j-1}$ and $v_j$ for all $j$. We say that $W$ is a $v_0$-$v_m$ walk of length $m$. We call $e_1$ the *first* edge of $W$ and $e_m$ the *last* edge of $W$. We say that $v_1,v_2,\ldots ,v_{m-1}$ are the *internal vertices* of the walk $W$. Note that since we are allowing repetitions, the vertices $v_0, v_m$ could also be internal vertices. The walk $W^R$ is the $v_m$-$v_0$ walk obtained by reversing the sequence (\[w\]). The characteristic vector of the walk $W$ is defined to be $\chi(W) =
\sum_{i=1}^m \chi(e_i)$.
The walk $W$ is said to be
when $v_0 = v_m$;
when the edges $e_1,\ldots ,e_m$ are distinct;
when the edges $e_1,\ldots ,e_m$ are distinct and the vertices $v_0,\ldots ,v_m$ are distinct;
when $W$ is closed, the edges $e_1,\ldots ,e_m$ are distinct, and the vertices $v_0,\ldots ,v_{m-1}$ are distinct.
We have defined paths and cycles as special classes of walks. However, sometimes it is more convenient to think of paths and cycles as subgraphs, as is done usually. This will be clear from the context. If $W_1$ is a $u$-$v$ walk and $W_2$ is a $v$-$w$ walk, then the *concatenation* of $W_1$ and $W_2$, denoted $W_1*W_2$, is the $u$-$w$ walk obtained by walking from $u$ to $v$ along $W_1$ and continuing by walking from $v$ to $w$ along $W_2$. Note that if $W_1$ and $W_2$ are trails, then $W_1*W_2$ is a trail whenever $W_1$ and $W_2$ have no edges in common.
Now let $(G,\mathcal{C})$ be a 2-colored graph. The walk $W$ in (\[w\]) is said to be
when $\mathcal{C}(e_j) \neq \mathcal{C}(e_{j+1})$ for each $j = 1,\ldots, m-1$;
when $W$ is internally alternating and if $W$ is closed we also have $\mathcal{C}(e_m) \neq
\mathcal{C}(e_1)$ (note that a walk can be closed and internally alternating without being alternating, but if $v_0 \neq v_m$, there is no distinction between internally alternating and alternating walks and we use the word alternating in this case); a closed alternating walk (respectively, trail) is abbreviated as *CAW* (respectively, *CAT*);
when $W$ is a cycle of even length and $W$ is alternating (Figure \[fig1\_2\] depicts even alternating cycles and their characteristic vectors); an even alternating cycle will also be called simply an *alternating cycle*;
when $W$ is a $v_0$-$v_0$ cycle of odd length and $W$ is internally alternating (Figure \[fig1\_3\] depicts odd internally alternating cycles);
when $W$ is alternating and is of the form $W = W_1 * P * W_2 * P^R$, where $W_1, W_2$ are odd internally alternating cycles, $P$ is a path between the bases of $W_1$ and $W_2$, and the internal vertices of $W_1$, $P$, and $W_2$ are disjoint (note that $W_1$ and $W_2$ may have the same base, in which case $P$ is empty; Figure \[fig1\_4\] depicts alternating bicycles and their characteristic vectors); clearly $\chi(W) =
\chi(W_1) + 2\chi(P) + \chi(W_2)$.
(-6,-2)(6,2) (-2,0)[m0]{} (0,0)[m1]{} (2;1)[n0]{} (2;2)[n1]{} (2;3)[n2]{} (2;4)[n3]{} (2;5)[n4]{} (2;6)[n5]{}
(2,-2)(18,2.2) (1;90)[n1]{}(1.4;90)[$v_0$]{} (1;210)[n2]{} (1;330)[n3]{} (2;90)[m1]{}(2.4;90)[$v_0$]{} (2;!360 7 div 90 add)[m2]{} (2;!720 7 div 90 add)[m3]{} (2;!3 360 mul 7 div 90 add)[m4]{} (2;!4 360 mul 7 div 90 add)[m5]{} (2;!5 360 mul 7 div 90 add)[m6]{} (2;!6 360 mul 7 div 90 add)[m7]{}
(-7,-1)(7,1) (1;0)[n1]{} (1;72)[n2]{} (1;144)[n3]{} (1;216)[n4]{} (1;288)[n5]{} (2.1756,0)[n6]{} (.7;60)[n7]{} (.7;180)[n8]{} (.7;300)[n9]{} (1;0)[m1]{} (1;72)[m2]{} (1;144)[m3]{} (1;216)[m4]{} (1;288)[m5]{} (2.1756,0)[m6]{} (3.35,0)[m7]{} (.7;60)[m8]{} (.7;180)[m9]{} (.7;300)[m10]{}
A CAW $W$ is said to be *irreducible* if $\chi(W)$ cannot be written as $\chi(W_1) + \chi(W_2)$ for any CAW’s $W_1$ and $W_2$. For instance, alternating cycles and bicycles are easily seen to be irreducible. Similarly, a CAT $T$ is said to be *irreducible* if $\chi(T)$ cannot be written as $\chi(T_1) + \chi(T_2)$ for any CAT’s $T_1$ and $T_2$. Figure \[fig1\_5\] depicts an irreducible CAW (with the direction of walk indicated by an arrow) and Figure \[fig1\_6\] depicts an irreducible CAT. Irreducibility is easily seen.
(-5,-3.5)(10,1)
(-.86603,.5)[n1]{} (-.86603,-.5)[n2]{} (0,0)[n3]{} (1,0)[n4]{} (2,0)[n5]{} (1.5,-.86603)[n6]{} (1,-1.73205)[n7]{} (2,-1.73205)[n8]{} (2.86603,.5)[n9]{} (2.86603,-.5)[n10]{} (3.73205,0)[n11]{} (4.73205,0)[n12]{} (5.5981,.5)[n13]{} (5.5981,-.5)[n14]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{}
(-4,-3)(6,3)
(0,.75)[n1]{} (-.5,1.61603)[n2]{} (-1,.75)[n3]{} (0,-.75)[n4]{} (-1,-.75)[n5]{} (-.5,-1.61603)[n6]{} (1.29904,0)[n7]{} (1.99,.95107)[n8]{} (1.99,-.95107)[n9]{} (3.10806,.5878 )[n10]{} (3.10806,-.5878)[n11]{} (3.60806,1.4538)[n12]{} (3.60806,-1.4538)[n13]{} (4.10806,.5878 )[n14]{} (4.10806,-.5878 )[n15]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{} [ ]{}
Section \[sec2\] considers the integral vectors, extreme rays, and dimension of the alternating cone. We use a simple alternating walk argument to show that the extreme rays of the alternating cone are the characteristic vectors of alternating cycles and bicycles, the integral vectors in the alternating cone are sums of characteristic vectors of irreducible CAW’s, and the $\{0,1\}$-vectors in the alternating cone are sums of characteristic vectors of irreducible CAT’s. Using the characterization of the extreme rays, we obtain that a simple graph $G$ is a threshold graph if and only if $\dim {\mathcal
A}({\widehat G}) = 0$ (this fact was our original motivation for defining the alternating cone). It is well-known that for a simple graph $G$, the property $\dim {\mathcal A}({\widehat G}) = 0$ (i.e., $G$ being threshold) depends only on the degree sequence of $G$. More generally, for any 2-colored graph $(G,\mathcal{C})$, we determine $\dim \mathcal{A}(G,\mathcal{C})$ in terms of the red degree sequence of $(G,\mathcal{C})$. We then relate this dimension to the concept of majorization (following [@ap]). Consider the set $D(n)$ of all ordered degree sequences $d=(d_1,d_2,\ldots
,d_n)$ of simple graphs on $n$ vertices, where $d_1 \geq d_2 \geq
\cdots \geq d_n$. Partially order $D(n)$ by majorization (the definition is recalled in Section \[sec2\] before Lemma \[mhl\]). It is well-known (see [@mp] and [@rg]) that the set of maximal elements of this poset is precisely the set of ordered degree sequences of threshold graphs. Define a map $A:
D(n) \rightarrow {\mathbb N}$ by $A(d) = \dim {\mathcal A}(\widehat
G)$, where $G$ is any simple graph with ordered degree sequence $d$. We show that $A$ is an order-reversing map ($d_1 \succeq d_2$ implies $A(d_1) \leq A(d_2)$). Thus, we can think of $A(d)$ as a kind of measure of how non-threshold the degree sequence $d$ is.
Section \[sec3\] is motivated by the following undirected analog of Hoffman’s circulation problem for directed graphs: let $G=(V,E)$, ${\mathcal C} : E \rar \{R,B\}$ be a 2-colored graph. Assume that we are given nonnegative integral lower and upper bounds $l,u : E \rar {\mathbb N}$ satisfying $l(e) \leq u(e)$ for all $e \in E$. We are interested in knowing whether there is a vector $y \in {\mathcal A}(G, {\mathcal C}) \cap {\mathbb N}^E$ satisfying $l(e) \leq y(e) \leq u(e)$ for all $e \in E$. We use residual graph techniques to reduce this problem to the *alternating reachability problem*: given distinct vertices $s,t$ in a 2-colored graph, is there an alternating $s$-$t$ trail? Recall that in the directed case, the circulation problem is reduced to the *directed reachability problem*: given distinct vertices $s,t$ in a directed graph, is there a directed $s$-$t$ path? This is solved by a breadth-first search algorithm, which either finds a directed $s$-$t$ path or produces an $s$-$t$ cut set. In [@BPS2] we give a polynomial-time algorithm to the alternating reachability problem generalizing the blossom forest algorithm of Edmonds for searching for an augmenting path with respect to a matching in a non-bipartite graph. The algorithm either finds an alternating $s$-$t$ trail or produces an $s$-$t$ Tutte set (which is an obstruction to the existence of an alternating $s$-$t$ trail. For the definition of a Tutte set, see [@BPS2]).
Circulations in directed graphs can be thought of in terms of flows. For example, the characteristic vector of a directed circuit corresponds to a unit of flow along the circuit. Such an interpretation is not available in the case of vectors in the alternating cone; the irreducible CAW of Figure \[fig1\_5\] does not correspond to a flow in an intuitive sense. On the other hand, the characteristic vector of an irreducible CAT *can* be thought of as a unit of flow around the trail. For a 2-colored graph $G=(V,E),\; {\mathcal C}: E\rar \{R,B\}$, it is thus natural to consider the convex polyhedral cone ${\mathcal T}(G,{\mathcal
C}) \subseteq {\mathbb R}^E$ generated by the characteristic vectors of the CAT’s in $(G,\mathcal{C})$. We call ${\mathcal
T}(G,{\mathcal C})$ the *cone of closed alternating trails*, or simply the *trail cone*, of $(G,\mathcal{C})$.
Consider a CAT in a 2-colored graph. Its characteristic vector satisfies the balance condition at every vertex. If we ignore the colors, the edge-set of the CAT is a disjoint union of the edge-sets of some cycles in the underlying graph. This shows that a nonnegative integral combination (that is to say, a linear combination with nonnegative integral *coefficients*) of characteristic vectors of CAT’s satisfies the balance condition at every vertex and can be written as a nonnegative integral combination of characteristic vectors of cycles in the underlying graph $G$. Let ${\mathcal Z}(G)$ denote the cone in ${\mathbb R}^E$ generated by the characteristic vectors of the cycles in $G$. The linear inequalities defining ${\mathcal Z}(G)$ were determined by Seymour [@s]. The observation above shows that ${\mathcal
T}(G,{\mathcal C}) \subseteq {\mathcal A}(G,{\mathcal C}) \cap
{\mathcal Z}(G)$. In [@BPS2] we prove that ${\mathcal
T}(G,{\mathcal C}) = {\mathcal A}(G,{\mathcal C}) \cap {\mathcal
Z}(G)$. The proof uses our solution to the alternating reachability problem.
We remark that in this paper we focus on graph-theoretical aspects of the alternating cone and not on algorithmic efficiency. We do consider algorithms, but always with a view to obtaining graph-theoretical results.
Extreme Rays and Dimension of the Alternating Cone {#sec2}
==================================================
A simple graph $G=(V,E)$ is said to be *threshold* if there are real vertex weights $c(v),\;v \in V$ such that every pair $e=\{u,v\} \in {\binom{V}{2}}$ satisfies $c(u) + c(v) > 0$ if $e
\in E$ and $c(u) + c(v) < 0$ if $e \notin E$. Our initial motivation for defining the alternating cone was the following observation. \[tg\] A simple graph $G=(V,E)$ is threshold if and only if $\dim {\mathcal A}({\widehat G}) = 0$. Given $e \in {\binom{V}{2}}$, let $\tau (e) = (\tau (e)(v) : v \in V)
\in {\mathbb R}^V$ denote the incidence vector of $e$, where $\tau
(e)(v)$ is $1$ if $v$ is an endpoint of $e$ and $0$ otherwise. Let ${\mathcal C}_R(G)$ denote the cone in ${\mathbb R}^V$ generated by the incidence vectors of the edges $E$, and let ${\mathcal
C}_B(G)$ denote the cone generated by the incidence vectors of the nonedges ${\binom{V}{2}} - E$. If we write (\[3\]) in matrix notation, the columns correspond to the incidence vectors of edges and the negatives of the incidence vectors of nonedges. It follows that ${\mathcal A}({\widehat G}) = \{0\}$ if and only if ${\mathcal C}_R(G)\, \cap\, {\mathcal C}_B(G) = \{0\}$.
**Only if:** Assume that the weights $c(v),\;v \in V$ satisfy the defining property of a threshold graph. This means that ${\mathcal
C}_R(G)$ and ${\mathcal C}_B(G)$ are on opposite sides of the hyperplane $\sum_{v \in V} c(v) x(v) = 0$. Hence ${\mathcal C}_R(G)
\,\cap\, {\mathcal C}_B(G) = \{0\}$.
**If:** Suppose ${\mathcal C}_R(G) \,\cap\, {\mathcal
C}_B(G) = \{0\}$. Then by the separation theorem of convex polyhedral cones, there is a hyperplane $\sum_{v \in V} c(v) x(v) =
0$ such that all nonzero vectors $(p(v): v \in V) \in {\mathcal
C}_R(G)$ satisfy $\sum_{v \in V} c(v) p(v) > 0$, and all nonzero vectors $(q(v): v \in V) \in {\mathcal C}_B(G)$ satisfy $\sum_{v
\in V} c(v) q(v) < 0$. Thus $\{u,v\} \in E$ implies $c(u)+c(v)>0$, and $\{u,v\} \notin E$ implies $c(u)+c(v)<0$.
We now determine the extreme rays of the alternating cone.
\[er\] Let $G=(V,E),\; {\mathcal C}:E \rar \{R,B\}$ be a 2-colored graph. Then
the extreme rays of the alternating cone ${\mathcal
A}(G,{\mathcal C})$ are the characteristic vectors of the alternating cycles and bicycles in $(G,\mathcal{C})$;
every integral vector in the alternating cone is a nonnegative integral combination of the characteristic vectors of irreducible CAW’s;
every $\{0,1\}$-vector in the alternating cone is a nonnegative integral combination of the characteristic vectors of irreducible CAT’s
the characteristic vector of an irreducible CAW is $\{0,1,2\}$-valued. (i) Clearly, the characteristic vectors of alternating cycles and bicycles are extreme. To show the converse, we will express any rational vector in the alternating cone as a nonnegative rational combination of the characteristic vectors of alternating cycles and bicycles.
Let $a=(a(e):e \in E)$ be a nonzero rational vector in ${\mathcal
A}(G,{\mathcal C})$. Pick $e_1 \in E$ with $a(e_1)\neq 0$. Without loss of generality we may assume that $e_1$ is colored red. Let $v_0$ and $v_1$ be the endpoints of $e_1$. Build an alternating trail as follows: choose a blue edge $e_2$ incident at $v_1$ with $a(e_2)\neq 0$ (this is possible by the balance condition). Let the other endpoint of $e_2$ be $v_2$. Now choose a red edge $e_3$ incident at $v_2$ with $a(e_3) \neq 0$, and so on. At some stage we will revisit an already visited vertex. Suppose this happens for the first time when we choose edge $e_{k+1}$, i.e., we have built an alternating trail (v\_0,e\_1,v\_1,e\_2,v\_2, …, e\_k,v\_k),k 1, where $v_0,v_1,\ldots ,v_k$ are distinct, $\mathcal{C}(e_k) \neq
\mathcal{C}(e_{k+1})$, and $e_{k+1}$ has endpoints $v_k$ and a vertex $u_0 \in\{v_0,v_1,\ldots ,v_{k-1}\}$. Then we have found either an alternating cycle $D$, or an odd internally alternating cycle $C$ with base $u_0$. In the first case, subtracting an appropriate multiple of $\chi(D)$ from $a$, we obtain another vector in the alternating cone whose support is strictly contained in the support of $a$. Thus, by induction on the size of the support, we are done.
In the second case, extend $C$ to an alternating trail $C*T$ as follows: $T$ starts with $T = (u_0,f_1,\ldots)$, where $f_1$ is an edge incident with $u_0$ and satisfies $a(f_1)\neq 0$ and $\mathcal{C}(f_1) \neq \mathcal{C}(e_{k+1})$. Let the other endpoint of $f_1$ be $u_1$. Now add to $T$ an edge $f_2$ incident with $u_1$ and satisfying $a(f_2)\neq 0$ and $\mathcal{C}(f_2) \neq
\mathcal{C}(f_1)$, and so on. At some stage we will revisit an already visited vertex of the trail $C*T$. Suppose this happens for the first time when we choose edge $f_{m+1}$, i.e., we have T & =& (u\_0,f\_1,u\_1,f\_2,u\_2, …, f\_m,u\_m),m 1, where $u_0,u_1,\ldots ,u_m$ are distinct, none of $\{u_1,\ldots
,u_m\}$ is on $C$ (see Figure \[fig2\_1\]), $\mathcal{C}(f_{m+1})
\neq \mathcal{C}(f_m)$, one endpoint of $f_{m+1}$ is $u_m$, and the other endpoint $v$ of $f_{m+1}$ is either in $T$ or is a vertex of $C$ different from $u_0$. Two cases arise:\
**Case (a):** $v$ is a vertex of $T$ (see Figure \[fig2\_2\]). We have found either an alternating cycle or an alternating bicycle, and we are done by induction on the size of the support, as in the previous paragraph.\
**Case (b):** $v$ is not a vertex of $T$ (see Figure \[fig2\_3\]). In this case, the edges $f_1,\ldots f_{m+1}$ together with an appropriate portion of $C$ determine an alternating cycle, and we are done.\
(-6,-2)(7.5,2) (-2.5,1)[n1]{} (-1.5,1)[n2]{} (-.5,1)[n3]{} (0,0)[n4]{} (-.5,-1)[n5]{} (-1.5,-1)[n6]{} (-2.5,-1)[n7]{} (1,0)[n8]{} (2,0)[n9]{} (3,0)[n10]{} (4,0)[n11]{} (-1.5,0)[$C$]{} \[65\](0,0)[$u_0$]{} \[90\](1,0)[$u_1$]{} \[90\](2,0)[$u_2$]{} \[90\](4,0)[$u_m$]{} \[90\](2,.35)[$T$]{}
(-6,-2)(9,2) (-2.5,1)[n1]{} (-1.5,1)[n2]{} (-.5,1)[n3]{} (0,0)[n4]{} (-.5,-1)[n5]{} (-1.5,-1)[n6]{} (-2.5,-1)[n7]{} (1,0)[n8]{} (2,0)[n9]{} (3,0)[n10]{} (4,0)[n11]{} (5,0)[n12]{} (-1.5,0)[$C$]{} \[65\](0,0)[$u_0$]{} \[90\](1,0)[$u_1$]{} \[90\](2,0)[$u_2$]{} \[90\](3,.025)[$v$]{} \[90\](5,0)[$u_m$]{} \[90\](2.5,.35)[$T$]{} \[270\](4,-0.415)[$f_{m+1}$]{}
(-6,-3.6)(8,2) (-2.5,1)[n1]{} (-1.5,1)[n2]{} (-.5,1)[n3]{} (0,0)[n4]{} (-.5,-1)[n5]{} (-1.5,-1)[n6]{} (-2.5,-1)[n7]{} (-3.08,0)[nn1]{} (1,0)[n8]{} (2,0)[n9]{} (3,0)[n10]{} (4,0)[n11]{} (-1.5,0)[$C$]{} \[65\](0,0)[$u_0$]{} \[90\](1,0)[$u_1$]{} \[90\](2,0)[$u_2$]{} \[90\](4,0)[$u_m$]{} \[90\](2,.35)[$T$]{} \[270\](1,-1.9)[$f_{m+1}$]{}
Essentially the same argument as given above appears in [@hip] (in the context of edges and non-edges).
\(ii) Let $a=(a(e):e \in E)$ be a nonzero integral vector in the alternating cone. Pick an edge $e_1$ with $a(e_1)\neq 0$ and with end points $v_0$ and $v_1$. Assume that we have an internally alternating walk W &=& (v\_0,e\_1,v\_1,e\_2,v\_2, …, e\_m,v\_m),m 1, with $\chi(W) \leq a$ (we can always start with the walk $(v_0,e_1,v_1)$). We show below that either we can extend $W$, or else there is a CAW (and hence an irreducible CAW) through $e_1$. Since we cannot extend indefinitely because of the condition $\chi(W) \leq a$, we are done. The following cases arise.\
**Case (a):** $v_m\neq v_0$. Then $\chi(W)$ does not satisfy the balance condition at $v_m$, but $a$ does, and since $\chi(W) \leq a$ and $a$ is integral, we can find an edge $e_{m+1}$ incident at $v_m$ with $\mathcal{C}(e_{m+1}) \neq \mathcal{C}(e_m)$ such that $\chi(W)(e_{m+1}) < a(e_{m+1})$. Extend $W$ by adding $e_{m+1}$ and the other end point of $e_{m+1}$.\
**Case (b):** $v_m = v_0$ and $\mathcal{C}(e_1) =
\mathcal{C}(e_m)$. We can extend $W$ just as in case (a).\
**Case (c):** $v_m = v_0$ and $\mathcal{C}(e_1) \neq
\mathcal{C}(e_m)$. In this case $W$ is a CAW.
\(iii) This is a special case of (ii): if $a$ is a $\{0,1\}$-vector, then the CAW’s in (ii) must be CAT’s.
\(iv) Consider a walk $W$ as in (\[w\]). This assigns a direction of traversal to each edge; for instance, the edge $e_2$ is traversed from $v_1$ to $v_2$. The direction of traversal may be different for two occurrences of the same edge. However, if a CAW $W$ traverses an edge three or more times, then two of these directions must be the same, and this can be used to write $W=W_1*W_2$ for two positive length CAW’s $W_1$ and $W_2$, so $W$ is not irreducible.
As a corollary of Theorem \[er\], we derive the following well-known characterization of threshold graphs. A simple graph $G$ is not threshold if and only if $\widehat
G$ contains an alternating cycle of length 4. **If:** Suppose $\{ \{i,j\},\{j,k\},\{k,l\},\{l,i\} \}$ is an alternating 4-cycle in $\widehat G$ with the pairs $\{i,j\},\{k,l\}$ red and the other two pairs blue. Assume that $G$ is threshold with vertex weights $c(v),\;v \in V$ satisfying the defining property. Since $\{i,j\},\{k,l\}$ are red, we have $c(i)+c(j) >0$, $c(k)+c(l)>0$ and therefore $c(i)+c(j)+c(k)+c(l) >
0$. Similarly, since $\{j,k\},\{l,i\}$ are blue, we have $c(i)+c(j)+c(k)+c(l) < 0$, a contradiction.
**Only if:** Since $G$ is not threshold, by Theorem \[tg\] ${\mathcal A}(\widehat G)$ has an extreme ray, which is an alternating cycle or an alternating bicycle by Theorem \[er\]. Suppose that this extreme ray is an alternating cycle of length greater than $4$. There is a chord of $\widehat G$ that splits this cycle into two even cycles (since $\widehat G$ is complete). Regardless of the color of the chord, one of these two cycles is alternating. Repeating this argument, we obtain an alternating cycle of length 4.
Now consider an extreme ray that is an alternating bicycle $W=W_1*P*W_2*P^R$. Let $u$ and $v$ be the bases of $W_1$ and $W_2$. Let $u'$ (respectively, $v'$) be any vertex of $W_1$ (respectively, $W_2$) different from $u$ (respectively, $v$). Consider the edge $\{u',v'\}$ of $\widehat G$. $W_1$ determines two alternating $u'$-$u$ paths, and one of them starts with an edge having color different from that of $\{u',v'\}$. Call this alternating path $P_1$. Similarly, using $W_2$, choose an alternating $v'$-$v$ path $P_2$ that starts with an edge having color different from that of $\{u',v'\}$. We now have the alternating cycle $P_1*P*P_2^R*(v',\{v',u'\},u')$, and we can use the argument of the preceding paragraph.
We now give a formula for the dimension of the alternating cone of a 2-colored graph.
A connected graph is said to be *odd unicyclic* if it contains precisely one cycle, and that cycle has odd length. In other words, an odd unicyclic graph is obtained from a tree by adding a new edge between two nonadjacent vertices of the tree so that the cycle created has odd length. A graph is a *pseudo forest* if each component of the graph is either acyclic or odd unicyclic. Pseudo forests are to be distinguished from $1$-forests, which are graphs whose connected components have at most one cycle, even or odd. The motivation for studying $1$-forests is combinatorial while pseudo forests have a linear algebraic origin (see Theorem \[qft\] below). Recall from the proof of Theorem \[tg\] that for an edge $e$, the vector $\tau(e) \in {\mathbb R}^V$ is the incidence vector of $e$, which is 1 in the two coordinates indexed by the endpoints of $e$, and is 0 elsewhere. The *incidence matrix* of a graph is the matrix whose columns are the incidence vectors of the edges. For a proof of the following result see [@gks]. \[qft\] For a graph $G=(V,E)$ and a set $X \subseteq E$, the set $\{ \tau(e) :
e \in X \}$ is linearly independent in ${\mathbb R}^V$ if and only if the graph $(V,X)$ is a pseudo forest. In particular, the rank of the incidence matrix of $G$ is equal to $\#V - \mbox{ number of
bipartite components of } G$. For a graph $G=(V,E)$ and an integer sequence $d=(d(v) :
v \in V)$, we use the notation $${\mathcal K}(d) = \{{\mathcal C}:E\rightarrow \{R,B\} : \mbox{the red degree of }
v \mbox{ is } d(v) \mbox{ for all } v \in V\},$$ i.e., ${\mathcal K}(d)$ denotes the set of all 2-colorings of $G$ having red degree sequence $d$. \[cl\] Consider the 2-colored graph $G=(V,E)$ with a coloring ${\mathcal
C} \in {\mathcal K}(d)$, and let $e \in E$. If $(G,\mathcal{C})$ has a CAW through $e$, then for each ${\mathcal C'} \in {\mathcal
K}(d)$, $(G,\mathcal{C'})$ has a CAW through $e$. Let ${\mathcal C'} \in {\mathcal K}(d)$ and consider the spanning subgraph $G'=(V,E')$ of $G$, where $E'$ consists of all the edges where $\mathcal{C}$ and $\mathcal{C'}$ disagree. Since the red degrees (and thus also the blue degrees) in $G$ agree under ${\mathcal C}$ and ${\mathcal C'}$, it follows that for each $v \in V$, the red degree of $v$ in $(G',\mathcal{C'})$ is equal to the blue degree of $v$ in $(G',\mathcal{C'})$. Thus the all-1 vector in ${\mathbb R}^{E'}$ is balanced in $(G',\mathcal{C'})$, i.e., is in $\mathcal{A}(G',\mathcal{C'})$. It follows from Theorem \[er\](iii) that for each $e \in E'$, $(G',\mathcal{C'})$ has a CAT through $e$, and therefore so does $(G,\mathcal{C'})$.
Now let $W$ be a CAW through $e$ in $(G,\mathcal{C})$. If ${\mathcal C}(e) \neq {\mathcal C'}(e)$, then we already know that $(G,\mathcal{C'})$ has a CAT through $e$, and we are done. So we may assume that ${\mathcal C}(e) = {\mathcal C'}(e)$. We will transform $W$ into a CAW $W'$ through $e$ in $(G,\mathcal{C'})$. Let $f$ be an edge in $W$ with endpoints $u$ and $v$. If ${\mathcal
C}(f)={\mathcal C'}(f)$, we do nothing. If ${\mathcal C}(f) \neq
{\mathcal C'}(f)$, then $f \in E'$ and $(G,\mathcal{C'})$ has a CAT through $f$. Dropping $f$ from this CAT, we obtain $u$-$v$ alternating trail $P$ in $(G,{\mathcal C'})$ whose first and last edges have the color ${\mathcal C}(f)$. We drop $f$ from $W$ and substitute the trail $P$ in its place. Doing this for every edge $f$ in $W$ with ${\mathcal C}(f)\neq {\mathcal C'}(f)$, we obtain a CAW $W'$ through $e$ in $(G,\mathcal{C'})$.
For a graph $G$ and an integer sequence $d$, let $E_d$ be the set of all edges $e$ of $G$ such that some 2-coloring in $\mathcal{K}(d)$ has a CAW through $e$ (equivalently by Lemma \[cl\], all 2-colorings in $\mathcal{K}(d)$ have a CAW through $e$).
\[dt\] Let $G=(V,E)$ be a graph and ${\mathcal C}$ a 2-coloring of $G$ with red degree sequence $d$. Then $$\dim {\mathcal A}(G,{\mathcal C}) = \#{E_d} - \#V +
b(V,E_d),$$ where $b(V,E_d)$ denotes the number of bipartite components of the graph $(V,E_d)$. An edge $e \in E$ is said to be *inessential* if $x(e)=0$ for all $x \in {\mathcal A}(G,{\mathcal C})$. From Theorem \[er\](ii) it follows that $e$ is inessential if and only if $(G,\mathcal{C})$ has no CAW through $e$. From basic polyhedral theory it now follows that $\dim {\mathcal A}(G,{\mathcal C})$ is equal to the nullity (i.e., number of columns minus rank) of the $\#V\times \#E_d$ vertex-edge incidence matrix of the graph $(V,E_d)$. The expression for the dimension now follows from Theorem \[qft\].
From Theorem \[dt\], $\dim {\mathcal A}(G,{\mathcal C})$ depends only on $G$ and the red degree sequence of $\mathcal{C}$. In the case of the associated 2-colored graphs of simple graphs we can say more. \[sdac\] Let $G_1$ and $G_2$ be simple graphs with degree sequences $d_1$ and $d_2$. If $d_1$ is a rearrangement of $d_2$ (so in particular $G_1$ and $G_2$ have the same number of vertices), then $\dim {\mathcal A}(\widehat G_1) = \dim {\mathcal A}(\widehat
G_2)$. Suppose that the permutation $\pi :V \rar V$ rearranges $d_1$ into $d_2$. The result follows from the fact that $\pi$ is an automorphism of the complete graph $\left(V,\binom{V}{2}\right)$.
Lemma \[sdac\] fails for 2-colored graphs that are not complete: Figures \[fig2\_4\] depicts two 2-colorings of a graph on the vertex set $\{1,2,\ldots ,7\}$ whose red degree sequences are permutations of each other (via the permutation $\pi$ that fixes $2,3,7$ and exchanges $1$ with $5$ and $4$ with $6$). However, it is easily seen that the dimensions of the alternating cones of the 2-colored graphs are 1 and 0, respectively. The permutation $\pi$ is not an automorphism of the underlying graph.
(-11,-2)(1,2) (0,-1)[n1]{} (2,-1)[n2]{} (2,1)[n3]{} (0,1)[n4]{} (3,-1)[n5]{} (3,1)[n6]{} (4.73,0)[n7]{} \[90\](0,1.125)[$1$]{} \[90\](2,1.125)[$2$]{} \[270\](2,-1.125)[$3$]{} \[270\](0,-1.125)[$4$]{} \[270\](3,-1.125)[$6$]{} \[90\](3,1.125)[$5$]{} \[0\](4.855,0)[$7$]{} (7,-1)[mn1]{} (9,-1)[mn2]{} (9,1)[mn3]{} (7,1)[mn4]{} (10,-1)[mn5]{} (10,1)[mn6]{} (11.73,0)[mn7]{} \[90\](7,1.125)[$1$]{} \[90\](9,1.125)[$2$]{} \[270\](9,-1.125)[$3$]{} \[270\](7,-1.125)[$4$]{} \[270\](10,-1.125)[$6$]{} \[90\](10,1.125)[$5$]{} \[0\](11.855,0)[$7$]{}
We now relate the dimension of the alternating cone to the concept of majorization. We begin with a few definitions.
Let $a=(a(1),\ldots ,a(n))$ and $b=(b(1),\ldots ,b(n))$ be real sequences of length $n$. Denote the $i$-th largest component of $a$ (respectively, $b$) by $a[i]$ (respectively, $b[i]$). We say that $a$ *majorizes* $b$, denoted by $a \succeq b$, if $$\sum_{i=1}^k a[i] \geq \sum_{i=1}^k b[i],\qquad k=1,\ldots ,n,$$ with equality for $k=n$. The majorization is *strict*, denoted by $a \succ b$, if at least one of the inequalities is strict, namely if $a$ is not a permutation of $b$. We recall a fundamental lemma about majorization in integer sequences, called Muirhead’s lemma. If $a=(a(1),\ldots ,a(n))$ is a sequence and there exist $i$ and $j$ such that $a(i) \geq a(j) + 2$, then the following operation is called a *unit transformation from $i$ to $j$ on $a$*: subtract 1 from $a(i)$ and add 1 to $a(j)$. Clearly, if $b$ is obtained from $a$ by a sequence of unit transformations, then $a\succ b$. The converse is also true for integer sequences.
\[mhl\] If $a$ and $b$ are integer sequences and $a\succ b$, then some permutation of $b$ can be obtained from $a$ by a sequence of unit transformations. For a proof see [@mp; @mo].
\[dml\] Let $G_1 = (V,E_1)$ and $G_2 = (V,E_2)$ be simple graphs with degree sequences $d_1$ and $d_2$. If $d_1 \succeq d_2$, then $\dim
{\mathcal A}(\widehat G_1) \leq \dim {\mathcal A}(\widehat G_2)$. If $d_2$ is a rearrangement of $d_1$, the result follows from Lemma \[sdac\], so we may assume that $d_1 \succ d_2$. By Muirhead’s lemma some permutation $d_2'$ of $d_2$ can be obtained from $d_1$ by a finite sequence $d_1 \succ d \succ \cdots \succ
d_2'$ of unit transformations. We will show that $d$ is the degree sequence of a simple graph $G$ satisfying $\dim {\mathcal
A}(\widehat G_1) \leq \dim {\mathcal A}(\widehat G)$. By Lemma \[sdac\] and induction on the number of unit transformations, the result will follow.
For notational convenience, let $V = \{1,2,\ldots,n\}$, and suppose $d$ is obtained from $d_1$ by a unit transformation from $i$ to $j$, so that $d_1(i) \geq d_1(j) + 2$. This implies that there exist distinct vertices $k,l \neq i,j$ such that $\{i,k\},
\{i,l\}$ are edges of $G_1$ and $\{j,k\}, \{j,l\}$ are not. Let $G$ be the graph with degree sequence $d$ obtained from $G_1$ by dropping the edge $\{i,k\}$ and adding the edge $\{j,k\}$ (see Figure \[fig2\_6\]).
(-4,-3)(4,2) (0,-1)[n1]{} (2,-1)[n2]{} (2,1)[n3]{} (0,1)[n4]{} (4,-1)[m1]{} (6,-1)[m2]{} (6,1)[m3]{} (4,1)[m4]{} \[90\](0,1.125)[$i$]{} \[90\](2,1.125)[$l$]{} \[270\](0,-1.125)[$j$]{} \[270\](2,-1.125)[$k$]{} \[90\](4,1.125)[$i$]{} \[90\](6,1.125)[$l$]{} \[270\](4,-1.125)[$j$]{} \[270\](6,-1.125)[$k$]{} (1,-2.5)[$G_1$]{} (5,-2.5)[$G$]{}
Consider the 2-colored graphs $\widehat{G}_1$ and $\widehat{G}$ with red degree sequences $d_1$ and $d$, respectively. We now show that $E_{d_1} \subseteq E_d$. By Theorem \[dt\], it will then follow that $\dim {\mathcal A}(\widehat G_1) \leq \dim {\mathcal
A}(\widehat G)$, since $\#(E_d - E_{d_1}) \geq b(V,E_{d_1}) -
b(V,E_d)$. This last inequality can be seen as follows: start with the graph $(V,E_{d_1})$ and add the edges $e \in E_d - E_{d_1}$ one at a time. If $e$ connects two components $C_1$ and $C_2$, the number of bipartite components decreases by one or stays the same, according as $C_1$ and $C_2$ are both bipartite or not; if $e$ connects two vertices in the same component $C$, the number of bipartite components stays the same if $C$ is nonbipartite, and it stays the same or decreases by one if $C$ is bipartite according to the parity of (any of the) cycles created by $e$.
Suppose $\{u,v\} \in E_{d_1}$. If $\{u,v\}$ is one of the pairs $\{i,k\},\{j,k\}$ that changed status by going from $G_1$ to $G$, then Figure \[fig2\_6\] depicts an alternating 4-cycle through $\{u,v\}$ in $\widehat G$, and thus $\{u,v\} \in E_d$ and we are done. So we may assume that $\{u,v\}$ is not one of these two pairs.
Since $\{u,v\} \in E_{d_1}$, $\widehat G_1$ has a CAW $W$ through $\{u,v\}$. Replace every occurrence of $$\begin{gathered}
\ldots,i,\{i,k\},k,\ldots,
\\
\ldots,k,\{i,k\},i,\ldots,
\\
\ldots,k,\{k,j\},j,\ldots,
\\
\ldots,j,\{j,k\},k,\ldots,
\end{gathered}$$ in $W$ by (respectively) $$\begin{gathered}
\ldots,i,\{i,l\},l,\{l,j\},j,\{j,k\},k,\ldots,
\\
\ldots,k,\{k,j\},j,\{j,l\},l,\{l,i\},i,\ldots,
\\
\ldots,k,\{k,i\},i,\{i,l\},l,\{l,j\},j,\ldots,
\\
\ldots,j,\{j,l\},l,\{l,i\},i,\{i,k\},k,\ldots,
\end{gathered}$$ keeping all other edges in $W$ fixed. This yields a CAW through $\{u,v\}$ in $G$, and thus $\{u,v\} \in E_d$.
As stated in the introduction, Theorem \[dml\] defines an order-reversing map $A: D(n) \rar {\mathbb N}$, which maps the degree sequence of a simple graph $G$ to the dimension of the alternating cone of the associated 2-colored graph $\widehat{G}$. Given $d=(d(1),\ldots ,d(n)) \in D(n)$, there is a well-known algorithm working only with the numbers $d(1),\ldots ,d(n)$ to determine whether $A(d)=0$ (see [@mp]). Motivated by this, we ask whether there is an algorithm working only with the numbers $d(1),\ldots ,d(n)$ for computing $A(d)$.
Intersection of the Alternating Cone with a Box {#sec3}
===============================================
Assume that we are given a 2-colored graph $G=(V,E)$, and for each $e \in E$ nonnegative integers $l(e),u(e)$ with $l(e) \leq u(e)$. We ask if there is a rational vector $x \in {\mathcal
A}(G,{\mathcal C})$ with $l(e) \leq x(e) \leq u(e)$ for all $e \in
E$. The next theorem restricts the search to half-integral $x$. \[hi\] Let $G=(V,E)$, ${\mathcal C}: E\rar \{R,B\}$ be a 2-colored graph, and $l,u : E\rightarrow {\mathbb N}$ maps with $l(e) \leq u(e)$ for all $e \in E$. If there exists a rational vector $x \in {\mathcal
A}(G,{\mathcal C})$ with $l(e) \leq x(e) \leq u(e)$ for all $e \in
E$, then there exists an integral $y \in {\mathcal A}(G,{\mathcal
C})$ with $2l(e) \leq y(e) \leq 2u(e)$ for all $e \in E$. We use elementary polyhedral theory. Since by assumption a feasible solution exists, there exists a basic feasible solution $\overline{x}
\in {\mathcal A}(G,{\mathcal C})$, with $l \leq \overline{x} \leq u$. In our case a basic feasible solution is obtained as follows. First choose a pseudo forest $(V,X)$ such that the columns corresponding to $X$ form a basis of the column space of the vertex-edge incidence matrix of $G$. For each $e \in E - X$ we have $\overline{x}(e)
= l(e)$ or $\overline{x}(e) = u(e)$. Now solve for the remaining $\overline{x}(e)$, $e \in X$ using the balance condition at every node. Since $l,u$ are integral and the determinant of the incidence matrix of an odd cycle is $\pm2$, the half-integrality of $\overline{x}$ easily follows, and $y = 2\overline{x}$ is as required.
Motivated by Theorem \[hi\], we want to improve half-integrality to integrality, so we are led to the following problem. Let a 2-colored graph $G=(V,E)$, ${\mathcal C}: E\rar
\{R,B\}$ and bounds $l,u : E \rightarrow {\mathbb N}$ be given. For $f \in \ac$, an edge $e$ is called *feasible w.r.t. $f$* if $l(e) \leq f(e) \leq u(e)$, and $f$ itself is called *feasible* if every edge is feasible w.r.t. $f$, *infeasible* otherwise. We ask if there is a feasible vector $f \in \ac \cap \mn^E$. We now reduce this problem to the problem of finding a CAT through a given edge in a 2-colored graph. This latter problem is easily reduced to the alternating reachability problem.
Let $f \in \ac \cap \mn^E$, $f$ not necessarily feasible. The *residual 2-colored graph $G(f)=(V,E(f))$ of $f$ w.r.t. $l,u$* is defined as follows. We take four disjoint copies $E_1,
E_2, E_3, E_4$ of $E$, and denote the copy of $e \in E$ in $E_i$ by $e_i$, $i=1,\ldots,4$. For each $e \in E$, we place $e_1$ in $E(f)$ with the color $\cc(e)$ when $f(e) \leq u(e) - 1$, place $e_2$ in $E(f)$ with the color $\cc(e)$ when $f(e) \leq u(e) - 2$, place $e_3$ in $E(f)$ with the color opposite $\cc(e)$ when $f(e) \geq
l(e) + 1$, and place $e_4$ in $E(f)$ with the color opposite $\cc(e)$ when $f(e) \geq l(e) + 2$.
Suppose that $G(f)$ has a CAT $T$. We extend the characteristic vector $\chi(T)$ by adding zero components at all elements of $E_1
\cup E_2 \cup E_3 \cup E_4 - E(f)$. By *augmenting $f$ along $T$* we mean replacing $f$ with $f_T$ given by $$f_T(e) = f(e) + \chi(T)(e_1) + \chi(T)(e_2) - \chi(T)(e_3) -
\chi(T)(e_4), \qquad e \in E.$$ Note that $f_T \in \ac \cap \mn^E$, and that in replacing $f$ with $f_T$, feasible edges remains feasible, the infeasible edges of $T$ move “in the right direction”, i.e., become feasible or move closer to feasibility, and of course the edges out of $T$ remain unchanged. \[at\] Suppose $f \in \ac\cap {\mn}^E$ is infeasible, but $\ac\cap {\mn}^E$ has a feasible vector. Then for each $e \in E$,
if $f(e) < l(e)$, then $G(f)$ has a CAT through $e_1$;
if $f(e) > u(e)$, then $G(f)$ has a CAT through $e_3$. We define a 2-colored subgraph $G'(f)=(V,E'(f))$ of $G(f)$ by letting $E'(f) = E(f) \cap (E_1 \cup E_3)$ and restricting the 2-coloring of $G(f)$ to $G'(f)$.
Let $g \in \ac \cap \mn^E$ be a feasible vector. We define $h : E'(f)\rar \mn$ as follows: for $e_1 \in E'(f)$, $$h(e_1) = \left\{ \ba{ll}
0 & \mbox{if } g(e) - f(e) < 0, \\
g(e) - f(e) & \mbox{if } g(e) - f(e) \geq 0,
\ea
\right.$$ and for $e_3 \in E'(f)$, $$h(e_3) = \left\{ \ba{ll}
0 & \mbox{if } g(e) - f(e) > 0,
\\
-(g(e) - f(e)) & \mbox{if } g(e) - f(e) \leq 0.
\ea
\right.$$ It is easy to check that $h$ is an integral vector in the alternating cone of $G'(f)$.
\(i) Assume that $e \in E$ with $f(e) < l(e)$. Then $e_1 \in
E'(f)$ and $h(e_1) > 0$ (since $g$ is feasible). By Theorem \[er\](ii), $h$ can be written as a sum of characteristic vectors of irreducible CAW’s in $G'(f)$, and thus $G'(f)$ has an irreducible CAW $W$ through $e_1$. By Theorem \[er\](iv), $\chi(W)$ is $\{0,1,2\}$-valued. Suppose $\chi(W)(a_1) = 2$ for some $a_1$ (respectively, $\chi(W)(a_3) = 2$ for some $a_3$). Then $g(a) - f(a) \geq 2$ (respectively, $f(a) - g(a) \geq 2$). Since $g$ is feasible, $a_2 \in E(f)$ (respectively, $a_4 \in E(f)$), and consequently $a_1 \in E(f)$ (respectively, $a_3 \in E(f)$) by the definition of $E(f)$. We then consider $W$ as a subset of $E(f)$ and replace the double occurrence of $a_1$ (respectively, $a_3$) in $W$ by a single occurrence of $a_1$ and of $a_2$ (respectively, of $a_3$ and of $a_4$). Doing this for all repeated edges in $W$ transforms it into a CAT in $G(f)$ through $e_1$.
\(ii) Similar to (i).
The problem of finding a CAT through a given edge $e$ in an edge-colored graph can be reduced to the alternating reachability problem as follows: let $e$ have endpoints $s$ and $t$. Remove $e$ from the graph, add two new vertices $s'$ and $t'$, add two new edges with color $\mathcal{C}(e)$, one between $s'$ and $s$ and one between $t'$ and $t$. Clearly the new graph has an alternating $s'$-$t'$ trail if and only if the original graph has a CAT through $e$.
We can now use the following familiar scheme to look for a feasible integral vector. Start with an integral balanced $f:E\rar \mn$, for example $f = 0$. If $f$ is infeasible, construct $G(f)$. Pick an edge $e \in E$ with $f(e)< l(e)$ (respectively, $f(e) > u(e)$), and find a CAT $T$ through $e_1$ (respectively, $e_3$) in $G(f)$ if one exists (using the alternating reachability algorithm in [@BPS2]), then augment $f$ along $T$. As noted in the proof of Theorem \[at\], $f_T$ is integral and balanced, feasible edges remain feasible, and in addition each infeasible edge in $T$, in particular $e$, has either become feasible or has moved closer to feasibility. Replace $f$ with $f_T$ and repeat. Since we are working with integral vectors, either we terminate with a feasible integral vector in time bounded by the total infeasibility, or else at some stage $G_f$ has no CAT through $e_1$ (respectively, $e_3$), in which case no feasible integral vector exists by Theorem \[at\].
As stated in the introduction, in this paper we are not dealing with efficiency issues but only with the graph-theoretic aspects of the alternating cone. Our discussion motivates the alternating reachability problem and the problem of determining the linear inequalities defining the trail cone. These two problems are considered in [@BPS2].
[**Acknowledgement:**]{} We thank the referees for their constructive suggestions that led to an improvement in the exposition.
[99]{}
S. R. Arikati and U. N. Peled, Degree sequences and majorization, : 179–211 (1994).
A. Bhattacharya, U. N. Peled, and M. Srinivasan, Alternating reachability, submitted for publication.
L. R. Ford, Jr. and D. R. Fulkerson, Princeton University Press (1962).
J. W. Grossman, D. M. Kulkarni, and I. E. Schochetman, Algebraic graph theory without orientation, : 289–308 (1994).
P. L. Hammer, T. Ibaraki and U. N. Peled, Threshold numbers and threshold completions, , Ed: P. Hansen, North-Holland, New York, Annals of Discrete Mathematics [**11**]{}: 125–145 (1981).
N. V. R. Mahadev and U. N. Peled, Annals of Discrete Mathematics [**56**]{} (1995).
A. W. Marshall and I. Olkin, Academic Press, New York, (1979).
R. Ruch and I. Gutman, The branching extent of graphs, : 286–295 (1979).
P. D. Seymour, Sums of Circuits, Eds: J. A. Bondy and U. S. R. Murty, Academic Press, New York, 341–355, (1979).
[^1]: UNP and MKS would like to thank Professor Martin Golumbic for his kind invitation to visit the Caesarea Edmond Benjamin de Rothschild Foundation Institute for Interdisciplinary Applications of Computer Science at the University of Haifa, Israel during May–June 2003, where part of this work was carried out. The warm hospitality and partial support of this visit from CRI is gratefully acknowledged.\
\* Corresponding author
|
---
address: 'Department of Physics, Iowa State University, Ames, IA 50011'
author:
- 'G. Valencia'
title: '$\Sigma^{+}\to p \mu^{+}\mu^{-}$: Standard Model or New Particle?'
---
Introduction
============
The HyperCP collaboration has observed three events for the mode $\Sigma^+\to p \mu^+ \mu^-$ [@Park:2005ek]. A striking feature of the result is that the three events have the same muon pair invariant mass, 214.3 MeV. HyperCP estimates the probability for this clustering at $0.8\%$ using a “form factor” distribution for the standard model expectations [@Bergstrom:1987wr].
This observation invites two calculations and we report on the results in this talk. First we present the best possible prediction for the Standard Model expectation. Since there are no known particles of mass 214 MeV, we do not expect a peak at that muon pair invariant mass. However, we need to know whether the SM distribution is narrower or wider than the form used by HyperCP to assess the significance of the clustering. Even if the three events represent new physics, it is necessary to know the SM level in order to determine if HyperCP should have seen events at other values of $m_{\mu\mu}$.
The second calculation involves assuming that the observed events are indeed evidence for a new particle and confronting this observation with existing constraints from kaon and B physics. In particular we study the conditions under which the observation is consistent with a light Higgs boson and find an explicit candidate for the new particle: the lightest CP-odd Higgs boson in the NMSSM, the $A_1^0$.
Standard Model Calculation
==========================
We first present the ingredients that enter the calculation within the SM [@He:2005yn]. The short distance contribution is too small to explain these events by four orders of magnitude, this decay is long distance dominated as is the case in similar kaon modes.
The long distance contributions to $\Sigma^{+}\to p \mu^{+}\mu^{-}$ can be pictured schematically as arising from the $\Sigma^{+}\to p \gamma^\star$ process. There are four independent form factors allowed by electromagnetic gauge invariance, $$\begin{aligned}
\label{M_BBg}
{\cal M}(B_i\to B_f\gamma^*)& =
&- e G_{F}^{}\, \bar{B}_f^{} \left[ i\sigma^{\mu\nu}q_\mu^{}(a+b\gamma_5^{})
+(q^2\gamma^\nu-q^\nu\!\!\not{\!q}) (c+d\gamma_5^{}) \right] B_i^{}\, \varepsilon_\nu^{} \,\,.\end{aligned}$$ Two of the form factors, $a(q^2)$ and $c(q^2)$, are parity conserving whereas $b(q^2)$ and $d(q^2)$ are parity violating. In addition, two of the form factors are non-zero at $q^2=0$ and contribute to the radiative decay $\Sigma^+\to p\gamma$: $a(0)$ and $b(0)$. All four form factors are complex and receive imaginary parts from $N\pi$ intermediate states.
We estimate these imaginary parts by taking the weak vertex $\Sigma^+ \to N\pi$ from experiment and using the $\,N\pi\to p\gamma^*\,$ scattering at lowest order in $\chi$PT (both conventional and heavy baryon). We check that our calculations agree with the existing ones at $q^2=0$.
To estimate the real part of the form factors we use $a(0)$ and $b(0)$, as determined from the width and decay distribution of the radiative decay $\Sigma \to p \gamma$ up to a discrete ambiguity. We then assume that value for the range of $q^2$ needed. This is consistent with our finding that the imaginary parts of the form factors are smooth and slowly varying over the $q^2$ range of interest. Finally, the real parts of $c(q^2)$ and $d(q^2)$ are obtained using a vector meson dominance model.
There is some uncertainty in the calculation, but the resulting range, $1.6 \times 10^{-8}
\leq \ \ {\cal B}(\Sigma^{+}\to p \mu^{+}\mu^{-})_{SM}\ \ \leq
9.0\times 10^{-8}$, is in good agreement with the measured rate, ${\cal B}(\Sigma^{+}\to p \mu^{+}\mu^{-})=(8.6^{+6.6}_{-5.4}\pm5.5)\times10^{-8}$ [@Park:2005ek]. The predicted $m_{\mu\mu}$ distribution shows no peaks near 214 MeV (or elsewhere) and is slightly flatter than the form factor used by HyperCP. This leads us to conclude that the probability of having the three events at the same invariant mass is about $0.5 \%$. Furthermore, the lower end of the range predicted for the rate is consistent with no events for HyperCP, allowing for the possibility of all three events being consistent with new physics.
A new Particle with mass 214 MeV?
=================================
We now turn to the interpretation of the 3 HyperCP events as a new particle [@Park:2005ek] with $M_{P^0}= 214.3~MeV$ and ${\cal B}(\Sigma^{+}\to p \mu^{+}\mu^{-})_{P^0} =
(3.1^{+2.4}_{-1.9}\pm 1.5)\times 10^{-8}
$. The observation implies that this hypothetical new light state, $P^0$, is short lived, does not interact strongly, is narrow and decays only into $\mu^+\mu^-$, $e^+e^-$ or $\gamma\gamma$, and has a $\Delta S =1$, $\Delta I = 1/2$ coupling to $\bar{s} d$ quarks. There are three questions to be answered and we address them in order. Why hasn’t it been seen before? Is there a candidate for such a state? Where else could it be observed?
Why hasn’t it been seen before?
-------------------------------
The most stringent constraint on a possible new particle $P^0$ is its non-observation in kaon decay. After all, the modes $K \to \pi \mu^+ \mu^-$ proceed via the same quark level transition as $\Sigma^+ \to p \mu^+ \mu^-$: $s\to d\mu^+ \mu^-$. Of the three experiments that have studied these modes: BNL865 [@Ma:1999uj], HyperCP [@Park:2001cv] and NA48 [@Batley:2004wg] the one with most statistics was BNL865 [@Ma:1999uj] with 430 events, 30 of which were in their lowest bin $2m_\mu \lsim m_{\mu\mu} \lsim 225$ MeV where the signal would have been observed. Their observation shows no peaks in the $m_{\mu\mu}$ distribution, which is consistent with long distance SM physics. On that basis, the most optimistic scenario for the new physics hypothesis is to assume that all the 30 events in the first bin were due to $P^0$ which leads to a 95% confidence limit bound ${\cal B}(K^+ \to \pi^+ P^0)\leq 8.7 \times 10^{-9}$ [@He:2006uu] (assuming that statistical errors dominate). This translates into a rate for $\Sigma^+ \to p P^0$ some 25 times too small to explain the HyperCP events. Similar results are obtained from the other kaon experiments, none of which saw a peak in their $m_{\mu\mu}$ distribution.
Another constraint arises from the non-observation of the hypothetical new particle in $b \to s \mu^+ \mu^-$. In this case both Belle and BaBar [@bbounds] have results that can be interpreted as a 95% confidence level bound [@He:2006uu] ${\cal B}(B\to X_s P^0) \leq 8 \times 10^{-8}$.
In Figure \[sketch\], we can see schematically how it is possible for the new state to be observed in $\Sigma$ decay while not in $K^+$ decay: the kaon decay modes with only one pion in the final state only constrain the effective $|\Delta S| =1$ scalar coupling of the new state whereas the $\Sigma$ decay is sensitive also to the effective $|\Delta S| =1$ pseudoscalar coupling. Any viable model for $P^0$ will then have an effective scalar coupling about 25 times smaller than the corresponding pseudoscalar coupling [@He:2005we].
In a similar manner, the constraints from $B$ decay require that the effective $bs$ coupling of $P^0$ be about an order of magnitude smaller than the corresponding $sd$ coupling scaled by $m_b/m_s$ and $(V_{ts}V_{tb}^\star) /(V_{ts}V_{td}^\star)$. The latter scaling is the appropriate one for one-loop Higgs penguins dominated by a top-quark and a $W$ boson in the intermediate state. A successful model for $P^0$ can not have these penguin diagrams dominating the effective FCNC of $P^0$ to down-type quarks.
We have also considered additional processes that can, in principle, constrain the interactions of the hypothetical $P^0$. $K-\bar{K}$ mixing allows an effective pseudoscalar coupling up to 50 times as large as required to explain the 3 HyperCP events. $K_L\to \mu^+\mu^-$ combined with the muon $g-2$ allow an effective pseudoscalar coupling as large as required. The muon $g-2$ allows a $P^0$ coupling to muons $g_{P\mu} \lsim 5 \times 10^{-4}$ which interestingly is about $m_\mu/v$ [@He:2005we; @others].
Is there a candidate for $P^0$?
-------------------------------
The possibility that $P^0$ is a light sgoldstino has been explored to some extent in the literature [@sgold]. Here, we pursue the possibility that $P^0$ is a light Higgs boson. For detailed phenomenology of kaon and hyperon decays involving a light Higgs particle it is necessary to recall that there are two types of contributions that are generally of similar size [@He:2006uu]. There are two-quark “Higgs penguin” contributions that arise at one loop order and depend on the details of the flavor changing sector of the model. There are also “four-quark” contributions arising from a tree-level, SM $W$ mediated $|\Delta S| =1$ decay, in which the light Higgs is radiated from any of the $u,d,s$ quarks or the $W$ boson via the tree-level flavor diagonal couplings of the Higgs. Both of these contributions can be calculated in chiral perturbation theory [@lighth], and we do so at leading order. Given our discussion in the previous section we concentrate on CP-odd or pseudoscalar Higgs bosons.
One possible candidate for $P^0$ is the $A_1^0$ of the NMSSM. The Higgs sector of the NMSSM contains the usual two Higgs doublets $H_D$ and $H_U$ that appear in the MSSM plus the Higgs singlet N. In the physical spectrum there are two CP-odd scalars, of which the $A_1^0$ is the lightest. It has been proposed in the literature that this $A_1^0$ can be naturally light due to a global $U(1)$ symmetry [@Dobrescu:2000yn].
The main features of the couplings of the $A_1^0$ to SM fields are as follows. Its coupling to $Zh$ ($h$ being the lightest CP even Higgs) is suppressed by $\tan\beta$ with respect to the MSSM $ZhA$ coupling allowing an evasion of LEP bounds in the large $\tan\beta$ regime. Its couplings to quarks are also suppressed by $\tan\beta$ with respect to those of the $A$ in the MSSM. This results, for large $\tan\beta$, in negligible couplings to up-type quarks. The couplings to down-type quarks are independent of $\tan\beta$ and can be written in terms of one parameter, $l_d$, which can be of order one [@Hiller:2004ii]: ${\cal L} =-l_d^{} m_d^{}\,\bar d\gamma_5^{}d(i A^0_1)/v
-l_d^{} m_\ell^{}\,\bar \ell\gamma_5^{}\ell(i A^0_1)/v +\cdots $.
The four-quark contributions to $A_1^0$ production in light meson and hyperon decay are thus proportional to $l_d$ and independent of other parameters in the model. It is then straightforward to compute these contributions to the HyperCP case. We find [@He:2006fr], ${\cal B}_{4q}(\Sigma^+\to p A_1^0) = 1.7 \times 10^{-7} |l_d|^2$, which matches the central value of the HyperCP result for $l_d\sim 0.4$. The bad news is that this then leads to ${\cal B}_{4q}(K^+\to \pi^+ A_1^0) \sim 10^{-6}$, two orders of magnitude larger than the limit from BNL E865. The conclusion illustrated by this calculation is that it is relatively easy to have a light Higgs that matches the HyperCP observation but it is very hard to avoid seeing it in kaon decay as well.
However, there are also the two-quark contributions to the amplitudes and it is possible to arrange a cancellation between amplitudes that satisfies the kaon bounds. The two-quark contributions are much more model dependent than the four-quark contributions, but also suffer from additional constraints due to non-observation of $P^0$ in $B$ decay. We have not performed a full parameter scan, but rather illustrated that it is possible to satisfy all constraints. To this effect we start with the specific model considered by Hiller [@Hiller:2004ii] and modify it accordingly. To suppress the FCNC in $B$ decay we consider $m_{\tilde{t}}=m_{\tilde{c}}$ and negligible squark mixing. The strength of the two-quark contribution to kaon decay is then tuned with $m_{\tilde{u}}-m_{\tilde{c}}$. We further consider (large) $\tan\beta = 30$, $m_{\tilde{t}}\sim 2.5$ TeV and $-\lambda x =150$ GeV to obtain neutralino masses in the 100-1500 GeV range [@He:2006fr].
. \[fig:result\]
In Figure \[fig:result\] we show our results [@He:2006fr]: the light shaded region corresponds to parameters that reproduce the HyperCP observation. The dark shaded region corresponds to those points that also satisfy the kaon bounds. As mentioned before the overlapping region is significantly smaller due to the cancellation required to satisfy the kaon bounds.
Where else can $P^0$ be observed?
---------------------------------
Finally, we explore other processes that can test the new particle hypothesis for the HyperCP result. We begin by considering only the effect of two-quark operators, assuming that the existing kaon bounds are bypassed because the effective $sd$ coupling is pseudoscalar. In this case the new state would show up in kaon decay modes with two pions in the final state and we can easily derive from the HyperCP measurement that (the errors reflect the experimental error only) [@He:2005we] $$\begin{aligned}
{\cal B}(K_L \to \pi^+\pi^- P^0) &\approx & (1.8^{+1.6}_{-1.4})\times 10^{-9} \nonumber \\
{\cal B}(K_L \to \pi^0\pi^0 P^0) &\approx & (8.3^{+7.5}_{-6.6})\times 10^{-9}.
\label{q2res}\end{aligned}$$ Both of these represent very significant enhancements over the corresponding SM rates and may be accessible to KTeV or NA48. In a similar manner this scenario results in [@He:2005we; @Deshpande:2005mb] $$\begin{aligned}
{\cal B}(\Omega^- \to \Xi^- P^0) &\approx & (2.0^{+1.6}_{-1.2})\times 10^{-6}.\end{aligned}$$ The best upper bound for this mode, also from HyperCP [@solomey], is $6.1 \times 10^{-6}$.
If the new state $P^0$ is a light Higgs, then there are other processes that are sensitive only to its flavor diagonal couplings [@Prades:1990vn] (or four-quark operators). For example the modes $V \to \gamma A_1^0$ have been proposed in the literature [@Mangano:2007gi]. The results are that ${\cal B}(\Upsilon_{1S}\to \gamma A_1^0)$ can reach about $1 \times 10^{-4} l_d^2$ and may be accessible to the B factories. Similarly ${\cal B}(\phi\to \gamma A_1^0$ can reach $1.4\times 10^{-8} l_d^2$ and may be accessible to DA$\Phi$NE [@Mangano:2007gi]. In a similar spirit we have proposed the modes $\eta \to \pi \pi A_1^0$ where we can predict [@He:2008zw] ${\cal B}(\eta \to \pi^+ \pi^- A_1^0) = 5.4 \times 10^{-7}l_d^2$, again possibly accessible to DA$\Phi$NE.
When the four-quark contributions are added to the two-quark contributions in the NMSSM (using parameters as in Hiller [@Hiller:2004ii] and Xiandong [@Xiangdong:2007vv]) the results of Eq. \[q2res\] are modified. An example of the resulting predictions for the rate of the kaon modes is shown in Fig. \[fig:fullresult\]. Full details can be found in the paper [@He:2008zw], but the $x$-axis is related to the strength of the two-quark contribution though an effective $g_P$ and the strength of the four-quark contribution is kept fixed. The dotted curves result from the two-quark contributions alone. The shaded (pink) bands indicate the allowed ranges of $C_L^{}$$-$$C_R^{}$ when the two and four-quark contributions have the same sign [@He:2008zw]. Each vertical (green) dashed line corresponds to the special case [@He:2006fr] of chargino dominated penguins.
Conclusions
===========
The decay $\Sigma^+ \to p \mu^+\mu^-$ within the SM is long distance dominated and the predicted rate is in the right range to explain the HyperCP observation. However, the predicted $m_{\mu\mu}$ distribution makes it unlikely to find the three events at the same mass ($P\lsim 0.8\%$). Existing constraints from kaon and B physics allow a new particle interpretation of the HyperCP result provided that the FCNC couplings of the new particle are mostly pseudoscalar and smaller for $b\to s$ transitions than naive scaling with CKM angles would predict.
The NMSSM has a CP-odd Higgs boson, the $A_1^0$ that could be as light as the required 214 MeV. Its diagonal couplings to quarks and muons in the large $\tan\beta$ limit can have the right size as well. There are several modes that can test this hypothesis independently from the details of the flavor changing sector of the model: $\Upsilon_{1S}\to \gamma A_1^0$, $\phi\to \gamma A_1^0$ and $\eta \to \pi\pi A_1^0$.
It is harder to suppress the effective scalar $sd$ coupling that appears in this model to the level required to satisfy the existing kaon bounds, but it is possible for certain values of the relevant parameters. The measurement of one of the modes $K_L \to \pi \pi \mu^+ \mu^-$ can confirm or refute this scenario.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was done in collaboration with Jusak Tandean and Xiao-Gang He. It was supported in part by DOE under contract number DE-FG02-01ER41155.
References {#references .unnumbered}
==========
[99]{} H. Park [*et al.*]{} \[HyperCP Collaboration\], Phys. Rev. Lett. [**94**]{}, 021801 (2005) \[arXiv:hep-ex/0501014\]. L. Bergstrom, R. Safadi, and P. Singer, Z. Phys. C [**37**]{}, 281 (1988). X.G. He, J. Tandean, and G. Valencia, Phys. Rev. D [**72**]{}, 074003 (2005) \[arXiv:hep-ph/0506067\]. H. Ma [*et al.*]{} \[E865 Collaboration\], Phys. Rev. Lett. [**84**]{}, 2580 (2000) \[arXiv:hep-ex/9910047\]. H.K. Park [*et al.*]{} \[HyperCP Collaboration\], [*ibid*]{}. [**88**]{}, 111801 (2002) \[arXiv:hep-ex/0110033\]. J.R. Batley [*et al.*]{} \[NA48/1 Collaboration\], Phys. Lett. B [**599**]{}, 197 (2004) \[arXiv:hep-ex/0409011\]. X.G. He, J. Tandean, and G. Valencia, Phys. Rev. D [**74**]{}, 115015 (2006) \[arXiv:hep-ph/0610274\], and references therein. B. Aubert [*et al.*]{} \[BABAR Collaboration\], Phys. Rev. Lett. [**93**]{}, 081802 (2004) \[arXiv:hep-ex/0404006\]; M. Iwasaki [*et al.*]{} \[Belle Collaboration\], Phys. Rev. D [**72**]{}, 092005 (2005) \[arXiv:hep-ex/0503044\]. X.G. He, J. Tandean, and G. Valencia, Phys. Lett. B [**631**]{}, 100 (2005) \[arXiv:hep-ph/0509041\]. Other studies leading to some of the conclusions in this section are: N.G. Deshpande, G. Eilam, and J. Jiang, Phys. Lett. B [**632**]{}, 212 (2006) \[arXiv:hep-ph/0509081\]. C.Q. Geng and Y.K. Hsiao, Phys. Lett. B [**632**]{}, 215 (2006) \[arXiv:hep-ph/0509175\], C.H. Chen and C.Q. Geng, Phys. Lett. B [**645**]{}, 189 (2007) \[arXiv:hep-ph/0612142\]; D.S. Gorbunov and V.A. Rubakov, Phys. Rev. D [**73**]{}, 035002 (2006) \[arXiv:hep-ph/0509147\]; S.V. Demidov and D.S. Gorbunov, JETP Lett. [**84**]{}, 479 (2007) \[arXiv:hep-ph/0610066\]. See for example, J.F. Gunion, H.E. Haber, G.L. Kane, and S. Dawson, SCIPP-89/13, and references therein. In particular, also B. Grzadkowski and J. Pawelczyk, Phys. Lett. B [**300**]{}, 387 (1993). B.A. Dobrescu and K.T. Matchev, JHEP [**0009**]{}, 031 (2000) \[arXiv:hep-ph/0008192\]. G. Hiller, Phys. Rev. D [**70**]{}, 034018 (2004) \[arXiv:hep-ph/0404220\]. X.G. He, J. Tandean, and G. Valencia, Phys. Rev. Lett. [**98**]{}, 081802 (2007) \[arXiv:hep-ph/0610362\]. N. Solomey, DPF 5-8 April 2003, Philadelphia, Pennsylvania, USA.
M.L. Mangano and P. Nason, Mod. Phys. Lett. A [**22**]{}, 1373 (2007) \[arXiv:0704.1719 \[hep-ph\]\]. X. G. He, J. Tandean and G. Valencia, arXiv:0803.4330 \[hep-ph\]. J. Prades and A. Pich, Phys. Lett. B [**245**]{}, 117 (1990).
G. Xiangdong, C.S. Li, Z. Li, and H. Zhang, arXiv:0712.0257 \[hep-ph\].
|
---
abstract: 'We present functional forms allowing a broader range of analytic solutions to common economic equilibrium problems. These can increase the realism of pen-and-paper solutions or speed large-scale numerical solutions as computational subroutines. We use the latter approach to build a tractable heterogeneous firm model of international trade accommodating economies of scale in export and diseconomies of scale in production, providing a natural, unified solution to several puzzles concerning trade costs. We briefly highlight applications in a range of other fields. Our method of generating analytic solutions is a discrete approximation to a logarithmically modified Laplace transform of equilibrium conditions.'
author:
- 'Michal Fabinger[^1]'
- 'E. Glen Weyl[^2]'
bibliography:
- 'FunctionalFormsForTractableEconomicModels.bib'
date: August 2018
title: |
Functional Forms for Tractable Economic Models\
and the Cost Structure of International Trade[^3]
---
Introduction {#SectionIntroduction}
============
Analytic solutions have played a major role in many fields of economics. They are useful both as closed-form, pencil-and-paper solutions to applied theory models, and as components (subroutines) of larger models, making them more computationally tractable.[^4] In this paper, we substantially expand the class of known analytic solutions to a broad class of standard economic models. [ ]{}We then illustrate the application of these ideas to a computationally-intensive model of international trade that helps resolve a long-standing puzzle about trade costs by allowing more realistic functional forms of such costs.[^5]
We observe that most frequently used functional forms that lead to analytic solutions in economics, namely linear and constant-elasticity functions, share a convenient property: They preserve functional forms under transformations that we refer to as [“]{}average-marginal transformations[”]{}. That is, the functional form of the average value of the function is the same as that of its derivative. [ ]{}Formally, we say that a functional form class is preserved by average-marginal transformations if for any function $F(q)$ the class also contains any linear combination of $F(q)$ and $q
F'(q)$.[^6] We then find all functions that have such property.
Within this class, we identify functional forms that have a given level of [“]{}algebraic tractability[”]{}, a property we define. These are linear combinations of power functions satisfying certain conditions. When used to represent demand and cost curves they lead to economic optimization conditions that may be transformed to polynomial equations of a degree smaller than 5. These, in turn, may be solved explicitly by the method of radicals. This substantially generalizes the simple analytic solutions that economists are familiar with in the case of constant marginal cost and either linear or constant-elasticity demand. Even beyond degree 5, precise solutions to such polynomial equations are available at minimal cost in standard mathematical software.
We show that elements of functional form classes preserved by average-marginal transformations also have advantageous properties when applied to aggregation over heterogeneous firms in monopolistically competitive models: They lead to closed-form aggregation integrals under very flexible assumptions. This means that a problem with a continuum of heterogeneous firms may be reduced to a set of explicit equations at the macroeconomic level.
In our method, the existence of closed-form solutions to optimization conditions sometimes requires parameter restrictions involving parameters both from the supply side and the demand side. These restrictions may or may not be approximately satisfied in a particular market. [ ]{}If they are not satisfied, one may be tempted to conclude that our method is not applicable. Most likely this is the reason why economists have not found (or have not attempted to find) the kind of solutions we discuss in our paper.
We explain, however, that the range of applicability of our method is larger than it may seem at first sight as this issue does not pose a large problem. Even if the parameter restrictions are not satisfied, analytic solutions [ ]{}at other parameter values may be used to construct an interpolation that covers parameter values of interest. In this way one can extend the usefulness of our analytic method. Another way is to realize that a given demand or cost function may be *approximated* by functions that satisfy our restrictions, in which case the restrictions are satisfied by choice.[^7]
While our approach is useful in many fields of economics, as we illustrate, the main application we focus on in this paper belongs to the field of international trade. Analytic tractability has been important for international trade to the extent that almost all models assumed constant marginal costs of both production and logistics/shipping. Under such assumptions trade models are much more straightforward to solve. Yet, as we discuss in detail, these assumptions contrast with models of cost used by the logistics managers that economists are presumably attempting to describe. As we show, our functional forms preserve analytical tractability while allowing the realism of matching such models.
Our primary application in this paper shows how such more realistic models of cost can help resolve the trade cost puzzle in a model of world trade flows with heterogeneous firms.[^8] Standard models of international trade attribute the observed rapid falloff of trade flows with distance to trade costs that increase dramatically with distance. But we have no reason to believe that such dramatic distance dependence of trade costs exists in the real world. Container shipping charges depend on distance only modestly, and in any case, represent only a tiny fraction of the value of the transported goods. [ ]{}A similar statement holds for the so-called iceberg trade costs, i.e., the damage of goods during transportation: We know goods typically do not get damaged during transport, and if they do, the damage probability is unlikely to strongly increase with the distance a shipping container traveled over the ocean. While trade costs may be sizeable, they are much more likely to be associated with shipment preparation and coordination or with loading and unloading, rather than with the distance traveled over the ocean. For this reason, the rapid falloff of trade with distance represents a puzzle from the point of view of standard models of international trade.[^9]
Our model resolves this puzzle in a very natural way. Firms find it costly to produce larger quantities due to increasing marginal costs of production. At the same time, they find it beneficial to concentrate their exports to a few destinations due to economies of scale in shipping. With this cost structure, even a small cost advantage of a particular destination will be enough to make the firm export there instead of other destinations. If trade costs are slightly smaller for closer destinations, this cost advantage will lead to substantially larger trade flows at smaller distances and substantially smaller trade flows at larger distances.
The model also resolves a puzzle related to firm entry into export markets. Although it is not as widely discussed as the trade cost puzzle, it is a clear empirical regularity that models with constant marginal costs cannot address in a natural way. In the data, one can often see two similar firms, say, from China, one exporting to, say, Portugal and not to Greece and the other exporting to Greece and not to Portugal. To reconcile such patterns with the assumption of constant marginal cost of production, standard international trade models introduce stochastic cost shocks specific to each firm-destination pair. These cost shocks have to be dramatically large. For the second firm they need to offset the entire profit the first firm makes from its exports to Portugal. In the absence of any real-world phenomenon that could lead to cost shocks of this kind, this represents a puzzle.[^10]
Our model explains this puzzle in a straightforward way. With increasing marginal costs of production and economies of scale in shipping, firms need to solve a combinatorially difficult problem of choosing export destinations.[^11] Different approximate solutions of this choice problem can lead to different sets of export destinations, even if the maximized profits are almost the same. One approximately optimal set of export destinations may include Portugal, while another one may include Greece.
We solve the model using an iterative method involving an outer loop and an inner loop. The outer loop requires an evaluation of firms[’]{} profit functions for many discrete choices of export destinations. Our functional forms bring a tractability advantage that makes these evaluations fast. In the inner loop, we solve for a general equilibrium of the world economy keeping the discrete choices fixed. There using our functional forms is helpful because it allows for an analytic calculation of derivatives that are needed for accelerated gradient descent algorithms.
Separately, we develop many other applications of the proposed functional forms. For the model of outsourcing decisions in a sequential supply chain constructed by @antras, we reformulate the theory to simplify the analysis and use this new formulation to apply our functional forms. This allows us to show that for more realistic demand functions, outsourcing occurs at both the early (viz.$\, $raw materials) and late (viz.$\, $final commercial sales) stages of production, while intermediate stages are performed in-house, corresponding to common observation of outsourcing patterns. For a model of labor bargaining by @stole [@stole2], we tractably generalize the closed-form solutions that have been found for linear or constant-elasticity demand and show that for realistic demand patterns the employment effects of bargaining have interesting and intuitive cyclical patterns. We also discuss applications to imperfectly competitive supply chains, two-sided platforms, selection markets, auctions, and, extensively, monopolistic competition.
Finally, we show that our method may be thought of as a discrete approximation to a logarithmically modified Laplace transform. It may also be thought of as a sieve method of non-parametric econometrics. In addition, the transformed variables reveal economic properties of demand functions that would appear accidental otherwise.
The next section provides a quick illustration of our functional forms with a focus on modeling demand under income inequality. Section \[sec:FunctionalFormsForAveragesAndMarginals\] presents our main theoretical results. Section \[SectionWorldTrade\] focuses on modeling world trade. Section \[SectionBreadthOfApplication\] discusses other applications. Section \[SectionArbitraryDemandAndCostFunctions\] develops the theory connecting our tractable functions to a logarithmically modified Laplace transform. The paper also includes an appendix and supplementary material.
Example: Replacing Constant-Elasticity Demand {#SectionReplacingConstantElasticityDemand}
=============================================
Constant-elasticity demand and its flexible replacement {#ConstantElasticityDemandAndItsFlexibleReplacement}
-------------------------------------------------------
The most canonical and widely-used demand form in economic analysis is the constant-elasticity specification, corresponding to inverse demand $P(q)=a
q^{-b}$. It is frequently used because of its analytic tractability. Historically, it appeared in the economic literature because in discrete-choice [ ]{}settings it is plausible that product[’]{}s valuations follow the income distribution and the income distribution was believed to be approximately Pareto, i.e., power-law.[^12] Modern data of income, however, led to different conclusions on the shape of the income distribution.[^13]
In this section we discuss another demand form that is also highly analytically tractable but has more flexibility. This flexibility allows us to get a much better match to the income distribution.[^14] As an illustration, we show that our proposed demand form leads to substantially different policy implications than the constant-elasticity form in the socially important case of bias of innovative technical progress.
![Comparing the fit of the best-fit lognormal and the best-fit constant-elasticity form to a double-Pareto lognormal estimation of the 2012 US income distribution, represented as a demand function (reversed quantile function). [ ]{}Dollars at any reversed quantile represent the income of the corresponding individual. [ ]{}On the left is the fit for the full income distribution, while the right shows the upper tail. [ ]{}We used a standard calibration of a double Pareto lognormal proposed by @reed and used the generalized method of moments to find the constant-elasticity demand function that best fits this throughout the full range of the income distribution. In the upper tail the constant elasticity approximation is a bit better of a fit than the lognormal. However, in the rest of the income distribution its fit is terrible, while the lognormal fits quite well (although in economic models it is harder to work with).[]{data-label="CEfigure"}](Figure1.pdf "fig:"){width="3.4in"} ![Comparing the fit of the best-fit lognormal and the best-fit constant-elasticity form to a double-Pareto lognormal estimation of the 2012 US income distribution, represented as a demand function (reversed quantile function). [ ]{}Dollars at any reversed quantile represent the income of the corresponding individual. [ ]{}On the left is the fit for the full income distribution, while the right shows the upper tail. [ ]{}We used a standard calibration of a double Pareto lognormal proposed by @reed and used the generalized method of moments to find the constant-elasticity demand function that best fits this throughout the full range of the income distribution. In the upper tail the constant elasticity approximation is a bit better of a fit than the lognormal. However, in the rest of the income distribution its fit is terrible, while the lognormal fits quite well (although in economic models it is harder to work with).[]{data-label="CEfigure"}](Figure1tail.pdf "fig:"){width="3.4in"}
![Comparing the fit of the best-fit lognormal and the best-fit quadratically solvable form to a double-Pareto lognormal estimation of the US Income distribution, represented as a demand function (reversed quantile function). [ ]{}Dollars at any reversed quantile represent the income of the corresponding individual. [ ]{}On the left is the fit for the full income distribution, while the right shows the upper tail.[]{data-label="ourfigure"}](Figure2.pdf "fig:"){width="3.4in"} ![Comparing the fit of the best-fit lognormal and the best-fit quadratically solvable form to a double-Pareto lognormal estimation of the US Income distribution, represented as a demand function (reversed quantile function). [ ]{}Dollars at any reversed quantile represent the income of the corresponding individual. [ ]{}On the left is the fit for the full income distribution, while the right shows the upper tail.[]{data-label="ourfigure"}](Figure2tail.pdf "fig:"){width="3.4in"}
Consider the task of representing the empirical income distribution using a corresponding demand function. Constant-elasticity demand fails this purpose, as illustrated in Figure \[CEfigure\]. This is because the income distribution is not Pareto but approximately double Pareto lognormal [@reedjorgensen; @doublepareto; @doubleparetolognormal]. [ ]{}Working with the double Pareto lognormal distribution (or with the more loosely fitting lognormal distribution) in economic models would be quite difficult.[^15] [ ]{}To overcome this difficulty, we propose a functional form that allows for the same basic shape, but leads to calculations almost as easy as for the constant-elasticity form:
$$P(q)=m-\\
ma\\_{-}\left(\frac{q}{q\\_0}\right)^{-b}-\\
ma\\_{+}\left(\frac{q}{q\\_0}\right)^b,\\
\, \, \, \, \, \, \,\\
\hspace*{0.5ex} \, \, \, \, \, \,\\
a\\_{-}\equiv 1-a\\_{+}.\\
\label{EquationQuadraticallyTractableFormForIncomeDistribution}$$
A set of parameter values that matches the income distribution (for the US in 2012) very well is $a_-=-1/2, a_+=5/2,
b=2/5$. We obtained these values by a generalized-method-of-moments fit and rounding the results. The match is illustrated in Figure \[ourfigure\].
Bias in technological progress {#BiasInTechnologicalProgress}
------------------------------
As a simple, illustrative application, we discuss the case of bias in technological progress described in @kremer [@worstcasebounds]. First, we do that for the case of constant-elasticity demand and explain why it is highly tractable. Then we turn to our proposed demand form in Equation \[EquationQuadraticallyTractableFormForIncomeDistribution\] and show that it preserves a high degree of tractability that constant-elasticity demand has.
When the private sector decides which products to develop, it chooses profit-making products, not necessarily those products that create the greatest social value. This bias in technological progress depends on the discrepancy between private and social gains. @kremer [@worstcasebounds] consider the fraction of the social gains from creating a new product that may be appropriated by a monopolist[^16], referred to as the appropriability ratio, and show that the maximal fraction of potential surplus that may be lost due to imperfect appropriability is equal to one minus this appropriability ratio. They compare different demand functions since they lead to different bias in research and development, but always assume no costs. Here we assume a fixed demand function and consider biases at different levels of marginal production cost. We walk quite didactically through the process of solving the model in order to illustrate the source of the tractability of the constant-elasticity form and why it carries over to our proposed generalized form but not to the lognormal distribution form. [ ]{}We then follow @worstcasebounds[’]{}s argument that a sensible demand function is one matching the world income distribution and use this as motivation for using our form to study the impact of cost on the appropriability ratio, which is very different under our form than under constant elasticity.
Consider a monopolist with a constant marginal cost $c$ and constant-elasticity (inverse) demand $P(q)=a q^{-b}$. Her marginal revenue is $P(q)+P'(q)q$. Under the constant elasticity form, $P'(q)q=-a b q^{-b}$, which has the same form as $P(q)$, just a different multiplicative constant out front. For this reason, the marginal revenue has the same form as well: $\text{MR}(q)=a (1-b) q^{-b}$ . The monopolist optimally equates it to the marginal cost, so the optimal quantity may be determined by solving the linear equation $a(1-b)x=c$ with $x\equiv q^{-b}$, yielding $q=(a(1-b)/c)^{1/b}$. From this it follows by substitution that the firm[’]{}s absolute markup is $\overline{\text{PS}}\equiv \text{PS}/q=c b/(1-b)$, where $\text{PS}$ stands for the producer surplus. Furthermore, the average consumer surplus also has the same form as $P(q)$, differing only by a multiplicative constant: $$\overline{\text{CS}}\equiv \frac{\text{CS}}{q}=\frac{\int _0^qP\left(\tilde{q}\right)d\tilde{q}-P(q)q}{q}=\frac{\frac{a}{1-b}q^{1-b}-a q^{1-b}}{q}=\frac{a
b}{1-b}q^{-b}.$$ Evaluated at the optimal quantity, the average consumer surplus is $\overline{\text{CS}}= c b(1-b)^{-2}$. The appropriability ratio, i.e., the ratio of producer surplus and the total surplus, may be evaluated as $\overline{\text{PS}}/\left(\overline{\text{PS}}+\overline{\text{CS}}\right)=(1-b)/(2-b)$, which is a constant independent of cost. Thus all products have precisely the same appropriability ratio, and cost is irrelevant to the bias of investments in research and development.
If we tried to investigate this problem in a tractable way for more general demand functions that have been used in the economic literature, we could use the Bulow-Pfleiderer demand introduced in the next section, which includes both constant-elasticity and linear demand as special cases. However, we would again find that the cost $c$ has no impact on the bias of technical progress.
If instead, we tried to investigate the implications of demand curves corresponding to other distributions of product valuations, such as the lognormal distribution or the double-Pareto lognormal distribution (which fits the income distribution), we would quickly find that such investigation cannot be carried out analytically.[^17]
Here we show that working with the demand form in Equation \[EquationQuadraticallyTractableFormForIncomeDistribution\] is much easier and elegant and leads to substantive economic results. Its marginal form $P'(q)q$ has the same functional form as $P(q)$ itself:$$P'(q)q= m b a_-\left(\frac{q}{q_0}\right)^{-b}-m b a_+\left(\frac{q}{q_0}\right)^b.$$ If we introduce the notation$$a_{n,-}\equiv (1-b)^na_-,\text{$\, \, \, \, \, \, $ }a_{n,+}\equiv (1+b)^na_+,\text{$\, \, \, \, \, $ }x \equiv \left(\frac{q}{q_0}\right)^b,$$ the monopolist[’]{}s first-order condition is just the quadratic equation$$-\text{ }a_{1,-}+\left(1-\frac{c}{m}\right) x-\text{ }a_{1,+}x^2=0.$$ This leads to the closed-form solution$$q=q_0x^{1/b},\, \, \, \, \, \, x=\frac{1}{2 a_{1,+}}\left(1-\frac{c}{m}+\sqrt{\left(1-\frac{c}{m}\right)^2-4 a_{1,-} a_{1,+}}\right).$$ The per-unit consumer and producer surplus again take the same functional form:$$\overline{\text{CS}}=- m b\text{ }\tilde{a}_-\left(\frac{q}{q_0}\right)^{-b}+m b\text{ }\tilde{a}_+ \left(\frac{q}{q_0}\right)^b\text{
},\, \, \, \, \, \, \overline{\text{PS}}=m-c-m a_-\left(\frac{q}{q_0}\right)^{-b} -m a_+ \left(\frac{q}{q_0}\right)^b .$$ The appropriability ratio is then$$\frac{\overline{\text{PS}}}{\overline{\text{PS}}+\overline{\text{CS}}}= \frac{m-c-m\text{ }a_-\left(q\left/q_0\right.\right)^{-b} -m\text{
}a_+\left(q\left/q_0\right.\right)^b }{m-c+m a_{-1,-}\left(q\left/q_0\right.\right)^{-b} +m a_{-1,+}\left(q\left/q_0\right.\right)^b }=\frac{\left(1-b^2\right)
\left(-a_-+a_+ \left(q\left/q_0\right.\right)^{2 b}\right)}{-\left(2+b-b^2\right) a_-+\left(2-b-b^2\right) a_+ \left(q\left/q_0\right.\right)^{2
b}},$$ where the last equality was obtained by substituting for the marginal cost from the first-order condition. [ ]{}Substituting the parameter values we specified right after Equation \[EquationQuadraticallyTractableFormForIncomeDistribution\] gives$$\frac{\overline{\text{PS}}}{\overline{\text{PS}}+\overline{\text{CS}}}=\frac{ 21+105\left(q\left/q_0\right.\right)^{4/5}}{56+180 \left(q\left/q_0\right.\right)^{4/5}}.$$ This equals $21/56\approx 37.5\%$ for $q=0$ (when the product serves a tiny fraction of the population) and monotonically increases in q to [ ]{}$53/118\approx 53.4\%$ for $q=q_0$ (when most of the population is served). [ ]{}This suggests a bias towards cheap, mass-market products and away from expensive products that mostly cater to the rich; of course, all this analysis is based, like @kremer[’]{}s, on aggregate surplus and might well reverse if distributional concerns were incorporated.$\quad $
While we focused here on biases from the appropriability ratio, it can be shown (in closed-form) that many other aspects of standard intellectual property policy differ substantially under our form from the results under the constant pass-through class. For example, under our form the ratio of consumer surplus to monopoly deadweight loss is much greater (usually by several times) than under the Bulow-Pfleiderer class so that patents are more desirable and optimal patent protection is greater than under the standard forms. Similarly allowing pharmaceutical producers to price discriminate often increases deadweight loss under the standard forms [@acv], while it is always beneficial under our form. Thus the standard forms are substantively misleading on a number of issues and the added complexity of using our form is minimal.
Central Results\[sec:FunctionalFormsForAveragesAndMarginals\]
=============================================================
In the previous section we focused on a particular functional form derived from our theory, a particular calibration target (the US income distribution) and a particular application. However, our approach applies much more broadly. We characterize all functional forms that have the useful property of the form above: namely that, in the language of demand curves, linear combinations of marginal revenue and inverse demand take the same form as inverse demand itself. Within these we then identify functional forms that lead to closed-form solutions utilizing power functions and the method of radicals.
Form preservation under the average-marginal transformation
-----------------------------------------------------------
Let us denote by $F\left(q\right)$ the average of an economic variable that depends on $q$, where a baseline interpretation of $q$ is a quantity of a good. The marginal variable is then $\left(qF\left(q\right)\right)'=F\left(q\right)+qF'\left(q\right)$. We now formally define what it means for these two variables to have the same functional form, as we alluded to in the previous section.
***(Form Preservation)*** We say that a functional form class $\mathcal{C}$ is [*form-preserving*]{} under average-marginal transformations if for any function $F(q)\mathcal{\in C}$, the class also contains any linear combination of $F(q)$ and $qF'\left(q\right)$. In other words, $F\mathcal{\in C}\Rightarrow\mathcal{\forall}\left(a,b\right)\in\mathbb{R}^{2}:\ aF+bqF'\in\mathcal{C}$. In economic terms, we interpret $F\left(q\right)$ as the average of the variable $qF\left(q\right)$, such as revenue or cost, and $F\left(q\right)+qF'\left(q\right)$ as its marginal counterpart. This definition thus states that any linear combination of the average and marginal variables belong to the defined class of functions.[^18]
Obviously, if $\mathcal{\mathcal{C}}$ is taken to be a sufficiently large (e.g. infinite-dimensional) class of functions it may be form-preserving in a fairly mechanical way. For example, if it is the set of all analytic functions with the domain $\left(0,\overline{q}\right)$ for some $\overline{q}$ then we know that $a F\left(q\right)+b q F'\left(q\right)$ is also analytic and has at least as large a domain. This observation is not very useful for the purposes of tractability because the set of all analytic functions with this domain contains many that, as we discussed in the previous section, are not tractable using standard analytic and computational methods.
Thus we will naturally wish to consider smaller classes. It is, therefore, useful to identify the most general set of finite-dimensional functional form classes that are form-preserving under the average-marginal transformations $F\rightarrow aF+bqF'$. Before stating the characterization theorem, let us briefly clarify what we mean by the dimensionality of a functional form class. For example, a functional form class $a_{1}e^{-a_{2}q}$, where $a_{1}$ and $a_{2}$ are continuously varying real numbers is two-dimensional, while $a_{1}e^{-a_{2}q}q^{-a_{3}}$ with continuously varying real $a_{1}$, $a_{2}$, and $a_{3}$ is three-dimensional.[^19]
\[formpreserve\]
***(Characterization of Form-Preserving Functions)*** Any real finite-dimensional functional form class with domain $(0,\infty)$ (or an open subinterval of it) that is form-preserving under average-marginal transformations must be a set of linear combinations of $$\left(\log q\right)^{a_{jk}}q^{-t_{j}},\quad a_{jk}=0,1,...,n_{j}^{(1)},\quad j=1,2,...,N_{1},$$ $$\left(\log q\right)^{b_{jk}}\cos\left(\tilde{t}_{j}\log q\right)q^{-\hat{t}_{j}},\quad b_{jk}=0,1,...,n_{j}^{(2)},\quad j=1,2,...,N_{2},$$ $$\left(\log q\right)^{c_{jk}}\sin\left(\tilde{t}_{j}\log q\right)q^{-\hat{t}_{j}},\quad c_{jk}=0,1,...,n_{j}^{(2)},\quad j=1,2,...,N_{2},$$ where $\left\{ t_{j}\right\} _{j=1}^{N_{1}}$, $\left\{ \tilde{t}_{j}\right\} _{j=1}^{N_{2}}$, and $\left\{ \hat{t}_{j}\right\} _{j=1}^{N_{2}}$ are fixed sets of real numbers and $N_1,N_2\in \mathbb N$ . If we exclude functions oscillating as $q\rightarrow0_{+}$, only the functions in the first row are allowed. In that case the most general form is the set of linear combinations of $$q^{-t_{j}},\quad q^{-t_{j}}\log q,\quad q^{-t_{j}}\left(\log q\right)^{2},\quad...\quad,q^{-t_{j}}\left(\log q\right)^{n_{j}},\quad j=1,2,...,N_{1}.$$
The proof is provided in Appendix \[AppendixProofsOfTheorems\].
Tractability
------------
We now provide a specific formal definition of “tractability” that allows us to characterize the class of form-preserving functional forms that have various levels of such tractability. While the term tractability is constantly invoked in economics papers to justify various “simplifying” assumptions, it is almost never defined formally.[^20] A potential reason for this is that there is no standard, clear definition within applied mathematics of the notion of tractability of the solution of mathematical equations. The theory of polynomial equations establishes that generic polynomial equations of degree at most four have solutions in terms of “the method of radicals” (roots of different orders) and that generic polynomial equations of higher degree have no such solutions, according to the Abel–Ruffini theorem. But this theory does not imply that one could not extend the list of “closed-form” functions by adding some other functions (other than roots) to provide solutions to higher order polynomials. In practice, polynomial equations of any reasonably low order (say less than a hundred) can be solved extremely rapidly by standard mathematical software [@allpolynomial].[^21]
For this reason, we use a specific definition of tractability, which we call *algebraic tractability*, that is very simplistic: an equation is algebraically tractable at some level $k$ if it can be solved using power functions and a solution to a polynomial equation of degree no greater than $k$. While this definition eliminates many other functions with known solutions, it does a good job capturing existing forms that are widely considered tractable while allowing an extension to richer forms in a pragmatic manner given the ease with which polynomial equations can be solved both analytically and computationally [@algebraic].
An important feature of the (non-oscillating) class of functional forms in Theorem \[formpreserve\] is that if we include terms with powers of logarithms we must also include all terms with powers of logarithms below this. That is, if the class includes linear combinations of $q\left(\log q\right)^{2}$ and $q^{-1/2}\left(\log q\right)^{2}$ it must also include linear combinations $q\log q$, $q^{-1/2}\log q$, $q$ and $q^{-1/2}$. With a small number of (explicitly enumerable) exceptions, classes of functional forms like this can rarely be solved in closed-form because of the combination of power and logarithmic terms.[^22]
On the other hand, the even-simpler class of sums of power functions nests all frequently-used tractable forms in the economic literature, namely constant-elasticity demand combined with constant marginal cost, linear demand combined with linear marginal cost as in @horizontal, and the “constant pass-through” demand of @bulow (henceforth BP) with constant marginal cost.[^23]$^{,}$[^24]As a result, we focus on functional form classes composed of linear combinations of power functions $q^{-t_{j}}$.[^25]
![Example of a bell-shaped-distribution-generated demand and U-shaped cost curve contributing to equilibrium conditions that can be solved linearly: $P(q)=3\left(q^{-0.3}-q^{10}\right)$ and $MC(q)=q^{-0.3}+10q^{10}$.[]{data-label="linearsolve"}](FigureNoConstant.pdf){width="4in"}
The BP demand corresponds to $P\left(q\right)=p_{0}+p_{t}q^{-t}$ for some real constants $t$, $p_{0}$ and $p_{t}$, not necessarily all positive. In a monopolist’s first-order condition, constant marginal cost enters in the same way as $-p_{0}$. In this sense, constant marginal cost is compatible with this demand side specification. (Similarly, linear marginal cost would be fully compatible with the demand side in the special case of $t=-1$, i.e., linear demand.)
Using the BP demand form with constant marginal cost leads both to tractability and to an important substantive implication: the constancy of the pass-through rate of the constant marginal cost to price. However, it is clearly possible to preserve the former property without the latter. For example, consider inverse demand and average cost of the form $P(q)=p_{s}q^{-s}+p_{t}q^{-t}$ and $AC(q)=ac_{s}q^{-s}+ac_{t}q^{-t}$.[^26] Then the monopolist’s first-order condition gives $$\left(p_{s}-ac_{s}\right)\left(1-s\right)q^{-s}+\left(p_{t}-ac_{t}\right)(1-t)q^{-t}=0\implies q=\left(-\frac{\left(p_{s}-ac_{s}\right)\left(1-s\right)}{\left(p_{t}-ac_{t}\right)(1-t)}\right)^{\frac{1}{s-t}}.$$ This more general form thus still leads to a closed-form solution but offers substantially more flexibility. For example, it can accommodate simultaneously U-shaped cost curves and demand curves generated by a bell-shaped valuation distribution (in the sense of discrete choice). Figure \[linearsolve\] provides an example. A disadvantage of this form, however, is that it does not include a constant term. A constant term would have been useful for studying the pass-through rate and similar comparative statics. Another disadvantage is the absence of an explicit expression for the direct demand $Q(p)=P^{-1}(p).$
It is thus useful to look beyond systems that lead to a linear equation (after a substitution using a power function). Quadratic, cubic and quartic equations also yield closed-form solutions by the method of radicals. Furthermore, polynomials of higher, but still small, order can be solved extremely quickly by most mathematical software without resorting to numerical search. For this reason, we define tractability in terms of the degree of polynomial solution a form admits.
\[DefinitionAlgebraicTractability\]
***(Tractability)*** We say that an economic problem involving a scalar $q$ is *algebraically tractable* *at level $k$* if a definite power of $q$ is the solution of a polynomial equation of order $k$. For short we often refer to this simply as “tractability” and use adverbial forms for low $k$ (e.g. linearly or quadratically tractable). By classical results of the theory of polynomial equations, only for $k\leq4$ can such an equation be explicitly solved by the method of radicals and thus we refer to economic problems that are algebraically tractable at level $k\leq4$ as *analytically tractable*.
We now characterize the set of functional forms from the power class that are tractable at level $k$ for any positive integer $k$. A very naive conjecture based on the above discussion is that this is simply the set of forms that can be written as the sum of $k+1$ powers. To see why this is wrong, consider the equation $$q+1+q^{-\nicefrac{1}{2}}=0.$$ This does not admit a quadratic solution, but can be solved cubically by defining $x\equiv q^{-\nicefrac{1}{2}}$, transforming the equation into $$x^{-2}+1+x=0\iff x^{3}+x^{2}+1=0.$$ While the quadratic solution fails here, the cubic succeeds, because the gap between the power of the first and second term ($1-0=1$) is not equal to that between the second and third term ($0-\left(-\nicefrac{1}{2}\right)=\nicefrac{1}{2}$); instead it is twice the second gap, implying that there is a “missing” term $q^{\nicefrac{1}{2}}$ in the equation. On the other hand, the equation $$q^{\nicefrac{1}{2}}+1+q^{-\nicefrac{1}{2}}=0$$ *is* quadratically tractable because the gap between the first and second powers equals that between the second and third. More broadly the number of such *evenly-spaced* powers sufficient to represent the class determines its level of tractability.
\[TheoremClosedFormSolutions\]
***(Closed-Form Solutions)*** A functional form class $\mathcal{\mathcal{C}}$ composed of all linear combinations of a finite set of powers of $q$ is algebraically tractable at level $k$ for generic linear coefficients if and only if the powers included are $\left\{ a+bi\right\} _{i\in J}$ for some fixed real numbers $a$ and $b$ and some fixed set of integers $J\subseteq\left\{ 0,\ldots,j\right\} $ for a fixed integer $j\leq k$. More informally, a class of sum of power laws is tractable at level $k$ if it consists of at most $k+1$ evenly-spaced powers of $q$.
[|c|c|c|c|c|]{} Form & Tractability properties & Flexibility & Special cases & Historical notes\
$F(q)=f_0+f_{-1}q$ &
---------------------
Linearly tractable
Linearly invertible
---------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
& Linear $MC$ & Constant $MC$ & @horizontal\
$F(q)=f_0 + f_{t} q^{-t}$ &
---------------------
Linearly tractable
Linearly invertible
---------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
&
--------------
Any constant
pass-through
--------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
&
---------------------
Linear
Constant elasticity
Exponential
---------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
&
------------------------------
BP
constant pass-through demand
------------------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
\
$F(q)=f_tq^{-t}+f_sq^{-s}$ & Linearly tractable &
--------------------------
Demand generated by
bell-shaped distribution
U-shaped cost
--------------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
& BP &
-----------------
@demanding
bi-power demand
-----------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
\
$F(q)=f_tq^{-t}+f_0+f_{-t}q^{t}$ &
--------------------------
Quadratically tractable
Quadratically invertible
--------------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
&
---------------------
Income distribution
U-shaped cost
---------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
& BP & This paper\
$F(q)= f_0+f_{t}q^{-t}+f_{2t}q^{-2t}$ &
--------------------------
Quadratically tractable
Quadratically invertible
--------------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
&
--------------------------
Demand generated by
bell-shaped distribution
U-shaped cost
--------------------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
& BP &
--------------
@demandforms
APT demand
--------------
: Various classes of linearly or quadratically tractable, form-preserving equilibrium systems discussed in this or previous papers. []{data-label="tractableforms"}
\
One example of applying this theorem was given in the previous section: our tractable form involves 3 evenly spaced power laws and thus is quadratically tractable. Table \[tractableforms\] summarizes a rich set of other possibilities covered by this theorem. The demand side of some of these has appeared in previous literature as we cite in the paper, though only in the case of @horizontal are we aware of authors harnessing the accompanying cost-side flexibility.
Aggregation over heterogeneous firms {#AggregationOverHeterogeneousFirms}
------------------------------------
Models of international trade involving firm heterogeneity frequently use the framework of @melitz or @melitzottaviano, which assume respectively constant elasticity and linear demand. While these forms clearly play a role in the tractability of those models, the models are not always explicitly solvable even under these forms. Instead, the key properties these allow is that the firms’ optimization problems may be solved explicitly and that aggregation integrals over heterogeneous firms may be expressed in closed form, assuming Pareto-distributed firm productivity.
We present a theorem that shows that substantial generalizations of these models can still lead to closed-form aggregation. We defer a full model set-up to Supplementary Material \[AppendixFlexibleMelitzModel\], but it may be thought of simply as the @melitz model with relaxed functional form assumptions on the shape of demand, supply, and firm productivity distributions.
\[TheoremAggregation\]
***(Aggregation)*** Suppose that the utility structure implies an inverse demand curve $P(q)$ and that firms have marginal cost functions $\text{MC}(q)=a \text{MC}_1(q)+\text{MC}_0(q)$, where $a$ is an idiosyncratic parameter influencing the firm[’]{}s productivity, distributed according to a cumulative distribution function $G(a)$. Assume that $P$, $\text{MC}_0$, $\text{MC}_1$, and $G$ are linear combinations of powers of their arguments, with the second order condition for the firm[’]{}s profit maximization satisfied. Furthermore, suppose that the powers are such that $\text{MC}_1$ and the difference between marginal revenue and $\text{MC}_0$ are both of the form $q^{\beta } N\left(q^{\alpha }\right)$ with common $\alpha $, but possibly differing $\beta $ and polynomials $N$. Then the aggregation integrals for the firms[’]{} revenue, cost, and profit may be performed explicitly. The resulting expressions may contain special functions, namely the standard hypergeometric function, the standard Appell function, or more generally Lauricella functions, and in the case of high-order polynomials (higher-order tractable specifications), increasingly high-degree polynomial root functions.
While this result is closely related to our general theory and our other applications (in particular, because this aggregation is possible when the relevant variables have our proposed forms), there are also a few differences worth noting. First, aggregation is still possible when the heterogeneous component $MC_1(q)$ of marginal cost is “shifted” (in the exponent space) by a uniform multiplicative power factor relative to $MR(q)-MC_0(q)$. This corresponds to the “possibly differing $\beta $” in the statement of the theorem. Second, our results here are about aggregation, not solution, and the resulting functions are not, therefore, solutions to polynomial equations but rather various functions that may be exotic to some economists, but are widely used in mathematics and related applied fields. Finally, as the complexity of the forms rises, it is the complexity of these functions that rises.
Closed-form aggregation is useful for at least three reasons. First of all, in the simplest cases the resulting aggregation integrals are just polynomial functions, which means that at the aggregate level the economic equations are relatively simple. Second, when the aggregation integrals lead to commonly used special functions, these are likely to be implemented in numerical software of the researcher’s choice. The researcher gets a fast and numerically reliable implementation of these functions and their derivatives without spending time on approximation methods. Third, it is possible to take advantage of the properties of these functions that have been studied in the mathematical literature.
Interpolation between solutions {#SubsectionInterpolationBetweenSolutions}
-------------------------------
We have discussed how to obtain closed-form solutions in economic modeling. We used linear combinations of power function and imposed conditions on their exponents. It is natural to ask what happens if these conditions are not satisfied. Suppose we have a computationally intensive model whose numerical solutions rely on closed-form solutions to its sub-problems. If we relax our assumptions on the exponents we just mentioned, the sub-problems will not be solvable in closed form and obtaining numerical solutions to the full problem may require an excessive amount of time. Here we would like to point out that in this case we have another way to proceed: we can solve the full problem at special loci where the conditions on exponents are satisfied, and then interpolate between the resulting solutions.
Let us illustrate this approach with a toy example that is not computationally intensive. Consider a monopolistic firm with marginal revenue $\text{MR}(q)=\text{MR}_0q^{-b_R}$, $b_R>0$ and marginal cost $\text{MC}(q)=\text{MC}_0+\text{MC}_1q^{-b_C}$, [ ]{}$\text{MC}_1>0$. After the substitution $x=q^{-b_R}$, the firm[’]{}s first-order condition becomes $\text{MC}_0+\text{MC}_1x^b=\text{MR}_0x$, with $b\equiv b_C/b_R$. This equation admits closed-form solutions by the method of radicals for $b\in$$\{$$-3$, $-2,-1,-\frac{1}{2},-\frac{1}{3},0,$$\frac{1}{4}$,$\frac{1}{3}$,$\frac{1}{2}$,1,2,3,4$\}$. The second-order condition is satisfied only for $b<1$, so we restrict our attention to the first 9 values. For illustration, consider the simple goal of finding numerical values of $q$ for $b$ between these points. Instead of solving the first-order conditions by usual numerical methods, we may interpolate between the closed-form solutions, say, using cubic splines. Figure \[FigureInterpolationBetweenAnalyticSolutions\] shows the result of such interpolation, as well as the true solutions to the first-order condition, for $\text{MC}_0=\text{MC}_1=\text{MR}_0=1$. The agreement is extremely good, with average absolute value of relative deviations equal to 0.00013 and maximum absolute value of relative deviations equal to 0.00056.[^27]
If variables of interest in large scale, computationally intensive problems are similarly well behaved, then clearly the interpolation method could save remarkable amounts of computation time and research budgets. There are other ways to extend the usefulness of the closed-form solutions to other parameter values. For example, it may be possible to perform a Taylor expansion around a given closed-form solution. Such approach may also be combined with the interpolation method.
More broadly, one can view our approach to economic modeling as resembling pragmatic approaches in Bayesian Statistics. In that literature, it is usually impossible to compute the posterior probability distribution associated with most prior distributions given the observed, often large, data set. [ ]{}It is therefore common to approximate the prior by one selected from a class of prior distributions which are known to update to another prior within that class in closed form as this minimizes the computational requirements of updating. [ ]{}In a similar manner, our tractable equilibrium forms may approximate arbitrary cost and demand curves, while allowing solutions in closed-form which allow nesting inside of computationally intensive models.
![Comparison of an interpolation between analytic solutions and the correct values. The blue dots represent analytic solutions. The blue solid line corresponds to an interpolation using cubic splines. The green dashed line represents correct values.\
[]{data-label="FigureInterpolationBetweenAnalyticSolutions"}](FigureInterpolationBetweenAnalyticSolutionsUsingSplines.pdf){width="4in"}
In the next two sections we explore concrete applications of our approach to closed-form solutions in economics. We will return to more theoretical matters in Section \[SectionArbitraryDemandAndCostFunctions\].
World Trade {#SectionWorldTrade}
===========
Overview {#SubsectionOverview}
--------
In this section we present a large-scale empirical application of our analytic approach to flexible functional forms in economics: a model of world trade with a realistic cost structure for heterogeneous firms.
International trade researchers almost always postulated constant marginal costs. Firms were assumed to have a constant marginal cost of production. They were also assumed to face constant marginal costs of trade, either in the [“]{}iceberg[”]{} form (i.e., damage of goods as they are transported) or in a per-unit form. Both of these assumptions are unrealistic. When we depart from them, we find an interpretation of world trade flows that is dramatically different from the conventional view. The model[’]{}s parameters and predictions take realistic values, which resolves empirical puzzles in the international trade literature.
Our model describes a world with multiple countries, with a general setup analogous to @melitz. Our two important modifications are as follows. First, in addition to the usual iceberg cost, we allow for a specific cost of trade that varies non-linearly with the traded quantity. Second, production is subject to increasing marginal cost, designed to capture the difficulty of scaling the firm, e.g. due to internal agency problems. After discussing computational considerations and, separately, the two economic cost effects, we return to the setup of the main model in Subsection \[SubsectionModelSetup\].
Computational considerations {#SubsectionComputationalConsiderations}
----------------------------
In applied fields of economics, such as the study of international trade, researchers can quickly reach the limits of what is computationally feasible because of the number of economic agents and the high dimensionality of their choice sets (or state spaces). In our case, we study trade flows between many countries involving heterogeneous firms, each of which is facing a combinatorially difficult decision problem. With powerful hardware, software, and efficient algorithms, we were able to get a model fit for given parameter values in about a month and at a non-trivial cost. We were utilizing our analytic solutions to sub-problems, without which the computation would be substantially longer and more costly.
Our functional forms help us in two ways: First, to evaluate firms[’]{} sales decisions conditional on the level of their marginal cost of production and export entry decisions, we just need to evaluate closed-form solutions. This is crucial [ ]{}for being able to quickly evaluate a large number of alternative sales patterns a firm may consider, and to find some of the best ones. Second, conditional of all firms[’]{} export entry decisions, we can solve for the resulting general equilibrium of world trade by accelerated gradient descent algorithms, the Adam algorithm in our case. For this algorithm to be useful, we need to be able to calculate gradients of candidate solutions[’]{} loss functions (error functions) analytically. Because of the scale of the problem, we do not perform the gradient calculation by hand. [ ]{}Instead, we rely on automatic analytic differentiation software, namely the neural-network optimization package of PyTorch, which allows us to run all computations in a highly parallel fashion on graphics processing units (GPUs).[^28]
Firm-level economies of scale in shipping: a generalized Economic Order Quantity model {#SubsectionFirmLevelEconomiesOfScaleInShipping}
--------------------------------------------------------------------------------------
Most models of international trade assume that the costs of trade are of the [“]{}iceberg[”]{} type: a fraction of all goods transported is assumed to be destroyed in transit. It seems implausible that most of true trade costs would scale with trade volume and value in this manner.[^29] A certain fraction of international trade papers, e.g. @melitzottaviano, allow for constant marginal per unit costs of trade.[^30] However, the adoption of standardized shipping containers has made such constant marginal per-unit costs of transportation extremely low relative to the trade costs necessary to explain the rates of global trade flows.
We work with the assumption that most important trade costs come from coordination (shipment preparation) costs and inventory costs, which is why the logistics literature focuses on them. These costs depend on the frequency of shipping. If a firm ships too infrequently, it will face large costs associated with idle inventory. If it ships too frequently, shipment preparation costs will add up to a large number. Knowing this trade-off, the firm will choose an optimal frequency of shipping that balances these two effects. The resulting effective cost of trade then exhibits economies of scale: a firm wishing to ship only a small quantity on average per unit of time will find shipping to be costly per unit of quantity.
To gain empirical insight into the scale economies of international trade, we estimate a model of optimal shipping frequency using monthly international shipment data. Our approach generalizes the classic Economic Order Quantity model of Ford W. Harris, which is widely taught in operations management courses in business schools and applied by logistics planning managers in corporations.[^31] Consider a firm that produces a single good in one country and wishes to ship to a different country quantity $q$ per year, on average. The firm faces a tradeoff between inventory costs and coordination costs associated with frequent shipping. The average annual inventory cost $C_i$ is linearly proportional to $q$ and to the time $T$ a typical unit of the good needs to remain in storage. If the size of each shipment is $q_s$, then $T$, in turn, is linearly proportional to $\left.q_s\right/q$, implying $C_i=\kappa _iq_s$, for some constant $\kappa _i$. The coordination cost $C_s$ of each shipment is proportional to its size: $C_s=\kappa _tq_s^{\gamma }$, $\gamma \in $\[0,1). (In addition, we could assume an additional term proportional to $q_s$, but this would not affect the optimal choice of $q_s$ for given $q$.) The resulting average annual coordination cost is $C_t=C_sq\left/q_s\right.=\kappa _tq q_s^{\gamma -1}$. [ ]{}Minimizing the sum of the inventory cost and the coordination cost leads to the optimal choice $q_s=\left(q (1-\alpha ) \kappa _t \kappa _i^{-1}\right)^{\frac{1}{2-\gamma }}$$\text{}$, the minimized value $(2-\gamma
) (1-\gamma )^{-\frac{1-\gamma }{2-\gamma }} \kappa _i^{\frac{1-\gamma }{2-\gamma }} \kappa _t^{\frac{1}{2-\gamma }} q^{\frac{1}{2-\gamma }}$, and the optimal frequency of shipping $f_s=$$q\left/q_s\right.$ equal to$$f_s=(1-\alpha )^{-\frac{1}{2-\gamma }} \kappa _i^{\frac{1}{2-\gamma }} \kappa _t^{-\frac{1}{2-\gamma }} q^{\frac{1-\gamma }{2-\gamma }}.$$ This result implies that we can infer the coordination cost exponent $\gamma $ by examining the relationship between the average annual quantity shipped and the frequency of shipping. If we regress the logarithm of shipping frequency $f_s$ on the logarithm of average annual quantity $q$, the resulting slope coefficient should equal $\beta \equiv (1-\gamma )/(2 - \gamma )$. The model predicts that this coefficient always lies between 0 and 1/2, since $\gamma \in $\[0,1).
Our simple model of shipping frequency choice nests two important extreme cases. The original Economic Order Quantity model, in which the cost per shipment is fixed, [ ]{}corresponds to $\gamma =0$ and $\beta =1/2$, implying effective cost of trade (here inventory and coordination) proportional to $\sqrt{q}$. The other extreme case has $\gamma \to 1$ and $\beta \to 0$ and corresponds to effective cost of trade linearly proportional to $q$, i.e., constant marginal cost of trade, as assumed in almost all of the international trade literature.
To estimate $\beta $ and to test the prediction that $\beta \in $(0,1/2\], we used a dataset on monthly shipments from China to Japan during years 2000-2006. [ ]{}We focus on firms in one narrowly-defined product category.[^32] [ ]{}Our point estimate of $\beta $ (averaged across industries) is 0.39 with a 95$\%$ confidence interval of \[0.36,0.42\].[^33] [ ]{}We can thus clearly reject the null hypothesis that $\gamma = 1$ and $\beta
= 0$, which would correspond to trade costs being linearly proportional to quantity shipped, as assumed in the vast majority of the international trade literature. We can also reject the original EOQ model, which would correspond to $\gamma =0$ and $\beta =1/2$. We see, however, that the original EOQ model is closer to reality than the linear proportionality assumption. [ ]{}
In our main trade model, we round the resulting value for $\beta $ from 0.39 to 0.4. This estimate implies that increasing quantity by 10$\%$ reduces (the variable component of) the marginal cost of trade by 4$\%$. [ ]{}We refer to the effective cost of trade as [“]{}cost of shipping[”]{}, remembering that it arises from per-shipment coordination costs and from inventory costs with optimally chosen shipping frequency. In the rest of this section, we use the notation $\nu _{\text{LT}}$ for what was $1-\beta =1/(2-\gamma )$ here, and set $\nu _{\text{LT}}=0.6$.
Export quantity determination {#SubsectionExportQuantityDetermination}
-----------------------------
For clarity of exposition, let us now consider the problem of export quantity determination for a firm that faces trade costs found in the previous subsection. Similar ingredients will appear also in our main model described in Subsection \[SubsectionModelSetup\].
A firm considers exporting to one foreign country. If it delivers quantity $q_f$ there, it will receive revenue $R\left(q_f\right)$, for which we choose the form [ ]{} $R\left(q_f\right)=\nu _R^{-1}\kappa _R q_f^{\nu _R}$, where $\nu _R=1-1/\sigma$ and $\sigma =5$. The elasticity of demand [ ]{}$\sigma =5$ is consistent with the typical range in the trade literature. The firm faces an iceberg trade cost factor $\tau $, meaning that it needs to send $\tau q_f$ in order for $q_f$ to arrive. The shipping requires $L_T\left(q_f\right)$ units of labor, which translates into a cost $w L_T\left(q_f\right)$. We choose $L_T(q)=\nu _{\text{LT}}^{-1} \kappa _{\text{LT}} q^{\nu _{\text{LT}}}$, with $\nu _{\text{LT}}=3/5$, in agreement with the previous subsection. In this illustrative example, we assume constant marginal cost $\text{MC}$.
The second derivative of the profit function $R\left(q_f\right)-\tau \text{MC} q_f+w L_T\left(q_f\right)$ is $\frac{2 }{5 }w \kappa _{\text{LT}}q_f^{-7/5}-\frac{1}{5
}\kappa _Rq_f^{-6/5}$, so the profit function is convex for $q_f\in \left(0,32 w^5 \kappa _{\text{LT}}^5\kappa _R^{-5}\right)$ and concave for $q_f\in \left(32 w^5 \kappa _{\text{LT}}^5\kappa _R^{-5},\infty \right)$. To identify the maximum, we just need to find potential local maxima in the second region and to check whether they are larger than zero. This is because at $q_f=0$ the profit is zero, and as $q_f\to \infty$ it goes to $-\infty$.
The firm[’]{}s first-order condition is$$R'\left(q_f\right)- \tau \text{MC}-w L_T'\left(q_f\right)=0\, \, \, \, \, \Longrightarrow \, \, \, \, \, -\frac{w \kappa _{\text{LT}}}{\tau }
q_f^{-\frac{2}{5}}+\frac{\kappa _R}{\tau } q_f^{-\frac{1}{5}}-\text{MC}=0.$$ We recognize that the function of $q_f$ on the left-hand side is one of our proposed tractable functional forms. We can, therefore, solve the first-order condition in closed-form, in this case using the quadratic formula. [ ]{}If $\text{MC}> \kappa _R^2/\left(4
w \tau \kappa _{\text{LT}}\right)$ [ ]{}there is no real solution and the firm will choose not to export. If $\text{MC}\leq \text{ }\kappa _R^2/\left(4
w \tau \kappa _{\text{LT}}\right)$, the solution that lies in the $\left.\left[32 w^5 \kappa _{\text{LT}}^5\kappa _R^{-5},\infty \right.\right)$ region equals$$q_f=\left(\frac{\kappa _R+\sqrt{\kappa _R^2-4 \tau \text{MC} \kappa _{\text{LT}} w}}{2 \tau \text{MC}}\right)^5.$$ Plugging this position of the local maximum into the profit function gives$$\frac{1}{192 \text{MC}^4 \tau ^4} \left(\kappa _R+\sqrt{\kappa _R^2-4 \text{MC} w \tau \kappa _{\text{LT}}}\right)^3 \left(-16 \text{MC} w \tau
\kappa _{\text{LT}}+3 \kappa _R \left(\kappa _R+\sqrt{\kappa _R^2-4 \text{MC} w \tau \kappa _{\text{LT}}}\right)\right)$$ The first two factors are positive, and the last one is positive if and only if $\text{MC} <15\kappa _R^2/\left(64 w
\tau \kappa _{\text{LT}}\right)\approx 0.234 \kappa _R^2/\left(w \tau \kappa _{\text{LT}}\right)$. If this condition is satisfied, the firm will export the quantity satisfying the first-order condition, otherwise, it will export zero.[^34] Thus for any level of marginal cost, the quantity chosen by the firm may be written compactly as$$q_f=\left(\frac{\kappa _R+\sqrt{\kappa _R^2-4 \tau \text{MC} \kappa _{\text{LT}} w}}{2 \tau \text{MC}}\right)^51_{\text{MC}<\frac{15 \kappa
_R^2}{64 w \tau \kappa _{\text{LT}}}},$$ where the second factor represents an indicator function. We see that the functional form allowed for a very simple and straightforward analysis. We also see that exporting may not be profitable even if there is no fixed cost of exporting. This implies that such model with constant elasticity of demand can generate an export cutoff without fixed costs of exporting.
In our main model described in Subsection \[SubsectionModelSetup\], which no longer assumes that the marginal cost of production is constant, we still benefit from the closed-form characterization of the solution to the first-order condition in terms of the level of marginal production cost. This is both for the evaluation of the solution and for taking derivatives of the solution, as needed by gradient descent algorithms. Of course, the degree of the benefit grows in proportion to the number of potential export destinations.
Increasing marginal cost of production {#SubsectionIncreasingMarginalCostOfProduction}
--------------------------------------
Economies of scale, modeled using fixed costs of production, are present in most models of firms in the international trade literature. By contrast, diseconomies of scale almost never appear in that literature. Yet there are many reasons to believe that increasing marginal costs of production are similarly important in shaping firms[’]{} behavior. This is presumably why introductory economics classes frequently illustrate increasing marginal cost schedules. Beyond short-to-medium term capacity constraints and adjustment costs usually discussed in such courses, even in the longer term if a company decides to scale up its production an order of magnitude, it needs to introduce an additional layer of management hierarchy, which brings with it non-trivial agency problems. In a large organization incentives are diluted, and maintaining motivation, discipline, and output quality becomes harder.[^35] Of course, managers of firms are intuitively aware of these problems, at least to some extent, and take them to account when shaping the structure of the firm.
Issues of this kind are the subject of interest to vast literature within organizational economics, which includes @williamson1967hierarchical, @calvo1978supervision, and @tirole1986hierarchies.[^36]
Estimating how much marginal costs increase with production volume is non-trivial since both economies and diseconomies of scale play a role in firm behavior. Our model provides a unique opportunity to obtain such estimates by matching firm-level multi-destination export data with world trade model solutions.
Model setup {#SubsectionModelSetup}
-----------
Apart from the cost structure of the firms, our model is closely analogous to @melitz, which many readers are familiar with. For this reason, we keep the description of the modeling setup succinct.
The world consists of [ ]{}$N_c$ countries, indexed by $k$. In each country, different varieties $\omega$ of a differentiated good are produced by monopolistically competitive heterogeneous single-product firms using a single factor of production, for simplicity referred to as labor.
Consider a firm located in country $k$ and identified by an index $i$. In order to produce a quantity $q_i$, the firm needs to pay a variable cost of [ ]{}$\frac{1}{1+\alpha } \kappa _{C,i} w_k q_i^{1+\alpha }$, where $w_k$ is the competitive wage the firm[’]{}s country $k$ and [ ]{}$\kappa
_{C,i}$ is a positive constant that depends on the firm. Importantly, the constant $\alpha $ [ ]{}determines how [ ]{}quickly marginal costs increase when any firm decides to scale up production; it is the elasticity of the marginal cost of production with respect to quantity. [ ]{}In addition to the variable cost, there is a fixed cost $f_o$ of operation and a fixed cost [ ]{}$f_x$ of exporting to a destination country $k_d$, expressed in units of domestic labor.[^37] [ ]{}
Entry into the industry is unrestricted, but involves a sunk cost of entry $f_e$, again in units of domestic labor. Only after the entry cost has been paid does the firm learn its variable cost parameter $\kappa _{C,i}$, drawn from a distribution with cumulative distribution function $\tilde{G}\left(\kappa
_C\right)$. When the value of $\kappa _{C,i}$ is revealed, the firm decides whether or not to exit the industry, and if it does not exit, whether to export to any of the other countries. In addition to endogenous exit, with a probability of $\delta _e$ per period the firm is exogenously forced to exit (starting from the end of the first period).
Trade costs have two components. The first corresponds to standard iceberg trade costs: in order of one unit of the good to arrive in the destination country $k_d$, $\tau _{k,k_d}$ units need to be shipped.[^38] The second component requires using an amount of labor given by $L_{T,k,k_d}(q)=\nu _{\text{LT}}^{-1} \kappa _{\text{LT},k,k_d} q^{\nu _{\text{LT}}}$, where we set $\nu _{\text{LT}}=\frac{3}{5}$ to be consistent with the empirical value, as in Subsection \[SubsectionExportQuantityDetermination\].[^39]
Consumers in each country have a CES utility function $U=(\int q_{\omega }^{1-\frac{1}{\sigma }}d\omega )^{\frac{\sigma }{\sigma -1}}$ that depends on the quantity $q_{\omega }$ of each variety $\omega $ consumed. As in Subsection \[SubsectionExportQuantityDetermination\], we set the elasticity of substitution $\sigma $ equal to 5, which is consistent with the typical range in the existing empirical literature of about 4 to 8. This exact choice is motivated by analytic tractability. Each country $k$ has an endowment of labor $L_{E,k}$, which is supplied at a country-specific competitive wage rate $w_k$ mentioned above.
The revenue a firm can earn by selling a quantity $q$ in a given market is $R_{k_d}(q)=\frac{\kappa _{R,k_d}}{\nu _R} q^{\nu _R}$, where $\nu
_R=1-\frac{1}{\sigma }$. The factor $\kappa _{R,k_d}$ is endogenously determined and depends on the price index and the consumption expenditures in the destination country.
The firm may choose to exit the industry (to save on the fixed cost $f_o$) or to operate and sell its product in a number of countries, earning a non-negative profit $\pi $ per period of operation. In expectation, an entrant needs to break even: $\delta _e f_e=\text{E$\pi $}$, which determines the equilibrium measure of firms in each country.[^40] Similarly, labor markets in each country $k$ need to clear, which means that the total labor demanded by firms at wage $w_k$ needs to equal the labor endowment $L_{E,k}$. If we impose balanced budget conditions, consumers[’]{} expenditures equal their wage earnings, as firms earn zero ex-ante profits.[^41]
The exporting firm[’]{}s problem {#SubsectionTheExportingFirmsProblem}
--------------------------------
Let us discuss the nature of the exporting firm[’]{}s problem. Increasing marginal costs will limit the scale of the firm[’]{}s production. Since trade is subject to decreasing marginal costs, the firm will concentrate its exports into a limited number of countries. The overall production level $q_i$ of firm $i$ as well as export market entry decisions are endogenous. For now let us consider the relation between of export quantities and $q_i$, conditional on having paid fixed costs of exporting to a number of countries.
The first-order condition for choosing the quantity $q_{f,i,k_d}$ that should reach a foreign market $k_d$ equates the marginal revenue and the comprehensive marginal cost that depends on the overall production level $q_i$:$$R_{k_d}'\left(q_{f,i,k_d}\right)=\tau _{k,k_d} \text{MC}_i\left(q_i\right)+w_k L_{T,k,k_d}'\left(q_{f,i,k_d}\right)\Rightarrow \frac{\kappa _{R,k_d}}{\tau
_{k,k_d}} q_{f,i,k_d}^{-\frac{1}{5}}=\text{MC}_i\left(q_i\right)+\frac{w_k \kappa _{\text{LT},k,k_d}}{\tau _{k,k_d}} q_{f,i,k_d}^{-\frac{2}{5}},$$ in analogy with Subsection \[SubsectionExportQuantityDetermination\]. The solution for $q_{f,i,k_d}$ given $q_i$ is:$$q_{f,i,k_d}=\frac{1}{\left(2 \tau _{k,k_d} \text{MC}_i\left(q_i\right)\right)^5} \left(\kappa _{R,k_d}+\sqrt{\kappa _{R,k_d}^2-4 w_k \kappa _{\text{LT},k,k_d}
\tau _{k,k_d} \text{MC}_i\left(q_i\right)}\right)^5$$ If the marginal cost of production $\text{MC}_i\left(q_i\right)$ exceeds $\kappa _{R,k_d}^2$/($4 w_k \kappa _{\text{LT},k,k_d}
\tau _{k,k_d}$), the first-order condition cannot be satisfied. For domestic sales we assume $\kappa _{\text{LT},k,k}=0$ and $\tau _{k,k}=1$, so the optimal quantity sold domestically is simply $q_{i,k}=\left(\kappa _{R,k}/\text{MC}_i\left(q_i\right)\right)^5$.
The total quantity $q_i$ produced should equal the total of quantity sold domestically and sent abroad: $q_i=q_{i,k}+\sum _{k_d\neq k}\tau _{k,k_d}
q_{i,k,k_d}$, with $q_{f,i,k_d}$ [ ]{}given by the formula above. This represents one equation for one unknown: $q_i$. Each root of this equation represents a candidate optimum for the firm.[^42] The profit-maximizing choice(s) of destinations may then be found by evaluating total profits at these candidate optima. For a small number of countries this is simple, but for large $N_c$ the problem becomes combinatorially difficult.[^43] For this reason, when we solve the model for a large number of countries, we use approximate algorithms instead of an exhaustive search.[^44]
Solution strategy {#SolutionStrategy}
-----------------
We solve the model using an iterative algorithm that has an outer loop and an inner loop.[^45] In the outer loop firms decide whether or not they pay fixed costs of operation and fixed costs of exporting and commit to their decision. In the inner loop, we then solve for the general equilibrium of the world economy given these fixed-cost decisions.
Finding this general equilibrium without the tractable functional forms is computationally difficult since a multi-level nested iteration is very time-consuming. However, thanks to the analytic nature of our model, we were able to obtain the general equilibrium much faster using accelerated gradient descent in a space parametrized by quantities $q$, wages $w$, measures of firms $M$, price-index related variables $\kappa _R$, and country-level expenditures $E$. We used the Adam optimizer of @kingma2014adam, as implemented in PyTorch, a neural network optimization software for GPU computing.[^46] The gradients are computed analytically by automatic differentiation (autograd, in this case) and backpropagation.[^47]$\, ^,\,$[^48]
Given a solution to the general equilibrium problem, we then let firms reconsider their fixed cost payments. For numerical stability, we do not update at once the fixed cost payment decisions of all firms. Instead, for each productivity level in a country we introduce $N_v=10$ versions (copies) of firms, which can differ by their fixed cost commitments. Updating fixed-cost commitments then proceeds in cohorts. In one iteration of the outer loop, version 1 firms will be able to reconsider the fixed cost payment. In the second iteration, version 2 firms will do so, etc. Keeping different versions of firms comes at a computational cost, of course, but we found this necessary.
Finding the best fixed cost decision is a combinatorially difficult problem. Given that there are $N_c-1$ potential export destinations, this leads to $2^{N_c-1}$ possibilities for the exports. With $N_c=100$, this is more than $10^{29}$. To obtain an approximate optimum, we use Algorithm [ ]{}2 of @buchbinder2015tight, which is stochastic in nature. [ ]{}We consider 9 (random) runs of that algorithm, and if the best of them is better than the firm[’]{}s previous fixed cost decision, we update it. After the update, we again solve for a new general equilibrium involving continuous variables.
Fitting the model {#FittingTheModel}
-----------------
We work with [ ]{}$N_c=100$ countries. [ ]{}This choice is motivated by data availability and parameter fit considerations: For a substantially larger number of countries, the trade data would be too noisy and unreliable. For a substantially smaller number, it would be impossible to read off the elasticity of the marginal production cost from the firms[’]{} export pattern using our method.
The labor endowment in the model is interpreted [ ]{}as an efficiency-adjusted number of units of a single production factor, which in practice would include labor, capital, and the related productivity. This effective labor endowment and the trade cost prefactors [ ]{}$\kappa _{\text{LT},k,k_d}$ are recovered by fitting the model to data on country GDP and world trade flows for the year 2006, as described below.
To match the typical empirical firm size distribution, which we take as Pareto distribution with Pareto index $\mu _R=1.05$, we choose [ ]{}the firm size distribution to be another Pareto distribution with Pareto index $\left.\mu _R(\sigma -1)\right/(1+\sigma \alpha )$.[^49] The productivity distribution is the same for every country in the model; any real-world overall firm productivity differences across countries are represented by adjustments to the countries[’]{} effective labor endowments.
For computational purposes we discretize the productivity distribution to $N_p=20$ discrete values, each representing the same probability mass.[^50]
In addition, we need to specify (the flow value of) the costs of entry, fixed costs of production, export market entry costs, as well as iceberg trade costs. We make these choices as simple as possible, independent of the country or country pair. Their values are given in Table \[tractableforms2\]. The flow value of the cost of entry is set to one half of the fixed cost of operation. The fixed cost of exporting is set to be negligible. The iceberg trade cost $\tau -1$ is non-zero but small enough to be consistent with prices firms in practice pay for insurance or as tariffs. In general, the parameters are chosen to reflect a long-term interpretation of the model, with timescales of many years.[^51]
$N_c$ 100 $N_p$ 20 $N_v$ 10 [ ]{} $\alpha$ multiple values
------------------ --------- ---------- ----------- -------------------- ----------- ---------- ----------------- -- --
$\sigma$ 5 [ ]{} $\nu _R$ 0.8 $\nu _{\text{LT}}$ 0.6 $\mu _R$ 1.05
$\delta _e$$f_e$ 0.05 $f_o$ 0.1 [ ]{} $f_x$ $10^{-5}$ $\tau$ 1.05
: Calibration parameter values []{data-label="tractableforms2"}
We use importer-reported data on international trade flows for the year 2006 from the UN Comtrade database. We select 100 countries/economies with the largest GDP, as reported by the IMF in its World Economic Outlook database, subject to trade and GDP data availability. We adjust the countries[’]{} GDP for tradability using the United Nations[’]{} gross value added database; see Appendix \[AppendixWorldTradeFlows\].
Elasticity of the marginal cost of production
---------------------------------------------
We solve for the model fit for different values of the parameter $\alpha $, the quantity-elasticity of the marginal cost of production. Then we compare the resulting pattern of firm trade with that of Chinese firm-level export data for 2006 in order to find what value of $\alpha $ leads to a good agreement.
We obtained fits to the data on world trade flows and adjusted GDP levels for values of $\alpha $ ranging from 0.15 to 0.3; see Figure \[FigureMedianSizeOfChineseFirmByDestinationCountry\]. In each case, we computed power-law best-fit curves that describe the dependence of the median size of Chinese firms that export to a destination as a function of the popularity rank of that export destination.[^52] The popularity is computed as the fraction of Chinese firms in the data that choose to export to the given destination. We also computed such best-fit curve for the data. The results are intuitive: For smaller $\alpha$, the difference between the median (log) size of firms exporting to unpopular destinations and to popular destinations is larger because in this case the most productive firms will dominate world trade, and if a less productive firm decides to export at all, it will choose a few of the popular destinations.
The data corresponds roughly to $\alpha \approx 0.225$ if we consider all 99 export destinations when computing the best-fit curves, or to $\alpha
\approx 0.25$, if we consider the first third of them by popularity rank. [ ]{}The first estimate has the advantage of taking into account a large range of export destinations. We include the second estimate because the top third of the destinations account for a vast majority of Chinese export and because the model[’]{}s precision is lower for very small countries. The values $\alpha \approx 0.225$ or [ ]{}$\alpha \approx 0.25$ would imply that if a firm decides to scale up production by an order of magnitude, its marginal cost increases by about 68$\%$ or 78$\%$, respectively. [ ]{}These values seem very realistic, given that such a dramatic expansion of the firm would require an additional layer of management hierarchy with related principal-agent problems. Note that these inefficiencies would be partially offset by savings on the fixed cost of production.
![Median revenue of Chinese exporting firm (base-10 log scale) by destination country for different values of $\alpha$ and for the observed values. For the first export destination (United States), the median log revenue is normalized to 0 to make visual comparisons easier. The top figure corresponds to all 99 export destinations in the model, while the bottom figure corresponds to the top third by export popularity.[]{data-label="FigureMedianSizeOfChineseFirmByDestinationCountry"}](FigureExportingFirmSizeByExportDestination99Destinations.pdf "fig:"){width="12cm"} ![Median revenue of Chinese exporting firm (base-10 log scale) by destination country for different values of $\alpha$ and for the observed values. For the first export destination (United States), the median log revenue is normalized to 0 to make visual comparisons easier. The top figure corresponds to all 99 export destinations in the model, while the bottom figure corresponds to the top third by export popularity.[]{data-label="FigureMedianSizeOfChineseFirmByDestinationCountry"}](FigureExportingFirmSizeByExportDestination33Destinations.pdf "fig:"){width="12cm"}
The gravity equation of trade and the dependence of trade costs on distance {#TheGravityEquationOfTradeAndTheDependenceOfTradeCostsOnDistance}
---------------------------------------------------------------------------
The model fit results have important implications for the gravity equation of trade and for the trade cost puzzle (discussed in detail by @disdier2008puzzling and @head2013separates).[^53] [ ]{}The gravity equation of trade implied by the data[^54] $$\log x_{i j}\approx -0.77 \log d_{i j}+1.12 \log y_i+1.10 \log y_j+\text{const.}$$ matches well the gravity equation implied by the fitted model[^55] $$\log x_{i j}\approx -0.71 \log d_{i j}+1.08 \log y_i+1.02 \log y_j+\text{const.}$$ Of course, this is not surprising given that the world trade flows were a target of our model fit; if the fit was perfect, the two equations would coincide. What is interesting, though, is that the trade cost prefactors (i.e. the factors $\kappa _{\text{LT},k,k_d}$ in $L_{T,k,k_d}(q)=\nu _{\text{LT}}^{-1} \kappa _{\text{LT},k,k_d} q^{\nu _{\text{LT}}}$) depend on distance only very weakly:$$\log \kappa _{\text{LT}}\approx 0.048 \log d_{i j}+0.032 \log y_i+0.048 \log y_j+\text{const.}$$ We see that trade flows decrease rapidly with distance despite only a very mild increase of trade cost prefactors $\kappa
_{\text{LT}}$ with distance. Although this may look surprising at first sight, there is clear intuition for this phenomenon: Due to increasing marginal costs of production, firms effectively have only a limited output to sell and due to economies of scale in shipping, they need to concentrate their exports to only a few countries. They choose close countries because the trade costs are slightly lower, which leads to strong effects for the decrease of trade with distance.[^56]
From the histogram in Figure \[HistogramOfLogTradeCostPrefactors\] we see that the dispersion of $\kappa _{\text{LT}}$ is very small, which is only possible with a very small dependence on distance. This small dispersion is very much consistent with sea shipping over large distances being only mildly more expensive than over short distances. This provides a very natural resolution to the trade cost puzzle in the international trade literature.
![Histogram of the (symmetrized) trade cost prefactors.[]{data-label="HistogramOfLogTradeCostPrefactors"}](FigureHistogramOfSymmetrizedTradeCostPrefactors.pdf){width="4.0in"}
Choice of export destinations {#ChoiceOfExportDestinations}
-----------------------------
In the Introduction, we briefly discussed an empirical pattern of firm entry into export markets that would seem puzzling in standard models of international trade. Our model naturally implies such pattern. Figure \[FigureExportDestinationsByTwoIdenticalFirms\] illustrates export market entry choices in the fitted model for pairs of identical firms, i.e. firms from the same country and with the same productivity.[^57] These would be impossible in a corresponding model with constant marginal cost unless we introduced unrealistically large firm-destination specific cost shocks (or other similar shocks).[^58] It is straightforward to see why this is the case. With constant marginal costs, the decision of whether or not to enter a particular export destination is independent of such decisions for other destinations, as long as the firm does not shut down. If there were no firm-destination specific shocks, then two identical firms with the same constant marginal costs would reach the same conclusions about the profitability of each export destination. In order to make one of the firms give up on a particular destination, we would have to introduce a firm-destination specific shock that would offset all the profit the firm was about to make from selling at that destination.
Our model naturally delivers the export destination choice pattern that would seem puzzling otherwise. With increasing marginal costs of production, a destination that is profitable for one firm may not be profitable for another identical firm, if that firm already serves other locations. Of course, if there were no economies of scale in shipping (and no significant export entry fixed costs), firms would dilute their exports over more destinations and would not face a combinatorial discrete choice problem. In that case, two identical firms would serve the same destinations, unless, again, there were firm-destination specific shocks. For this reason, both increasing marginal costs of production and economies of scale in shipping are crucial for our model[’]{}s ability to resolve the export destination choice puzzle.
A mechanism of this kind also has the potential to explain why personal relationships can play a large role in international trade. Just like shorter distance, knowing someone trustworthy to cooperate with at a potential export destination can provide a mild profit advantage for exporting there instead of other destinations. This modest advantage may then have a large effect on trade flows, given increasing marginal costs of production and economies of scale in shipping.[^59]
The subject of the export destination choice pattern is, of course, very rich and calls for an in-depth empirical and theoretical investigation, which will be provided in a separate, monothematic paper.
![The top map highlights export destinations of two Chinese firms in the fitted model. Specifically, for two identical firms the map shows destinations to which both firms export (yellow), destinations to which only firm 1 exports (red), destinations to which only firm 2 exports (purple), and the country of origin (green). The bottom map shows similar information for two firms from the Czech Republic.[]{data-label="FigureExportDestinationsByTwoIdenticalFirms"}](FigureExportDestinationsByTwoIdenticalFirmsFromChina.pdf "fig:"){width="12cm"} ![The top map highlights export destinations of two Chinese firms in the fitted model. Specifically, for two identical firms the map shows destinations to which both firms export (yellow), destinations to which only firm 1 exports (red), destinations to which only firm 2 exports (purple), and the country of origin (green). The bottom map shows similar information for two firms from the Czech Republic.[]{data-label="FigureExportDestinationsByTwoIdenticalFirms"}](FigureExportDestinationsByTwoIdenticalFirmsFromCzechia.pdf "fig:"){width="12cm"}
Implications for modeling international trade {#SubsectionImplicationsForModelingInternationalTrade}
---------------------------------------------
A vast majority of models of international trade (and spatial economics) assume constant marginal cost of production within firm, even though empirical evidence for such constancy is lacking and even though organizational economics is telling us that scaling up a firm is highly nontrivial, given all the internal agency problems. [ ]{}An obvious reason for making the assumption of constant marginal cost is that it decouples firms[’]{} behavior in different export destinations and makes the models easy to solve. Without such decoupling we need to solve combinatorial discrete choice problems, which are hard in the case of submodular function maximization (corresponding to increasing marginal cost).
We have seen that working directly with increasing marginal costs leads to a dramatically different perspective on quantitative and qualitative aspects of international trade. It is computationally challenging, but the results are worth it. Some of the puzzles are no longer puzzling, as trade costs behave the way we would expect. [ ]{}
In the future, working with trade models that impose constant marginal production costs will not be as appealing to us as before. It suddenly has a flavor of the proverbial searching for keys under a streetlight. Once we accept the idea of working with models with increasing marginal costs, there are many questions to address. It would be good to re-think many topics in international trade, such as the impact of various policies, interventions or technological changes on the equilibria and on the welfare of different agents in the economy.[^60] The model we worked with is very parsimonious, but including multiple locations per country, multiple sectors, supply chains, and/or foreign direct investment would be desirable. Some of these ingredients would bring their own combinatorial discrete choice problems. The models will certainly be even more computationally intensive than the one we worked with. Improvements in algorithms and hardware, hopefully, will make solving the models feasible.
Breadth of Application {#SectionBreadthOfApplication}
======================
Overview of applications
------------------------
In this section we provide a brief overview of numerous other applications.
### Supply chains with hold-up [@antras]\[SubsectionSupplyChainsWIthHoldup\]
@antras develop a model of continuum sequential supply chains where a main firm organizing its production needs to decide whether to outsource or insource (i.e. perform in-house) each stage of the production process. Production requires relationship-specific investment, which leads to a hold-up problem in the spirit of [ ]{} @grossmanhart. Outsourcing a production stage has the advantage of giving high-powered incentives to the producers, while insourcing has the advantage of mitigating the hold-up problem.
The paper works with constant-elasticity demand and concludes that there can be only one production stage at which the main firm switches production mode; depending on the parameter values, either all of the upstream or all of the downstream (but not both) is outsourced and the rest is insourced. This, of course, clashes with the fact that for many manufacturing supply chains both the upstream (say, elementary components) and the downstream (say, retail) are outsourced, while the core of the production process is insourced.
In Appendix \[AppendixSupplyChainsWithHoldup\] and in Supplementary Material \[AppendixSupplyChainsWithHoldupDetails\], we introduce a transformation of economic variables that makes the mathematics of the model dramatically simpler, in particular connecting the analysis to the classical monopsony problem, whose cost-side aspects are analogous to the demand-side aspects of a monopoly problem.[^61] This allows us to observe by insights analogous to ours above that constant-elasticity demand may be replaced by our tractable functional forms without almost any loss of analytic power. We find that for a realistic functional form of this kind (where the product has a [“]{}saturation point[”]{} in terms of quality), the model implies that both upstream and downstream parts of the supply chain are optimally outsourced, while the middle (core) of the supply chain is optimally insourced, as our intuition suggests in many real-world cases.
### Labor bargaining without commitment (@stole) \[SubsectionLaborBargainingWithoutCommitment\]
Stole-Zwiebel bargaining, as introduced in @stole [@stole2], has become one of the standard ways of modeling labor bargaining in relation to unemployment. In their model, if an employee leaves the firm after an unsuccessful wage bargaining, the remaining employees may renegotiate their wage, and they will choose to do so since the firm[’]{}s bargaining position is weakened. For this reason, the firm will choose to employ more workers than it would if labor markets are competitive; employing an additional worker lowers negotiated wages for the others.
While this model appears to differ from previous examples we considered, as it is not a straightforward monopsony model, we show that behavior under the Stole-Zwiebel model corresponds to a [“]{}partial[”]{} application of the marginal-average transformation ([“]{}partial monopolization[”]{}) and thus remains tractable under our forms. Thus it is common to use standard, form-preserving tractable forms to analyze this model, especially constant-elasticity. The downside of the assumption is that interesting effects are suppressed: the overemployment ratio (ratio of actual employment and employment under competitive labor markets) is a constant independent of economic conditions.
We introduce richer functional forms that preserve the tractability of the model. We find that for a plausible parameterization, changes in the overemployment ratio can account for a non-trivial fraction of employment changes over the business cycle. These results are discussed in Supplementary Material \[AppendixLaborBargainingWithoutCommitment\].
### Imperfectly competitive supply chains \[SubsectionImperfectlyCompetitiveSupplyChains\]
Imperfectly competitive supply chains, as described in @salinger, are a very natural and popular way of modeling multi-stage production. We find that models of this kind may be solved in closed form not only for linear or constant-elasticity demand but also for our proposed, much more flexible functional forms. Intuitively, behavior at each level of the supply chain is derived by applying the marginal-average transformation to behavior at the preceding level, as each step of the supply chain forms the demand for the level above it. We discuss this application in Appendix [ ]{} \[sub:Imperfectly-competitive-supply\] and provide the details in Supplementary Material \[sequential\].
### Two-sided platforms [\` a]{} la @rt2003\[SubsectionTwoSidedPlatforms\]
@rt2003 developed a model of two-sided platforms that allows for understanding pricing decisions for the two sides of the market and their surplus consequences. The model used linear demand. We find that our more flexible functional forms preserve the tractability of the model. This can lead to very different conclusions, as discussed in Supplementary Material \[AppendixTwoSidedPlatforms\].
### Auction Theory\[SubsectionAuctionTheory\]
#### Symmetric independent private values first-price auctions. {#SymmetricIndependentPrivateValuesFirstPriceAuctions}
First price auctions with symmetric independent private values may be solved explicitly for uniform or Pareto value distributions. We find that the tractable functional forms we propose still lead to closed-form solutions, and at the same time they allow for more realistic (i.e. bell-shaped) value distributions. We discuss these results in Supplementary Material \[AppendixSymmetricIndependentPrivateValuesFirstPriceAuctions\].
#### Auctions v. posted prices [@onlineauctions]. {#AuctionsVPostedPrices}
@onlineauctions develop a model in which online sellers choose either auctions or posted prices. They use a uniform distribution in their model. We find that our proposed functional forms also lead to tractable models but allow a richer set of possibilities for the sellers[’]{} optimal behavior that better match the data. We explain the details in Supplementary Material \[AppendixAuctionsVPostedPrices\].
### Selection markets {#SelectionMarkets}
In selection markets (markets with adverse or advantageous selection) as in @imperfectcomp[’]{}s generalization of @einav2010estimating and @einav2011selection, the equilibrium conditions are such that again our proposed tractable functional forms lead to closed-form solutions. This allows for modeling possibilities that provide a better match to the empirical evidence, as explained in Supplementary Material \[AppendixSelectionMarkets\].
### Monopolistic competition {#SelectionMarkets}
Tractable functional forms are very useful in the case of monopolistic competition beyond what we discussed in the previous section. Supplementary Material \[AppendixMonopolisticCompetition\] contains an extensive discussion of other possible modeling choices that generalize, say, the Melitz model or the Krugman model. These calculations may be used as a basis for new research projects on international trade.
General Approximation and the Laplace-Log Transform\[SectionArbitraryDemandAndCostFunctions\]
=============================================================================================
In most of the examples in the previous sections, we have focused on average-marginal form-preserving classes of relatively low dimensions that are tractable at low orders. While these are useful in many applications and reasonably flexible, they have limits in their ability to fit arbitrary equilibrium systems. In this section we show that this limitation arises from the desired tractability of these forms, rather than any underlying rigidity of our average-marginal form-preserving classes. Under weak conditions we formulate here, arbitrary (univariate) equilibrium forms can be approximated arbitrarily well by members of form-preserving classes. The limit of this approximation is the inverse Laplace-log transform of the equilibrium condition. Highly tractable forms may thus be seen as ones with “simple” inverse Laplace-log transforms. We show how the special, policy-relevant features of many common demand forms can be characterized in terms of their transforms. Proofs of the theorems in this, more abstract, section appear in Appendix \[AppendixProofsOfTheorems\]. A number of these proofs are straightforward adaptations of theorems in the existing literature. We include those theorems here for completeness and for the reader’s convenience.
In the next subsection we provide definitions of the Laplace-log transform, utilizing existing mathematical literature. Identifying the most important connections between what is useful in economics and the mathematical literature is non-trivial. While a reasonable number of economists are familiar with Laplace transform based on the Riemann-Stieljes integral, a theory based on that integral would exclude, say, the exponential demand function, which is a popular modeling choice in the economics literature. For a more complete theory we need to utilize the distribution theory by Laurent Schwartz, which has not been used in economics.
The Laplace-log transform and arbitrary approximation
-----------------------------------------------------
Under quite general conditions, univariate equilibrium conditions may be expressed as linear combinations of average-marginal form-preserving functions. To make this statement precise, we focus on the demand side here and write an inverse demand curve of interest as $P\left(q\right)=U'\left(q\right)$, where $U(q)$ is a function primitive to $P\left(q\right)$. We assume that $P\left(q\right)$ is non-increasing, which implies that such primitive function exists. Depending on the model of choice, $U(q)$ may or may not be proportional to the utility of an agent, but to keep the terminology simple, here we refer to $U\left(q\right)$ as the utility.[^62] Even though we explicitly discuss the demand side here, the mathematical theorems below apply to the cost side as well, with a straightforward reinterpretation.
\[completeness\]We observe that virtually all shapes of demand functions that are useful in economics may be associated with a utility function of the form[^63]
$$U\left(q\right)=\intop_{-\infty}^{0}u\left(t\right)q^{-t}dt,\label{eq:UtilityAsASymbolicIntegral}$$
for an appropriate $u\left(t\right)$, where we work on some arbitrarily chosen finite interval $\left[0,\bar{q}\right]$. This integral may be interpreted as a Laplace transform in terms of the variable $s\equiv\log q$, and for this reason, we refer to $u(t)$ as the inverse Laplace-log transform of $U(q)$.[^64]$^,$[^65] At the same time, the integral may be thought of as expressing $U\left(q\right)$ as a linear combination of form-preserving functions of Theorem \[formpreserve\].
**Technical Clarification (Integral Definition).**[^66] Here we define the integral (\[eq:UtilityAsASymbolicIntegral\]) to be the Riemann-Stieltjes integral $$U\left(q\right)=\intop_{-\infty}^{0}q^{-t}du_{I}\left(t\right)\label{eq:UtilityAsRiemannStieljesIntegral}$$ for some function $u_{I}\left(t\right)$, not necessarily nonnegative, such that the integral converges. If this function is differentiable, its derivative $u'_{I}\left(t\right)$ is the $u\left(t\right)$ that appears on the right-hand side of (\[eq:UtilityAsASymbolicIntegral\]). If $u_{I}\left(t\right)$ is only piecewise differentiable, then $u\left(t\right)$ is not an ordinary function but involves Dirac delta functions (i.e. point masses) at the points of discontinuity of $u_{I}\left(t\right)$.
The corresponding inverse demand curve is $P\left(q\right)=U'\left(q\right)=-\intop_{-\infty}^{0}t\ u\left(t\right)q^{-t-1}dt$, or $$P\left(q\right)=\intop_{-\infty}^{1}p\left(t\right)q^{-t}dt,\label{eq:InverseDemandAsASymbolicIntegral}$$ where we defined $p\left(t\right)\equiv\left(1-t\right)u\left(t-1\right)$. We see that $P\left(q\right)$ is a linear combination of form-preserving functions of Theorem \[formpreserve\]. The following theorem summarizes convenient properties of this approach to demand curves: uniqueness, inclusion of linear combinations of power functions, approximations to arbitrary functions, and analyticity.
\[TheoremLaplaceTransformWithRiemannStieltjesIntegrals\]
***(Laplace-log Transform with Riemann-Stieltjes Integrals)***\
***(A)*** For each function $U\left(q\right)$ that may be represented in the form (\[eq:UtilityAsASymbolicIntegral\]) in the sense of (\[eq:UtilityAsRiemannStieljesIntegral\]), there exists just one normalized[^67] function $u_{I}\left(t\right)$ such that (\[eq:UtilityAsRiemannStieljesIntegral\]) holds. ***(B)*** Any polynomial utility function may be written in the form (\[eq:UtilityAsASymbolicIntegral\]). ***(C)*** All functions of the form (\[eq:UtilityAsASymbolicIntegral\]) are analytic. In particular, their derivatives of any order exist. ***(D)*** An arbitrary utility function $\tilde{U}\left(q\right)$ continuous on an interval $\left[0,\bar{q}\right]$ may be approximated with an arbitrary precision by utility functions of the form (\[eq:UtilityAsASymbolicIntegral\]), in the sense of uniform convergence[^68] on $\left[0,\bar{q}\right]$.
Note that part D of this theorem is a simple consequence of the Weierstrass approximation theorem.[^69] The reader may ask why we do not instead work simply with polynomials in $q$ and use them as approximations. Even though this would be possible in principle, it would not be practical. This is because in economics we often need flexibility in the $q\rightarrow 0_+$ limit behavior of the inverse demand function. With any (finite-order) polynomial, we would always get finite $\lim_{q\rightarrow 0_+} P(q)$, i.e., a choke price; to allow for $\lim_{q\rightarrow 0_+} P(q) = \infty$, we could not stay within a finite-order approximation.
Theorem \[formpreserve\] allowed for functions other than linear combinations of power functions, such as $q^{-\alpha}(\log q)^n$ or $\log q$, that are also useful in economics.[^70] Although according to part D of the last theorem, such functions may be approximated by functions of the Riemann-Stieltjes interpretation (\[eq:UtilityAsRiemannStieljesIntegral\]) of (\[eq:UtilityAsASymbolicIntegral\]), it is convenient to be able to write them *exactly* in the form (\[eq:UtilityAsASymbolicIntegral\]) by using a more powerful definition of the integral. This is achieved by the following counterpart of Theorem \[TheoremLaplaceTransformWithRiemannStieltjesIntegrals\], which goes beyond the theory of the Riemann-Stieltjes integral and instead discusses Laplace transform of generalized functions based on the distribution theory by Laurent Schwartz. In the following, $\bar{s}$ is a real number smaller than $\log\bar{q}$.
\[TheoremLaplaceTransformWithSchwartzIntegrals\]
***(Laplace-log Transform with Schwartz Integrals)*** A function $U\left(q\right)$ such that the related function $U_{_{\left[s\right]}}\left(s\right)\equiv U\left(e^{s}\right)$ considered in the half-complex-plane domain $\mathbb{C}_{\bar{s}}^{-}\equiv\left\{ s|\mbox{Re }s<\bar{s}\right\} $ is analytic (i.e. holomorphic) and bounded by a polynomial function may be expressed in the form (\[eq:UtilityAsASymbolicIntegral\]) with $u$ representing a distribution, i.e. a generalized function, or more precisely an element of $\mathcal{D}'$ as defined by @zemanian1965distribution.[^71] This distribution is unique. Conversely, for any Laplace-transformable distribution $u$, the integral (\[eq:UtilityAsASymbolicIntegral\]) viewed as a function of $s\equiv\log q$ in the domain $\mathbb{C}_{\bar{s}}^{-}$ is analytic and bounded by a polynomial of $s$.
***(Laplace Versions of Economic Variables)\[DefinitionLaplaceAsAnAdjective\]*** For a variable $V\left(q\right)$ that may be expressed as an integral of the form $V\left(q\right)=\intop_{a}^{b}v\left(t\right)q^{-t}dt$, we use the adjective *Laplace* to refer to $v\left(t\right)$. For example, $u\left(t\right)$ of (\[eq:UtilityAsASymbolicIntegral\]) would be referred to as *Laplace utility*, and $p\left(t\right)$ of (\[eq:InverseDemandAsASymbolicIntegral\]) as *Laplace inverse demand* or *Laplace price*.
Here we present a theorem describing the relationship of the integral and its discrete approximation. Its proof is constructed using the Euler-Maclaurin formula related to the trapezoidal rule for numerical integration. Following the same logic, it is possible to derive and prove other approximation theorems by adapting numerous theorems on numerical integration that exist in the applied mathematics literature.
\[TheoremDiscreteApproximation\] ***(Discrete Approximation)*** The Laplace-log transform of a function $f(t)$ may be expressed as
$$\int_{-\infty }^{t_{\max }} q^{-t} f(t) \, dt=\text{$\Delta $t} \sum _{t\in T} q^{-t} f(t)-\frac{1}{2} q^{-t_{\max }} \text{$\Delta $t} f\left(t_{\max
}\right)-\frac{1}{2} q^{-t_{\min }} \text{$\Delta $t} f\left(t_{\min }\right)+R,$$ where $T\equiv \left\{t_{\min }, t_{\min }+\text{$\Delta $t}, \text{...} , t_{\max }\right\}$ is an evenly spaced grid with at least two points, $m$ is an integer such that $f$ is $(2m+1)$-times continuously differentiable on $\left[t_{\min },t_{\max }\right]$ and where the remainder $R$ is described below.
The remainder in the theorem consists of three parts: $R\equiv R_1+R_2+R_3$. The first part $R_1$ is simply the difference of $\int_{-\infty
}^{t_{\max }} q^{-t} f(t) \, dt$ and $\int_{t_{\min }}^{t_{\max }} q^{-t} f(t) \, dt$, and can be made very small, since $\int_{-\infty }^{t_{\min
}} q^{-t} f(t) \, dt=q^{-t_{\min }} \int_{-\infty }^0 q^{-t} f\left(t+t_{\min }\right) \, dt$, which is exponentially suppressed for $t_{\min }$ chosen sufficiently negative and for a well-behaved $f(t)$. The second part $R_2$ may be expressed using derivatives of $h(t)\equiv f(t) q^{-t}\text{}$ at $t_{\min }$ and $t_{\max }$: $$R_2=\sum _{k=1}^m \frac{B_{2 k}}{(2 k)!} \left(\text{$\Delta $t}^{2 k} h^{(2 k-1)}\left(t_{\min }\right)-\text{$\Delta $t}^{2 k} h^{(2 k-1)}\left(t_{\max
}\right)\right),$$ where $B_{2 k}$ represent Bernoulli numbers. These terms are suppressed by powers of $\Delta $t as well as by the factorial in the denominator.[^72] The last part $R_3$ may be expressed and bounded using integrals of high derivatives of $h(t)$: $$\begin{array}{lll}
R_3=-\frac{\text{$\Delta $t}^{2 m+1}}{(1+2 m)!} \int_{t_{\min }}^{t_{\max }} P_{1+2 m}(t) h^{(1+2 m)}(t) \, dt & \text{, } & | R_3 |\leq \frac{2
\zeta (2 m+1)\text{$\Delta $t}^{2 m+1}}{(2 \pi )^{2 m+1}} \int_{t_{\min }}^{t_{\max }} | h^{(2 m+1)}(t) | \, dt \\
\end{array}
,$$ where $\zeta $ is the Riemann zeta function and $P_{1+2 m}$ are periodic Bernoulli functions.
Note that this theorem provides a prescription for the weights of the power terms that approximate the integral and gives a bound for the associated error. Of course, by leaving the weights flexible and fitting them using a generalized method of moments, it is possible to get a better approximation with a smaller error. It is also possible to use alternative prescribed weights that correspond to other numerical integration methods. The fact that very different weight choices can all give good approximations is related to the fact that the problem of finding optimal weights is a case of so-called ill-posed problems, for which regularization is typically used in the applied mathematics and econometrics literature.[^73]
Complete monotonicity and pass-through behavior\[sub:CompleteMonotonicity\]
---------------------------------------------------------------------------
Continuous representations of inverse demand functions introduced in the previous subsection provide more conceptual clarity than discrete approximations, which have their idiosyncrasies depending on precisely how many terms are included. These representations in terms of inverse Laplace-log transform can provide useful intuition. For example, if a researcher wishes to find a good discrete approximation to a particular inverse demand function, the researcher may compute the exact inverse Laplace-log transform (or consult Supplementary Material \[sub:LaplaceInverseDemandFunctions\]) to see where the Laplace inverse demand function $p(t)$ is positive or negative. Choosing a few evenly spaced mass points with a similar positivity/negativity pattern is then likely to lead to a tractable approximation to the original inverse demand function that has similar qualitative properties.
Inverse Laplace-log transform representations of inverse demand functions are useful also for another reason: Many demand curves have economic properties (determining many policy implications) that are easily understood in terms of the inverse Laplace-log transform. To develop the related theory, we start with a standard definition of completely monotone functions and then discuss relations between complete monotonicity, the form of Laplace inverse demand, and economic consequences for the pass-through rate.[^74] We classify many commonly used demand functions using this property, given that, as we discussed in the previous section, many policy questions turn on properties of the pass-through rate tied down by complete monotonicity.
***(Completely Monotone Function)*** A function $f\left(x\right)$ is completely monotone iff for all $n\in\mathbb{N}$ its $n$th derivative exists and satisfies *$\left(-1\right)^{n}f^{\left(n\right)}\left(x\right)\ge0$.*
It turns out that many commonly used demand functions are such that the consumer surplus is completely monotone as a function of negative log quantity. For this reason, we make the following definition.
***(Complete Monotonicity of the Demand Specification)***[^75]
We say that a demand function (or a utility function) satisfies the complete monotonicity criterion iff the associated consumer surplus is a completely monotone function of $-s$, i.e. for all $n\in\mathbb{N}$, *$CS_{_{\!\left[s\right]}}^{\left(n\right)}\left(s\right)\ge0$,* or equivalently**[^76] *$U_{_{\!\left[s\right]}}^{\left(n\right)}\left(s\right)-U_{_{\!\left[s\right]}}^{\left(n+1\right)}\left(s\right)\ge0$.* Strict complete monotonicity criterion then refers to these inequalities being strict.
\[TheoremNonnegativityOfLaplaceConsumerSurplus\]
***(Nonnegativity of Laplace Consumer Surplus)*** A (single-product) utility function is bounded below and satisfies the complete monotonicity criterion iff the Laplace consumer surplus $cs\left(t\right)$ is nonnegative and supported on $\left(-\infty,0\right)$, i.e. $CS\left(q\right)=\int_{-\infty}^{0}$$cs\left(t\right)q^{-t}dt$ for some $cs\left(t\right)\ge0$.**
\[TheoremMonotonicityOfThePassThroughRate\]
***(Monotonicity of the Pass-Through Rate)*** The complete monotonicity criterion for demand functions implies the pass-through rate decreasing with quantity in the case of constant-marginal-cost monopoly. The only exception is BP demand, for which the pass-through rate is constant.
\[TheoremCompleteMonotonicityOfDemandSpecification\]
***(Complete Monotonicity of Demand Specification)*** The following demand functions satisfy the complete monotonicity criterion:[^77]\
Pareto/constant elasticity ($\epsilon>1$), BP ($\epsilon>1$), logistic distribution, log-logistic distribution ($\gamma>1$), Gumbel distribution ($\alpha>1$), Weibull distribution ($\alpha>1$), Fr' echet distribution ($\alpha>1$), gamma distribution ($\alpha>1$), Laplace distribution[^78], Singh-Maddala distribution ($a>1$), Tukey lambda distribution ($\lambda<1$), Wakeby distribution ($\beta>1$), generalized Pareto distribution ($\gamma<1$), Cauchy distribution.
[**Corollary. (Monotonicity of the Pass-Through Rate)**]{} *The last two theorems imply that the demand functions listed in Theorem \[TheoremCompleteMonotonicityOfDemandSpecification\] lead to constant-marginal-cost pass-through rate decreasing in quantity, with the exception of Pareto/constant elasticity as well as the more general BP demand, which are known to lead to constant pass-through.*
\[TheoremAbsenceOfCompleteMonotonicityOfDemandSpecification\]
***(Absence of Complete Monotonicity of Demand Specification)*** The following demand functions do **not** satisfy the complete monotonicity criterion: normal distribution, lognormal distribution, constant superelasticity (Klenow and Willis), Almost Ideal Demand System (either with finite or infinite surplus), log-logistic distribution ($\gamma<1$), Fr' echet distribution ($\alpha<1$), Weibull distribution ($\alpha<1$), Gumbel distribution ($\alpha<1$), Pareto/constant elasticity ($\varepsilon>1$), gamma distribution ($\alpha<1$), Singh-Maddala distribution ($a<1$), Tukey lambda distribution ($\lambda>1$), Wakeby distribution ($\beta<1$), generalized Pareto distribution ($\gamma>1$).
In our Supplementary Material \[forms\] we provide a more complete taxonomy of pass-through properties of some of the demand forms mentioned here. Interestingly, the normal distribution has economic properties close to those of forms that satisfy the complete monotonicity criterion, since the non-complete monotonicity manifests itself only for very high-order derivatives.[^79] The lognormal distribution is not quite so well-behaved, but the more realistic income model (the double Pareto lognormal) behaves similarly for calibrated parameter values.
Conclusion
==========
We have shown that the set of analytic solutions to many common economic problems is substantially richer than typically assumed. They let economists work with flexible, realistic models, instead of imposing restrictive, unrealistic assumptions in order to get analytic solutions of traditional kinds. Our approach to getting analytic solutions is also useful when applied to sub-problems of larger economic models. In those cases it leads to the ability to solve those models numerically in a much more efficient way, as in our international trade application.
The international trade model provides a perspective on the gravity equation of trade that is completely different from the rest of the literature. The model resolves economic puzzles related to the cost of trade since its parameters take realistic values and at the same time the model matches well firm-level and country-level trade patterns.
Of course, there are many other applications of our method, some of which we briefly discussed here, some of which we will report in separate papers, and some of which, hopefully, the reader will develop on his/her own.
Proofs of Theorems {#AppendixProofsOfTheorems}
==================
**Proof of Theorem \[formpreserve\] (Characterization of Form-Preserving Functions).** Here we present a constructive proof of the theorem that exactly traces the steps we first used to derive the theorem’s statement. It is instructive for readers familiar with Fourier transform or Laplace transform because it highlights the properties of functions we emphasize in this paper and shows how using the transforms, calculations may be conveniently performed just in a couple of lines. Other readers may prefer reading Supplementary Material \[AppendixDiscussionOfTheCharacterizationTheorem\], where we discuss how the theorem may be proven without functional transforms.
Here we derive the structure of $m$-dimensional functional form classes $\mathcal{C}$ that are invariant under average-marginal transformations. We take as the domain of the functions an open interval $I$ of positive real numbers, which may include all positive real numbers.[^80] For convenience we express the (infinitely differentiable) functions $F\left(q\right)$ on $I$ in terms of functions $G\left(s\right)$ defined in the corresponding logarithmically transformed domain, with the identification $s\equiv\log q$, $F\left(q\right)\equiv G\left(\log q\right)$. Consider a function $F\left(q\right)\mathcal{\in C}$ and its counterpart $G\left(s\right)$. In terms of $G$, the average-marginal form-preservation requires that the counterpart of $aG+bG'$ belong to the class $\mathcal{C}$, if the counterpart of $G$ does so. For technical reasons, we will work with $G\left(s\right)$ multiplied by the characteristic function $1_{S}\left(s\right)$ of an arbitrarily chosen finite non-empty interval $S\equiv\left(s_{1},s_{2}\right)\in I$, i.e. with $G_{S}\left(s\right)\equiv G\left(s\right)1_{S}\left(s\right)$.[^81] We denote by $\hat{G}_{S}\left(\omega\right)$ the Fourier transform of $G_{S}\left(s\right)$, which in turn may be expressed as the inverse Fourier transform $G_{S}\left(s\right)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}\hat{G}_{S}\left(\omega\right)e^{-i\omega s}d\omega$.[^82]
By iterating the defining property of average-marginal form-preservation, we know that the class $\mathcal{C}$ contains also counterparts of the derivatives $G^{\left(n\right)}\left(s\right)$. We will consider the first $m$ of them, in addition to $G\left(s\right)$. For $n=1,2,...,m$, we denote by $G_{S}^{\left(n\right)}\left(s\right)$ the truncation of $G^{\left(n\right)}\left(s\right)$ to the interval $S$, i.e. $G_{S}^{\left(n\right)}\left(s\right)\equiv G^{\left(n\right)}\left(s\right)1_{s\in S}$. Inside the interval $S$, $$G_{S}^{\left(n\right)}\left(s\right)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}\left(-i\omega\right)^{n}\hat{G}_{S}\left(\omega\right)e^{-i\omega s}d\omega,\quad\mbox{for}\;s\in S,\;n\in\{0,1,2,...,m\}.\label{eq:NThDerivative}$$ The $m+1$ functions $G_{S}\left(s\right),G_{S}^{\left(1\right)}\left(s\right),G_{S}^{\left(2\right)}\left(s\right),...,G_{S}^{\left(m\right)}\left(s\right)$ span a vector space with dimensionality $m+1$ or less. Dimensionality equal to $m+1$ would contradict the assumption of having an $m$-dimensional functional form class, which implies that the set of functions $G_{S}\left(s\right),G_{S}^{\left(1\right)}\left(s\right),G_{S}^{\left(2\right)}\left(s\right),$ $...,G_{S}^{\left(m\right)}\left(s\right)$ must be linearly dependent on the interval $S$. As a result, there must exist a polynomial $T_{0}\left(.\right)$ (with real coefficients), such that $$\int_{-\infty}^{\infty}T_{0}\left(-i\omega\right)\hat{G}_{S}\left(\omega\right)e^{-i\omega s}d\omega\label{eq:T0GS}$$ is zero for any $s\in S$. This expression vanishes not only for $s\in S\equiv\left(s_{1},s_{2}\right)$, but also for $s\in\left(-\infty,s_{1}\right)$ and $s\in\left(s_{2},\infty\right)$. This is because the right-hand-side of (\[eq:NThDerivative\]) when extended to arbitrary $s\in\mathbb{R}$ represents the $n$th derivative of $G_{S}\left(s\right)$ in the sense of the Schwartz distribution theory, and given that $G_{S}\left(s\right)$ vanishes for $s\in\left(-\infty,s_{1}\right)$ and $s\in\left(s_{2},\infty\right)$, so must its $n$th derivative. Given that the expression (\[eq:T0GS\]) is a generalized function[^83] of $s$ that gives zero when integrated against any test function[^84] supported on $(-\infty,s_{1}-\epsilon]\cup\left[s_{1}+\epsilon,s_{2}-\epsilon\right]\cup[s_{2}+\epsilon,\infty)$ for any $\epsilon>0$, we may write it as a linear combination of Dirac delta functions and a finite number of their derivatives located at $s_{1}$ and $s_{2}$. By computing its Fourier transform we find that $T_{0}\left(-i\omega\right)\hat{G}_{S}\left(\omega\right)$ must be of the form $
T_{1}\left(\omega\right)e^{is_{1}\omega}+T_{2}\left(\omega\right)e^{is_{2}\omega}
$ with some polynomials $T_{1}\left(\omega\right)$ and $T_{2}\left(\omega\right)$, with complex coefficients in general. Consequently, $\hat{G}_{S}\left(\omega\right)$ may be written as $$\hat{G}_{S}\left(\omega\right)=\frac{T_{1}\left(\omega\right)}{T_{0}\left(-i\omega\right)}e^{is_{1}\omega}+\frac{T_{2}\left(\omega\right)}{T_{0}\left(-i\omega\right)}e^{is_{2}\omega}.$$ The polynomial $T_{0}\left(-i\omega\right)$ may have a common factor with $T_{1}\left(\omega\right)$ or $T_{2}\left(\omega\right)$ or both. If we cancel these common factors, we may rewrite the expression as
$$\hat{G}_{S}\left(\omega\right)=\frac{T_{3}\left(\omega\right)}{T_{5}\left(\omega\right)}e^{is_{1}\omega}+\frac{T_{4}\left(\omega\right)}{T_{6}\left(\omega\right)}e^{is_{2}\omega}\label{eq:GHatS}$$
for some polynomials $T_{3}$, $T_{4}$, $T_{5}$, and $T_{6}$, such that $T_{3}$ has no common divisors with $T_{5}$ and similarly for $T_{4}$ with $T_{6}$. Let us compute the inverse Fourier transform of the last expression for $\hat{G}_{S}\left(\omega\right)$ using the residue theorem. To perform the integration, we consider each of the two terms in (\[eq:GHatS\]) separately and specialize to $s\in S$. We close the integration contour by semicircles at infinity of the complex plane, correctly chosen so that their contribution to the integral vanishes. The integral value is then equal to the sum of the pole (residue) contributions, which give exponentials of $s$ multiplied by polynomials of $s$. We see that for $s\in S$, $
G_{S}\left(s\right)=\sum_{j=1}^{N}D_{j}\left(s\right)e^{-ist_{j}},
$ for some integer $N$, complex numbers $t_{j}$ and polynomials $D_{j}\left(s\right)$. Since the interval $S$ was chosen arbitrarily, not just $G_{S}\left(s\right)$, but also $G\left(s\right)$ itself must take this form. In the last expression the constants may be complex. Without loss of generality, we can assume that the first $N_{1}$ numbers $t_{j}$ are real and the remaining ones have an imaginary part. By combining individual terms into real contributions so that $G\left(s\right)$ is real, we get $$G\left(s\right)=\sum_{j=1}^{N_{1}}A_{j}\left(s\right)e^{-st_{j}}+\sum_{j=1}^{N_{2}}\left(B_{j}\left(s\right)\cos\tilde{t}_{j}s+C_{j}\left(s\right)\sin\tilde{t}_{j}s\right)e^{-\hat{t}_{j}s},$$ where $A_{j}\left(s\right)$, $B_{j}\left(s\right)$, and $C_{j}\left(s\right)$ are polynomials, and $N_{1}+2N_{2}=N$. This form of $G\left(s\right)$ translates into the following form of $F\left(q\right)$: $$F\left(q\right)=\sum_{j=1}^{N_{1}}A_{j}\left(\log q\right)q^{-t_{j}}+\sum_{j=1}^{N_{2}}\left(B_{j}\left(\log q\right)\cos\left(\tilde{t}_{j}\log q\right)+C_{j}\left(\log q\right)\sin\left(\tilde{t}_{j}\log q\right)\right)q^{-\hat{t}_{j}}.\label{eq:AverageMarginalInvariantFunction}$$ If we wish to exclude the possibility of oscillations, e.g. in economic applications where we allow the functional form to be valid arbitrarily close to $q=0$, we can set the polynomials $B_{j}$ and $C_{j}$ to zero and consider only functions of the form $
F\left(q\right)=\sum_{k=1}^{N_{1}}A_{j}\left(\log q\right)q^{-t_{j}}.
$ An example of functional forms of this kind is $aq^{-t}+bq^{-u}+cq^{-u}\log q+dq^{-u}(\log q)^{2}$. The reader can easily verify that this is a four-dimensional functional form class invariant under average-marginal transformations. In general, it is now straightforward to check that the result (\[eq:AverageMarginalInvariantFunction\]) implies the statement of the theorem.
**Proof of Theorem \[TheoremClosedFormSolutions\] (Closed-Form Solutions).** The proof is straightforward. By assumption, there exists some definite power $b$ such that $x \equiv q^b$ satisfies an algebraic equation of order $k$: $P_k(x)=0$, where $P_k(x)$ is a polynomial of order at most $k$. For this to be true, all elements of the functional form class must factorize as $q^a P_k(q^b)$ for some definite $a$. When expanded, the powers of $q$ in individual terms lie on the grid ${a, a+b, ... , a + b k }.$
#### Proof of Theorem \[TheoremAggregation\]
**(Aggregation)**. The firm[’]{}s revenue $q P(q)$, cost $\int \text{MC}(q) \, dq$, and profit are all linear combinations of powers of $q$. For this reason, it suffices to show that it is possible to perform explicitly aggregation integrals $\mathcal{I}$ for powers of $q$ (the quantity optimally chosen by a firm with productivity parameter $a$): $\mathcal{I}\equiv \int q(a)^{\gamma _1} \, dG(a)$. Changing the integration variable to $q$ gives: $\mathcal{I}=\int q^{\gamma _1} G'(a(q)) a'(q) \, dq$. The firm[’]{}s first-order condition equates the marginal revenue $R'(q)=P(q)+q P'(q)$ to the marginal cost $\text{MC}_0(q)+a \text{MC}_1(q)$ and implies $$a=\frac{R'(q)-\text{MC}_0(q)}{\text{MC}_1(q)}\Rightarrow a'(q)=\frac{R''(q)-\text{MC}_0'(q)}{\text{MC}_1(q)}-\frac{R'(q)-\text{MC}_0(q)}{\text{MC}_1(q){}^2}
\text{MC}_1'(q).$$ Substituting these expressions into the integral gives $$\mathcal{I}=\int q^{\gamma _1} \left(\frac{R''(q)-\text{MC}_0'(q)}{\text{MC}_1(q)}-\frac{R'(q)-\text{MC}_0(q)}{\text{MC}_1(q){}^2} \text{MC}_1'(q)\right)
G'\left(\frac{R'(q)-\text{MC}_0(q)}{\text{MC}_1(q)}\right) \, dq.$$ Since $G'(a)$ is a mixture of powers of $a$, and $\left(R'(q)-\text{MC}_0(q)\right) \text{MC}_1'(q)$ and [ ]{}$R''(q)-\text{MC}_0'(q)$ are mixtures of powers of $q$, the integral on the right-hand side may be written as a linear combination of integrals of the type $$\int q^{\gamma _5} \text{MC}_1(q){}^{\gamma _7} \left(-\text{MC}_0(q)+R'(q)\right){}^{\gamma _6} \, dq,$$ where $\gamma _7$ equals $-\gamma _6-1$ or $-\gamma _6-2$. Given our assumptions, up to a known multiplicative constant this integral equals $\int q^{\gamma _8} N_1\left(q^{\alpha }\right){}^{\gamma _9} N_2\left(q^{\alpha }\right){}^{\gamma _{10}} \, dq$. If we change the integration variable to $x\equiv q^{\alpha }$, the problem reduces to computing the integral $\int x^{\gamma _{11}} N_1(x){}^{\gamma _{12}} N_2(x){}^{\gamma _{13}} \, dx$. To complete the proof, it suffices to examine the structure of this intergral for different structures of the polynomials. Depending on the structure of the polynomials, the following six non-exclusive cases may arise:
\(1) If the polynomials $N_1$ and $N_2$ are trivial, the integral reduces to a power function of $q$, without any special functions.
\(2) If either $N_1$ or $N_2$ is trivial and the other polynomial is linear, the integral leads to the standard hypergeometric function, denoted $\text{}_2F_1$, since up to an additive constant $$\int x^{\gamma _{11}} \left(1+\gamma _{14} x\right){}^{\gamma _{13}} \, dx=\frac{x^{1+\gamma _{11}}}{1+\gamma _{11}} \, _2F_1\left(1+\gamma _{11},-\gamma
_{13};2+\gamma _{11};-x \gamma _{14}\right)$$
\(3) If both $N_1$ and $N_2$ are linear, the integral leads to the standard Appell function, denoted $F_1$, since up to an additive constant $$\int x^{\gamma _{11}} \left(1+\gamma _{18} x\right){}^{\gamma _{12}} \left(1+\gamma _{19} x\right){}^{\gamma _{13}} \, dx=\frac{x^{1+\gamma _{11}}}{1+\gamma
_{11}} F_1\left(1+\gamma _{11};-\gamma _{12},-\gamma _{13};2+\gamma _{11};-x \gamma _{18},-x \gamma _{19}\right)$$
\(4) If either $N_1$ and $N_2$ is trivial and the other polynomial is quadratic, the integral again leads to the standard Appell function, denoted $F_1$: $$\int x^{\gamma _{11}} \left(1+\gamma _{14} x+\gamma _{15} x^2\right){}^{\gamma _{13}} \, dx=\text{}$$
$$\frac{\gamma _{15}^{\gamma _{13}} x^{1+\gamma _{11}}}{1+\gamma _{11}} \left(\frac{1+x \gamma _{14}+x^2 \gamma _{15}}{\gamma _{15}+x \gamma _{14}
\gamma _{15}+x^2 \gamma _{15}^2}\right){}^{\gamma _{13}} F_1\left(1+\gamma _{11};-\gamma _{13},-\gamma _{13};2+\gamma _{11};\gamma _{16} x,\gamma
_{17} x\right)$$ where $\gamma _{16}=-2 \gamma _{15} \left(\gamma _{14}+\sqrt{\gamma _{14}^2-4 \gamma _{15}}\right){}^{-1}$, and $\gamma _{17}=2 \gamma _{15} \left(-\gamma _{14}+\sqrt{\gamma _{14}^2-4 \gamma _{15}}\right){}^{-1}$.
\(5) If $N_1$ and $N_2$ are both of order less than five, we can factorize them into products of linear polynomials with the factorization performed in closed form by the method of radicals. The resulting integral may be expressed using Lauricella functions. In particular, by the fundamental theorem of algebra, $N_1$ and $N_2$ may be written as products of linear functions. This means that up to a multiplicative constant, $x^{\gamma _{11}}
\left(1+\gamma _{18}x\right){}^{\gamma _{12}} \left(1+\gamma _{19}x\right){}^{\gamma _{13}}$ equals $x^{b-1} \left(1-u_1x\right){}^{-b_1} \ldots
\left(1-u_nx\right){}^{-b_n}$, where $u_i$ represent the reciprocals of the roots of the polynomials. These roots, as well the constants $b$, $b_1$, ..., $b_n$ may be found explicitly using the standard formulas for solutions to quadratic, cubic, or quartic equations. Up to an additive constant, the corresponding integral equals $$\int x^{b-1} \left(1-u_1 x\right){}^{-b_1} \ldots \left(1-u_n x\right){}^{-b_n} \, dx=\frac{x^b}{b} F_D{}^{(n)}\left(b,b_1,\ldots ,b_n,b+\text{1;}
u_1 x,\ldots ,u_n x\right)$$ This is because in general the Lauricella function $F_D{}^{(n)}$ is defined as $$F_D{}^{(n)}\left(b,b_1,\ldots ,b_n,c ; x_1,\ldots ,x_n\right)=\frac{\Gamma (c)}{\Gamma (b) \Gamma (c-b)} \int_0^1 y^{b-1} (1-y)^{c-b-1} \left(1-x_1
y\right){}^{-b_1} \ldots \left(1-x_n y\right){}^{-b_n} \, dy$$ with $\Gamma$ denoting the standard gamma function, and in the special case of $c=b+1$ this definition becomes $$F_D{}^{(n)}\left(b,b_1,\ldots ,b_n,b+1 ; x_1,\ldots ,x_n\right)=b \int_0^1 y^{b-1} \left(1-x_1 y\right){}^{-b_1} \ldots \left(1-x_n y\right){}^{-b_n}
\, dy$$ Substituting $y\to \left.x_0\right/x, x_1\to u_1x$ and $x_n\to u_nx$ then leads to the desired result for the integral: $$\int_0^x x_0^{b-1} \left(1-u_1 x_0\right){}^{-b_1} \ldots \left(1-u_n x_0\right){}^{-b_n} \, dx_0=\frac{x^b}{b} F_D{}^{(n)}\left(b,b_1,\ldots
,b_n,b+1 ; u_1 x,\ldots ,u_n x\right)$$
\(6) Finally, if either $N_1$ or $N_2$ is of order five or higher, the factorization involves root functions, since the method of radicals can no longer be used. However, the resulting integral may still be expressed using Lauricella functions as described above.
We conclude that the structure of the resulting expressions for the integral agrees with the statement of Theorem 3.
**Proof of Theorem \[TheoremLaplaceTransformWithRiemannStieltjesIntegrals\] (Laplace-log Transform with Riemann-Stieltjes Integrals).** **(A)** This follows from Theorem I.6.3 of of @widder2010laplace. **(B)** If we choose $u_{I}\left(t\right)$ appearing in Equation \[eq:UtilityAsRiemannStieljesIntegral\] from the paper to be piecewise constant with a finite number $N$ of points of discontinuity $\left\{ t_{j},j=1,2,...,N\right\} $, the integral becomes $
U\left(q\right)=\sum_{j=1}^{N}a_{j}q^{-t_{j}},
$ where $a_{j}$ is the (signed) magnitude of the discontinuity at point $t_{j}$, i.e. the magnitude of the mass that $u\left(t\right)$ has at point $t_{j}$. If we choose $t_{j}$ to be nonpositive integers, $U\left(q\right)$ will be a polynomial of $q$. By appropriate choices of $N$ and $a_{j}$, any polynomial of $q$ may be expressed in this way. **(C)** Given that polynomials are included in Equation \[eq:UtilityAsASymbolicIntegral\] from the paper, the theorem follows from the Weierstrass approximation theorem, which states that polynomials are dense in the space of continuous functions on a compact interval. For a constructive proof of the theorem due to Bernstein, see e.g. Section VII.2 of @feller2008introduction. **(D)** This follows from Theorem I.5a of @widder2010laplace.
**Proof of Theorem \[TheoremLaplaceTransformWithSchwartzIntegrals\] (Laplace-log Transform with Schwartz Integrals).** The three sentences of the theorem are implied by the following statements in @zemanian1965distribution: (1) ** Theorem 8.4-1 and Corollary 8.4-1a, (2) Theorem 8.3-1a, (3) Theorem 8.3-2 and the text following Corollary 8.4-1a.
**Proof of Theorem \[TheoremDiscreteApproximation\] (Discrete Approximation).** This theorem follows straightforwardly from Theorem 4 of @apostol1999elementary. That theorem provides in its Equation 25 a convenient form of the Euler-Maclaurin formula, which may be written, after a small change of notation, as:
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$\sum _{k=1}^{n_T} F(k)=\int_1^{n_T} F(x) \, dx+\mathcal{C}(F)+E_F\left(n_T\right),$
$\mathcal{C}(F)=\frac{1}{2} F(1)-\sum _{r=1}^m \frac{B_{2 r}}{(2 r)!} F^{(2 r-1)}(1)+\frac{1}{(2 m+1)!} \int_1^{\infty } P_{2 m+1}(x) F^{(2 m-1)}(x)
\, dx,$
$E_F\left(n_T\right)=\frac{1}{2} F\left(n_T\right)-\sum _{r=1}^m \frac{B_{2 r}}{(2 r)!} F^{(2 r-1)}\left(n_T\right)+\frac{1}{(2 m+1)!} \int_{n_T}^{\infty
} P_{2 m+1}(x) F^{(2 m-1)}(x) \, dx.$
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We can use this form of the Euler-Maclaurin formula to prove the discrete approximation theorem. The relationship we want to prove is
$\sum _{t\in T} q^{-t} f(t)=\frac{1}{\text{$\Delta $t}} \int q^{-t} f(t) \, dt+\frac{1}{2} q^{-t_{\min }} f\left(t_{\min }\right)+\frac{1}{2} q^{-t_{\max
}} f\left(t_{\max }\right)-\frac{R_1+R_2+R_3}{\text{$\Delta $t}},$
where $T\equiv \left\{t_{\min }, t_{\min }+\text{$\Delta $t}, \text{...} , t_{\max }\right\}$ and $n_T$ is the number of points in the grid $T$. Equivalently,
$\sum _{t\in T} q^{-t} f(t)=\frac{1}{\text{$\Delta $t}} \int_{t_{\min }}^{t_{\max }} q^{-t} f(t) \, dt+\frac{1}{2} q^{-t_{\min }} f\left(t_{\min
}\right)+\frac{1}{2} q^{-t_{\max }} f\left(t_{\max }\right)-\frac{R_2+R_3}{\text{$\Delta $t}}.$
If we use the notation $$F(k)\equiv q^{-t_{\min }-k \text{$\Delta $t}} f\left(t_{\min }+(k-1) \text{$\Delta $t}\right)$$ we can rewrite the individual terms in the desired formula as
------------------------------------------------------------------------------------------------------------------------------------------------------
$\sum _{t\in T} q^{-t} f(t)=\sum _{k=1}^{n_T} F(k),$
$\frac{1}{2} q^{-t_{\min }} f\left(t_{\min }\right)+\frac{1}{2} q^{-t_{\max }} f\left(t_{\max }\right)=\frac{F(1)}{2}+\frac{F\left(n_T\right)}{2},$
$\frac{R_2}{\text{$\Delta $t}}=\sum _{r=1}^m \frac{B_{2 r}}{(2 r)!} \left(F^{(-1+2 r)}(1)-F^{(-1+2 r)}\left(n_T\right)\right),$
$\frac{R_3}{\text{$\Delta $t}}=-\frac{\text{$\Delta $t}^{2 m}}{(1+2 m)!} \int_{t_{\min }}^{t_{\max }} P_{1+2 m}(t) h^{(1+2 m)}(t) \, dt=-\frac{1}{(2
m+1)!} \int_1^n P_{1+2 m}(x) F^{(1+2 m)}(x) \, dx.$
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By comparing these expressions with those of Theorem 4 of @apostol1999elementary, we see that the main statement of Theorem \[TheoremDiscreteApproximation\] is valid. The bound on $R_3$ then simply follows from the formula $| P_{2 m+1}(x) |\leq 2 (2 m+1)! (2 \pi )^{-2 m-1}$ ; see p. 538 of @lehmer1940maxima.\
**Proof of Theorem \[TheoremNonnegativityOfLaplaceConsumerSurplus\] (Nonnegativity of Laplace Consumer Surplus).** This theorem follows from Bernstein’s theorem on completely monotone functions, formulated e.g. as Theorem IV.12a of @widder2010laplace or Theorem 1.4 of @schilling2012bernstein.
**Proof of Theorem \[TheoremMonotonicityOfThePassThroughRate\] (Monotonicity of the Pass-Through Rate).** Constant marginal cost monopoly pass-through rate may be expressed as $
\rho={CS_{_{\!\left[s\right]}}'\left(s\right)}/{CS_{_{\!\left[s\right]}}''\left(s\right)},
$ which is straightforward to verify from the basic definitions. For a completely monotone problem, Laplace consumer surplus $cs\left(t\right)$ is nonnegative. For this reason, the inverse of $\rho$ may be expressed as a weighted average of $t$ with nonnegative weight $
w\left(t,s\right)\equiv{t\:cs\left(t\right)e^{-st}}/{\int_{-\infty}^{0}t\,cs\left(t\right)e^{-st}dt}
$ as follows $$\frac{1}{\rho}=\frac{CS_{_{\!\left[s\right]}}''\left(s\right)}{CS_{_{\!\left[s\right]}}'\left(s\right)}=-\frac{\int_{-\infty}^{0}t^{2}cs\left(t\right)e^{-st}dt}{\int_{-\infty}^{0}t\ cs\left(t\right)e^{-st}dt}=-\int_{-\infty}^{0}t\,w\left(t,s\right)dt.$$ In response to an increase in $s$, the weight gets shifted towards more negative $t$,[^85] and $1/\rho$ decreases. We conclude that $\rho$ is decreasing in $q$. Only if $t\,cs\left(t\right)$ is supported at one point will there be no shift in weight and $\rho$ remains constant. That case corresponds to BP demand.
![Confidence intervals for the cost exponent $\gamma$ for individual industries at the 95% level. For visualization purposes, the industries are ordered by the standard deviation of $\gamma$ and stacked vertically.[]{data-label="FigureConfidenceIntervalsForBeta"}](FigureConfidenceIntervalsForGamma.pdf)
**Proof of Theorem \[TheoremCompleteMonotonicityOfDemandSpecification\] (Complete Monotonicity of Demand Specification).** The complete monotonicity properties follow by straightforwardly recognizing that in these cases $t\,p\left(t\right)$ is nonnegative and supported on $\left(-\infty,1\right)$, with the corresponding Laplace inverse demand functions $p\left(t\right)$ listed in our Supplementary Material \[sub:LaplaceInverseDemandFunctions\], which also contains additional discussion. Note that for most of the inverse demand functions listed in the theorem, it is also possible to prove complete monotonicity using Theorems 1–6 of @miller2001completely.
**Proof of Theorem \[TheoremAbsenceOfCompleteMonotonicityOfDemandSpecification\] (Absence of Complete Monotonicity of Demand Specification) .** The statement of the theorem follows by inspection of the Laplace inverse demand functions, as in the previous proof. Additional discussion may be found in Supplementary Material \[sub:LaplaceInverseDemandFunctions\].
Details of the Generalized EOQ Model Estimation {#AppendixDetailsOfTheGeneralizedEOQModelEstimation}
===============================================
Here we provide additional details of the estimation of the cost parameter $\beta =(1-\gamma )/(2-\gamma )$. As mentioned in the main text, we selected industries that included at least 10 firms satisfying our criteria. The corresponding confidence intervals corresponding to individual industries are plotted in Figure \[FigureConfidenceIntervalsForBeta\].
In principle, the value of average estimated $\beta $ could be sensitive to the cutoff on the number of firms per industry. Table \[TableSensitivityToCutoff\] summarizes the dependence of the resulting average $\beta $ on the choice of the cutoff. It turns out that the average $\beta $ remains roughly the same even for large changes of the cutoff on the number of firms.
$N_{f,\min }$ $N_I$ $\beta$ $\sigma _{\beta }$
--------------- ------- --------- --------------------
5 192 0.39 0.20
10 70 0.39 0.12
15 45 0.39 0.10
20 23 0.41 0.10
25 14 0.39 0.10
30 11 0.42 0.07
35 9 0.42 0.08
: Sensitivity to the cutoff $N_{f,\min }$ of the number of firms per industry. The cutoff influences the number of industries $N_I$ that satisfy the sample selection criteria and the resulting mean $\beta$ and the corresponding standard deviation $\sigma _{\beta }$.[]{data-label="TableSensitivityToCutoff"}
The estimated value of $\gamma$ could be, in principle, also influenced by seasonality patterns. To investigate this issue, we construct a measure of seasonality of individual industries. In particular, we calculate a Herfindahl-like seasonality index based on the shares of trade in individual months of the year, defined as $H_s=\sum
_{i=1}^{12} v_i^2$, where $v_i$ is the average share of month $i$ in the average annual trade value. A high value of the index means that trade flows are very unevenly distributed across months. Then we regress $\gamma$ on this measure. We find that the 95% confidence interval of the slope coefficient is \[-0.69,1.21\] and the corresponding p-value is 0.58. For robustness, we change the cutoff to 5 firms, getting [ ]{}the confidence interval \[-0.91,0.30\] and the p-value of 0.32. In both cases we do not reject the hypothesis that the slope coefficient is zero. The data is plotted in Figure \[FigureInfluenceOfSeasonality\].
[.5]{}
![The relationship of the cost exponent $\alpha$ for specific industries and the industry seasonality index $H_s$. Figure (a) corresponds to the sample used for the main estimation, which is based on industries with at least 10 firms satisfying the sample selection criteria. We do not observe any systematic pattern relating $\alpha$ and $H_s$. Figure (b) corresponds to a cutoff set to 5 firms as a robustness check. Also, in this case the values of $\alpha$ do not seem to be influenced by $H_s$.[]{data-label="FigureInfluenceOfSeasonality"}](FigureInfluenceOfSeasonalityFor10FirmCutoff.pdf)
[.5]{}
![The relationship of the cost exponent $\alpha$ for specific industries and the industry seasonality index $H_s$. Figure (a) corresponds to the sample used for the main estimation, which is based on industries with at least 10 firms satisfying the sample selection criteria. We do not observe any systematic pattern relating $\alpha$ and $H_s$. Figure (b) corresponds to a cutoff set to 5 firms as a robustness check. Also, in this case the values of $\alpha$ do not seem to be influenced by $H_s$.[]{data-label="FigureInfluenceOfSeasonality"}](FigureInfluenceOfSeasonalityFor5FirmCutoff.pdf)
World Trade {#AppendixWorldTradeFlows}
===========
Details of data construction {#AppendixWorldTradeFlowsDetailsOfDataConstruction}
----------------------------
Here we provide details of the data construction for Section \[SectionWorldTrade\]. The economies used to fit our model are, in descending order by 2006 GDP, United States, Japan, Germany, China, United Kingdom, France, Italy, Canada, Spain, Brazil, Russia, South Korea, Mexico, India, Australia, Netherlands, Turkey, Switzerland, Sweden, Belgium, Saudi Arabia, Norway, Poland, Austria, Denmark, Greece, South Africa, Iran, Argentina, Ireland, Nigeria, United Arab Emirates, Thailand, Finland, Portugal, Hong Kong, Venezuela, Malaysia, Colombia, Czech Republic, Chile, Israel, Singapore, Pakistan, Romania, Algeria, Hungary, New Zealand, Kuwait, Peru, Kazakhstan, Bangladesh, Morocco, Vietnam, Qatar, Slovakia, Croatia, Ecuador, Luxembourg, Slovenia, Dominican Republic, Oman, Belarus, Tunisia, Bulgaria, Syria, Sri Lanka, Serbia/Serbia and Montenegro, Lithuania, Guatemala, Kenya, Costa Rica, Lebanon, Latvia, Azerbaijan, Cyprus, Ghana, Uruguay, Yemen, Tanzania, El Salvador, Bahrain, Trinidad and Tobago, Panama, Cameroon, Ivory Coast, Iceland, Estonia, Ethiopia, Jordan, Macau, Zambia, Bosnia and Herzegovina, Bolivia, Jamaica, Uganda, Honduras, Paraguay, Gabon, and Senegal. These countries were selected based on data availability. We computed the tradable share (percentage) of GDP by selecting tradable sectors from United Nations gross value added database. We fit the GDP in the model to the tradable portion of GDP, computed as GDP reported by IMF World Economic Outlook database multiplied by the tradable share of GDP.[^86] [ ]{}This means that, for example, education revenue and education expenditures are not counted towards the model[’]{}s GDP and expenditures, which is appropriate for a model designed to capture manufacturing and similar industries. Multi-sector extensions including services are, of course, possible. Also, note that we exclude re-imports and re-exports from the trade flows data.
Related literature {#AppendixWorldTradeFlowsRelatedLiterature}
------------------
Here we briefly discuss connections of the results of Subsection \[TheGravityEquationOfTradeAndTheDependenceOfTradeCostsOnDistance\] to related issues in the literature, as mentioned in Footnote \[FootnotePointingToAppendixWorldTradeFlowsRelatedLiterature\]. @helpman2008estimating [ ]{}studied the role of the extensive margin of trade for the estimation of the distance-dependence of trade costs based on world trade flows. The authors found the distance effect to be 27 to 30 percent smaller than in benchmark estimates based on the gravity equation of trade without extensive margin effects. Although this is an important correction, it is not enough to resolve the trade cost puzzle. We get much stronger effects because of the increasing marginal cost of production. Moreover, unlike that paper we do not need unrealistically high export market entry costs that would be inconsistent with the everyday experience that even sole entrepreneurs with very limited capital (for example, 25,000 USD) are able to start an import/export business, a fact that is explained in many resources, such as @importexportbusiness .
Separately, @arkolakis2010market [ ]{}builds an elegant model of international trade where fixed costs of exporting are indeed negligible (and marginal costs of production are constant). Even though the demand is CES, some firms choose not to export to a particular destination because before serving a customer, they need to pay a sizeable per-customer advertising cost, which can make serving that customer unprofitable. An argument against this mechanism is that it would not work if targeted advertising was possible. Empirical evidence in the industrial organization literature shows that the main portion of observed aggregate demand elasticity comes from heterogeneity in the consumers[’]{} valuation of products, not from elasticity of demand by a given individual; an individual[’]{}s demand is quite inelastic in the data. If firms could reach high-valuation customers and advertize directly to them, they would export to that destination. Especially in recent years targeted advertising via the Internet is quite easy and widespread, so it is hard to justify the modeling assumption that it is impossible. For this reason, it is better to think of the insightful paper @arkolakis2010market [ ]{}in a more abstract way: as an investigation of situations where effective demand departs from CES. In principle, we could remove economies of scale in shipping from our model and instead modify the demand. In this case, again, we could combine this with our assumption of increasing marginal costs of production, and using our proposed tractable functional forms for demand we could proceed with computations in the same way. But of course, we already have empirical evidence on the economies of scale in shipping, and we know that logistics costs as a proportion of world GDP are very large. Note that the influential study of export decisions @eaton2011anatomy [ ]{}also uses the @arkolakis2010market [ ]{}mechanism in theoretical modeling.
Firm export patterns {#AppendixWorldTradeFlowsFirmExportPatterns}
--------------------
Here we mention other possible mechanisms potentially leading to patterns similar to those in Figure \[FigureExportDestinationsByTwoIdenticalFirms\] of Subsection \[ChoiceOfExportDestinations\], as referenced in Footnote \[FootnotePointingToAppendixWorldTradeFlowsFirmExportPatterns\]. If the countries significantly differ and we break the symmetry between the firms (in terms of how their products enter utility functions), it is possible to explain patterns resembling those in Figure \[FigureExportDestinationsByTwoIdenticalFirms\]. For example, windows imported by Finland are likely to be very different from windows imported by Portugal. If a firm specializes in only one kind of windows, it is natural for them to export to only one of these destinations. Another possible phenomenon that could lead to similar patterns in the data would be distribution centers in export destinations. For example, a firm may serve both Spain and Portugal from one distribution center based in Spain. In that case international trade flow data would not record such sales in Portugal as exports to Portugal, but instead as exports to Spain and then exports from Spain to Portugal. Yet another possibility is the case of very large firms. If these firms were so large that monopolistic competition description of the market was inappropriate and we needed to model it as an oligopoly, there could be an alternative explanation for choosing different export destinations. In this case strategic effects of market entry could potentially play a role. A firm may not choose to serve Greece because Greece is already served by its rival and the market the is not profitable enough for two firms to enter. The puzzle would still remain for smaller firms that cannot influence the entire industry. More generally, these three explanations may be valid in some cases but are not powerful enough to explain the majority of the empirical regularity in the data, especially in the case of smaller firms that directly export goods that are not geographically specialized. Case studies of individual exporters also make it clear that the export pattern is typically not explained by those three explanations. A detailed investigation of these issues will be reported separately.
Applications {#AppendixBreadthOfApplication}
============
Supply chains with hold-up [@antras]\[AppendixSupplyChainsWithHoldup\]
----------------------------------------------------------------------
We consider a generalization of the supply chain model of @antras [henceforth AC]. Instead of the variables introduced in the original paper, we use a different set of variables that makes the mathematics and intuition substantially simpler.[^87]
A firm produces a final good by sequentially using a continuum of customized inputs each provided by a different supplier indexed by $j\in[0,1]$, with small $j$ representing initial stages of production (upstream) and large $j$ representing final stages (downstream). If production proceeds smoothly, the [*effective*]{} quality-adjusted quantity $q$ of the final good is the integral of the effective quality-adjusted quantity contributed by intermediate input $j$, which we denote $q_{s}\left(j\right)$: $q=\int_{0}^{1}q_{s}(j)\, dj$. This effective quantity represents both the quantity of the good and its quality level. But we will refer to it simply as “quality”, since this will make the discussion sound more natural. If production is “disrupted” by the failure of some supplier $\overline{j}\in[0,1)$ to cooperate, then only the quality accumulated to that point in the chain is available, with all further quality-enhancement impossible: $q=\int_{0}^{\overline{j}}q_{s}(j)\, dj$. The firm faces an inverse demand function $P(q)$, which does not necessarily have to be decreasing because, for example, consumers may have little willingness-to-pay for an improperly finished product. If there is no disruption in production, $q=q(1)$.
Following the property rights theory of the firm [@grossmanhart; @hartmoore; @antrasalone], input production requires relationship-specific investments. The marginal revenue from additional quality brought by supplier $j$, $MR\left(q(j)\right)q_{s}(j)$ is therefore split between the firm and supplier $j$, where $MR=P+P'q$.[^88] In particular, the supplier receives a fraction $1-\beta(j)$ (its bargaining power).
The cost of producing quality $q_s(j)$ is homogeneous across suppliers and equal to $C(q_{s}(j))$, which is assumed strictly convex.[^89] Thus the first-order condition of supplier $j$ equates the share of marginal revenue she bargains for with her marginal cost: $$MC\left(q_{s}(j)\right)\equiv C'\left(q_{s}(j)\right)=\left[1-\beta\left(j\right)\right]MR\left(q\left(j\right)\right).\label{eq:SupplierFOC}$$ The cost to the firm of obtaining a contribution $q_{s}(j)$ from supplier $j$ is, therefore, the surplus it must leave in order to induce $q_{s}(j)$ to be produced, $q_{s}MC\left(q_{s}(j)\right)$. The firm chooses $\beta\left(j\right)$ through the nature of the contracting relationship optimally for each supplier to maximize its profits. Following AC and @antrashelpman1 [@antrashelpman2], we mostly focus on the *relaxed* problem where $\beta\left(j\right)$ may be adjusted freely and continuously. This provides most of the intuition for what happens when the firm is constrained to choose between two discrete levels of $\beta$ corresponding to outsourcing (low $\beta$) and insourcing (high $\beta$) and may be more realistic given the complexity of real-world contracting [@boundaries]. Note that by convexity, $MC'>0$, while each $q_{s}$ makes a linearly separable contribution to $q$. Thus for any fixed $q$ the firm wants to achieve, it does so most cheaply by setting all $q_{s}=q$ by Jensen’s Inequality. Thus Equation \[eq:SupplierFOC\] becomes, at any optimum $q^{\star}$, $$\beta^{*}\left(j\right)=1-\frac{MC\left(q^{\star}\right)}{MR\left(jq^{\star}\right)}.\label{betasolution}$$ From this we immediately see that $\beta^{\star}$ is co-monotone with $MR$: in regions where marginal revenue is increasing, $\beta^{\star}$ will be rising and conversely when marginal revenue is decreasing. The marginal revenue associated with constant elasticity demand is in a constant ratio to inverse demand. This implies AC’s principal result that when revenue elasticity is less than unity the firm will tend to outsource upstream and when revenue elasticity is less than unity the firm will tend to outsource downstream. However, it seems natural to think that $P(q)$ would initially rise, as consumers are willing to pay very little for a product that is nowhere near completion, and would eventually fall as the product is completed according to the standard logic of downward-sloping demand. We now solve in an equally-simple form a model allowing this richer logic.
Equation \[betasolution\] implies that the surplus left to each supplier is $q_{s}MC(q)$ and thus total cost is $qMC(q)$. The problem reduces to choosing $q$ to maximize revenue $qP(q)$ less cost $qMC(q)$, giving first-order condition $$MR(q)=MC\left(q\right)+q\, MC'\left(q\right).\label{firmfoc}$$ This differs from the familiar neoclassical first-order condition $MR\left(q\right)=MC\left(q\right)$ only by the presence of the (positive) term $q\, MC'\left(q\right)$. Note that $MC+qMC'$ bears the same relationship to $MC$ that $MC$ bears to $AC$; this equation therefore similarly inherits the tractability properties of the standard monopoly problem. The reason is that the hold-up makes multi-part tariff pricing impossible, creating a linear-price monopsony structure by forcing the firm to pay suppliers the marginal cost of the last unit of quality for all units produced.
Let us now consider $P(q)=p_{-t}q^{t}+p_{-u}q^{u}$ and $MC(q)=mc_{-t}q^{t}+mc_{-u}q^{u}$. This includes AC’s specification as the special case when $p_{-t}=0$ and $mc_{-u}=0$ so that each has constant elasticity.[^90] However, let us focus instead on the case when $t,u,mc_{-u},p_{-t}>0=mc_{-t}>p_{-u}$ and $u>t$ so that the first term of the inverse demand dominates at small quantities while the second dominates at large quantities. The expression resulting for $\beta^{*}\left(j\right)$ is:
$$\beta^{*}\left(j\right)=1-\frac{1}{(1+u)\left[\left(1-\frac{p_{-u}}{mc_{-u}}\right)j^{t}+\frac{p_{-u}}{mc_{-u}}j^{u}\right]}.\label{eq:ResultingEffectiveBargainingPower}$$
Note that because $mc_{-u}>0>p_{-u}$, the first denominator term is positive and the second denominator term is negative. This implies that at small $j$ (where $j^{t}$ dominates), $\beta^{\star}$ increases in $j$, while at large $j$, it decreases in $j$. In the AC complements case when $p_{-u}=0$, or even if $p_{-u}$ is sufficiently small, this large $j$ behavior is never manifested and all outsourcing (low $\beta^{\star}$) occurs at early stages. Also note that only the ratio of coefficients $\frac{p_{-u}}{mc_{-u}}$ matters for the sourcing pattern; $p_{-t}$ is irrelevant, as the joint level of $p_{-u}$ and $mc_{-u}$.
![Optimal relaxed and restricted $\beta^{\star}$ in the AC model when $t=0.35,u=0.7,\frac{p_{-u}}{mc_{-u}}=-4$.[]{data-label="antrasfigure"}](FigureAntrasChorBargainingRestrictedBeta.pdf){width="3.7in"}
However, for many parameters an inverted U-shape emerges. For example, Figure \[antrasfigure\] shows the case when $t=0.35,u=0.7,p_{-t}=1.8,\frac{p_{-u}}{mc_{-u}}=-4$. The curve corresponds to the shape of the relaxed solution. Depending on precisely which values of $\beta$ we take insourcing and outsourcing to correspond to, this can lead to insourcing in the middle of the production and outsourcing at either end. In Supplementary Material \[AppendixSupplyChainsWithHoldupDetails\] we study in detail the constrained problem using largely closed-form methods for the case when outsourcing gives $\beta_{O}=0.8$ and insourcing gives $\beta_{I}=0.4$. This is illustrated by the lines in Figure \[antrasfigure\], which show the constrained optimum. This gives the same qualitative answer as the relaxed problem, as expected.
Imperfectly competitive supply chains\[sub:Imperfectly-competitive-supply\]
---------------------------------------------------------------------------
The models that founded the field of industrial organization were @cournot’s of symmetric oligopoly and complementary monopoly. Equilibrium in these models is characterized by $$P+\theta P'q=MC.$$ Under Cournot competition, $\theta=\nicefrac{1}{n}$, where $n$ is the number of competing firms and $MC$ is interpreted as the common marginal cost of all producers. Under Cournot complements (which does not require symmetry) $\theta=m$, where $m$ is the number of complementary producers and $MC$ is interpreted as the aggregated marginal cost of all producers.[^91] Note that $P+\theta P'q$ is just a linear combination of $P$ and $P'q$ and thus has the same form as either of these components in a form-preserving class of functional forms. Thus either problem yields exactly the same characterization of tractability as the monopoly problem.
In the last half-century a variant on @cournot’s complementary monopoly problem proposed by @spengler has been more commonly used. In this model one firm sells an input to another who in turn sells to a consumer. The difference from @cournot’s model is principally in the timing; namely the “upstream” firm is assumed to set her price prior to the downstream firm. In this case the upstream firm effectively sets part of the downstream firm’s marginal cost. Her first-order condition is $$P+P'q=MC+\hat P,$$ where $\hat P$ is the sales price set by the upstream firm. Thus the effective inverse demand faced by the upstream firm is $\hat{P}(q)\equiv P(q)+P'(q)q-MC(q)$. The upstream firm then solves a monopoly problem with this inverse demand. This yields an upstream marginal revenue curve bearing the same relationship to $\hat{P}$ that $MR$ bears to $P$. Because the form-preserving feature may be applied an arbitrary number of times, however, this transformation does not change our characterization of tractability. Thus a form-preserving class has the same tractability characterization in @spengler’s model as in the standard @cournot model.
We can go further and allow for many layers of production and arbitrary imperfect competition (or complements) at each later as in @salinger. The same characterization of tractability continues to apply. In Supplementary Material \[sequential\] we provide an explicit expression for the coefficients in the polynomial equation for any tractable form. @adachi2014cost [@adachi2014double] argue that flexible functional forms are particularly important in such models because many important and policy-relevant properties are imposed by standard tractable forms. For example, the markup of the upstream firm in @spengler’s model is identical to that of the two firms if they merged under the BP demand class, but the upstream firm will typically charge a lower markup than an integrated firm under reasonable conditions (bell-shaped-distribution-generated demand and U-shaped cost curves).
$\ $
**Supplementary Material**
Laplace Inverse Demand Functions {#sub:LaplaceInverseDemandFunctions}
================================
The following table contains Laplace inverse demand functions corresponding to inverse demand functions used in the literature. Although for most Laplace inverse demand functions we include only a few terms, closed-form expressions for all terms exist. Here $p_{a}$ refers to a mass-point of magnitude $p_{a}$ at location $a$. In the alternative notation on the lower lines, $\delta(x-a)$ refers to a mass-point of magnitude 1 at location $a$, i.e. to a Dirac delta function centered at $a$. We use standard notation for special functions: $\Gamma$ stands for the gamma function and $W$ for the Lambert W function.
$$\begin{array}{l}
\begin{array}{ccc}
\text{Constant elasticity / Pareto:} & q(P)=\left(\frac{P}{\beta}\right)^{-\epsilon} & P(q)=\beta q^{-1/\epsilon}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{c}
p_{\frac{1}{\epsilon}}=\beta\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \beta\delta\left(t-\frac{1}{\epsilon}\right)\end{array}\\
\begin{array}{ccc}
\text{Constant pass-through / BP:} & q(P)=\left(\frac{P-\mu}{\beta}\right)^{-\epsilon} & P(q)=\mu+\beta q^{-1/\epsilon}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccc}
p_{0}=\mu, & p_{\frac{1}{\epsilon}}=\beta\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \beta\delta\left(t-\frac{1}{\epsilon}\right)+\mu\delta(t)\end{array}\\
\begin{array}{ccc}
\text{Gumbel distribution:} & q(P)=\exp\left(-\exp\left(\frac{P-\alpha}{\beta}\right)\right) & P(q)=\alpha+\beta\log(-\log(q))\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccc}
p_{0}=\mu, & p(t)=-\frac{\beta}{t} & \text{for} & t<0\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \alpha\delta(t)-\frac{\beta1_{t<0}}{t}\end{array}\\
\begin{array}{ccc}
\text{Weibull distribution:} & q(P)=e^{-\left(\frac{P}{\beta}\right)^{\alpha}} & P(q)=\beta(-\log(q))^{\frac{1}{\alpha}}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccc}
\frac{(-1)^{\frac{1}{\alpha}}\beta t^{-\frac{1}{\alpha}-1}}{\Gamma\left(-\frac{1}{\alpha}\right)} & \text{for} & t<0\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \frac{(-1)^{\frac{1}{\alpha}}\beta1_{t<0}t^{-\frac{1}{\alpha}-1}}{\Gamma\left(-\frac{1}{\alpha}\right)}\end{array}\\
\begin{array}{ccc}
\text{Fr{\' e}chet distribution:} & q(P)=1-e^{-\left(\frac{P-\mu}{\beta}\right)^{-\alpha}} & P(q)=\mu+\beta(-\log(1-q))^{-1/\alpha}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccc}
p_{0}=\mu, & p_{\frac{1}{\alpha}}=\beta, & p_{\frac{1}{\alpha}-1}=-\frac{\beta}{2\alpha}, & p_{\frac{1}{\alpha}-2}=\frac{\beta}{8\alpha^{2}}-\frac{5\beta}{24\alpha}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \left(\frac{\beta}{8\alpha^{2}}-\frac{5\beta}{24\alpha}\right)\delta\left(t-\frac{1}{\alpha}+2\right)+\beta\delta\left(t-\frac{1}{\alpha}\right)-\frac{\beta\delta\left(t-\frac{1}{\alpha}+1\right)}{2\alpha}+\mu\delta(t)+\dots\end{array}\\
\begin{array}{ccc}
\text{Logistic distribution:} & q(P)=\left(\exp\left(\frac{P-\mu}{\beta}\right)+1\right)^{-1} & P(q)=\mu-\beta\log\left(\frac{1}{1-q}-1\right)\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccccccc}
p_{0}=\mu, & p_{0}^{\text{(1)}}=-\beta, & p_{-1}=-\beta, & p_{-2}=-\frac{\beta}{2}, & p_{-3}=-\frac{\beta}{3}, & p_{-4}=-\frac{\beta}{4}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & -\beta\sum_{j=1}^{\infty}\frac{\delta(j+t)}{j}+\mu\delta(t)-\beta\delta'(t)\end{array}\\
\begin{array}{ccc}
\text{Log-logistic distribution:} & q(P)=\left(\left(\frac{P}{\sigma}\right)^{\gamma}+1\right)^{-1} & P(q)=\sigma\left(\frac{q}{1-q}\right)^{-1/\gamma}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccc}
p_{\frac{1}{\gamma}}=\sigma, & p_{\frac{1}{\gamma}-1}=-\frac{\sigma}{\gamma}, & p_{\frac{1}{\gamma}-2}=\frac{\sigma}{2\gamma^{2}}-\frac{\sigma}{2\gamma}, & p_{\frac{1}{\gamma}-3}=-\frac{\sigma}{6\gamma^{3}}+\frac{\sigma}{2\gamma^{2}}-\frac{\sigma}{3\gamma}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \left(\frac{\sigma}{2\gamma^{2}}-\frac{\sigma}{2\gamma}\right)\delta\left(t-\frac{1}{\gamma}+2\right)+\sigma\delta\left(t-\frac{1}{\gamma}\right)-\frac{\sigma\delta\left(t-\frac{1}{\gamma}+1\right)}{\gamma}+\dots\end{array}\\
\begin{array}{ccc}
\text{Laplace distribution (}q<\frac{1}{2}\text{):} & q(P)=\frac{1}{2}\exp\left(\frac{\mu-P}{\beta}\right) & P(q)=\mu-\beta\log(2q)\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccc}
p_{0}=\mu-\beta\log(2), & p_{0}^{\text{(1)}}=-\beta\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \delta(t)(\mu-\beta\log(2))-\beta\delta'(t)\end{array}\\
\begin{array}{ccc}
\text{Laplace distribution (}q>\frac{1}{2}\text{):} & q(P)=1-\frac{1}{2}\exp\left(\frac{P-\mu}{\beta}\right) & P(q)=\mu+\beta\log(2(1-q))\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccccc}
p_{0}=\beta\log(2)+\mu, & p_{-1}=-\beta, & p_{-2}=-\frac{\beta}{2}, & p_{-3}=-\frac{\beta}{3}, & p_{-4}=-\frac{\beta}{4}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \delta(t)(\beta\log(2)+\mu)-\beta\sum_{j=1}^{\infty}\frac{\delta(j+t)}{j}\end{array}\\
\begin{array}{ccc}
\text{Normal distribution:} & q(P)=\text{erfc}\left(\frac{P-\mu}{\sqrt{2}\sigma}\right) & P(q)=\mu-\sqrt{2}\sigma\text{erfc}^{-1}(2-q)\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccc}
p_{0}^{\text{(1)}}=-\sqrt{\frac{\pi}{2}}\sigma, & p_{0}^{\text{(2)}}=-\frac{1}{2}\sqrt{\frac{\pi}{2}}\sigma, & p_{0}^{\text{(3)}}=\frac{1}{24}\left(-\sqrt{2}\pi^{3/2}-2\sqrt{2\pi}\right)\sigma, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & -\sqrt{\frac{\pi}{2}}\sigma\delta'(t)-\frac{1}{2}\sqrt{\frac{\pi}{2}}\sigma\delta''(t)+\frac{1}{24}\left(-\sqrt{2}\pi^{3/2}-2\sqrt{2\pi}\right)\sigma\delta^{(3)}(t)+\dots\end{array}\\
\begin{array}{ccc}
\text{Lognormal distribution:} & q(P)=\text{erfc}\left(\frac{\log(P)-\mu}{\sqrt{2}\sigma}\right) & P(q)=\exp\left(\mu-\sqrt{2}\sigma\text{erfc}^{-1}(2-q)\right)\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccc}
p_{0}^{\text{(1)}}=\sqrt{\frac{\pi}{2}}\left(-e^{\mu}\right)\sigma\delta'(t), & p_{0}^{\text{(2)}}=\frac{1}{4}\pi e^{\mu}\sigma^{2}-\frac{1}{2}\sqrt{\frac{\pi}{2}}e^{\mu}\sigma, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \left(\frac{1}{4}\pi e^{\mu}\sigma^{2}-\frac{1}{2}\sqrt{\frac{\pi}{2}}e^{\mu}\sigma\right)\delta''(t)-\sqrt{\frac{\pi}{2}}e^{\mu}\sigma\delta'(t)+\dots\end{array}
\end{array}$$ $$\begin{array}{l}
\begin{array}{ccc}
\text{Almost Ideal Demand System:} & q(P)=\frac{\alpha+\beta\log(P)}{P} & P(q)=-\frac{\beta W\left(-\frac{qe^{-\frac{\alpha}{\beta}}}{\beta}\right)}{q}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccccc}
p_{0}=e^{-\frac{\alpha}{\beta}}, & p_{-1}=\frac{e^{-\frac{2\alpha}{\beta}}}{\beta}, & p_{-2}=\frac{3e^{-\frac{3\alpha}{\beta}}}{2\beta^{2}}, & p_{-3}=\frac{8e^{-\frac{4\alpha}{\beta}}}{3\beta^{3}}, & p_{-4}=\frac{125e^{-\frac{5\alpha}{\beta}}}{24\beta^{4}}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \frac{125e^{-\frac{5\alpha}{\beta}}\delta(t+4)}{24\beta^{4}}+\frac{8e^{-\frac{4\alpha}{\beta}}\delta(t+3)}{3\beta^{3}}+\frac{3e^{-\frac{3\alpha}{\beta}}\delta(t+2)}{2\beta^{2}}+e^{-\frac{\alpha}{\beta}}\delta(t)+\frac{e^{-\frac{2\alpha}{\beta}}\delta(t+1)}{\beta}+\dots\end{array}\\
\begin{array}{ccc}
\text{Constant superelasticity:} & q(P)=\left(\epsilon\log\left(\frac{\theta-1}{\theta P}\right)+1\right)^{\frac{\theta}{\epsilon}} & P(q)=\frac{(\theta-1)e^{\frac{1}{\epsilon}-\frac{q^{\epsilon/\theta}}{\epsilon}}}{\theta}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccc}
p_{0}=e^{\frac{1}{\epsilon}}-\frac{e^{\frac{1}{\epsilon}}}{\theta}, & p_{-\frac{\epsilon}{\theta}}=\frac{e^{\frac{1}{\epsilon}}}{\theta\epsilon}-\frac{e^{\frac{1}{\epsilon}}}{\epsilon}, & p_{-\frac{2\epsilon}{\theta}}=\frac{e^{\frac{1}{\epsilon}}}{2\epsilon^{2}}-\frac{e^{\frac{1}{\epsilon}}}{2\theta\epsilon^{2}}, & p_{-\frac{3\epsilon}{\theta}}=\frac{e^{\frac{1}{\epsilon}}}{6\theta\epsilon^{3}}-\frac{e^{\frac{1}{\epsilon}}}{6\epsilon^{3}}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \left(\frac{e^{\frac{1}{\epsilon}}}{2\epsilon^{2}}-\frac{e^{\frac{1}{\epsilon}}}{2\theta\epsilon^{2}}\right)\delta\left(t+\frac{2\epsilon}{\theta}\right)+\delta(t)\left(e^{\frac{1}{\epsilon}}-\frac{e^{\frac{1}{\epsilon}}}{\theta}\right)+\left(\frac{e^{\frac{1}{\epsilon}}}{\theta\epsilon}-\frac{e^{\frac{1}{\epsilon}}}{\epsilon}\right)\delta\left(t+\frac{\epsilon}{\theta}\right)+\dots\end{array}\\
\begin{array}{ccc}
\text{Cauchy distribution:} & q(P)=\frac{\tan^{-1}\left(\frac{a-P}{b}\right)}{\pi}+\frac{1}{2} & P(q)=a+b\tan\left(\pi\left(\frac{1}{2}-q\right)\right)\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccccccc}
p_{1}=\frac{b}{\pi}, & p_{0}=a, & p_{-1}=-\frac{\pi b}{3}, & p_{-3}=-\frac{\pi^{3}b}{45}, & p_{-5}=-\frac{2\pi^{5}b}{945}, & p_{-7}=-\frac{\pi^{7}b}{4725}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & a\delta(t)+\frac{b\delta(t-1)}{\pi}-\frac{1}{3}\pi b\delta(t+1)-\frac{1}{45}\pi^{3}b\delta(t+3)-\frac{2}{945}\pi^{5}b\delta(t+5)-\frac{\pi^{7}b\delta(t+7)}{4725}+\dots\end{array}\\
\begin{array}{ccc}
\text{Singh Maddala distribution:} & q(P)=\left(\left(\frac{P}{b}\right)^{a}+1\right)^{-\tilde{q}} & P(q)=b\left(q^{-\frac{1}{\tilde{q}}}-1\right)^{\frac{1}{a}}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccc}
p_{\frac{1}{a\tilde{q}}}=b, & p_{-\frac{a-1}{a\tilde{q}}}=-\frac{b}{a}, & p_{-\frac{2a-1}{a\tilde{q}}}=\frac{b}{2a^{2}}-\frac{b}{2a}, & p_{-\frac{3a-1}{a\tilde{q}}}=-\frac{b}{6a^{3}}+\frac{b}{2a^{2}}-\frac{b}{3a}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \left(\frac{b}{2a^{2}}-\frac{b}{2a}\right)\delta\left(\frac{2a-1}{a\tilde{q}}+t\right)+b\delta\left(t-\frac{1}{a\tilde{q}}\right)-\frac{b\delta\left(\frac{a-1}{a\tilde{q}}+t\right)}{a}\end{array}+\dots\\
\begin{array}{ccc}
\text{Tukey lambda distribution:} & q(P)=P^{(-1)}(P) & P(q)=\frac{(1-q)^{\lambda}-q^{\lambda}}{\lambda}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccccccccc}
p_{-\lambda}=-\frac{1}{\lambda}, & p_{0}=\frac{1}{\lambda}, & p_{-1}=-1, & p_{-2}=\frac{\lambda}{2}-\frac{1}{2}, & p_{-3}=-\frac{\lambda^{2}}{6}+\frac{\lambda}{2}-\frac{1}{3}, & \text{...}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \left(-\frac{\lambda^{2}}{6}+\frac{\lambda}{2}-\frac{1}{3}\right)\delta(t+3)+\frac{\delta(t)}{\lambda}+\left(\frac{\lambda}{2}-\frac{1}{2}\right)\delta(t+2)-\frac{\delta(t+\lambda)}{\lambda}-\delta(t+1)+\dots\end{array}\\
\begin{array}{ccc}
\text{Wakeby distribution:} & q(P)=P^{(-1)}(P) & P(q)=\mu-\frac{\gamma\left(1-q^{-\delta}\right)}{\delta}+\frac{\alpha\left(1-q^{\beta}\right)}{\beta}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \begin{array}{ccccc}
p_{0}=\frac{\alpha}{\beta}-\frac{\gamma}{\delta}+\mu, & p_{-\beta}=-\frac{\alpha}{\beta}, & p_{\delta}=\frac{\gamma}{\delta}\end{array}\end{array}\\
\begin{array}{ccc}
\text{} & p(t): & \delta(t)\left(\frac{\alpha}{\beta}-\frac{\gamma}{\delta}+\mu\right)-\frac{\alpha\delta(t+\beta)}{\beta}+\frac{\gamma\delta(t-\delta)}{\delta}+\dots\end{array}
\end{array}\label{TableOfInverseLaplaceTransforms}$$
Here we provide clarification of some of the expressions in the table above. In many cases the terms in the Laplace inverse demand were obtained utilizing series expansions, as discussed in Subsection \[UsingTaylorSeriesExpansion\] of this supplementary material. More explicitly, we utilized the following series representations of the Laplace inverse demand.
$$\begin{array}{ll}
\text{Fr{\' e}chet distribution:} & P(q)=\mu +\beta q^{-1/\alpha } \sum _{k=0}^{\infty } \binom{-\frac{1}{\alpha }}{k} \left(\sum _{j=2}^{\infty
} j^{-1} q^{j-1}\right){}^k \\
\text{Log-logistic distribution:} & P(q)=\sigma q^{1/\gamma } \sum _{n=0}^{\infty } (-1)^n \binom{\frac{1}{\gamma }}{n} q^n \\
\text{Almost Ideal Demand System:} & P(q)=\beta \sum _{n=0}^{\infty } \frac{(-1-n)^n}{(1+n)!} \left(-\frac{e^{-\frac{\alpha }{\beta }}}{\beta }\right)^{1+n}
q^n \\
\text{Constant superelasticity:} & P(q)=\sum _{n=0}^{\infty } \frac{e^{1/\epsilon } (-\epsilon )^{-n} (\theta -1)}{\theta n!} q^{\frac{n \epsilon
}{\theta }} \\
\text{Cauchy distribution:} & P(q)=a+\frac{b}{\pi q}+b \sum _{k=1}^{\infty } \frac{(-1)^k 2^{2 k} \pi ^{-1+2 k} B_{2 k}}{(2 k)!} q^{-1+2 k} \\
\text{Singh-Maddala distribution: } & P(q)=b q^{-\frac{1}{a \tilde{q}}} \sum _{n=0}^{\infty } (-1)^n \binom{\frac{1}{a}}{n} q^{\frac{n}{\tilde{q}}}
\end{array}$$
$$\text{}$$ We used the standard notation for generalized binomial coefficients and denoted Bernoulli numbers by $B_{2 k}$. Constant superelasticity refers to the inverse demand function introduced by @klenowwillis. The Gumbel distribution is also known as the type I extreme value distribution, the Fr[' e]{}chet distribution as type II extreme value, and the Weibull distribution as type III extreme value.
In the case of the Gumbel distribution, the Laplace inverse demand is not an ordinary function, but a distribution (generalized function) in the sense of the distribution theory by Laurent Schwartz. For this reason, we use regularization to give a precise meaning to integrals involving the Laplace inverse demand function that was schematically written in the previous table. We provide three regularization prescriptions and illustrate them for the case of Laplace inverse demand itself. [ ]{}The first prescription is
$$P(q)=\lim_{a\to 0^+} \, \left(\int _{-\infty }^{\infty }(\alpha -\beta (\gamma +\log (a))) \delta (t)dt+\int_{-\infty }^a \frac{\beta q^{-t}}{t}
\, dt\right).$$ Here we moved the upper bound in the second integral beyond zero and added the regularization term proportional to $\log
(a)$. The second integral is to be interpreted in the sense of principal value.[^92] [ ]{}It is straightforward to verify that this expression leads to the correct expression for $P(q)$. First, we evaluate the integrals to get
$$P(q)=\lim_{a\to 0^+} \, (\alpha -\gamma \beta +\beta \text{Ei}(-a \log (q))-\beta \log (a)).$$ Here $\gamma $ is the Euler gamma, $\gamma \approx 0.577216$, and Ei stands for the special function called exponential integral. Evaluating the limit then leads to the correct expression
$$P(q)=\alpha +\beta \log (-\log (q)).$$
The second prescription is analogous and shifts the upper bound of the second integral to negative numbers:
$$P(q)=\lim_{a\to 0^+} \, \left(\int _{-\infty }^{\infty }(\alpha -\beta (\gamma +\log (a))) \delta (t)dt+\int_{-\infty }^{-a} \frac{\beta q^{-t}}{t}
\, dt\right).$$ Evaluating the integral gives
$$P(q)=\lim_{a\to 0^+} \, (\alpha -\gamma \beta -\beta \Gamma (0,-a \log (q))-\beta \log (a)),$$ and taking the limit leads again to the correct expression. Here $\Gamma $ is the incomplete gamma function.
The third prescription is computationally most convenient because it does not involve taking a limit. The regularizing term is expressed in the form of an integral
$$P(q)=\int _{-\infty }^{\infty }(\alpha -\beta \gamma ) \delta (t)dt+\beta \int_{-\infty }^0 \frac{q^{-t}-1_{t>-1}}{t} \, dt.$$ Here $1_{t>-1}$ is an indicator function. The integral may then be computed directly, again leading to the correct expression. The same methods may be used when interpreting other integrals involving generalized functions that behave as $1/t$ close to $t=0$.
For the normal and lognormal distributions the Laplace inverse demand functions are again not ordinary functions. They were obtained using Taylor series expansions of the error function. Expressions involving these Laplace inverse demand functions may need to be summed using the Euler summation method to ensure proper convergence.
The expressions above may be used to straightforwardly derive Theorems \[TheoremCompleteMonotonicityOfDemandSpecification\] and \[TheoremAbsenceOfCompleteMonotonicityOfDemandSpecification\] of the main paper. An alternative, but sometimes less straightforward way is to utilize Theorems 1–6 of @miller2001completely. If we are interested in the monotonicity properties of the pass-through rate, we can use the corollary in Section \[SectionArbitraryDemandAndCostFunctions\]. However, we also identified more direct ways to prove monotonicity properties of the pass-through rate for certain demand functions. These proofs are included in Section \[forms\] of this supplementary material.
Evaluation of Inverse Laplace Transform
=======================================
In the main text of the paper we used the term Laplace-log transform to emphasize that this is Laplace transform in terms of the logarithm of an economic quantity. In this section, which focuses on mathematical issues, we use the term Laplace transform, keeping the economic interpretation implicit.
Numerical Evaluation of Inverse Laplace Transform
-------------------------------------------------
There exist many methods for numerical evaluation of inverse Laplace transform, now usually integrated into mathematical and statistical software. For a summary and important references, see, e.g., Chapter 6 of @egonmwan2012numerical. Note that just like other types of non-parametric methods, numerical inversion of Laplace transform requires some regularization, such as the @tikhonov1963solution regularization. This is because Laplace transform inversion is a so-called *ill-posed* problem, which means that there exist large changes in the inverse Laplace transform that lead to only small changes in the original function in the domain of interest. For a classic discussion, see @bellman1966numerical.
Analytic Evaluation of Inverse Laplace Transform
------------------------------------------------
Mathematical software allows for symbolic inversion of Laplace transform.[^93] However, it is often more convenient to evaluate the inverse Laplace transform using more direct methods.
### Using Taylor series expansion {#UsingTaylorSeriesExpansion}
We would like to emphasize that finding analytic expressions for Laplace inverse demand is often much simpler than it seems since it many cases it only requires finding a Taylor series expansion of a definite function. Consider, for example, the case of log-logistic distribution of valuations included in the Supplementary Material \[sub:LaplaceInverseDemandFunctions\], which corresponds to inverse demand $P\left(q\right)=\sigma(\frac{q}{1-q})^{-1/\gamma}$. This may be written as $P\left(q\right)=\sigma q^{-1/\gamma}\times\left(1-q\right)^{1/\gamma}$, i.e. a product of a power function and a function that has a well-defined Taylor expansion at $q=0$: $$\left(1-q\right)^{1/\gamma}=1-\frac{1}{\gamma}\ q+\frac{1-\gamma}{2\gamma^{2}}\ q^{2}+...=\sum_{n=0}^{\infty}(-1)^{n}\binom{\frac{1}{\gamma}}{n}q^{n},$$ where the $n$th term contains a generalized binomial coefficient. This immediately translates into $$P\left(q\right)=\sigma q^{-\frac{1}{\gamma}}-\frac{\sigma}{\gamma}\ q^{1-\frac{1}{\gamma}}+\frac{\left(1-\gamma\right)\sigma}{2\gamma^{2}}\ q^{2-\frac{1}{\gamma}}+...=\sum_{n=0}^{\infty}(-1)^{n}\binom{\frac{1}{\gamma}}{n}q^{n-\frac{1}{\gamma}},$$ and from here we can read off the masses at points $t=\frac{1}{\gamma},\frac{1}{\gamma}-1,\frac{1}{\gamma}-2,$... that together constitute the Laplace inverse demand included in Supplementary Material \[sub:LaplaceInverseDemandFunctions\].
### Using the traditional inverse Laplace transform formula
The readers may be familiar with the traditional inverse Laplace transform formula based on the Bromwich integral in the complex plane:[^94] $$f\left(t\right)=\frac{1}{2\pi i}\lim_{T\rightarrow\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}f_{\mathcal{L}}\left(s\right)ds,\mbox{\ where \ensuremath{f_{\mathcal{L}}\left(s\right)\equiv\int_{0}^{\infty}e^{-st}f\left(t\right)dt.}}\label{eq:BromwichIntegral}$$ For the purposes of this paper we did not actually need it. We obtained the Laplace inverse demand function listed in Supplementary Material \[sub:LaplaceInverseDemandFunctions\] by simpler methods.
### Piecewise inverse Laplace transform
Readers familiar with Fourier transform but not with Laplace transform may potentially be concerned about applicability of our approach to the case of linear demand. Our prescription is simple: If the inverse demand takes the form $P\left(q\right)=a-bq$, we restrict our attention to $q\in\left(0,\frac{a}{b}\right)$, without affecting the form of Laplace inverse demand $p\left(t\right)$. The reason why this is possible is that to evaluate $p\left(t\right)$ using, say, Equation \[eq:BromwichIntegral\], we do not need the values of $P\left(q\right)\equiv P\left(e^{s}\right)$ for $s\in\left(-\infty,\infty\right)$, as a superficial analogy with Fourier transform might suggest. Instead, the integral in Equation \[eq:BromwichIntegral\] is in the imaginary direction. Writing the inverse demand as $P\left(e^{s}\right)=a-be^{s}$ for $\mbox{Re }s<\log\frac{a}{b}$ and $P\left(e^{s}\right)=0$ for $\mbox{Re}\ s>\frac{a}{b}$ and working with each piece separately will not make the Laplace inverse demand complicated. We will just have two different Laplace inverse demand functions, each valid for a range of $q$.
Solving Cubic and Quartic Equations Simply
==========================================
The readers may have seen general formulas for solutions to cubic and quartic equations that looked very complicated. It turns out that the intimidating look is caused just by shifts and rescalings of variables. Solving these equations is actually very straightforward:
#### Cubic equations.
To solve the equation $x^{3}+3ax+2=0$, we substitute $x\equiv y^{1/3}-ay^{-1/3}$, which leads to the quadratic equation $y^{2}+2y-a^{3}=0$ with solutions $y=\pm\sqrt{1+a^{3}}-1$. Given this result, the solutions to any other cubic equation may be obtained by rescaling and shifting of $x$.[^95]
#### Quartic equations.
A quartic equation of the form $x^{4}+ax^{2}+bx+1=0$ is equivalent to $(x^{2}+\sqrt{\alpha}x+\beta)(x^{2}-\sqrt{\alpha}x+\beta^{-1})=0$ with $\alpha\ge0$ and $\beta$ chosen to satisfy $\beta+\beta^{-1}=a+\alpha$ and $\beta^{-1}-\beta=\frac{b}{\sqrt{\alpha}}$ so that the coefficients of different powers of $x$ match. If we substitute the right-hand-side expressions into the trivial identity $(\beta+\beta^{-1})^{2}-(\beta-\beta^{-1})^{2}=4$, we get a cubic equation for $\alpha$, which we know how to solve. With the help of the quadratic formula, a solution for $\alpha$ then translates into a solution for $\beta$, and consequently for $x$. Given these results, the solutions to any other quartic equation may be obtained simply by rescaling and shifting of $x$.
Discussion of the Characterization Theorem {#AppendixDiscussionOfTheCharacterizationTheorem}
==========================================
Here we present a discussion of the logic behind Theorem \[formpreserve\] that only requires knowledge of very elementary calculus, i.e. simple differentiation and integration, instead of assuming knowledge of complex analysis and functional transforms.[^96]
One-dimensional functional form classes
---------------------------------------
Consider a real function $P(q)$ (on an open interval of positive $q$) that belongs to a one-dimensional functional form class invariant under (i.e., preserved by) average-marginal transformations. It must be the case that [ ]{}$a P(q)+b q P'(q)$ also belongs to this class. For this reason, [ ]{}if $q P'(q)$ were not a multiple of $P(q)$, the class would not be one-dimensional, which would be a contradiction. Denoting the coefficient of proportionality as $A$, we get the differential equation$$q P'(q)=A P(q),$$ which implies$$\frac{dP}{P}=A \frac{dq}{q}\, \, \, \, \, \Longrightarrow \, \, \, \, \, \log \left| P\right| =A \log q+\text{const.}\, \, \, \, \, \Longrightarrow
\, \, \, \, \, P(q)=c_1 q^A,$$ with some real constant $c_1$. We conclude that the one-dimensional functional form classes invariant under the average-marginal transformations are the classes of power functions with a fixed exponent.
In the next subsection we work in terms of the variable $s=\log q$. For clarity, let us repeat the computation above using $s$, with the identification $H(s)\equiv P(q)$ for $s=\log q$. The differential equation becomes $H'(s)=A H(s)$ and leads to the same result as above:$$\frac{dH}{H}=A\, ds\, \, \, \, \, \Longrightarrow \, \, \, \, \, \log \left| H\right| =A s+\text{const.}\, \, \, \, \, \Longrightarrow \, \, \,
\, \, H(s)=c_1 e^{A s}.$$
Two-dimensional functional form classes
---------------------------------------
Any member $H(\log (q))=P(q)$ of a two-dimensional class invariant under average-marginal transformations must satisfy the following differential equation:$$H''(s)=A H(s)+B H'(s).$$ The logic is analogous to the one-dimensional case in the previous subsection. If $H'(s)$ and $H(s)$ are proportional to each other, the equation is automatically satisfied. Consider the case where $H'(s)$ and $H(s)$ are not proportional to each other. By the invariance property, any linear combination of $H(s)$ and $H'(s)$ must belong to the class. Consequently, any linear combination of $H(q)$, $H'(s)$, and $H''(s)$ must belong to the class. These linear combinations span a three-dimensional space, [ ]{}unless $H''(s)$ is a linear combination of [ ]{}$H(q)$ and $H'(s)$, i.e. unless the differential equation is satisfied for some $A$ and $B$. This establishes the validity of the differential equation.
Denote by $r_1$ and $r_2$ the two roots of the quadratic equation$$x^2=A +B\text{ }x,$$ namely$$r_1=\frac{1}{2} \left(B+\sqrt{4 A+B^2}\right),\, \, \, \, \, r_2=\frac{1}{2} \left(B-\sqrt{4 A+B^2}\right).$$ It is straightforward to check that the differential equation $H''(s)=A H(s)+B H'(s)$ may be alternatively written as$$\left(1-r_1\frac{d}{ds}\right)\left(\left(1-r_2\frac{d}{ds}\right)H(s)\right)=0.$$ The function $\left(1-r_2\frac{d}{ds}\right)H(s)$, which we denote $f(s)$, therefore satisfies the differential equation$$\left(1-r_1\frac{d}{ds}\right)f(s)=0.$$ This differential equation gives the solution$$f(s)-r_1f'(s)=0\Longrightarrow ds=r_1\frac{df}{f}\Longrightarrow \log |f|=\frac{s}{r_1}+\text{const}.\Longrightarrow f(s)=\tilde{c}_1 e^{\frac{s}{r_1}},$$ which means$$\left(1-r_2\frac{d}{ds}\right)H(s)= \tilde{c}_1 e^{\frac{s}{r_1}}.$$ To get the final expression for $H(q)$, we will solve this differential equation in two alternative cases.
### The case of two distinct roots
Let us consider the case where the two roots, $r_1$ and $r_2$, are not equal. To solve this last differential equation, let us perform the substitution $H(s)=e^{\frac{s}{r_1}}g(s)+\frac{1 }{r_1-r_2}\tilde{c}_1r_1e^{\frac{s}{r_1}}$. The differential equation then becomes$$\left(1-r_2\frac{d}{ds}\right)\left(e^{\frac{s}{r_1}}g(s)+\frac{e^{\frac{s}{r_1}} \tilde{c}_1r_1}{-r_1+r_2}\right)= \tilde{c}_1 e^{\frac{s}{r_1}},$$ or after canceling terms proportional to $\tilde{c}_1$, $$\left(1-r_2\frac{d}{ds}\right)\left(e^{\frac{s}{r_1}}g(s)\right)=0.$$ This differential equation has the following solution:$$\left(1-\frac{r_2}{r_1}-r_2\frac{d}{ds}\right)g(s)=0\, \, \, \, \, \Longrightarrow \, \, \, \, \, g(s)=c_2 e^{\left(1-\frac{r_2}{r_1}\right)\frac{s}{r_2}}\,
\, \, \, \, \Longrightarrow \, \, \, \, \, g(s)=c_2 e^{s \left(\frac{1}{r_2}-\frac{1}{r_1}\right)}.$$ For the function $H(s)$ this implies$$H(s)=c_2 e^{ \frac{s}{r_2}}+\frac{ \tilde{c}_1r_1}{r_1-r_2}e^{\frac{s}{r_1}}.$$ After introducting the notation $c_1=\frac{ 1}{r_1-r_2}\tilde{c}_1r_1$, the expression for $H(s)$ becomes$$H(s)=c_1e^{\frac{s}{r_1}}+c_2 e^{ \frac{s}{r_2}}.$$ If both roots $r_1$ and $r_2$ are real, then this is the desired final form. If they have a non-zero imaginary part, we would like to manipulate the expression further. In this case $\frac{1}{r_1}$ and $\frac{1}{r_2}$ are complex conjugates, which means we may write them as $$\frac{1}{r_1}=a_R+a_I i,\text{$\, \, \, \, \, $ }\frac{1}{r_2}=a_R-a_I i.$$ For $H(s)$ to be real, we also need $c_1$ and $c_2$ to be complex conjugates:$$c_1=c_R-c_I i,\text{$\, \, \, \, \, $ }c_2=c_R+c_I i.$$ $H(s)$ then becomes$$H(s)=\left(c_R-c_I i\right)e^{a_Rs}\left(\cos \left(a_Is\right)+i \sin \left(a_Is\right)\right)+\left(c_R+c_I i\right) \left(\cos \left(a_Is\right)-i
\sin \left(a_Is\right)\right),$$ which after canceling terms gives the desired final form$$H(s)=c_Re^{a_Rs}\cos \left(a_Is\right)+c_I e^{a_Rs} \sin \left(a_Is\right).$$
### The case of a double root
Now let us consider the case where the two roots are equal: $r_1=r_2$. In this case, we need to solve the equation$$\left(1-r_1\frac{d}{ds}\right)H(s)= \tilde{c}_1 e^{\frac{s}{r_1}}$$ We perform the substitution [ ]{}$H(s)=e^{\frac{s}{r_1}}g(s)$, which leads to$$\left(1-r_1\frac{d}{ds}\right)\left(e^{\frac{s}{r_1}}g(s)\right)= \tilde{c}_1 e^{\frac{s}{r_1}}\, \, \, \, \Longrightarrow \, \, \, \, \, -r_1\frac{d}{ds}g(s)=
\tilde{c}_1\text{ $\, \, \, \, $}\Longrightarrow \, \, \, \, \, g(s)=- \frac{\tilde{c}_1}{r_1} s + \text{const}.$$ The result for $H(s)$ is $H(s)=e^{\frac{s}{r_1}}(- \frac{\tilde{c}_1}{r_1} s + \text{const}.)$, which, after renaming the constants, gives the desired final form$$H(s)=e^{\frac{s}{r_1}}\left(c_2+c_1s\right).$$
We see that overall, the resulting characterization of two-dimensional classes of functional forms invariant under average-marginal transformations is consistent with the statement of Theorem \[formpreserve\].
Higher-dimensional functional form classes
------------------------------------------
The same method may be used to derive higher-dimensional form-preserving classes of functional forms. The differential equations one needs to solve are standard and have known solutions that utilize the properties of the *characteristic equations* of the differential equations, which are analogous to the equation $x^2=A +B\text{ }x$ we used above. The proof of Theorem \[formpreserve\] described in Appendix \[AppendixProofsOfTheorems\] may be thought of as using such differential equations, just represented in a different, transformed way as the vanishing of the expression in Equation \[eq:T0GS\] inside the interval $S$. Solving the differential equations using transforms is much quicker and more convenient.\
Applications {#+apps}
=============
Supply chains with hold-up: the restricted problem {#AppendixSupplyChainsWithHoldupDetails}
--------------------------------------------------
Extending our analysis of the Antràs and Chor model in Appendix \[AppendixSupplyChainsWithHoldup\], we consider the solution of the restricted AC model in the case where the firm is restricted to two discrete levels of bargaining power corresponding to “out-sourcing” and “in-sourcing”. As in the relaxed solution, consider the optimal choice of a path for $\beta$ subject to producing a total quality $\hat{q}$. Note that $q(j;\beta)$ is a strictly increasing function of $j$ for any path of $\beta$ achieving $\hat{q}$ by definition. Thus it is equivalent, instead of solving for the optimal restricted $\beta$ for each $j$, to solve for the optimal $\beta^{\star\star}$ for each $q\left(j;\beta\right)\in\left[0,\hat{q}\right]$ and then invert the resulting $q\left(j;\beta^{\star\star}\right)$ function to recover the value optimal $\beta$ at each $j$. This method preserves the separability we used in the relaxed problem and thus greatly simplifies the restricted problem. Wherever it does not create confusion we suppress as many arguments as possible, especially the dependence on $\beta$, to preserve notational economy.
By the same arguments as in the restricted case, the cost of production $\hat{q}$ is $C\left(\hat{q};\beta\right)$ where $$C\left(\hat{q};\beta\right)=\int_{0}^{\hat{q}}\left[1-\beta(q)\right]MR(q)dq,$$ where $\beta(q)$ is a notationally-abusive contraction of $\beta\left(j\left(q;\beta\right)\right)$. However, to actually produce $\hat{q}$, we need $$\int_{0}^{1}S\left(\left[1-\beta\left(q(j)\right)\right]MR\left(q(j)\right)\right)dj=\hat{q},$$ where $S=MC^{-1}$, the supply curve, exists because of our assumption that $MC$ is strictly monotone increasing. Changing variables so that both integrals are taken over $j$: $$C\left(\beta\right)=\int_{0}^{1}\left[1-\beta\left(q(j)\right)\right]MR\left(q(j)\right)S\left(\left[1-\beta\left(q(j)\right)\right]MR\left(q(j)\right)\right)dj.$$
Thus the firm solves a Lagrangian version of this problem that is separable in each $j$, or equivalently $q$: $$\max_{\beta}\int_{0}^{1}\lambda S\left(\left[1-\beta\left(q(j)\right)\right]MR\left(q(j)\right)\right)-\left(\left[1-\beta(q)\right]MR(q)S\left(\left[1-\beta\left(q(j)\right)\right]MR\left(q(j)\right)\right)\right)dj-\lambda\hat{q}.$$ At each $q$ this is a simple maximization problem. The firm chooses the value of $\beta$ maximizing $$\lambda S\left(\left[1-\beta\left(q\right)\right]MR\left(q\right)\right)-\left[1-\beta(q)\right]MR(q)S\left(\left[1-\beta\left(q\right)\right]MR\left(q\right)\right),$$ the difference between the total value of the production by that firm and the total cost of that production. Clearly, both terms are decreasing in $\beta$ given that $MR>0$ in any range where the firm would consider producing, so given that the firm chooses between only two values of $\beta$, $\beta_{I}>\beta_{O}$, the firm will strictly choose in-sourcing if and only if $$MR(q)>\frac{\lambda\left[S\left(\left[1-\beta_{O}\right]MR\left(q\right)\right)-S\left(\left[1-\beta_{I}\right]MR\left(q\right)\right)\right]}{\left[1-\beta_{O}\right]S\left(\left[1-\beta_{O}\right]MR\left(q\right)\right)-\left[1-\beta_{I}\right]S\left(\left[1-\beta_{I}\right]MR\left(q\right)\right)}.\label{restrictedconditionbeta}$$ If the sign here is equality (which generically occurs on a set of measure $0$ so long as the functions are nowhere constant relative to one another) then the firm is indifferent and if the inequality is reversed the firm strictly chooses in-sourcing. As $\lambda$ rises, the firm will in-source less and produce more; thus varying $\lambda$ over all positive numbers traces out all potentially optimal solutions. Note that this could easily be extended to a situation where the firm has any simple restricted choice of $\beta$, not just two values.
Furthermore, once $\beta(q)$ is set, we can easily recover the optimal $\beta^{\star\star}$ for each $j$ by noting that the optimal value of $\beta^{\star\star}$ at $\tilde{j}$ is the optimal value at $\tilde{q}$ satisfying the production equation $$\int_{0}^{\tilde{j}}S\left(\left[1-\beta^{\star\star}\left(q(j)\right)\right]MR\left(q^{\star\star}(j)\right)\right)dj=\tilde{q}.$$ This implies the differential equation $q'(j)=S\left(\left[1-\beta^{\star\star}\left(q(j)\right)\right]MR\left(q^{\star\star}(j)\right)\right)$ and thus the inverse differential equation $j'(q)=\frac{1}{S\left(\left[1-\beta^{\star\star}\left(q\right)\right]MR\left(q^{\star\star}\right)\right)}$ which together with the boundary condition $j(0)=0$ yields $j(q)$ and thus $\beta^{\star\star}$ at each $j$.
It remains only to pin down the optimal value of $\lambda$. To do this, denote the set of $q$ on which Inequality \[restrictedconditionbeta\] is satisfied $B_{I}(\lambda)$ and on which it is reversed $B_{O}(\lambda)$.[^97] Total production is $$q_{\lambda}=\int_{j\in(0,1):q(j)\in B_{I}(\lambda)}S\left(\left(1-\beta_{I}\right)MR\left(q(j)\right)\right)dj+\int_{j\in(0,1):q(j)\in B_{O}(\lambda)}S\left(\left(1-\beta_{O}\right)MR\left(q(j)\right)\right)dj,$$ while total cost $C_{\lambda}=$ $$\int_{B_{I}(\lambda)\cap\left(0,q_{\lambda}\right)}\left[1-\beta_{I}\right]MR\left(q\right)dq+\int_{B_{O}(\lambda)\cap\left(0,q_{\lambda}\right)}\left[1-\beta_{O}\right]MR\left(q\right).$$ Profit is $$R\left(q_{\lambda}\right)-C_{\lambda}$$ and the first-order condition for its maximization is $$MR\left(q_{\lambda}\right)\frac{\partial q_{\lambda}}{\partial\lambda}-\frac{\partial C_{\lambda}}{\partial\lambda}=0\implies MR\left(q_{\lambda}\right)=\frac{\frac{\partial C_{\lambda}}{\partial\lambda}}{\frac{\partial q_{\lambda}}{\partial\lambda}}=\lambda,$$ because $\lambda$ is defined as the shadow cost of relaxing the constraint on production.
Now we consider obtaining as close as possible to an explicit solution. Note that, to do so, we must be able to characterize $S,B_{O}$ and $B_{I}$ explicitly. $S$ is the inverse of $MC$ and thus $MC$ must admit an explicit inverse. To characterize $B_{O}$ and $B_{I}$ explicitly requires solving Inequality \[restrictedconditionbeta\] with equality to determine the relevant thresholds, which, as we will see, requires marginal revenue to have an explicit inverse.
One of the simplest forms satisfying these conditions and yet yielding our desired non-monotonicity is $P(q)=p_{0}+p_{-t}q^{t}+p_{-2t}q^{2t}$ and $MC(q)=mc_{-t}q^{t}$, where $t,p_{0},p_{-t},mc_{-t}>0>p_{-2t}$. In this case $S(p)=\left(\frac{p}{mc_{-t}}\right)^{\frac{1}{t}}$. Thus the equality version of Inequality \[restrictedconditionbeta\] becomes $$MR(q)=\frac{\lambda\left(\left[\frac{\left(1-\beta_{O}\right)MR(q)}{mc_{-t}}\right]^{\frac{1}{t}}-\left[\frac{\left(1-\beta_{I}\right)MR(q)}{mc_{-t}}\right]^{\frac{1}{t}}\right)}{\left(1-\beta_{O}\right)\left[\frac{\left(1-\beta_{O}\right)MR(q)}{mc_{-t}}\right]^{\frac{1}{t}}-\left(1-\beta_{I}\right)\left[\frac{\left(1-\beta_{I}\right)MR(q)}{mc_{-t}}\right]^{\frac{1}{t}}}\implies$$ $$\implies MR(q)=\frac{\lambda\left[\left(1-\beta_{O}\right)^{\frac{1}{t}}-\left(1-\beta_{I}\right)^{\frac{1}{t}}\right]}{\left(1-\beta_{O}\right)^{\frac{1+t}{t}}-\left(1-\beta_{I}\right)^{\frac{1+t}{t}}}\equiv\lambda k,$$ where $k$ is the relevant collection of constants. Note that this is an extremely simple threshold rule in terms of marginal revenue. Given that we have chosen a form of marginal revenue that admits an inverse, it is simple to solve out for the threshold rule in terms of quantities; this is why we needed marginal revenue to have an inverse solution. $$p_{0}+(1+t)p_{-t}q^{t}+(1+2t)p_{-2t}q^{2t}=\lambda k\implies$$ $$q=\left(\frac{-p_{-t}(1+t)\pm\sqrt{p_{-t}(1+t)^{2}+4\left(p_{0}-k\lambda\right)p_{-2t}(1+2t)}}{2p_{-2t}(1+2t)}\right)^{\frac{1}{t}}.$$ Between these two roots, in-sourcing is optimal; outside them, outsourcing is optimal.[^98]
This provides closed-form solutions as a function of $\lambda$, but $\lambda$ remains to be determined. This is, unfortunately, where things start to get a bit messier. The integral determining $q_{\lambda}$ can be explicitly taken, but only in terms of the less-standard Appell Hypergeometric function. The equation for $MR\left(q_{\lambda}\right)=\lambda$ therefore cannot be solved explicitly for $\lambda$. However, it is a single explicit equation. Once $\lambda$ has been determined, optimal sourcing is determined in closed-form as described above. We plot this and the relaxed optimal $\beta$, in Figure \[closedformAC\], in the same format as in the paper for the case when $p_{0}=.2,p_{-t}=2,p_{-2t}=-4,mc_{-t}=.5,t=.5,\beta_{I}=.8,\beta_{O}=.3$. Clearly, we obtain similar, non-monotone results, but now these require only a single call of Newton’s method to solve an otherwise explicit equation, as opposed to the two-dimension search we required to solve the case presented in the paper.
![Relaxed and restricted solutions to the AC model when $P(q)=0.2+2q^{\frac{1}{2}}-4q$, $MC(q)=\frac{q^{\frac{1}{2}}}{2}$, $\beta_{O}=0.3$ and $\beta_{I}=0.8$.[]{data-label="closedformAC"}](FigureClosedFormRestrictedAC.pdf){width="4in"}
We do not discuss second-order conditions here, but they can easily be derived and checked to hold for this example as well as for the example in the paper. A grossly sufficient condition is that marginal revenue is declining over the solution range, as is the case in both of these examples.
Labor bargaining without commitment [@stole; @stole2] {#AppendixLaborBargainingWithoutCommitment}
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@stole [@stole2 henceforth SZ] consider a model of labor market bargaining where contracts cannot commit workers. Each worker is, therefore, able to extract a share of the surplus the firm gains from a marginal worker. However, that surplus is determined by the profits the firm would earn if that worker were to leave, in which case the firm would bargain with other workers for a share of the remaining surplus. This causes (a) wages to depend on infra-marginal profits and (b) firms to over-employ workers relative to a standard labor market since having reserve workers decreases the marginal value of any given employee, lowering equilibrium wages and raising profits.
The setup of the SZ model is as follows. At the beginning of a period, a firm hires workers, each of whom supplies one unit of labor if employed.[^99] When this process has been completed but before production takes place, the workers are free to bargain over their wages for this period. At that time the firm cannot hire any additional workers, so if any bargaining is not successful and any worker leaves the firm, fewer workers will be available for production in this period. Moreover, after the worker’s departure, the remaining employees are free to renegotiate their wages, and in principle the process may continue until the firm loses all its employees. Assuming its revenues are concave in labor employed, this gives the firm an incentive to “over-employ” or [*hoard*]{} workers as hiring more workers makes holding a marginal worker less valuable to the firm and thus reduces workers’ bargaining power.
If the bargaining weight of the worker relative to that of the firm’s owner is $\lambda$, then the relationship surplus splitting condition is $S_{w}$=$\lambda S_{f}$. The worker’s surplus is simply the equilibrium wage corresponding to the current employment level minus the outside option: $S_{w}=W\left(l\right)-W_{0}$, where $W$ is the wage as a function of $l$, the labor supplied. For expositional simplicity, we assume the firm transforms labor into output one-for-one, though analytic solutions also exist for any power law production technology when $\lambda=1$ and in other cases. Thus we assume $q=l$ and henceforth use $q$ as our primary variable analysis for consistency with previous sections.
The firm faces inverse demand $P(q)$ and thus its profits are $\Pi(q)=\left[P(q)-W(q)\right]q$. The firm’s surplus from hiring an additional worker is then $\Pi'(q)$. This gives the differential equation $$W\left(q\right)-W_{0}=\lambda MR\left(q\right)+\lambda\left(W\left(q\right)q\right)'\Rightarrow\lambda(W\left(q\right)q^{1+\frac{1}{\lambda}})'=q^{\frac{1}{\lambda}}\left(\lambda MR\left(q\right)+W_{0}\right),\label{eq:SZwage}$$ where $MR(q)\equiv P(q)+P'(q)q$ and the implication can be verified by simple algebra and is a standard transformation for an ordinary differential equation of this class. Integrating both of the sides of the equation, imposing the boundary condition that the wage bill shrinks to $0$ at $q=0$, and solving out yields wages and profits
$W\left(q\right)=q^{-\left(1+\frac{1}{\lambda}\right)}\intop_{0}^{q}x^{\frac{1}{\lambda}}MR(x)dx+\frac{W_{0}}{1+\lambda},\quad \Pi\left(q\right)=P(q)q-q^{-\frac{1}{\lambda}}\intop_{0}^{q}x^{\frac{1}{\lambda}}MR(x)dx-\frac{W_{0}}{1+\lambda}$.
While the wage equation is intractable in general, the operation on the right-hand side does not change the functional form of any element of the average-marginal form-preserving class. To gain intuition for this, note that as $\lambda\rightarrow0$, the model converges to the neoclassical model because the worker has no bargaining power; thus the equation becomes $MR(q)=W_{0}$. On the other hand as $\lambda\rightarrow\infty$, the equation converges to $P(q)=W_{0}$ as workers capture all revenue and divide it equally. Thus for intermediate $\lambda$ the marginal-average transformation is effectively applied “partially” to $P(q)$. To see this mathematically, suppose $MR(q)=aq^{-b}$. Then the integral term in Equation \[eq:SZwage\] becomes
$$\frac{(1+\lambda)a\int_{0}^{q^{\star}}x^{\frac{1}{\lambda}}x^{-b}dx}{\lambda (q^{\star})^{1+\frac{1}{\lambda}}}
=\frac{(1+\lambda)a}{(q^{\star})^{1+\frac{1}{\lambda}}}\ \frac{(q^{\star})^{\frac{1+\lambda-b\lambda}{\lambda}}}{1+\lambda-b\lambda}
=\frac{1+\lambda}{1+\lambda-b\lambda}\ a(q^{\star})^{-b}.$$ More generally, for $MR(q)$ a linear combination of power terms, each term of becomes multiplied by ${(1+\lambda)}/{(1+\lambda+t\lambda)}$, where $t$ is the power on the term. This tractability under form-preserving classes, but general intractability, has led researchers to study the SZ model almost exclusively under linear and constant elasticity demand.
While this class can yield important insights, it also has significant limitations. In particular, in the rest of this subsection we show that under this class the percentage over-employment relative to the neoclassical benchmark is constant as a function of the prevailing wage and multiplicative demand shifters. Thus proportional over-employment does not vary, for example, over the business cycle as consumers become richer and employment grows overall. By contrast in a calibrated model with equal bargaining weights ($\lambda=1$), using demand derived from the US income distribution as in Section \[SectionReplacingConstantElasticityDemand\], Equation \[EquationQuadraticallyTractableFormForIncomeDistribution\], we find that over a reasonable business cycle range over-employment should shift by roughly 0.4% of total employment. While quite small in absolute terms, this could account for a non-trivial fraction of cyclic variation in employment and is ruled out by the standard model. Furthermore, this model is quadratically tractable, nearly as tractable as the standard constant elasticity or linear specifications that are linearly tractable. It thus seems a natural alternative to make future analysis of labor bargaining more realistic without losing significant tractability.
To carry out this calculation, we note that the firm’s optimal $q$ solves its first-order condition, $\Pi'\left(q\right)=0$, which, after some algebraic manipulations, is $$\frac{(1+\lambda)\intop_{0}^{q}x^{\frac{1}{\lambda}}MR(x)dx}{\lambda q^{1+\frac{1}{\lambda}}}=W_{0}.\label{eq:StoleZwiebelFOC}$$
Let us define (relative) labor hoarding as $h\equiv {(q^{\star}-q^{\star\star})}/{q^{\star\star}}$, where $q^{\star}$ is SZ employment and $q^{\star\star}$ is the employment level that a neoclassical firm with identical technology would choose: $MR\left(q^{\star\star}\right)=W_{0}$. Combining these definitions with Equation \[eq:StoleZwiebelFOC\] gives a useful condition for $h$ in terms of the equilibrium employment level $q^{\star}$: $$MR\left(\frac{q^{\star}}{1+h}\right)=\frac{\left(1+\lambda\right)\intop_{0}^{q^{\star}}x^{\frac{1}{\lambda}}MR(x)dx}{\lambda\left(q^{\star}\right)^{1+\frac{1}{\lambda}}}.\label{eq:StoleZwiebelOveremploymentCondition}$$ Note that this equation, and Equation \[eq:StoleZwiebelFOC\], involves only (a) marginal revenue and (b) integrals of it multiplied by a power of $q$ and then divided by a power of $q$ higher by 1. It can easily be shown that the support of the Laplace marginal revenue is preserved by this transformation using essentially the same argument we used in the paper to show this support was shifted by exactly one unit when consumer surplus is calculated. This implies that Equations \[eq:StoleZwiebelFOC\] and \[eq:StoleZwiebelOveremploymentCondition\] have precisely the same tractability characterization as does the basic monopoly model we studied in Section \[SectionReplacingConstantElasticityDemand\] of the paper.[^100]
Given the complexity of Equation \[eq:StoleZwiebelOveremploymentCondition\] from any perspective other than our tractable forms, we investigate it using these forms, following @inequality who study the model under constant elasticity demand. First consider the BP class, $P(q)=p_{0}+p_{t}q^{-t}$, which nests the constant elasticity case when $p_{0}=0$. Solving Equation \[eq:StoleZwiebelOveremploymentCondition\] for $h$ yields $$h=\left(\frac{1+\lambda}{1+\lambda-t\lambda}\right)^{\frac{1}{t}}-1.\label{SZBP}$$ Therefore hoarding is constant in $q^{\star}$ and consequently in $W_{0}$. Thus under the BP class of demand, including constant elasticity, the economic cycle (the nominal outside option) has [*no effect*]{} on relative hoarding. It can easily be shown that $h$ monotonically increases in $t$, so that the less concave demand (and thus profits) are, the more hoarding occurs. We found this counterintuitive, as we believed, building off the intuition supplied by SZ about the relationship between the “front-loading” that drives hoarding and concavity, that labor hoarding was driven by concavity in the firm’s profit function.[^101] Instead it appears that the reverse is the case. This shows one advantage of considering explicit functional forms: they help correct false intuitions. In particular, because $t$ clearly parameterizes concavity the comparative static has a natural interpretation.
This new intuition suggests that the hoarding may not be constant over the economic cycle if, during that cycle, the curvature of firm profits change. For example, if during booms broad parts of the population are served and during recessions only wealthier individuals are served, then labor hoarding should be counter-cyclical as the distribution of income among the wealthy is more convex than among the middle-class and poor.
To analyze this we used our proposed functional form from Equation \[EquationQuadraticallyTractableFormForIncomeDistribution\] in Section \[SectionReplacingConstantElasticityDemand\], in the version where it actually represents the income distribution as this is appropriately normalized to our assumption of $q=l$ (willingness-to-pay for a unit of labor): $$P(q)= 50000\left(\frac{1}{2} q^{-\frac{2}{5}}+2- \frac{5}{2}q^{\frac{2}{5}}\right).$$ (in US dollars). Plugging this into Equation \[eq:StoleZwiebelOveremploymentCondition\] and $MR\left(q^{\star\star}\right)=W_{0}$, assuming to match the convention in the literature that $\lambda=1$ (though this plays no role in the simple form of our solution) and rounding to the second significant digit yields: $$h=1.6\left(\frac{1+\sqrt{1+\frac{1.2\cdot10^9}{\left(100000-W_0\right)^2}}}{1+\sqrt{1+\frac{1.1\cdot10^8}{\left(100000-W_0\right)^2}}}.\right)^{\nicefrac{5}{2}}-1,\label{hoardingresult}$$
![Relative labor hoarding in the Stole-Zwiebel model with $\lambda=1$ and demand given by the approximation for $W_{0}\in$ \[10000,50000\] (in USD).[]{data-label="SZcalibration"}](FigureSZ.pdf){width="3.2in"}
We interpret a reduction in $W_{0}$, or equivalently a multiplicative scaling up of $P$, to be a boom (as it leads to higher production) and a rise in $W_{0}$ to be a recession. The expression on the right-hand side of Equation \[hoardingresult\] can easily be shown to be increasing in $W_0$ (in fact, this is true quite broadly beyond this particular calibration). Thus a recession hoarding rises, contrasting with the standard intuition that unions exacerbate recessions by creating nominal wage rigidity and suggesting the effects of individual workers’ bargaining may have qualitatively different comparative statics than collective bargaining does. Figure \[SZcalibration\] shows the results quantitatively. Hoarding is large ($\approx$ 59%), but its comparative statics are less pronounced. It rises by a bit less than one percentage point when the outside option rises from \$30k to \$50k, a reasonable range of variation over the economic cycle. Thus, while the BP approximation of constancy appears not to be very far off these effects may of a similar magnitude to cyclic shifts in employment and are thus worth considering.
Imperfectly competitive supply chains {#sequential}
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Consider the model of imperfectly competitive supply chains where each stage of production strategically anticipates the reactions of the subsequent stage proposed by @salinger. There are $m$ stages of production interacting via linear pricing. Producers at each stage act simultaneously and the stages act in sequence. We solve by backwards induction.
Producers at stage $m$ take an input from producers at stage $m-1$ and sell it to final consumers, facing inverse demand $P_{m}$. The $n_{m}$ firms at stage $m$ are symmetric Cournot competitors with average cost $AC_{m}$. The linear price clearing the market between stage $m-1$ and $m$ is $\hat{P}_{m-1}$. Using the standard first-order condition for Cournot competition and dropping arguments, the first-order equilibrium conditions are $$P_{m}+\frac{1}{n_{m}}P_{m}'q=\hat{P}_{m-1}+AC_{m}+\frac{1}{n_{m}}AC_{m}'q\iff$$ $$\hat{P}_{m-1}=P_{m}+\frac{1}{n_{m}}P_{m}'q-AC_{m}-\frac{1}{n_{m}}AC_{m}'q.$$ Thus the effective inverse demand facing the firms at stage $m-1$ is $$P_{m-1}\equiv P_{m}+\frac{1}{n_{m}}P_{m}'q-AC_{m}-\frac{1}{n_{m}}AC_{m}'q,$$ as all output produced at stage $m-1$ is used as an input at stage $m$. Effectively the inverse demand at stage $m-1$ is the (competition-adjusted) marginal profit (competition-adjusted marginal revenue less marginal cost) at stage $m$.
This analysis may be back-propagated up the supply chain to obtain a first-order condition at the first stage determining the quantity in the industry. However, at each stage one higher derivative of $P_{m}$, at least and also of some of the cost curves, enters the first-order conditions. Thus the implicit equation for the first-order conditions characterizing the supply chain is usually quite elaborate and is both difficult to analyze in general and highly intractable, even computationally, for many functional forms. For example, @cable use this computational tractability concern to justify their focus on simultaneous decisions upstream and downstream in a related vertical contracting model.
However, we now derive a simple explicit transformation of the Laplace inverse demand and average cost characterizing the supply chain and discuss how this can be used to overcome these difficulties. Note that $$P_{m}+\frac{1}{n_{m}}P_{m}'q=\left(1-\frac{1}{n_{m}}\right)P_{m}+\frac{1}{n_{m}}MR_{m},$$ where $MR_{m}=P_{m}+P_{m}'q$. Let $p_{m}$ be the Laplace inverse demand. From Section \[SectionArbitraryDemandAndCostFunctions\] we have that the Laplace marginal revenue is $(1-t)p_{m}$ and thus that the inverse Laplace-log transform of $\left(1-\frac{1}{n_{m}}\right)P_{m}+\frac{1}{n_{m}}MR_{m}$ is just $\left(1-\frac{t}{n_{m}}\right)p_{m}$. By the same logic, if we denote the Laplace average cost by $ac_{m}$ the inverse Laplace-log transform of $AC_{m}+\frac{1}{n_{m}}AC_{m}'q$ is $\left(1-\frac{t}{n_{m}}\right)ac_{m}$.
Iterating this process, one obtains that the Laplace first-order condition at the initial stage, which we denote $f_{1}$, is $$p_{m}\prod_{i=1}^{m}\left(1-\frac{t}{n_{i}}\right)-\sum_{i=1}^{m}\left[ac_{i}\prod_{j=1}^{i}\left(1-\frac{t}{n_{j}}\right)\right].$$ This obviously differs only in its (trivially computed) coefficients and not in its support from the $ac_{i}$’s and $p_{m}$ that make it up. Thus if all $ac_{i}$’s and $p_{m}$ are chosen to have the same tractable support (with the desired number of evenly spaced mass points to achieve desired tractability) then the full will be equally tractable. Beyond this, even if $p_{m}$ and the $ac_{i}$’s are specified in an arbitrary manner, the resulting Laplace first-order condition can be trivially computed from the inverse Laplace-log transforms of each of these inputs and then either solved directly by applying the Laplace transform or approximated using a small number of evenly spaced mass points for tractability. In either case, this approach significantly reduces the complexity of computing and representing the system.
Two-sided platforms à la @rt2003\[AppendixTwoSidedPlatforms\]
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@rt2003 propose a model of a two-sided platform motivated by the credit card industry. Sellers and buyers are randomly matched and independently decide whether they want to accept credit cards and whether they want to use them conditional on cards being accepted. These decisions are driven by the price charged (or subsidy paid) to each side. In particular, in order for a fraction of sellers $q_{S}$ to wish to accept cards, the price that must be charged to sellers is $P_{S}\left(q_{S}\right)$, and similarly for buyers.
Let $U_{I}\left(q_{I}\right)\equiv\int_{0}^{q_{I}}P_{I}(x)dx$ be the gross utility on side $I$. Because $U_{I}'\left(q_I\right)=P_{I}\left(q_I\right)$, the average gross utility $\overline{U}_{I}\left(q_{I}\right)\equiv\nicefrac{U_{I}\left(q_{I}\right)}{q_{I}}$ has the average-marginal relationship to inverse demand. Thus average consumer surplus $\overline{V}_{I}\left(q_{I}\right)=\overline{U}_{I}\left(q_{I}\right)-P_{I}\left(q_{I}\right)$ has the same functional form as $P_I'q_I$ for a form-preserving functional form class.
@rt2003 show that, when there is a constant and symmetric marginal cost of clearing transactions $c$, imperfectly competitive equilibrium between symmetric firms is characterized by $$P_{S}\left(q_{S}\right)+P_{B}\left(q_{B}\right)-c=-\theta P_{S}'\left(q_{S}\right)q_{S}=-\theta P_{B}'\left(q_{B}\right)q_{B}$$ for some constant $\theta<1$.[^102] On the other hand, they show that Ramsey pricing (which nests the unconstrained social planner’s problem as a special case) is characterized by $$P_{S}\left(q_{S}\right)+P_{B}\left(q_{B}\right)-c=-\theta\overline{V}_{S}\left(q_{S}\right)=-\theta\overline{V}_{B}\left(q_{B}\right)$$ for some constant $\theta$, equal to unity in the case of the unconstrained social optimum and approaching $0$ as Ramsey pricing is required to break even. Thus if inverse demand on both sides of the market is specified within the same form-preserving class (@rt2003 assume linear demand in their example) then our characterization of tractability applies here as well.
Again the added flexibility of our forms is important in this context. For example, @ramslind considers how platforms would choose an “interchange fee” between two sides of the market, holding fixed the overall level of prices. He demonstrates that if both sides have BP demand, then users on both sides of the market and profit maximization [*all*]{} agree on the same optimal interchange fee. However, this is generally false and thus assuming BP demand trivializes the wide-ranging regulatory debate over interchange fees. In fact under plausible (bell-shaped) demand forms, perhaps surprisingly, both sides in aggregate prefer to face higher prices (consumers prefer lower interchange, merchants prefer higher) to subsidize use on the other side of the market. From a social perspective, the more heterogeneous side and/or the side which has more complete adoption should be taxed to subsidize the other side more than will be in the interest of a profit-maximizing platform, even for fixed aggregate prices.
Auction theory {#AppendixAuctionTheory}
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### Symmetric independent private values first-price auctions {#AppendixSymmetricIndependentPrivateValuesFirstPriceAuctions}
Consider $N$ symmetric bidders with privately-known values $v_{i}$ for a single object drawn independently and identically from a distribution with differentiable CDF $F$. Let $V(q)\equiv F^{-1}(q)$ be the quantile function of $F$. Let $b_{\star}$ be a symmetric-equilibrium bid function mapping values to bids in a first-price auction in which the highest bidder wins and pays her bid value; any such equilibrium bid function can be shown to be strictly monotone increasing under weak conditions. The probability that the bid of any individual bidder is below $x$ is then $G_{\star}(x)\equiv F\left(b_{\star}^{-1}(x)\right)$. Thus the probability that bidder $i$ wins if she submits a bid of $x$ is, by symmetry, $\left[G_{\star}(x)\right]^{N-1}$.
The expected utility a bidder with value $v$ thus earns from a bid of $x$ is $$\left(v-x\right)\left[G_{\star}(x)\right]^{N-1}=\left(v-B_{\star}(q)\right)q^{N-1},$$ where $q$ is the fraction of other bidders with (weakly) lower bids and $B_{\star}(q)\equiv G_{\star}^{-1}(q)$ is the quantile function of the equilibrium bid distribution. A necessary condition for her optimization is therefore $$\left(v-B_{\star}(q)\right)(N-1)q^{N-2}+B_{\star}'(q)q^{N-1}=0\iff v=B_{\star}(q)+\frac{1}{N-1}qB_{\star}'(q).$$ For this to be a symmetric, monotone equilibrium for the posited bid distribution, it must be that a bidder with value at reversed quantile $q$ of the value distribution chooses to bid (weakly) higher than precisely a fraction $q$ of her rivals. Thus a necessary condition for a symmetric equilibrium is
$V(q)=B_{\star}(q)+\frac{1}{N-1}qB_{\star}'(q)$.
Sufficient conditions, which we omit here, are well-known in the literature. Note that the right-hand side of this expression involves the marginal and average forms of $B_{\star}$. Thus, by simple coefficient matching, if $V$ is chosen to be from a form-preserving class then there is always an equilibrium $B_{\star}$ from the same class. This may be used directly to analytically relate the values and bids at various quantiles, which is all that is necessary for many analytic problems.
However if one wishes to obtain a closed form for $b_{\star}$ itself, then one must choose the class to be tractable at the level of complexity of the desired closed form and include a constant (a power of $0$) in the class. By definition, $G_{\star}=F\circ b_{\star}^{-1}$, so $b_{\star}^{-1}=V\circ G_{\star}$ and consequently $b_{\star}=B_{\star}\circ F$. Thus if $F$ and $V$ have forms tractable at level $k$, then so does $b_{\star}$. Evidently uniform and exponential distributions, which have linear and logarithmic $V$ respectively, are linearly tractable, explaining why they are ubiquitously used for examples in symmetric first-price auction models.
However these forms are quite restrictive in that they cannot, for example, have the bell shape usually found in empirical studies of valuation distributions in auctions [@hailetamer; @hortacsu]. Our forms can easily generate such shapes and thus allow tractable examples with realistic value distributions.
### Auctions v. posted prices [@onlineauctions] {#AppendixAuctionsVPostedPrices}
@onlineauctions consider the trade-off a seller faces between using an auction and setting a posted price in an online retail market. They assume sellers of goods know the common (positive) hassle cost $\lambda$ for buyers to participate in an auction, but may still use an auction because they do not know their common value $v$ of the good. The seller has an opportunity cost of selling $c$, and $v$ is drawn from a distribution $F$ that the seller knows. Assuming, as the authors do, that at least two bidders participate, the auction guarantees that the seller gets value $v-\lambda$ as long as $v-\lambda\geq r$, where $r$ is the reserve price the seller sets. Alternatively, the seller may set a posted price $p$, in which case she will sell the good if $v\geq p$.
Let $P(q)\equiv F^{-1}(1-q).$ If a seller sells the good with probability $q$, then in an auction with the reserve price set to $P(q)-\lambda$ she will receive an average price $\overline{U}(q)-\lambda$, where $\overline{U}(q)\equiv \nicefrac{\int_{0}^{q}P(x)dx}{q}$ by the same logic as in Supplementary Material \[AppendixTwoSidedPlatforms\]. If the seller sells the good with probability $q$ with a posted price by setting price $P(q)$, she will receive price $P(q)$ with certainty. Thus the region in which she wishes to use an auction rather than a posted price is when she wishes to sell with probability $q$ such that $\overline{U}(q)>P(q)+\lambda$. As noted in Supplementary Material \[AppendixTwoSidedPlatforms\], $\overline{U}$ has the average-marginal relationship to $P$. For this reason, if $P$ is specified according to an average-marginal form-preserving class including a constant term (power $0$ term), then the resulting optimal cut-off rules for using a definite mechanism are tractable at the level of tractability of the class (in terms of both the cost and the desired probability of sale, which is more directly observed in the authors’ data).
@onlineauctions present such an example, by assuming a uniform distribution and thus a linear form for $P$. In this case $\overline{U}-P$ uniformly grows in $q$. This implies that sellers with a low cost (low opportunity cost of sale), such as impatient private individuals clearing old property out of the house, who wish to achieve sale with high probability (quickly) will use auctions. On the other hand, those who have a high cost, such as professional vendors, who want to achieve a sale with low probability (slowly) to wait will set a high posted price. However this is not generally true. If $P$ takes a constant elasticity form, for example, the reverse pattern holds: low-cost sellers set a low posted price and sell quickly while high cost (patient) sellers run an auction.
For the bell-shaped demands that appear to fit @onlineauctions’s data best, the gap between $\overline{U}$ and $P$ is actually non-monotone, first declining an then rising. This suggests auctions should be polarized into goods that sell with very low and very high probability; that is among those clearing out their houses and among the most professional sellers. This is in fact what the authors find; they cannot even measure the posted-price demand curve at very low sale probabilities as they do not observe sufficiently many items selling that infrequently with posted prices, while the same is true at very high probabilities. This suggests richer classes of tractable, form-preserving demand may be more useful in modeling this trade-off than is the uniform distribution.
Selection markets {#AppendixSelectionMarkets}
-----------------
@akerlof1970market analyzed markets where the cost of providing a service differs by the identity of the consumer to whom it is provided. He studies a case that he labels “adverse selection” in which consumers differ in only a single characteristic and in which raising this one dimension increases both consumers’ willingness-to-pay for the product and the cost of serving them. @einav2010estimating and @einav2011selection maintain @akerlof1970market’s assumption of a single product but allow consumers to differ along multiple dimensions that may impact their willingness to pay and cost in potentially rich ways.
@einav2010estimating define an inverse demand curve $P(q)$ for $q\in(0,1)$ as the willingness to pay of the individual in the $(1-q)$th quantile of the willingness-to-pay distribution. They define average cost $AC(q)$ as the average cost of individuals who are in the quantiles above $1-q$ of the willingness-to-pay distribution. They argue that perfectly competitive equilibrium requires $AC(q)=P(q)$ while social optimization requires $MC(q)=P(q)$, where $MC$ has the average-marginal relationship to $AC$. @imperfectcomp extend this framework to nest a variety of models of imperfect competition using a conduct parameter $\theta$ as in Subsection \[sub:Imperfectly-competitive-supply\] above and show that equilibrium is characterized by $\theta MC(q)+(1-\theta)AC(q)=(1-\theta)P(q)+\theta MR(q)$.
As is clear by now, both sides of this equation are tractable for any value of $\theta$ at whatever level the cost and demand side are specified if these are chosen to be part of a form-preserving class. Many analyses have assumed linear forms on both the cost and demand side [@cutler; @einav2010estimating; @einav2011selection], partly for tractability. As @scheuersmetters highlight, this assumption rules out many interesting phenomena, such as selection that is “advantageous” (higher willingness to pay correlating with lower cost) over some range but adverse over other ranges or multiple local competitive equilibria that @scheuersmetters argue may have challenged the introduction of the Affordable Care and Patient Protection Act in the United States. Broader tractable form-preserving classes, especially those with bell-shaped demand and cost curves, allow these possibilities and appear to fit existing empirical evidence more closely.
Monopolistic competition {#AppendixMonopolisticCompetition}
------------------------
### Tractable generalizations of the Dixit-Stiglitz framework with separable utility
In the simplest monopolistic competition model, consumers derive their utility from a continuum of varieties $\omega\in\Omega$ of a single heterogeneous good in a separable way: $%
U_{\Omega}=\int_{\Omega}u_{\omega}\left(q_{\omega}\right)\, d\omega.\ \label{eq:SeparableUtility}
$ In the original Dixit-Stiglitz model specialized to the case of constant elasticity of substitution $\sigma$, $u_{\omega}(q_{\omega})$ is a power of the consumed quantities $q_{\omega}$: $u_{\omega}(q_{\omega})\propto q_{\omega}^{1-{1}/{\sigma}}$. In our generalization we wish to be able to apply Theorem \[TheoremClosedFormSolutions\], so we let $u\left(q_{\omega}\right)$ be a linear combination of different powers of $q_{\omega}$. More explicitly, consumer optimization requires that marginal utility of extra spending is equalized across varieties: $u_{\omega}'\left(q_{\omega}\right)=\lambda P_{\omega}$, where $P_{\omega}$ is the price of variety $\omega$ and $\lambda$ is a Lagrange multiplier related to consumers’ wealth. To ensure tractability, we let the residual inverse demand $P_{\omega}\left(q_{\omega}\right)={u_{\omega}'\left(q_{\omega}\right)}/{\lambda}$ and the corresponding revenue $R_{\omega}\left(q_{\omega}\right)$ be linear combinations of equally-spaced powers of $q_{\omega}$: $%
P_{\omega}\left(q_{\omega}\right)=\sum_{t\in T}p_{\omega,t}q_{\omega}^{-t},\quad R_{\omega}\left(q_{\omega}\right)=\sum_{t\in T}p_{\omega,t}q_{\omega}^{1-t}\
$for some finite and evenly-spaced set $T$, with the number of elements of $T$ determining the precise degree of tractability. For convenience of notation, we choose a num' eraire in a way that keeps $P_{\omega}\left(q_{\omega}\right)$ for a given $q_{\omega}$ independent of macroeconomic circumstances.
Each variety of the differentiated good is produced by a single firm. We assume that the marginal cost and average cost of production can be written as $%
MC_{\omega}(q)=\sum_{t\in T}mc_{\omega,t}q_{\omega}^{-t},\quad AC_{\omega}(q)=\sum_{t\in T\cup\left\{ 1\right\} }ac_{\omega,t}q_{\omega}^{-t},\
$where $mc_{\omega,t}=\left(1-t\right)ac_{\omega,t}$. A constant component of average cost (and marginal cost) would correspond to $ac_{\omega,0}$ and a fixed cost would correspond to $ac_{\omega,1}$. However, given the generality possible here we do not necessarily have to assume that these components are present in all models under consideration.
With this specification, Theorem \[TheoremClosedFormSolutions\] applies and the firm’s problem has closed-form solutions unless $T$ has six elements or more. Moreover, if firms are heterogeneous in their productivity, then \[TheoremAggregation\] leads to closed-form aggregation integrals for suitable choices of the productivity distribution, as in the case of a generalized Melitz model discussed below.
### Tractable generalizations of the D-S framework with non-separable utility
Here we briefly discuss tractable monopolistic competition in the case of non-separable utility.[^103] The utility has the very general form $$U_{\Omega}\equiv F\left(U_{\Omega}^{\left(1\right)},U_{\Omega}^{\left(2\right)},...,U_{\Omega}^{\left(m\right)}\right),\quad U_{\Omega}^{\left(i\right)}\equiv\int_{\Omega}U^{\left(i,\omega\right)}\left(q_{\omega}\right)\, d\omega.$$ In order to preserve tractability, we assume that $U^{\left(i,\omega\right)}\left(q_{\omega}\right)$ are linear combinations[^104] of equally-spaced powers of $q_{\omega}$ and that the set of exponents does not depend on $i$ or $\omega$. For example, we could specify $U_{\Omega}\equiv U_{\Omega}^{\left(1\right)}+\kappa_{1}(U_{\Omega}^{\left(1\right)})^{\xi_{1}}+\kappa_{2}(U_{\Omega}^{\left(2\right)})^{\xi_{2}}$, $U_{\Omega}^{\left(1\right)}\equiv\int_{\Omega}q_{\omega}^{\gamma_{1}}\, d\omega$, and $U_{\Omega}^{\left(2\right)}\equiv\int_{\Omega}q_{\omega}^{\gamma_{2}}\, d\omega$, with $(\gamma_{1}+1)/(\gamma_{2}+1)$ equal to the ratio of two small integers. In the language of heterogeneous-firm models, the choice $\kappa_{1}=\kappa_{2}=0$ corresponds to the @melitz model, while the choice $\xi_{1}=2$, $\xi_{2}=1$, $\gamma_{1}=1$, and $\gamma_{2}=2$ gives the @melitzottaviano model, which is based on a non-homothetic quadratic utility. Our general specification allows also for homothetic non-separable utility functions that feature market toughness effects analogous to those in the @melitzottaviano model.
It is straightforward to verify that as in the separable-utility case, Theorems \[TheoremClosedFormSolutions\] and \[TheoremAggregation\] still apply and lead to closed-form solutions to the firm’s problem and closed-form aggregation. This is because the structure of the firm’s problem is unchanged. Non-separability only makes the resulting system of equations for macroeconomic aggregates more complex. The system itself may still be written in closed form due to Theorem \[TheoremAggregation\], under appropriate assumptions on the productivity distribution.
### Tractable generalizations of the Dixit-Stiglitz framework
In the baseline monopolistic competition model consumers derive their utility from a continuum of varieties $\omega\in\Omega$ of a single heterogeneous good: $$U_{\Omega}=\int_{\Omega}u_{\omega}\left(q_{\omega}\right)\, d\omega.\label{eq:SeparableUtility}$$ In the original Dixit-Stiglitz model with constant elasticity of substitution $\sigma$, $u_{\omega}(q_{\omega})$ is a power of the consumed quantities $q_{\omega}$: $u_{\omega}(q_{\omega})\propto q_{\omega}^{1-\nicefrac{1}{\sigma}}$. In our generalization $u\left(q_{\omega}\right)$ is assumed to be a function of a combination of different powers of $q_{\omega}$ . More explicitly, consumer optimization requires that marginal utility of extra spending is equalized across varieties: $u_{\omega}'\left(q_{\omega}\right)=\lambda P_{\omega}$, where $P_{\omega}$ is the price of variety $\omega$ and $\lambda$ is a Lagrange multiplier related to consumers’ wealth. To ensure tractability, we let the residual inverse demand $P_{\omega}\left(q_{\omega}\right)=\nicefrac{u_{\omega}'\left(q_{\omega}\right)}{\lambda}$ and the corresponding revenue $R_{\omega}\left(q_{\omega}\right)$ be linear combinations of equally-spaced powers of $q_{\omega}$: $$P_{\omega}\left(q_{\omega}\right)=\sum_{t\in T}p_{\omega,t}q_{\omega}^{-t},\quad R_{\omega}\left(q_{\omega}\right)=\sum_{t\in T}p_{\omega,t}q_{\omega}^{1-t}$$ for some finite and evenly-spaced set $T$, with the number of elements of $T$ determining the precise degree of tractability. For the convenience of notation, we choose a numéraire in a way that keeps $P_{\omega}\left(q_{\omega}\right)$ for a given $q_{\omega}$ independent of macroeconomic circumstances.
Each variety of the differentiated good is produced by a single firm. We assume that the marginal cost and average cost of production can be written as $$MC_{\omega}(q)=\sum_{t\in T}mc_{\omega,t}q_{\omega}^{-t},\quad AC_{\omega}(q)=\sum_{t\in T\cup\left\{ 1\right\} }ac_{\omega,t}q_{\omega}^{-t},$$ where $mc_{\omega,t}=\left(1-t\right)ac_{\omega,t}$. A constant component of average cost (and marginal cost) would correspond to $ac_{\omega,0}$ and a fixed cost would correspond to $ac_{\omega,1}$. However given the generality possible here we do not necessarily have to assume that these components are present in all models under consideration.
### Flexible @krugman model
The @krugman model of trade, featuring monopolistic competition and free entry of identical single-product firms, may be solved explicitly for the tractable demand and cost functions mentioned above, not just constant-elasticity demand and constant marginal cost specified in the original paper. Here we consider these solutions in the case of two symmetric countries, which leads to a symmetric equilibrium.
There is a continuum of identical consumers with preferences as in Equation \[eq:SeparableUtility\] who earn labor income. The amount of labor a firm needs to hire in order to produce quantity $q$ may be split into a fixed part $f$ and a variable part $L\left(q\right)$ that vanishes at zero quantity. Both $L\left(q\right)$ and the revenue function $R\left(q\right)$ are assumed to allow for a linear term. The firm only uses labor for production, so its total cost is $w\left(L\left(q\right)+f\right)$, where $w$ is the competitive wage rate. Having produced quantity $q$, the firm splits it into $q_{d}$ to be sold domestically, and $\tau q_{x}$ to be shipped abroad. Due to ** iceberg-type ** trade costs ($\tau\ge1$), a fixed fraction of the shipped good is lost during transport, and only quantity $q_{x}$ is received in the other country. (Non-iceberg trade costs are considered in the appendix.) Let us denote the equilibrium level of marginal cost, measure of firms, international trade flows, and welfare by $MC$$^{\star}$, $N^{\star}$, $X^{\star}$, and $W^{\star}$, respectively, and similarly for other variables. The total labor endowment of one of the two symmetric economies is $L_{E}$.
There exists an explicit map $MC^{\star}\rightarrow\left(f,q_{d}^{\star},q_{x}^{\star},w^{\star}\right)$ and an explicit map $\left(MC^{\star},L_{E}\right)\rightarrow\left(N^{\star},X^{\star},W^{\star}\right)$. These relationships represent a closed-form solution to the model in terms of $MC^{\star}$ and exogenous parameters.
To see briefly why this is the case, it is convenient to express the model’s equations in terms of the equilibrium level of marginal cost $MC^{\star}$.[^105] Output optimally designated for the domestic market and the export market will satisfy $R'(q_{d})=MC^{\star}$ and $R'(q_{x})=\tau\, MC^{\star}$, respectively, and therefore may be solved for in closed form in terms of $MC^{\star}$ for tractable specifications of the revenue function (or consumer preferences).[^106] The same is true for wages since $w=MC^{\star}/L'(q_{d}+\tau q_{x})$.
For a chosen $MC^{\star}$ we may compute the level of fixed cost $f$ consistent with it using the free-entry condition: $R(q_{d})+R(q_{x})=wL(q_{d}+\tau q_{x})+wf$. The equilibrium number (measure) $N^{\star}$ of firms in each economy then satisfies $N^{\star}=L_{E}/(L\left(q_{d}+q_{x}\right)+f)$, where $L_{E}$ is the labor labor endowment one of the two economies.[^107] Other variables of interest, e.g. trade flows or welfare, are then simply functions of the ones discussed above.
#### Krugman model with non-iceberg and iceberg international trade costs. {#krugman-model-with-non-iceberg-and-iceberg-international-trade-costs. .unnumbered}
Although the Krugman model with non-iceberg trade costs is not our main focus here, we mention it for completeness. Let us assume the presence of non-iceberg international trade costs that require hiring labor $L_{T}\left(q_{x}\right)$ in order for $q_{x}$ to reach its destination in the other country.[^108] The export FOC is now $R'(q_{x})-wL'_{T}(q_{x})=\tau MC^{\star}$, while the free entry condition becomes $R(q_{d})+R(q_{x})=wL(q_{d}+\tau q_{x})+wL_{T}(q_{x})+wf$. The resulting number (measure) of firms is $N^{\star}=L_{E}/((L\left(q_{d}+q_{x}\right)+f)+L_{T}(q_{x}))$. The model may be solved explicitly along the same lines in terms of chosen $MC^{\star}$ and $w$, with $f$ and $\tau$ treated as derived quantities.
### Flexible @melitz model {#AppendixFlexibleMelitzModel}
The @melitz model is again based on monopolistic competition and assumes a constant elasticity of substitution between heterogeneous-good varieties. Relative to the @krugman model, it introduced a novel channel for welfare gains from trade, namely increased average firm productivity resulting from trade liberalization or analogous decreases in trade costs. Here we generalize the model to allow for more flexible demand functions, non-constant marginal costs of production, and trade costs that may have components that are neither iceberg-type nor constant per unit.
#### Single country. {#single-country. .unnumbered}
For clarity of exposition, we first describe the flexible and tractable version of the @melitz model in the case of a single country and later discuss its generalization. Just like the @krugman model, it involves two types of agents: monopolistic single-product firms and identical consumers, who supply their labor in a competitive labor market and consume the firms’ products.[^109]
Labor is the only factor of production: all costs have the interpretation of labor costs and are proportional to a competitive wage rate $w$. Each heterogeneous-good variety is produced by a unique single-product firm, which uses its monopolistic market power to set marginal revenue equal to marginal cost. Demand and costs are specified tractably as discussed above; this time we do not need to assume that variable cost and revenue functions allow for a linear term.
If the firm is not able to make positive profits, it is free to exit the industry. In situations of main interest, this endogenous channel of exit is active: there exist firms that are indifferent between production and exit. There is also an exogenous channel of exit: in every period with probability $\delta_{e}$ the firm is forced to permanently shut down.
Entry into the industry is unrestricted but comes at a fixed one-time cost $wf_{e}$. Only after paying this fixed cost, the entering firm observes a characteristic $a$, drawn from a distribution with cumulative distribution function $G\left(a\right),$ that influences the firm’s cost function. In the original @melitz model the constant marginal cost of production is equal to $wa$. Here we leave the specification more general while maintaining the convention that increasing $a$ increases the firm’s cost at any positive quantity $q$. In expectation, the stream of the firm’s profits must exactly compensate the (risk-neutral) owner for the entry cost, which implies the *unrestricted entry condition* $wf_{e}=\mathbb{E}\Pi\left(q;a\right)/\delta_{e}$, with the profit $\Pi\left(q;a\right)$ evaluated at the optimal quantity.[^110]
The amount of labor needed to produce quantity $q$ is $L\left(q;a\right)+f$, where $L\left(q;a\right)$ corresponds to variable cost ($L\left(0;a\right)=0$) and $f$ to a fixed cost. $L\left(q;a\right)$ is assumed to be tractable with respect to $q$, but also with respect to $a$.[^111] In terms of the labor requirement function $L\left(q;a\right)$, the firm profit maximization condition and the z*ero cutoff profit condition* are $R'\left(q\right)=wL'\left(q;a\right)$ and $R(q_{c})=wL(q_{c};a_{c})+wf$, where $q_{c}$ and $a_{c}$ correspond to a *cutoff firm*, i.e. a firm that is in equilibrium indifferent between exiting and staying in the industry. We denote by $L_{E}$, $M^{\star}$,$M_{e}^{\star}$, $W^{\star}$ the labor endowment, and the equilibrium measure of firms, measure of entering firms, and level of welfare, respectively.
The firm profit maximization condition and the free entry condition are
$$\begin{gathered}
R'\left(q\right)=wL'\left(q;a\right),\label{eq:MelitzOneCountryA}\\
R(q_{c})=wL(q_{c};a_{c})+wf.\label{eq:MelitzOneCountryB}\end{gathered}$$
A convenient solution strategy is to choose $q_{c}$ and then calculate $f_{e}$ as a derived quantity. For a chosen $q_{c}$ we can find $a_{c}$ explicitly by combining (\[eq:MelitzOneCountryA\]) and (\[eq:MelitzOneCountryB\]) into $R'\left(q_{c}\right)\left(L(q_{c};a_{c})+f\right)$= $R(q_{c})L'\left(q_{c};a_{c}\right)$, since $L\left(q;a\right)$ is assumed to be tractable also with respect to $a$. Wages are then given recovered from (\[eq:MelitzOneCountryB\]): $w=R(q_{c})/(L(q_{c};a_{c})+f)$.
Now we need to show how to calculate the fixed cost of entry $f_{e}$ and the measure of firms. The fixed cost of entry consistent with the chosen cutoff quantity is given simply by the unrestricted entry condition: $$w\delta_{e}f_{e}=\bar{\Pi}=\intop_{q\geq q_{c}}\left(R\left(q\right)-wL\left(q;a\right)-wf\right)dG\left(a\left(q\right)\right).$$ Here $a\left(q\right)$ is the firm’s productivity parameter as an explicit function of the optimally chosen quantity $q$ that results from using (\[eq:MelitzOneCountryA\]). For Pareto $G$, and $L$ and $R$ tractable from the point of view of $q$ (but not necessarily having a linear term) and $L\left(q;a\right)$ linear in $a$, there exist closed-form expressions for this integral in terms of special functions, which are straightforward to derive, especially if one uses symbolic manipulation software such as Mathematica. If the shape parameter of the Pareto distribution is a negative integer, the integrals actually reduce to simple power functions.
If $M_{e}$ denotes the measure of firms that enter each period (in one country), then the measure of operating firms is $M=G\left(a_{c}\right)M_{e}/\delta_{e}$. The total labor used in the economy is given by $L_{E}=M_{e}f_{e}+Mf+M\bar{L}$, where $\bar{L}=G\left(a_{c}\right)^{-1}\intop_{q\geq q_{c}}L\left(q;a\right)dG\left(a\left(q\right)\right)$ is the labor on average hired for the variable cost of production. Under the same assumptions, the integral again has an explicit form in terms of special functions. We see that in these cases we can get fully explicit expressions for $f_{e}$ and $M$ in terms of chosen $q_{c}$ and $L_{E}$.
Other quantities of interest, such as trade flows or welfare, may be found in an analogous, straightforward fashion.
#### Two countries with non-iceberg and iceberg international trade costs.
Just like in the case of the flexible Krugman model, it is convenient to write the model in terms of equilibrium marginal cost, which this time is firm-specific and also depends on the firm’s chosen export status. For tractability we will need the revenue function $R\left(q\right)$ and the production labor requirement function $L\left(q;a\right)$ to allow for a linear term. The same is true for labor corresponding to the non-iceberg trade costs, here denoted by $L_{T}\left(q_{x}\right)$. As in the original @melitz paper, we consider equilibria characterized by two cutoffs, here denoted $a_{1}$ and $a_{2}$, such that least productive firms with $a>a_{1}$ exit, more productive firms with $a\in(a_{2},a_{1}]$ serve only their domestic market, and most productive firms with $a\leq a_{2}$ serve both countries. In general, we denote the equilibrium marginal cost of a non-exporting firm as $MC_{n}^{\star}$ and that of an exporting firm as $MC_{x}^{\star}$. Variables corresponding to the two cutoffs are distinguished by subscripts $1$ and $2$, so for example $MC_{1n}^{\star}$ is the optimal marginal cost of a firm with $a=a_{1}$, and $MC_{2x}^{\star}$ and $MC_{2n}^{\star}$ are optimal marginal costs of a firm with $a=a_{2}$ that decides to export or not to export, respectively. We denote by $M_{x}^{\star}$ and $X^{\star}$ the equilibrium measure of exporting firms and international trade flows.
Our solution strategy is to treat $MC_{1n}$ and $MC_{2x}$ as given and to express other variables of the model in terms to these two chosen parameters. In particular, we will show how to derive explicit expressions for the fixed cost of exporting $f_{x}$ and cost of entry $f$$_{e}$. The (variable-cost) labor requirement $L\left(q;a\right)$ is assumed to be a tractable combination of equidistant powers of $a$, with coefficients that in general depend on $q$. Firms’ profit maximization leads to the set of equations:
$$\begin{gathered}
MC_{n}=R'(q_{n})\label{eq:MelitzTwoCountriesA}\\
MC_{n}=wL'\left(q_{n};a\right)\label{eq:MelitzTwoCountriesB}\\
MC_{x}=R'(q_{d})\label{eq:MelitzTwoCountriesC}\\
MC_{x}=\begin{array}{c}
\frac{1}{\tau}\end{array}\!\! R'(q_{f})-\begin{array}{c}
\frac{1}{\tau}\end{array}\!\! wL_{T}'(q_{f})\label{eq:MelitzTwoCountriesD}\\
MC_{x}=wL'(q_{d}\!+\!\tau q_{f};a)\label{eq:MelitzTwoCountriesE}\\
R(q_{1n})-wL(q_{1n};a_{1})=f\label{eq:MelitzTwoCountriesF}\\
R(q_{2d})+R(q_{2f})-wL(q_{2d}+\tau q_{2f};a_{2})-wL_{T}(q_{2f})=f+f_{x}\label{eq:MelitzTwoCountriesG}\end{gathered}$$
Here $q_{n}$ is the quantity sold by a non-exporting firm, while $q_{d}$ and $q_{f}$ represent quantities that reach domestic and foreign customers of an exporting firm, respectively. In addition to exporting cost $wL_{T}(q_{f})$, we allow for an iceberg trade cost factor $\tau\ge1$.
For a chosen $MC_{1n}$, we can calculate $q_{1n}$ from (\[eq:MelitzTwoCountriesA\]). The corresponding $a_{1}$ may be found by solving a linear equation that results from combining (\[eq:MelitzTwoCountriesB\]) and (\[eq:MelitzTwoCountriesF\]) in a way that eliminates wages. Wages then may be recovered by substituting back to (\[eq:MelitzTwoCountriesB\]).
For a chosen $MC_{2x}$, we can derive $q_{2d}$ from (\[eq:MelitzTwoCountriesC\]) and $q_{2f}$ from (\[eq:MelitzTwoCountriesD\]). The value of $a_{2}$ is then determined by (\[eq:MelitzTwoCountriesE\]). We find $q_{2n}$ by solving (\[eq:MelitzTwoCountriesA\]) and (\[eq:MelitzTwoCountriesB\]) with $MC_{2n}$ eliminated, and then in turn use one of these to find $MC_{2n}$. This means that we know the marginal cost at the cutoffs. The fixed cost of exporting $f_{e}$ is then identified from (\[eq:MelitzTwoCountriesG\]).
For a given marginal cost, we can find the corresponding quantities and productivity parameters $a$ by a similar method from (\[eq:MelitzTwoCountriesA\]-\[eq:MelitzTwoCountriesE\]), this time treating $w$ as known. We denote the resulting functions $q_{n}\left(MC_{n}\right)$, $q_{d}\left(MC_{x}\right)$, $q_{x}\left(MC_{x}\right)$, $a_{n}\left(MC_{n}\right)$, and $a_{x}\left(MC_{x}\right)$. Using these functions we can now determine the entry labor requirement $f_{e}$ from the unrestricted entry condition: $$w\delta_{e}f_{e}=\bar{\Pi}=\intop_{y\in S_{n}}\Pi\left(q_{n}\left(y\right);a_{n}\left(y\right)\right)dG\left(a_{n}\left(y\right)\right)+\intop_{y\in S_{x}}\Pi\left(q_{x}\left(y\right);a_{x}\left(y\right)\right)dG\left(a_{x}\left(y\right)\right),$$ where $\Pi$ is the profit function (revenue minus cost), $G\left(a\right)$ is the cumulative distribution function of $a$, and the integration ranges are $S_{n}\equiv\left(MC_{2n},MC_{1n}\right)$ and $S_{x}\equiv\left(0,MC_{1n}\right)$. Under various assumptions these integrals may be evaluated in closed form, often involving special functions. If a measure $M_{e}$ of firms enters each period (in one of the countries), then the equilibrium measure of operating firms is $M=M_{e}G\left(a_{1}\right)/\delta_{e}$ and that of exporting firms is $M_{x}=M_{e}G\left(a_{2}\right)/\delta_{e}$. These measures may be calculated from the labor market clearing condition $M_{e}f_{e}+Mf+M_{x}f_{x}+(M-M_{x})\bar{L}_{n}+M_{x}\bar{L}_{x}=L_{E},$ where
$$\bar{L}_{n}\equiv\!\!\begin{array}{c}
\frac{1}{G\left(a_{1}\right)-G\left(a_{2}\right)}\end{array}\!\!\!\intop_{y\in S_{n}}\!\! L\left(q_{n}\left(y\right);a_{n}\left(y\right)\right)dG\left(a_{n}\left(y\right)\right)\!,\,\bar{L}_{x}\equiv\!\!\begin{array}{c}
\frac{1}{G\left(a_{2}\right)}\end{array}\!\!\!\intop_{y\in S_{x}}\!\! L\left(q_{x}\left(y\right);a_{x}\left(y\right)\right)dG\left(a_{x}\left(y\right)\right)\!.$$ Under the same assumptions as before, these integrals may be evaluated in closed form. Again, other variables of interest, such as trade flows or welfare, may be obtained in a similar way.
### Flexible @melitz/Melitz-Ottaviano model with non-separable utility
While a significant part of the international trade literature relies on separable utility functions, there exist realistic economic phenomena what are more easily modeled with non-separable utility. An instantly classic alternative to the @melitz model that uses non-separable utility is the model of @melitzottaviano, which assumes that with a greater selection of heterogeneous-good varieties available to consumers, the marginal gain from an additional variety decreases relative to the gains from increased quantity. Trade liberalization leads to tougher competition, which results not only in higher productivity but also in the decrease of markups charged by a given firm.
Here we briefly discuss a generalization of the flexible Melitz model where the utility function is allowed to be non-separable. This generalized model contains as special cases both the @melitz model and the @melitzottaviano model.[^112] The utility is of the form $$U_{\Omega}\equiv F\left(U_{\Omega}^{\left(1\right)},U_{\Omega}^{\left(2\right)},...,U_{\Omega}^{\left(m\right)}\right),\quad U_{\Omega}^{\left(i\right)}\equiv\int_{\Omega}U^{\left(i,\omega\right)}\left(q_{\omega}\right)\, d\omega.$$ In order to preserve tractability, we assume that $U^{\left(i,\omega\right)}\left(q_{\omega}\right)$ are linear combinations[^113] of equally-spaced powers of $q_{\omega}$ and that the set of exponents does not depend on $i$ or $\omega$. For example, we could specify $U_{\Omega}\equiv U_{\Omega}^{\left(1\right)}+\kappa_{1}(U_{\Omega}^{\left(1\right)})^{\xi_{1}}+\kappa_{2}(U_{\Omega}^{\left(2\right)})^{\xi_{2}}$, $U_{\Omega}^{\left(1\right)}\equiv\int_{\Omega}q_{\omega}^{\gamma_{1}}\, d\omega$, and $U_{\Omega}^{\left(2\right)}\equiv\int_{\Omega}q_{\omega}^{\gamma_{2}}\, d\omega$, with $(\gamma_{1}+1)/(\gamma_{2}+1)$ equal to the ratio of two small integers. The choice $\kappa_{1}=\kappa_{2}=0$ corresponds to the @melitz model, while the choice $\xi_{1}=2$, $\xi_{2}=1$, $\gamma_{1}=1$, and $\gamma_{2}=2$ gives the @melitzottaviano model, which is based on a non-homothetic quadratic utility. Our general specification allows also for homothetic non-separable utility functions that feature market toughness effects analogous to those in the @melitzottaviano model.
It is straightforward to verify that just like the flexible @melitz model with separable utility, this more general version leads to tractable optimization by individual firms, as well as for tractable aggregation under the same conditions. The reason for the tractability of the firm’s problem is simple: the firm’s first-order condition will have the same structure as previously, a linear combination of equidistant powers (with an additional dependence of the coefficients of the linear combination on *aggregate* variables of the type $\int_{\Omega}q_{\omega}^{\gamma}\, d\omega$ for some constants $\gamma$). Given that the nature of the firm’s problem is unchanged, it follows that being able to explicitly aggregate over heterogeneous firms does not require any additional functional form assumptions relative to the separable utility case.
Demand Forms {#forms}
============
Curvature properties {#curvature}
--------------------
Table \[taxonomy\] provides a taxonomy of the curvature properties of demand functions generated by common statistical distributions and the single-product version of the Almost Ideal Demand System. Following @socialnalebuff [@imperfectnalebuff], we define the curvature of demand as $$\kappa(p)\equiv\frac{Q''(p)Q(p)}{\left[Q'(p)\right]^{2}}.$$ @cournot showed that the pass-through rate of a constant marginal cost monopolist is $$\frac{1}{2-\kappa}$$ and thus that a) that the comparison of $\kappa$ to unity determines the comparison of pass-through to unity in this case and b) that if $\kappa'(p)>0$ that pass-through rises with price (falls with quantity), and conversely if $\kappa$ declines with price (rises with quantity). The comparison of $\kappa$ to unity also determines whether a demand is log-convex and its sign whether demand is convex. The comparison of $\kappa$ to $2$ determines whether demand has declining marginal revenue, a condition also known as @myerson’s regularity condition.
For probability distribution $F$, the corresponding demand function $Q(p)=s\left(1-F\left(\frac{p-\mu}{m}\right)\right)$ where $s$ and $m$ are stretch parameters [@wt] and $\mu$ is a position parameter. Note that in this case $$\kappa(p)=-\frac{\frac{s^{2}}{m^{2}}F''\left(\frac{p-\mu}{m}\right)\left(1-F\left(\frac{p-\mu}{m}\right)\right)}{\frac{s^{2}}{m^{2}}\left[F'\left(\frac{p-\mu}{m}\right)\right]^{2}}=-\frac{F''\left(\frac{p-\mu}{m}\right)\left(1-F\left(\frac{p-\mu}{m}\right)\right)}{\left[F'\left(\frac{p-\mu}{m}\right)\right]^{2}}.$$ Note, thus, that neither global level nor slope properties of $\kappa$ are affected by $s,m$ or $\mu$. We can thus analyze the properties of relevant distributions independently of their values, as represented in the table and the following proposition.
The most prominent conclusion emerging from this taxonomy is that the vast majority of forms used in practice in computational, statistical models such as @blp have monotonically increasing curvature and most have curvature below unity. This suggests two conclusions. The first, highlighted in the paper, is that, to the extent we believe these forms are more realistic than tractable forms, they have properties systematically differing from the BP class and thus it is important to derive tractable forms capable of matching their central property of monotonically increasing in price/decreasing in quantity curvature.
A second possible conclusion is that, to the extent that in some cases these properties are [*not*]{} empirically relevant, such as in the data of @levinisgod, standard forms rule out observed behavior and thus analysts may wish to consider more flexible forms along these dimensions, such as those we derive in the paper. To the extent there are not strong theoretical reasons to believe in the restrictions imposed by standard statistically based forms (which, in many cases, there are) allowing such relaxation is important because in many contexts the properties of firm demand and equilibrium are inherited directly from the demand function, at least with constant marginal cost [@weylfabinger; @gabaixetal; @quint]. Which conclusion is most appropriate will obviously depend on the empirical context and the views of the analyst.
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: A taxonomy of some common demand functions[]{data-label="taxonomy"}
\[categorization\] Table \[taxonomy\] summarizes global properties of the listed statistical distributions generating demand functions. $\alpha$ is the standard shape parameter in distributions that call for it.
Characterization of the curvature level (comparisons of $\kappa$ to unity) follow from classic classifications of distributions as log-concave or log-convex as in @logconcave, except in the case of AIDS in which the results are novel.[^114] Note that our discussion of stretch parameters in the paper implies we can ignore the scale parameter of distributions, normalizing this to $1$ for any distributions which has one. A similar argument applies to position parameter: because this only shifts the values where properties apply by a constant, it cannot affect global curvature or higher-order properties. This is useful because many of the probability distributions we consider below have scale and position parameters that this fact allows us to neglect. We will denote this normalization by [*Up to Scale and Position*]{} (USP).
We begin by considering the first part of the proof, that for any shape parameter $\alpha<1$ the Fréchet, Weibull and Gamma distributions with shape $\alpha$ violate DMR at some price. We show this for each distribution in turn:
1. Type II Extreme Value (Fréchet) distribution: USP, this distribution is $F(x)=e^{-x^{-\alpha}}$ with domain $x>0$. Simple algebra shows that $$\kappa(x)=\frac{(e^{x^{-\alpha}}-1)x^{\alpha}(1+\alpha)+\left(1-e^{x^{-\alpha}}\right)\alpha}{\alpha}.$$ As $x\rightarrow\infty$ and therefore $x^{-\alpha}\rightarrow0$ (as shape is always positive), $e^{x^{-\alpha}}$ is well-approximated by its first-order approximation about $0$, $1+x^{-\alpha}$. Therefore the limit of the above expression is the same as that of $$\frac{x^{-\alpha}x^{\alpha}(1+\alpha)-x^{-\alpha}\alpha}{\alpha}=\frac{1+\alpha+x^{-\alpha}\alpha}{\alpha}\rightarrow\frac{1}{\alpha}+1$$ as $x\rightarrow\infty$. Clearly, this is greater than $2$ for $0<\alpha<1$ so that for sufficiently large $x$, $\kappa>2$.
2. Weibull distribution: USP, this distribution is $F(x)=1-e^{-x^{\alpha}}$. Again algebra yields: $$\kappa(x)=\frac{1-\alpha}{\alpha x^{\alpha}}+1.$$ Clearly, for any $\alpha<1$ as $x\rightarrow0$ this expression goes to infinity, so that for sufficiently small $x$, $\kappa>2$.
3. Gamma distribution: USP, this distribution is $F(x)=\frac{\gamma(\alpha,x)}{\Gamma(\alpha)}$ where $\gamma(\cdot,\cdot)$ is the lower incomplete Gamma function, $\Gamma(\cdot,\cdot)$ is the upper incomplete Gamma function and $\Gamma(\cdot)$ is the (complete) Gamma function: $$\kappa(x)=\frac{e^{x}(1-\alpha+x)\Gamma(\alpha,x)}{x^{\alpha}}.\label{gammak}$$ By definition, $\lim_{x\rightarrow0}\Gamma(\alpha,x)=\Gamma(\alpha)>0$ so $$\lim_{x\rightarrow0}\kappa(x)=+\infty$$ as $1-\alpha>0$ for $\alpha<1$. Thus clearly for small enough $x$, the Gamma distribution with shape $\alpha<1$ has $\kappa>2$.
We now turn to the categorization of demand functions as having increasing or decreasing pass-through. As price always increases in cost, this can be viewed as either pass-through as a function of price or pass-through as a function of cost.
1. Normal (Gaussian) distribution: USP, this distribution is given by $F(x)=\Phi(x)$, where $\Phi$ is the cumulative normal distribution function; we let $\phi$ denote the corresponding density. It is well-known that $\Phi''(x)=-x\phi(x)$. Thus $$\kappa(x)=\frac{x\left[1-\Phi(x)\right]}{\phi(x)}.$$ Taking the derivative and simplifying yields $$\kappa'(x)=\frac{\left[1-\Phi(x)\right]\left(1+x^{2}\right)-x\phi(x)}{\phi(x)},$$ which clearly has the same sign as its numerator, as $\phi$ is a density and thus everywhere positive. But a classic strict lower bound for $\Phi(x)$ is $\frac{x}{1+x^{2}}\phi(x)$, implying $\kappa'>0$.
2. Logistic distribution: USP, this distribution is $F(x)=\frac{e^{x}}{1+e^{x}}$. Again algebra yields $$\kappa'(x)=e^{-x}>0.$$ Thus the logistic distribution has $\kappa'>0$.
3. Type I Extreme Value (Gumbel) distribution: USP, this distribution has two forms. For the minimum version it is $F(x)=1-e^{-e^{x}}$. Algebra shows that for this distribution $$\kappa'(x)=e^{-x}.$$ Note that this is the same as for the logistic distribution; in fact $\kappa$ for the Gumbel minimum distribution is identical to the logistic distribution. This is not surprising given the close connection between these distributions [@mcfadden].
For the maximum version it is $F(x)=e^{-e^{-x}}$. Again algebra yields $$\kappa'(x)=e^{-x}\big(e^{2x}[e^{e^{-x}}-1]-e^{e^{-x}}[e^{x}-1]\big).$$ For $x<0$ this is clearly positive as both terms are strictly positive: $1>e^{x}$ and because $e^{-x}>0$, $e^{e^{-x}}>1$. For $x>0$ we can rewrite $\kappa'$ as $$e^{e^{-x}}\left(e^{x}-1\right)+e^{-x}\left(e^{e^{-x}}-1\right),$$ which again is positive as $e^{x}>1$ for $x>0$ and $e^{e^{-x}}>1$ by our argument above.
4. Laplace distribution: USP, this distribution is $$F(x)=\left\{ \begin{array}{cc}
1-\frac{e^{-x}}{2} & x\geq0,\\
\frac{e^{x}}{2} & x<0.
\end{array}\right.$$ For $x>0$, $\rho=1$ (so in this range pass-through is not strictly increasing). For $x<0$ $$\kappa'(x)=2e^{-x}>0.$$ So the Laplace distribution exhibits globally weakly increasing pass-through, strictly increasing for prices below the mode. The curvature for this distribution is $1-2e^{-x}$ as opposed to $1-e^{-x}$ for Gumbel and Logistic. However, these are very similar, again pointing out the similarities among curvature properties of common demand forms.
5. Type II Extreme Value (Fréchet) distribution with shape $\alpha>1$: From the formula above it is easy to show that the derivative of the pass-through rate is $$\kappa'(x)=x^{-(1+\alpha)}\Big([1+\alpha]\big[x^{2\alpha}(e^{x^{-\alpha}}-1)-e^{x^{-\alpha}}x^{\alpha}\big]+\alpha e^{x^{-\alpha}}\Big)>0,$$ which can easily be shown to be positive as follows. Let us multiply the inequality by the positive factor $\frac{e^{-x^{-\alpha}}}{\alpha+1}$. Denoting $X\equiv x^{-\alpha}$, the inequality becomes $$\left(\frac{\alpha}{\alpha+1}-\frac{1}{2}\right)+\left(\frac{1}{X^{2}}-\frac{e^{-X}}{X^{2}}-\frac{1}{X}+\frac{1}{2}\right)>0.$$ The first term is positive because $\alpha>1$. The second term is positive because $e^{-X}<1-X+\frac{1}{2}X^{2}$ for any $X>0$. Thus this distribution, as well, has $\kappa'>0$.
6. Type III Extreme Value (Reverse Weibull) distribution: USP, this distribution is $F(x)=e^{-(-x)^{\alpha}}$ and has support $x<0$. Algebra shows $$\kappa'(x)=(-x)^{\alpha-1}\alpha^{2}\left[1-\alpha+e^{(-x)^{\alpha}}\Big([1-\alpha]\big[(-x)^{\alpha}-1\big]+[-x]^{2\alpha}\alpha\Big)\right],$$ which has the same sign as $$1-\alpha+e^{(-x)^{\alpha}}\Big([1-\alpha]\big[(-x)^{\alpha}-1\big]+[-x]^{2\alpha}\alpha\Big).\label{type3firstderiv}$$ Note that the limit of this expression as $x\rightarrow0$ is $$1-\alpha-(1-\alpha)=0$$ and its derivative is $$\frac{e^{(-x)^{\alpha}}(-x)^{2\alpha}\alpha\big(1+\alpha+[-x]^{\alpha}\alpha\big)}{x},$$ which is clearly strictly negative for $x<0$. Thus Expression \[type3firstderiv\] is strictly decreasing and approaches $0$ as $x$ approaches $0$. It is therefore positive for all negative $x$, showing that again in this case $\kappa'>0$.
7. Weibull distribution with shape $\alpha>1$: As with the Fréchet distribution algebra from the earlier formula shows $$\kappa'(x)=x^{\alpha-1}(\alpha-1)\alpha^{2},$$ which is clearly positive for $\alpha>1$ as the range of this distribution is positive $x$. Thus the Weibull distribution with $\alpha>1$ has $\kappa'>0$.
8. Gamma distribution with shape $\alpha>1$: Taking the derivative of Expression \[gammak\] yields: $$\kappa'(x)=\frac{\alpha-1-x+\frac{e^{x}}{x^{\alpha}}\big(x^{2}-2x[\alpha-1]+[\alpha-1]\alpha\big)\Gamma(\alpha,x)}{x},$$ which has the same sign as $$\alpha-1-x+\frac{e^{x}}{x^{\alpha}}\big(x^{2}-2x[\alpha-1]+[\alpha-1]\alpha\big)\Gamma(\alpha,x),\label{gammaexpression}$$ given that $x>0$. Note that as long as $\alpha>1$ $$x^{2}+(\alpha-2x)(\alpha-1)=x^{2}-2(\alpha-1)x+\alpha(\alpha-1)>x^{2}-2(\alpha-1)x+(\alpha-1)^{2}=\left(x+1-\alpha\right)^{2}>0.$$ Therefore so long as $x\leq\alpha-1$ this is clearly positive. On the other hand when $x>\alpha-1$ the proof depends on the following result of @gamma:
Let a be a positive parameter, and let q(x) be a function, differentiable on $(0,\infty)$ , such that $lim_{x\rightarrow\infty}x^{\alpha}e^{-x}q(x,\alpha)=0$. Let
$$T(x,\alpha)=1+(\alpha-x)q(x,\alpha)+x\frac{\partial q}{\partial x}(x,\alpha).$$
If $T(x,\alpha)>0$ for all $x>0$ then $\Gamma(\alpha,x)>x^{\alpha}e^{-x}q(x,\alpha)$.
Letting
$$q(x,\alpha)\equiv\frac{x-(\alpha-1)}{x^{2}+(\alpha-2x)(\alpha-1)},$$
$$T(x,\alpha)=\frac{2(\alpha-1)x}{\big(\alpha^{2}+x[2+x]-\alpha[1+2x]\big)^{2}}>0$$
for $\alpha>1,x>0$. So $\Gamma(\alpha,x)>x^{\alpha}e^{-x}q(x,\alpha)$. Thus Expression \[gammaexpression\] is strictly greater than
$$\alpha-1-x+x-(\alpha-1)=0$$
as, again, $x^{2}+(\alpha-2x)(\alpha-1)>0$. Thus again $\kappa'>0$.
This establishes the second part of the proposition. Turning to our final two claims, algebra shows that the curvature for the Fréchet distribution is
$$\kappa(x)=\frac{\alpha-e^{x^{-\alpha}}\big(\alpha-x^{\alpha}[1+\alpha]\big)-x^{\alpha}(1+\alpha)}{\alpha}=\frac{\left(1-e^{x^{-\alpha}}\right)\left[\alpha-x^{\alpha}\left(1+\alpha\right)\right]}{\alpha}.$$
Note for any $\alpha>1$ this is clearly continuous in $x>0$. Now consider the first version of the expression. Clearly, as $x\rightarrow0$, $x^{\alpha}\rightarrow0$ and $e^{x^{-\alpha}}\rightarrow\infty$ so the expression goes to $-\infty$. So for sufficiently small $x>0$, $\kappa(x)<1$. On the other hand, consider the second version of the expression. Its numerator is
$$\left(1-e^{x^{-\alpha}}\right)\left[\alpha-x^{\alpha}\left(1+\alpha\right)\right].$$ By the same argument as above with the Fréchet distribution the limit of the above expression as $x\rightarrow\infty$ is the same as that of $$\left(-x^{-\alpha}\right)\left(-x^{\alpha}\left(1+\alpha\right)\right)$$ as $x\rightarrow\infty$. Thus $$\lim_{x\rightarrow\infty}\kappa(x)=\frac{1+\alpha}{\alpha}>1$$ and thus for sufficiently large $x$ and any $\alpha>1$, this distribution has $\kappa>1$.
Finally, consider our claim about AIDS. First note that for this demand function
$$\kappa(p)=2+\frac{b\big(a-2b+b\log p\big)}{\big(a-b+b\log p\big)^{2}}<1$$
as $b<0$ and $p\leq e^{-\frac{a}{b}}<e^{2-\frac{a}{b}}$. This is less than $1$ if and only if $$a^{2}+2ab\big(\log p-2\big)+b^{2}\Big(1+\big[\log(p)-2\big]\log p\Big)<b^{2}\big(2-\log p\big)-ab$$ or $$\big(a+b\log p\big)^{2}-b^{2}\big(\log p+1\big)<0.$$ Clearly, as $p\rightarrow0$ the second term is positive; therefore there is always a price at which $\kappa(p)>1$. On the other hand as $p\rightarrow e^{-\frac{a}{b}}$ this expression goes to $$0-b^{2}\bigg(1-\frac{a}{b}\bigg)=b(a-b)<0.$$ Thus there is always a price at which $\kappa(p)<1$. $$\kappa'(p)=b^{2}-\big(a-2b+b\log p\big)^{2},$$ which has the same sign as $$b^{2}-\big(a-2b+b\log p\big)^{2}<b^{2}-(2b)^{2}=-3b^{2}<0.$$ Thus $\kappa'<0$.
We now turn to two important distributions, which are typically used to model the income distribution, whose behavior is more complex and which, to our knowledge, have not been analyzed for their curvature properties. We focus only on the two that we believe to be most common (the first), best theoretically founded (both) and to provide the most accurate match to the income distribution (the second). Namely, we analyze the lognormal and double Pareto-lognormal (dPln) distributions, the latter of which was proposed by @reed and @reedjorgensen. Other common, accurate models of income distributions which we have analyzed in less detail, appear to behave in a similar fashion.
We begin with the lognormal distribution, which is much more commonly used, and for which we have detailed, analytic results. However, while most of the arguments for the below proposition are proven analytically, some simple points are made by computational inspection.
For every value $\sigma$, there exist finite thresholds $\overline{y}(\sigma)>\underline{y}(\sigma)$ such that
1. If $y\geq\overline{y}(\sigma)$ then $\kappa'\leq0$, and similarly with strict inequalities or if the directions of the inequalities both reverse.
2. If $y\geq\underline{y}(\sigma)$ then $\kappa\geq1$, and similarly with strict inequalities or if the directions of the inequalities both reverse.
Both $\overline{y}$ and $\underline{y}$ are strictly decreasing in $\sigma$.
![Curvature of a lognormal distribution calibrated to the US income distribution: parameters are $\mu=10.5$ and $\sigma=0.85$.[]{data-label="lnincome"}](FigureLogNormalCurvature.pdf){width="5in"}
Under the lognormal distribution, the behavior depends critically on the amount of inequality or equivalently the standard deviation of the logarithm of the distribution: there is famously a one-to-one relationship between the Gini coefficient associated with a lognormal distribution and its logarithmic standard deviation. If inequality is not high, the behavior of curvature like a normal distribution occurs except at fairly high incomes levels; for a Gini of $.34$, for example, monotonicity of $\kappa$ is preserved until the top 1% of the income distribution and log-concavity outside of the top 30%. However, if inequality is sufficiently high, in particular if the Gini coefficient is above about $.72$, then the lognormal distribution has $\kappa>2$ over some range and then $\kappa$ converges back to $1$ for very large incomes. This result is not discussed in the proposition but can easily be seen by inspecting a graph of the expression for $\kappa$ given in the proof of the proposition for various values of $\sigma$ yielding Gini coefficients of various magnitudes around $.72$.
For intermediate levels of inequality between these, like that seen in nearly every country, the lognormal distribution has curvature that rises from $-\infty$ to above unity before gradually returning towards unity. For an example calibrated to the US income distribution (Figure \[lnincome\]), the crossing to above unity occurs at an income of about \$33k, between the mode and the median and the downward slope begins at about \$100k. Despite this, curvature never falls below unity again and in fact is at each quantile increasing in $\sigma$ (again, not discussed in the proposition). Again taking the example of the US income-calibrated distribution, curvature peaks at about $1.21$ and only falls to $1.20$ by \$200k, eventually leveling out to about $1.1$ for the extremely wealthy.[^115] Thus, in practice, curvature is closer to flat at the top than significantly declining.
For a lognormal distribution with parameters $(\mu,\sigma)$, $F(x)=\Phi\left(\frac{\log(x)-\mu}{\sigma}\right)$, so that $$Q(p)=1-\Phi\left(\frac{\log(p)-\mu}{\sigma}\right),Q'(p)=-\frac{\phi\left(\frac{\log(p)-\mu}{\sigma}\right)}{\sigma p}$$ and $$Q''(p)=-\frac{\phi'\left(\frac{\log(p)-\mu}{\sigma}\right)}{\sigma^{2}p^{2}}+\frac{\phi\left(\frac{\log(p)-\mu}{\sigma}\right)}{\sigma p^{2}}=-\frac{\phi\left(\frac{\log(x)-\mu}{\sigma}\right)}{\sigma^{2}p^{2}}\left(\sigma+\frac{\log(x)-\mu}{\sigma}\right).$$ where the second equality follows from the identities regarding the normal distribution from the previous proof and $y\equiv\frac{\log(p)-\mu}{\sigma}$. Thus $$\kappa\left(p(y)\right)=\frac{\left(y+\sigma\right)\left[1-\Phi(y)\right]}{\phi(y)}.\label{lognormalcurve}$$ Note that we immediately see, as discussed above, that $\kappa$ increases in $\sigma$ at each quantile as the inverse hazard rate $\frac{1-\Phi}{\phi}>0$; similarly, for any quantile associated with $y$, $\kappa\rightarrow\infty$ as $\sigma\rightarrow\infty$ so it must be that the set of $y$ for which $\kappa>1$ a) exists for sufficiently large $\sigma$ and b) expands monotonically in $\sigma$. This implies that, if point 2) of the proposition is true, $\underline{y}$ must strictly decrease in $\sigma$. This also implies that for sufficiently large $\sigma$, $\kappa>2$ for some $y$.
Now note that $\lim_{y\rightarrow\infty}\frac{y\left[1-\Phi(y)\right]}{\phi(y)}=1$. To see this, note that both the numerator and denominator converge to $0$ as $1-\Phi$ dies super-exponentially in $y$. Applying l’Hospital’s rule: $$\lim_{y\rightarrow\infty}\frac{y\left[1-\Phi(y)\right]}{\phi(y)}=\lim_{y\rightarrow\infty}\frac{1-\Phi(y)-\phi(y)y}{\phi'(y)}=\frac{y\phi(y)-\left[1-\Phi(y)\right]}{y\phi(y)}=\frac{0}{0}.$$ where the first equality follows from the identity for $\phi'$ we have repeatedly been using, and from here on we no longer note the use of. Again applying l’Hospital’s rule: $$\lim_{y\rightarrow\infty}\frac{y\left[1-\Phi(y)\right]}{\phi(y)}=\lim_{y\rightarrow\infty}\frac{\phi(y)+y\phi'(y)+\phi(y)}{\phi(y)+y\phi'(y)}=\lim_{y\rightarrow\infty}\frac{2\phi(y)-y^{2}\phi(y)}{\phi(y)-y^{2}\phi(y)}=\lim_{y\rightarrow\infty}\frac{2-y^{2}}{1-y^{2}}=1.$$ The same argument, but one step less deep, shows that $\lim_{y\rightarrow\infty}\frac{\sigma\left[1-\Phi(y)\right]}{\phi(y)}=0$. Together these imply that $\lim_{y\rightarrow\infty}\kappa\left(p(y)\right)=1$ and thus that, if $\kappa>1$ at some point, it must eventually decrease to reach $1$.
Similar methods may be used to show, as discussed in the paper, that $\kappa\rightarrow-\infty$ as $y\rightarrow-\infty$. Furthermore, we know from the proof for the normal distribution above that $\frac{y\left[1-\Phi(y)\right]}{\phi(y)}$ is monotone increasing and that $\frac{\sigma\left[1-\Phi(y)\right]}{\phi(y)}$ is monotone decreasing. The latter point implies that the set of $y$ for which $\kappa$ is decreasing must be strictly increasing in $\sigma$ and thus that, if point 1) of the proposition is true, then $\overline{y}$ must strictly decrease in $\sigma$.
All that remains to be shown is that $\kappa$’s comparison to unity and the sign of $\kappa'$ obey the threshold structure posited. Note that we only need to show the cut-off structure for $\kappa'$ and that this immediately implies the structure for $\kappa$, given the smoothness of all functions involved, because if $\kappa$ increases up to some threshold and then decreases monotonically while reaching an asymptote of unity, it must lie above unity above some threshold. Otherwise, if it ever crossed below unity, it would have to be increasing in some region to asymptote to unity at very large $p$, violating the threshold structure for $\kappa'$. Furthermore, the same logic implies that the region where $\kappa>1$ must be strictly larger than the region where $\kappa'<0$ (that $\overline{y}>\underline{y}$) as $\kappa$ must rise strictly above unity before sloping strictly down towards it.
![The figure shows the value, in logarithmic scale, of the left-hand side of Inequality \[lognormalcalculations\].[]{data-label="prooffigure"}](FigureLogNormalCalculations.pdf){width="3in"}
We drop arguments wherever possible in what follows to ease readability. We use the symbol $\propto$ to denote expressions having the same sign, not proportionality as is typical. $$\kappa'=\frac{\left(1-\Phi\right)\phi-(y+\sigma)\phi^{2}-(y+\sigma)(1-\Phi)\phi'}{\phi^{2}}=\frac{1-\Phi-(y+\sigma)\left[\phi-y(1-\Phi)\right]}{\phi}\propto$$ $$1-\Phi-(y+\sigma)\left[\phi-y(1-\Phi)\right]\propto\frac{1-\Phi}{\phi-y(1-\Phi)}-y-\sigma.$$ where the last sign relationship follows by the common inequality that $\phi(y)>y\left[1-\Phi(y)\right]$. Thus $\kappa'>0$ if and only if $$\frac{1-\Phi}{\phi-y(1-\Phi)}-y>\sigma.\label{lognormalcalculations}$$ Figure \[prooffigure\] shows that the left-hand side of this inequality is strictly decreasing. We have not found a simple means to prove this formally, but it is clearly true by inspection of the figure. Thus the left-hand side of Inequality \[lognormalcalculations\] must cross $\sigma$ at most once, and this must be from above to below.
It only remains to show that this expression does, in fact, make such as single crossing for all values of $\sigma$. It suffices to show that the small $y$ limit of the left-hand side of inequality \[lognormalcalculations\] is $\infty$ and that its large $y$ limit is $0$. We show these in turn.
The first claim is easy: clearly $-y\left(1-\Phi\right)\rightarrow\infty$, while $1-\Phi$ is finite, as $y\rightarrow-\infty$. Thus the first term approaches $0$ and the second $\infty$ as $y\rightarrow-\infty$.
The second claim is more delicate. The expression is the same as $$\frac{\left(1-\Phi\right)\left(1+y^{2}\right)-y\phi}{\phi-y\left(1-\Phi\right)}.$$ This asymptotes to the indefinite expression $\frac{0}{0}$ as $y\rightarrow\infty$ as it is well-known that $\lim_{y\rightarrow\infty}\frac{\phi}{y\left(1-\Phi\right)}=1$. Applying l’Hospital’s rule yields $$\lim_{y\rightarrow\infty}\frac{\left(1-\Phi\right)\left(1+y^{2}\right)-y\phi}{\phi-y\left(1-\Phi\right)}=\lim_{y\rightarrow\infty}\frac{-\phi\left(1+y^{2}\right)+2y\left(1-\Phi\right)-\phi-y\phi'}{\phi'-\left(1-\Phi\right)+y\phi}=$$ (applying now-familiar tricks) $$\lim_{y\rightarrow\infty}2\frac{\phi-y(1-\Phi)}{1-\Phi}=\frac{0}{0}.$$ Again, we apply l’Hospital’s rule: $$\lim_{y\rightarrow\infty}2\frac{\phi-y(1-\Phi)}{1-\Phi}=\lim_{y\rightarrow\infty}2\frac{\phi'-(1-\Phi)+y\phi}{\phi}=\lim_{y\rightarrow\infty}-\frac{1-\Phi}{\phi}=0.$$
Even the slight decline in the lognormal distribution’s curvature at very high incomes is an artifact of its poor fit to incomes distributions at very high incomes. It is well-known that at very high incomes the lognormal distribution fits poorly; much better fit is achieved by distributions with fatter (Pareto) tails, especially in countries with high top-income shares like the contemporary United States [@topincomes]. A much better fit is achieved by the dPln distribution [@reed]. Figure \[income\]’s left panel shows curvature as a function of income for the parameters @reed estimates (for the 1997 US income distribution). Curvature monotonically increases up the income distribution.
![Curvature of the double-Pareto lognormal distribution lognormal under parameters estimated by [@reed] (left) and by updated by us (right); parameters in the former case are $\alpha=22.43,\beta=1.43,\mu=10.9,\sigma=0.45$ ad in the latter case are $\alpha=3,\beta=1.43,\mu=10.9,\sigma=0.5$. The x-axis has a logarithmic scale in income.[]{data-label="income"}](FigureLNDPCurvature.pdf "fig:"){width="3in"} ![Curvature of the double-Pareto lognormal distribution lognormal under parameters estimated by [@reed] (left) and by updated by us (right); parameters in the former case are $\alpha=22.43,\beta=1.43,\mu=10.9,\sigma=0.45$ ad in the latter case are $\alpha=3,\beta=1.43,\mu=10.9,\sigma=0.5$. The x-axis has a logarithmic scale in income.[]{data-label="income"}](FigureDPLNCurvatureGFit.pdf "fig:"){width="3in"}
However, it levels off at quite moderate income (it is essentially flat beyond \$100k) and at a lower level ($\approx1.04$) than under the lognormal calibration, except at exorbitant incomes, where the lognormal distribution has thin tails. Thus it actually has a [*thinner*]{} tail, except at the very extreme tail, than the lognormal calibration, paradoxically. This is because @reed calibrated only to the mid-section of the US income distribution, given that the survey he used is notoriously thin and inaccurate at higher incomes; this led him to estimate a very high (thin-tailed) Pareto coefficient in the upper tail of $22.43$. Consensus economic estimates, for example @progressive, suggest that 1.5-3 is the correct range for the Pareto coefficient of the upper tail of the income distribution in the 2000’s.
We therefore construct our own calibration consistent with that finding. To be conservative we set the upper tail Pareto coefficient to $3$, maintain $\beta=1.43$ to be consistent with @reed and because the lower-tail is both well-measured in his data and has not changed dramatically in the last decade and a half [@striking]. We then adjust $\mu$ and $\sigma$ in the unique way, given these coefficients, to match the latest US post-tax Gini estimates ($.42$), using a formula derived by @dPlnglobal, and average income (\$53k). This yields the plot in the right panel of Figure \[income\]. There curvature continues to monotonically increase at a significant rate up to quite high incomes: at \$50k it is $.87$, at \$100k it is $1.19$ and by \$200k it has leveled off at $1.31$, near its asymptotic value of $1+\frac{1}{\alpha}=\frac{4}{3}$. It is this last calibration that we use to represent the dPln calibration US income distribution in the paper.
Moreover, the monotone increasing nature of curvature is not only true in the US data. While we have not been able to prove any general results about this four-parameter class, we have calculated similar plots to Figure \[income\] for every country for which a dPln income distribution has been estimated, as collected by @dPlnglobal. In every case curvature is monotone increasing in income, though in some cases it levels off at a quite low level of income (typically when the Gini is high relative to the upper tail estimate). Even this leveling off seems to us likely to be a bit of an artifact, arising from the lack of reliable top incomes tax data in many of the developing countries on which @dPlnglobal focus. In any case, it appears that a “stylized fact” is that a reasonable model of most country’s income distributions has curvature that is significantly below unity among the poor, rises above unity for the rich and monotone increasing over the full range so long as top income inequality is significant relative to overall inequality.
[^1]: Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan and CERGE-EI, Prague, the Czech Republic: fabinger@e.u-tokyo.ac.jp.
[^2]: Microsoft Research New York City, 641 Avenue of the Americas, New York, 10011 USA and Department of Economics, Princeton University: glenweyl@microsoft.com.
[^3]: This paper replaces its previous working versions “Pass-Through and Demand Forms”/“A Tractable Approach to Pass-Through Patterns”/“The Average-Marginal Relationship and Tractable Equilibrium Forms”. We are grateful to many colleagues and seminar participants for helpful comments. We appreciate the research assistance of Konstantin Egorov, Eric Guan, Franklin Liu, Eva Lyubich, Yali Miao, Daichi Ueda, Ryo Takahashi, Huan Wang, and Xichao Wang. This research was funded by the Kauffman Foundation, the Becker Friedman Institute for Research in Economics, the Japan Science and Technology Agency and the Japan Society for the Promotion of Science to which we are grateful. We are particularly indebted to Jeremy Bulow for detailed discussion and for inspiring this work and to James Heckman for advice on relevant theorems in duration analysis and nonparametric estimation. All errors are our own.
[^4]: This later type of use is particularly important in the closely allied computationally-intensive field of Bayesian statistics where closed-form tractable priors are typically used to approximate otherwise computationally intractable probability models.
[^5]: In this main, computationally intensive application we find that analytic characterization of the solutions of sub-problems in larger-scale models is particularly useful in conjunction with analytic-differentiation software, graphics processing units (GPUs), and related optimization algorithms. GPU computing has witnessed dramatic developments over the last few years, which our work benefitted from.
[^6]: A simple economic interpretation would be to identify $F(q)$ with the price $P(q)$ of a good sold by a monopolist, i.e. with the *average* revenue the firm receives per unit sold, in which case $F(q)+q F'(q)=P(q)+q
P'(q)$ is the *marginal* revenue. The name of the transformation is chosen to be consistent with this and similar examples. The Bulow-Pfleiderer demand class [@bulow] discussed later is also invariant under average-marginal transformations.
[^7]: Yet way of extending the usefulness of the solutions is to use Taylor series expansions around them, which may be useful for certain types of models.
[^8]: Even though we do need considerable computational power to fit our model to the data, without the tractability of our functional forms the computations would be significantly harder and we would not have attempted to obtain a calibration of world trade flows that we discuss below.
[^9]: This puzzle in various forms has been discussed by many authors; see @disdier2008puzzling for an overview and @head2013separates for an in-depth discussion of the problem.
[^10]: We discuss alternative mechanisms in Section \[SectionWorldTrade\].
[^11]: In economics there are many combinatorially difficult problems, and we expect our methods to be useful there.
[^12]: In cases where each individual can consume at most one unit of an indivisible product, the inverse demand function equals the reversed quantile function of the distribution of valuations (willingness to pay), up to constant rescaling. The reversed income quantile function here refers to the function that maps a given quantile $q$ measured starting at the top of the income distribution to the corresponding valuation level. Note also that, of course, we do not wish to say that the most important property of constant-elasticity demand lies in the context in which it first appeared. We are merely using this example as an illustration of our approach to demand functions.
[^13]: The origin of constant-elasticity demand historically appears to be the argument by @saybook that willingness to pay for a typical discrete-choice product is likely to be proportional to income, and thus that the distribution of the willingness to pay should have the same shape as the income distribution. [ ]{}(Say[’]{}s assumption is likely to be approximately correct for example for products that save a fixed amount of time to the owner, independently of their wealth.) Since [ ]{}early probate measurements of top incomes exhibited power laws (i.e., Pareto distributions) [@garnier; @say], by extrapolation @dupuit and @Mill suggested that demand would have a constant elasticity. [ ]{}This observation appears to be the origin of the modern focus on constant elasticity demand form [@dupuithistory; @cobborigins]. However evidence on broader income distributions that became available in the 20th century as the tax base expanded [@capital] shows that, beyond the top incomes that were visible in 19th century data, the income distribution is roughly lognormal through the mid-range and thus has a probability density function that is bell-shaped, rather than power-law. [ ]{}Distributions that accurately match income distributions throughout their full range [@reedjorgensen; @doublepareto; @doubleparetolognormal] have a similar bell shape but incorporate the Pareto tail measured in the 19th century data.
[^14]: Similarly, this flexibility could allow us to get a better match to a distribution of valuations in cases where it differs from the exact income distribution.
[^15]: See Footnote \[FootnoteLognormalDistributionAnalyticDifficulty\].
[^16]: @budishroinwilliams studied this problem recently in a different context.
[^17]: \[FootnoteLognormalDistributionAnalyticDifficulty\]For a lognormal distribution with mean $\mu _{\ell }$ and standard deviation $\sigma _{\ell }$ of the exponent, the inverse demand is $P(q)=\exp
\left(\sigma _{\ell } \Phi ^{-1}(1-q)+\mu _{\ell }\right)$, where $\Phi$ is the standard normal cumulative distribution function and where we normalized maximum demand to 1. There is no analytic solution to the monopolist[’]{}s optimization condition $\text{MR}=c$ because the following expression is too complicated: $P'(q)q=-\sqrt{2\pi }\sigma _{\ell } q \exp \left(\sigma _{\ell } \Phi ^{-1}(1-q)+\mu _{\ell }+\frac{1}{2}\left[
\Phi ^{-1}(1-q)\right]^2\right)=-P(q)\sqrt{2\pi }\sigma _{\ell } q \exp \left(\frac{1}{2}\left[\Phi ^{-1}(1-q)\right]^2\right)$. The more realistic double-Pareto lognormal distribution leads to even more complicated expressions. [ ]{}Clearly, if demand functions of this kind were used inside larger models, the absence of analytic tractability could quickly become a significant obstacle.
[^18]: Note that any form-preserving class is also form-preserving under multiple applications of operators of this type.
[^19]: While this intuitive description is sufficient for practical purposes, more formally we say that an *$m$-dimensional functional form class* is a subset of a space of functions (of a scalar, continuous variable) that is homeomorphic to an $m$-dimensional manifold, possibly with a boundary. Such manifold, with or without a boundary, is often referred to as the *moduli space*.
[^20]: Of course, in other contexts the word “tractability" may have other meanings that are also useful. We specify below what we mean by “tractability" in this paper.
[^21]: Of course, the notion of “tractability" and “closed-form solutions" is subjective to some extent. Equations whose solutions may be expressed in terms of functions that are familiar enough are often said to have closed-form solutions. That does not imply, however, that such notion is meaningless. Familiar functions are easier to work with for researchers thanks to existing intuition, as well as thanks to their implementation in symbolic or numerical software. In this paper we made definite choices to resolve the terminological ambiguity.
[^22]: The most notable exception is the case when only a single power of $q$ is used which can be divided out of the equation to yield a polynomial in $\log q$. While this class is of some interest, we do not focus on it here because it has the unappealing property that if one wishes to include a constant term (which is often desirable as we discuss below) one is limited to a small number of powers of logarithms and all other parameters are set. There are other specific exceptions and exploring the use of these is an interesting direction for future research, but none offers the flexibility afforded by power functions that we focus on below. This is likely why they have formed the basis of so much prior work. We thus see the logarithm-based forms instead as limits of the power forms that are worth including but not focusing on.
[^23]: In this section, for expositional purposes, we discuss tractability from the point of view of monopoly problems. But it is worth noting that tractability considerations would be exactly the same for Cournot oligopoly and very similar in the many applications we discuss in this paper.
[^24]: The BP demand, defined below, gives constant pass-through rate of specific taxes to monopolist’s prices only in the case of a constant marginal cost. For this reason, we prefer to use the term Bulow-Pfleiderer (BP) demand, instead of the frequently used term “constant pass-through demand”.
[^25]: There are a few cases not nested in the forms of Theorem \[formpreserve\] for which the firm’s first-order condition may be solved. Hyperbolic demand curves used by @simonovska2015income are one of them. Cases where the solution is in terms of the Lambert W function are the exponential utility function of @newapproach [@behrens] and single-product versions of the Almost Ideal Demand System and of translog demand.
[^26]: @demanding studied the properties such bi-power form applied to inverse demand functions in combination with constant marginal cost. Their goal was not to obtain closed-form solutions.
[^27]: The corresponding values for Mathematica[’]{}s Hermite polynomial interpolation were 0.00069 and 0.0021.
[^28]: PyTorch is an open source software framework developed by Facebook primarily for deep learning in artificial neural networks. Its first version was released in 2017.
[^29]: Tariffs would depend on value in the same way as iceberg trade costs, although the details of their impact would be different, as goods are not destroyed and governments collect tariff revenue. That said, most trade costs modeled as iceberg trade costs in the literature are not supposed to represent tariffs and we will not focus on tariffs in this paper, although, of course, they may be incorporated in our model.
[^30]: Per-unit costs of trade seem more realistic than costs of trade proportional to the goods[’]{} value, as documented by, e.g., @hummelsskiba. Note that this reference did not allow for non-linearity of trade costs.
[^31]: Despite not appearing in the international trade literature, the Economic Order Quantity (EOQ) model [@harris] is perhaps the most classical model of trade costs in the operations research literature and is regularly taught to business students as a method of optimizing their inventory decisions; see e.g. @cardenas2014celebrating for highlights of its importance. Judging from the absence of citations, the academic international trade community is largely unaware of Harris[’]{} publication. When fixed costs per shipment are included in international trade models, they are incorporated in theoretical models with different structures. Those models are similar in spirit, but do not strictly speaking contain the EOQ model or its generalizations [@kropf2014fixed; @hornok2015administrative]. These papers provide very useful insights into shipment frequency issues, and so does the purely empirical paper @hornok2015per. Note also that economies of scale in shipping were studied by @anderson2014gravity and @forslid2016big, but those approaches were not based on shipping frequency and in the former case involved external (i.e., not within-firm) economies of scale.
[^32]: We selected firms by requiring that they specialize in one product category (one 8-digit HS code). The exporting firm had to be active for more than two years to be included in our estimation sample. We selected industries that included at least 10 firms meeting these criteria, in order to work with industries that allow for a precise estimate of $\beta$. We were also careful to take into account potential effects of seasonality, which could affect our estimates. We constructed a measure of seasonal variations of exports for individual industries. Our estimates of $\beta$ did not differ almost at all between industries with larger and smaller seasonality. We discuss more details in Appendix \[AppendixDetailsOfTheGeneralizedEOQModelEstimation\].
[^33]: The confidence interval corresponds to a simple statistical model in which $\beta$ for different industries is drawn from a normal distribution.
[^34]: If exporting leads to zero profit just like not exporting, we specify that the firm chooses not to export.
[^35]: The restaurant industry is an obvious example: few people would associate chain restaurants with outstanding culinary experience. Another fairly obvious example is the automobile industry: there are many automakers in the world, each having a relatively small market share, very stable over time, even though cars produced by different automakers are highly substitutable from costumers[’]{} perspective. With constant marginal costs of production this would require a remarkably small dispersion of marginal costs across firms, which is especially hard to rationalize given the large observed fluctuations of currency exchange rates. Also, the increasing nature of marginal costs of production was one of the reasons why socialist economies were unsuccessful: state-controlled monopolies avoid duplication of effort in product design and other fixed costs of production, yet they suffer from severe agency problems that private sector competition can mitigate. Although here we emphasize increasing marginal costs of production in the long term, they are also interesting at short time scales; see @almunia2018venting and references therein.
[^36]: Oliver E. Williamson’s Nobel lecture (@williamsonnobellecture) provides an excellent, compact discussion of the many things that may go wrong in a large organization. For a related discussion, see @tirole1988theory.
[^37]: In general, we can allow for country-dependence of these costs: $f_{o,k}$ and $f_{x,k,k_d}$. We chose to make them country-independent for simplicity, not for tractability or computational feasibility reasons.
[^38]: Including also per-unit trade costs would not affect the computational feasibility of the model.
[^39]: The cost $L_{T,k,k_d}(q)$ is associated with coordination/shipment preparation tasks and with inventory costs. Its form is motivated by the empirical results of Subsection \[SubsectionFirmLevelEconomiesOfScaleInShipping\].
[^40]: The model has no explicit discounting of future utility, but $\delta _e$ plays a role similar to a discount rate.
[^41]: In our empirical setting we allow for budget imbalances that reflects similar imbalances in the data.
[^42]: Mathematically, the firm[’]{}s choice of destinations in order to maximize profit is a submodular function maximization. This is because serving an additional set of markets $A$ is less attractive if the initial set of markets $S_l$ is larger:\
$\pi \left(S_2\cup A\right)-\pi \left(S_2\right)\leq \pi \left(S_1\cup A\right)-\pi \left(S_1\right)$ for [ ]{}$S_1\subseteq S_2$ and $A\cap
S_2=\emptyset$. Here $\pi (S)$ denotes the optimal profit a firm can earn if it serves a set of markets $S$. If instead our problem was supermodular function maximization, it would be algorithmically easy. International trade papers such as @antras2017margins take advantage of supermodular function maximization being straightforward.
[^43]: For an in-depth discussion of combinatorial discrete choice problems in economics, see @eckert2017combinatorial. The method that @eckert2017combinatorial propose would be useful for us if the number of countries we consider were substantially smaller. This is because the method reduces the exponent of an exponentially difficult problem, but does not change its exponential nature; submodular function maximization is NP-hard in general.
[^44]: We should clarify that even conditional on having made export fixed-cost payments, the number of candidate optima is still combinatorially large. This is because for some destinations it may be impossible to satisfy the FOC and in those cases we allow the firm to export zero amount there. To avoid this difficulty, [ ]{}when we consider candidate optima, we restrict attention to those that satisfy a particular ordering condition, without loss of generality. We rank export destinations by the level of (constant) marginal cost that would make them a profitable destination, in descending order. Then we require that if a firm exports a positive amount to a given destination, it also exports to all preceding destinations. Imposing this condition is without loss of generality because if a firm decides to export zero amount to a destination, it should not have paid the associated fixed cost of exporting in the first place.
[^45]: Due to its combinatorial nature, the exact version of our model is computationally extremely difficult. It may seem natural to try to obtain approximate solutions by first fixing aggregate variables in the model, solving for firm decisions given these aggregates, and then updating the aggregate variables based on the firms[’]{} behavior. We attempted to do that, but could not get results within a reasonable amount of time and budget. This is because for any values of aggregates, we needed to solve separate discrete choice problems by many firms, which requires a lot of time. For this reason, we used a different nesting of loops: we moved all discrete choice decisions into an outer loop of an iterative algorithm, and given these discrete choices, we solved for all continuous variables in an inner loop.
[^46]: We tried several accelerated and non-accelerated gradient descent algorithms. Adam performed the best.
[^47]: Our model, as detailed in the next subsection, had more than 20,000 variables and described 2 million potential trade flows. Newton[’]{}s method would not be feasible here, given that the Hessian of the loss function would have 400 million entries, although light-weight second order methods, such as L-BFGS, could potentially be useful. They would again benefit from the analytic nature of our model. For an overview of optimization algorithms, see the excellent book by @goodfellow2016deep.
[^48]: Given firms[’]{} sunk cost decisions, we need to solve for the general equilibrium of the world economy, i.e., we need to solve for wages, price indexes, and the measure of firms of each type in each country, as well as for production levels of each firm. What makes our calculation fast is the fact that we have explicit formulas for quantities sent to individual destinations conditional on the firm[’]{}s marginal cost, and that these formulas and their derivatives may be evaluated extremely fast.
[^49]: The value of 1.05 for the Pareto index of the firm size distribution has empirical support in @aoyama2010econophysics, at least for the advanced economies studied there. In our open-economy model, there is no simple closed-form expression for the firm size (revenue) as a function of firm productivity. For this reason we use a formula that would hold exactly for closed economies, as well as in the absence of trade costs for the world. Simple algebra shows that the required value of the [ ]{}productivity Pareto index is [ ]{}$\left.\mu _R(\sigma -1)\right/(1+\sigma
\alpha )$ and we use this value. The calibration results in a good match for the firm size distribution of Chinese exporters (both for all firms and for single-HSID firms, as these have the same empirical shape). Given this encouraging result, we have not explored other specifications for the productivity distribution.
[^50]: Initially, we tried $N_p=10$, but such crude discretization led to numerical errors that were too large. Also note that even though for simplicity we sometimes refer to the probability masses as [“]{}firms[”]{}, they really represent collections of firms in monopolistic competition, not a few discrete firms in an oligopoly model.
[^51]: We do not attempt to model high-frequency phenomena in international trade (except that shipping frequency considerations provide a micro-foundation for our trade costs). For studying month-to-month or year-to-years changes, it would not be appropriate to assume that the sunk fixed cost of exporting is fairly negligible.
[^52]: More precisely, we use the generalized method of moments to fit functions of the form $c_0\, (\text{rank})^{-c_1}+c_2$.
[^53]: See @head2014gravity for a recent overview of the literature on the gravity equation of trade. Our purpose here is to highlight the consequences of our model[’]{}s mechanisms, so we focus on the baseline gravity equation that describes the dependence of trade flows on distance and effective GDPs of countries. A comprehensive, in-depth investigation of our model that parallels detailed studies in the gravity-equation literature will be reported separately. It is, of course, worth investigating gravity equations with added controls, such as common language. Similarly, it is good to account for the [“]{}multilateral resistance[”]{} phenomenon (i.e.more isolated countries being more eager to trade with a given partner) already when designing the regression/estimation equations to study. Our model provides very different structural equations than other models, so the matter of multilateral resistance is quite involved. In addition, it is good to explicitly consider trade flow zeros in constructing the regression/estimation equations, although that makes little difference here as almost all trade flows are non-zero in our sample of 100 economies.
[^54]: This is for 30 largest economies. For 100 economies the results would be more noisy.
[^55]: Here we used $\alpha =0.225$.
[^56]: We briefly discuss related literature and mechanisms in Appendix \[AppendixWorldTradeFlowsRelatedLiterature\].\[FootnotePointingToAppendixWorldTradeFlowsRelatedLiterature\]
[^57]: The choice of countries for the figure is not completely arbitrary. China was chosen for the figure because it is a large country and we see patterns of this kind in its firm-level data. We chose the Czech Republic since it is a small country with many neighbors and we know of patterns of this kind based on a series of interviews with Czech exporters featured in *Hospodarske noviny*, a newspaper.
[^58]: Although for identical firms this would be impossible if we introduce differences between the firms, there are other phenomena that may play a role. We briefly discuss them in Appendix \[AppendixWorldTradeFlowsFirmExportPatterns\].\[FootnotePointingToAppendixWorldTradeFlowsFirmExportPatterns\]
[^59]: Similarly, the mechanism may help explain why in the long run trade liberalization can have dramatic effects on trade flows, as for example in the case of the 2001 US-Vietnam trade liberalization; see @mccaig2018export.
[^60]: For example, for welfare consequences we can no longer use simple, elegant formulas such as those derived by @arkolakis2012new.
[^61]: Using our transformed variables would have saved at least 10 pages of the original paper @antras. But of course, relative to these authors we have the benefit of hindsight.
[^62]: $U\left(q\right)$ would literally be a term in the utility function $U\left(q\right)+\tilde{q}\tilde{P}$ in a model with two goods $q$ and $\tilde{q}$, where $\tilde{q}$ is treated as a num' eraire good with price $\tilde{P}$ normalized to 1. In this case the marginal utility of $q$ equals its price $P\left(q\right)$.
[^63]: The Laplace-log representation (\[eq:UtilityAsASymbolicIntegral\]) of a given utility function $U\left(q\right)$ exists under various conditions. Theorem 18b in Section VII.18 of @widder2010laplace states general necessary and sufficient conditions on $U\left(q\right)$ for the existence of $u_{I}\left(t\right)$ such that (\[eq:UtilityAsRiemannStieljesIntegral\]) is satisfied; *almost all* utility functions we may encounter in economic applications *do satisfy these conditions*. Sections VII.12-17 of @widder2010laplace provide conditions that guarantee that $u_{I}\left(t\right)$ exists and has certain properties, such as being of bounded variation, nondecreasing, or belonging to the functional space $L^{p}$. Additional conditions may be found in Chapter 2 of the book by @arendt2011vector, which contains recent developments in the theory. In situations when utility unbounded below is desired, e.g. for constant demand elasticity smaller than 1, we can depart from (\[eq:UtilityAsASymbolicIntegral\]) and instead use the bilateral specification $U\left(q\right)=\intop_{-\infty}^{\infty}u\left(t\right)q^{-t}dt$. However this generalization requires the use of more technically involved bilateral Laplace transforms and thus we do not discuss it in greater detail here, though analogous results are available on request.
[^64]: Our use of $t$ for exponents throughout the text and our use of $s\equiv\log(q)$ here match the standard notation in the literature on Laplace transforms.
[^65]: After an extensive literature search of hundreds of articles and talking to numerous economists, including highly accomplished econometricians, we concluded that this is almost certainly the first time (inverse) Laplace transform in log quantity is used in the economic literature. Note, however, that a different transform, namely (inverse) Laplace transform in quantity, as opposed to log quantity, has been used in economics. These transforms have different properties and should not be confused. Note also that the way we use Laplace transform is different from, say, engineering fields in the sense that, because of the additional logarithm, functions of main interest for us in economics typically would not be of interest in engineering, and vice versa. For this reason, books containing detailed tables of Laplace transform were not of help to us. Except for trivial cases, we needed to derive the transforms listed in Supplementary Material \[sub:LaplaceInverseDemandFunctions\] by ourselves.
[^66]: Note that in certain parts of the paper we need a more general definition of the integral (\[eq:UtilityAsASymbolicIntegral\]) than the definition (\[eq:UtilityAsRiemannStieljesIntegral\]). In those cases, e.g. in the proof of Theorem \[formpreserve\], we use the Schwartz distribution theory instead of the Riemann-Stieltjes integral theory.
[^67]: Normalization here means that $u_{I}\left(0+\right)=0$ and $u_{I}\left(t\right)=(u_{I}\left(t+\right)+u_{I}\left(t-\right))/2.$ See Section I.6 of @widder2010laplace.
[^68]: By *uniform convergence* we mean that for any continuous $\tilde{U}\left(q\right)$ there exists a sequence $\left\{ U_{j}\left(q\right),j\in\mathbb{N}\right\} $ of functions of the form (\[eq:UtilityAsASymbolicIntegral\]) such that for any $\epsilon>0$, all elements of the sequence after some position $n_{\epsilon}$ satisfy $\sup_{q\in\left[0,\bar{q}\right]}|\tilde{U}\left(q\right)-U_{j}\left(q\right)|<\epsilon$.
[^69]: There is also a related, more powerful theorem, the M[ü]{}ntz-Szász theorem. @barnett1983muntz use it to propose to write direct demand as M[ü]{}ntz-Szász polynomials of prices. Here we write inverse demand as polynomials of powers of quantities (times possibly another power of quantity), but the same logic would apply here: we could use M[ü]{}ntz-Szász polynomials.
[^70]: For example, $P(q) = a - b \log q$ corresponds to exponential demand, studied by many authors, including @acv. Similarly, inverse demand functions $P(q) = a - b (\log q)^n$ have interesting implications for market failure in sequential supply chains such as Cournot’s multiple-marginalization problem.
[^71]: Here “bounded by a polynomial” refers to the absolute value of $U_{_{\left[s\right]}}\left(s\right)$ being no greater than the absolute value of some polynomial of *$s$* in the domain $\mathbb{C}_{\bar{s}}^{-}$.
[^72]: Moreover, it is possible to rescale $q$ by a constant factor to keep log $q$ small in absolute value for the range of quantities of interest.
[^73]: As mentioned above, the validity of such approximations may be proved along the lines of the proof given here.
[^74]: @brockett1987class also discuss relations between complete monotonicity and a type of Laplace transform. The Laplace transform used there is in terms of quantity $q$, whereas in our discussion, it is in terms of the logarithm of quantity. These two transforms are distinct and should not be confused. Similarly, the mathematical notion of complete monotonicity has very different economic manifestations in @brockett1987class and in our work.
[^75]: In principle, it is possible to empirically test whether an empirical demand curve satisfies the complete monotonicity criterion. The relevant empirical test has been developed by @heckman1990testing. It would just have to be translated from the duration analysis context to our demand theory context.
[^76]: The fact that these definitions are equivalent may be seen as follows: With the marginal utility of the outside good normalized to one and $U\left(0\right)$ is set to zero, we have $CS\left(q\right)=-qP\left(q\right)+\int_{0}^{q}P\left(q_{1}\right)dq_{1}=-qU'\left(q\right)+\int_{0}^{q}U'\left(q_{1}\right)dq_{1}=U\left(q\right)-qU'\left(q\right)$. This translates into $CS_{_{\!\left[s\right]}}\left(s\right)=U_{_{\!\left[s\right]}}\left(s\right)-U_{_{\!\left[s\right]}}'\left(s\right)$, where we use the subscript $[s]$ to emphasize that the variable is to be treated as a function of $s$. The equivalence for any $n\in\mathbb{N}$ then follows by differentiation.
[^77]: The parameter names are chosen as in Mathematica.
[^78]: Each half of the distribution separately, or the full distribution smoothed by arccosh to ensure the existence of the derivatives.
[^79]: In particular we found that the normal distribution of consumer values has properties very close to those satisfying the complete monotonicity criterion: constant-marginal-cost pass-through is increasing in price (as we show below), and low-order derivatives of $CS\left(s\right)$ with respect to $-s$ are positive. We concluded that the complete monotonicity criterion is not satisfied based on examining the sign on the tenth derivative of $CS\left(s\right)$. The absence of complete monotonicity is consistent with our expression to the corresponding Laplace inverse demand, which does not seem to satisfy $t\,cs\left(t\right)\ge0$. In most economic applications, the difference from completely monotone problems is inconsequential because it manifests itself only in very high derivatives of $CS\left(s\right)$.
[^80]: If functions of negative numbers were of interest, we could simply switch to working in terms of $(-q)$ instead of $q$ and derive analogous results.
[^81]: If we worked with infinite intervals, the convergence of the integrals below would not be always guaranteed.
[^82]: The Fourier transform used in the proof is equivalent to the Laplace transform with imaginary $s$. Both transforms may be thought of as parts of the holomorphic Fourier-Laplace transform.
[^83]: By a generalized function we mean an element of the space $\mathcal{S}'\left(\mathbb{R}\right)$ of distributions.
[^84]: A test function here refers to an element of the space $\mathcal{S}\left(\mathbb{R}\right)$ of space of rapidly decreasing functions.
[^85]: In the same mathematical sense as in the definition of first order stochastic dominance.
[^86]: In the model, all imports are consumed domestically, which implies that exports cannot be larger than tradable GDP. However, such situation might arise for small, highly open economies. To avoid this discrepancy, when the calculated portion of tradable GDP that goes to domestic consumption is smaller than five percent of the tradable GDP in the data, we increment it so that it reaches that level. This is done by correspondingly increasing both the (adjusted) tradable GDP and the (adjusted) consumption in the economy. This criterion was satisfied for just one economy, Hong Kong. Of course, a more realistic way of modeling this situation is to include multi-stage production and/or multi-stage transportation in the model. This will require some additional research work, but it is a clear direction to pursue.
[^87]: The relationship between our variables introduced in the next paragraph (in a notation compatible with the rest of this paper) and the variables in @antras is as follows. Let us use the symbol $\tilde{q}$ to refer to a quantity measure denoted $q$ in AC, which is *distinct* from what we call effective quality-adjusted quantity $q$. In order to recover AC’s original model as a special case, we identify their output $\tilde{q}$ with $q^{1/\alpha}$, where $\alpha\in\left(0,1\right)$ is a constant defined there. For the present discussion we do not need $q$ to be linearly proportional to the number of units produced. It is just some measure of the output, which may or may not be quite abstract. A similar statement applies to the customized intermediate input. Our measure $q_{s}\left(j\right)$ of a particular input is related to AC’s measure $x$$\left(j\right)$ by $q_{s}\left(j\right)=\theta^{\alpha}(x\left(j\right))^{\alpha}$, where $\theta$ is a positive productivity parameter defined in their original paper.
[^88]: See AC’s Subsection 3.1 for a discussion of why only marginal revenue, and not the full-downstream revenue, is the pie that is bargained over and an alternative micro-foundation of this model.
[^89]: The AC model corresponds to $C(q_{s})=(q_{s})^{1/\alpha}c/\theta$, where $c$ and $\theta$ are positive constants defined in their paper. In our notation, the suppliers’ cost is convex but their contributions towards the final output are linear. In the original paper the suppliers’ cost is linear, but their contributions towards the final output have diminishing effects. These are two alternative interpretations of the same economic situation from the point of view of two different systems of notation. As mentioned before, in our interpretation, the product of a supplier is $q_{s}$, whereas in the original paper the supplier’s product is $x$, related to $q_{s}$ by $q_{s}\left(j\right)=\theta^{\alpha}(x\left(j\right))^{\alpha}$.
[^90]: In particular, in their notation, AC have $t=\frac{1}{\alpha}$, $u=1+\frac{\rho}{\alpha}$, $mc_{-t}=\nicefrac{c}{\alpha\theta}$ and $p_{-u}=A^{1-\rho}$, where $\theta$ and $\rho$ is are positive constants defined in AC, not to be confused with the pass-through rate denoted by $\rho$ or the conduct parameter denoted by $\theta$ in other parts of this paper.
[^91]: With constant marginal cost, and in some other special cases, the asymmetric Cournot competition model may also be solved if both demand is specified in an appropriate form. To maintain the generality of our analysis we do not discuss this solvable, asymmetric special case.
[^92]: In Mathematica, principal value integrals may be computed by choosing the option *PrincipalValue* *$\to$* *True* for the *Integrate* function.
[^93]: The corresponding functions are *InverseLaplaceTransform* in Mathematica, *ilaplace* in MATLAB, or *inverse\_laplace\_transform* in Python (SymPy).
[^94]: Here $i$ is the imaginary unit and $\gamma$ is a real number large enough to ensure that $F\left(s\right)$ is holomorphic in the half-plane $\mbox{Re\ }s>\gamma$ (or has a holomorphic analytic continuation to this half-plane).
[^95]: What we described here is a version of Vieta’s substitution that we customized to avoid cluttering of various rational factors.
[^96]: We are grateful to an anonymous referee for the excellent suggestion that this is possible and worth doing. In the original version of the paper we only included the proof of Theorem \[formpreserve\] based on functional transforms. That proof is quick and easy, but less pedagogical because it requires knowledge of more advanced mathematics.
[^97]: We ignore the generically $0$-measure set on which it is an equality.
[^98]: Actually if $\lambda k<p_{0}$ then the lower root should be interpreted as $0$.
[^99]: The model is formally dynamic but is usually studied in its steady state as described here.
[^100]: Note that Equation \[eq:StoleZwiebelFOC\] also involves a constant and thus only our tractable forms with a constant term will maintain their tractability in this model. This is why we focus on this class below.
[^101]: However, it is worth noting that another source of profit convexity, fixed costs, has an opposite effect and is a natural element to include in the model. This can be done in a straightforward way using our technology given our previous discussion, but we omit it here for brevity.
[^102]: @doublemarg2sms extends this characterization to the case of complements when $\theta>1$. For analogous reasons to the previous applications all results here may be extended to arbitrary imperfectly competitive supply chains.
[^103]: In the case of heterogeneous firms, this generalization contains as special cases both the @melitz model and the @melitzottaviano model. To be more precise, let us note that in addition to the heterogeneous-good varieties explicitly considered here, the @melitzottaviano model includes a homogeneous good. In our discussion, the homogeneous good is absent, but adding it to the model is straightforward.
[^104]: Of course, without loss of generality we could assume that $U^{\left(i,\omega\right)}\left(q_{\omega}\right)$ are power functions and let the function $F$ combine them into any desired linear combinations. However, for clarity of notation it is preferable to keep the number $m$ of different expressions $U_{\Omega}^{\left(i\right)}$ small.
[^105]: The case of a single country corresponds to the Dixit-Stiglitz model. It may be obtained from our two-country discussion by setting $\tau\rightarrow\infty$ and $q_{x}=0$. In this case one does not have to express the model’s equations in terms of the equilibrium level of marginal cost $MC^{\star}$ as we do below. Instead, for tractable functions $R\left(q\right)$ and $L\left(q\right)$ one can solve for equilibrium quantity $q^{\star}$ in closed form (in terms of the fixed cost of production $f$) from an equation that combines profit maximization and free entry: $(L(q)+f)R'(q)=R(q)L'(q)$.
[^106]: As mentioned in the paper, a convenient choice of numéraire allows us to keep the revenue function $R\left(.\right)$ independent of economic circumstances.
[^107]: $L_{E}$ may be exogenous, as in the original Krugman model, but even for endogenous labor supply, it is possible to obtain fully explicit solutions to the model in terms of the parameter $MC^{\star}$.
[^108]: In a symmetric equilibrium it does not matter how this labor is split between the countries, as long as the symmetry of the model is maintained. For asymmetric countries, we could assume that the transport requires labor from both countries. The model may be solved in terms of marginal costs of serving each market.
[^109]: For simplicity, consumers do not discount the future, although it would be easy to incorporate an explicit discount factor. Formally, the model includes an infinite number of periods, but it may be thought of as a static model because the equilibrium is independent of time.
[^110]: In the case of a single country, the profit is simply $\Pi\left(q;a\right)=q\left[P\left(q;a\right)-AC\left(q;a\right)\right]$. Also, note that the unrestricted entry condition is often referred to as the *free entry condition*, but here we avoid this term since there is a positive entry cost.
[^111]: For example, the function $L\left(q;a\right)$ could be linear in $a$, as would be the case in the original @melitz model. A simple example of a tractable choice of functional forms is $L\left(q\right)=\tilde{L}\left(q\right)+a\hat{L}\left(q\right)$, $\hat{L}\left(q\right)\equiv q^{t}$, $\tilde{L}\left(q\right)\equiv\tilde{\ell}_{t}q^{-t}+\tilde{\ell}_{u}q^{-u}$, and $R\left(q\right)=r_{t}q^{-t}+r_{u}q^{-u}$.
[^112]: In addition to the heterogeneous-good varieties explicitly considered here, the @melitzottaviano model includes a homogeneous good. In our discussion, the homogeneous good is absent but adding it to the model is straightforward.
[^113]: Of course, without loss of generality we could assume that $U^{\left(i,\omega\right)}\left(q_{\omega}\right)$ are power functions and let the function $F$ combine them into any desired linear combinations. However, for clarity of notation it is preferable to keep the number $m$ of different expressions $U_{\Omega}^{\left(i\right)}$ small.
[^114]: We do not classify the slope of pass-through for demand functions violating declining marginal revenue as this is such a common assumption that we think such forms would be unlikely to be widely used and because it is hard to classify the slope of pass-through when it is infinite over some ranges.
[^115]: Note, however, that in the true limit as $y\rightarrow\infty$, $\kappa\rightarrow1$. However, in practice this occurs at such high income levels that the asymptote to a bit above $1$ is a more realistic representation.
|
---
abstract: 'Up to now, planet search programs have concentrated on main sequence stars later than spectral type F5. However, identifying planets of early type stars would be interesting. For example, the mass loss of planets orbiting early and late type stars is different because of the differences of the EUV and X-ray radiation of the host stars. As an initial step, we carried out a program to identify suitable A-stars in the CoRoT fields using spectra taken with the AAOmega spectrograph. In total we identified 562 A-stars in IRa01, LRa01, and LRa02.'
author:
- Daniel Sebastian
- 'Eike W. Guenther'
title: 'Identifying A-stars in the CoRoT-fields IRa01, LRa01, LRa02'
---
[ address=[Thüringer Landessternwarte Tautenburg, 07778 Tautenburg, Germany]{} ]{}
[ address=[Thüringer Landessternwarte Tautenburg, 07778 Tautenburg, Germany]{}]{}
Introduction
============
CoRoT is a satellite mission launched in December 2006. It is specialized on the detection of extrasolar planets and for studying the pulsations of stars by means of ultra-precise photometric measurements. The photometric accuracy achieved is 10 to 100 times better than what be achieved from the ground ($10^{-4}$ in the exoplanet channel). In the so-called long runs, the fields are observed continuously for 150 days. About 500 stars are observed with the full sampling rate of 32 seconds in the exoplanet channel. All other stars, typically about 6000, are observed with a sampling rate of 8.5 minutes. Up to now, CoRoT has discovered more than 15 extrasolar planets. Amongst them is the first transiting rocky planet found outside the solar system (CoRoT-7b), a planet of a young, active star (CoRoT-2b), a temperate planet (CoRoT-9b), and the first transiting brown dwarf orbiting a normal star (CoRoT-3b). The CoRoT objects open up a new window for studying extrasolar planets. However, the host stars of all these planets are F, G, and K-stars.
As outlined in detail in Guenther et al. (2010) it would be very interesting to detect planets of earlier type stars. For example, the mass loss of planets orbiting early and late type stars is different because of the differences in the EUV and X-ray radiation of the host stars. By comparing the properties of planets orbiting A-stars and late-type stars, we can find out what the influence of the central star for the planet is. However, detecting planets of A-type stars by means of transit observations is difficult, because the transits are shallower than for smaller stars. Additionally, many A-stars oscillate, which makes the analysis of the light curves rather difficult. In order to make progress in this field of research it is thus essential to identify the A-star first. This is the aim of this work.
Identifying candidates and observations
=======================================
Prior to the launch of the satellite, many CoRoT fields were observed using multi-colour photometry (B, V, R, I). Almost all stars that CoRoT observed are also in the 2MASS data-base, so that J, H, K magnitudes are available. The whole photometric data is accessible through EXODAT (Deleuil et al. 2006).
However, identifying A-type stars only on the basis of the broad-band photometry is difficult because of the reddening. We thus take a two-step approach. In the first step we pre-select the stars based on photometry. As a criterion we use the B$-$V colors, and select all stars with B$-$V$= -0.16^{\rm m}$ to $0.42^{\rm m}$, corresponding to an un-reddened B5V to F5V star (Binney & Merrifield 1998). The second step is then the proper determination of the spectral types based on spectroscopy. While the colour-selection approach reduces dramatically the number of stars that we have to study, we may loose a few highly reddened A-stars in this process. Thus, our survey does not aim at detecting [*all*]{} A-stars observed by CoRoT but it aims at finding a sample of A-stars that can be studied.
Although we preselected the targets, we still have to take spectra of several hundred stars. Luckily, as part of the ground-based follow-up observations the CoRoT fields IRa01, LRa01, and LRa02 were observed with the multi-object spectrograph AAOmega mounted on the AAT (Anglo-Australian-Telescope when the observations were taken, now renamed to Australian-Astronomical-Telescope). AAOmega is ideal for our purposes, as this instrument allows to take spectra with up to 350 stars in a field of (Saunders et al. 2004; Smith et al. 2004). The CoRoT fields IRa01 and LRa01 have a size of 1.4$\times$ 2.8, and LRa02 1.4$\times$ 1.4. Mounted in the prime focus of this telescope is a fibre positioned that feeds the AAOmega spectrograph. Using the AAT together with the AAOmega spectrograph we have obtained more than 20000 spectra of stars in the CoRoT-fields.
IRa01, LRa01, and LRa02 are located in the so-called anti-center “eye“ of the CoRoT-mission (RA 6h to 7h and DEC $-$10to 10). The data was obtained in two campaigns. The first campaign was carried out from the $13^{\rm th}$ to the $20^{\rm th}$ of January 2008. Unfortunately, observations could only be obtained on the $13^{\rm th}$ and $14^{\rm th}$ of January. Nevertheless, 4112 spectra of stars in IRa01 and LRa01 were taken. The second campaign was carried out from the $28^{\rm th}$ of December 2008 to the $4^{\rm th}$ of January 2009. Observations were carried out in all eight nights, and we took spectra of 14187 stars in all three fields.
We used “Configure”, the target allocation software in order to find the optimum configuration of the fibres. In each setting we typically managed to place 350 fibres onto target stars, and 25 fibres onto the sky background. In order to optimize the exposure time we minimized the spread in brightness of the stars observed in each setting. We started our observations with the fields containing the brightest stars and subsequently used setting of fainter stars.
Our targets have V-magnitudes in the range 10 to 15, corresponding to the bright part of the CoRoT/Exoplanet targets. Down to m$_{\rm V}$ = 14.5, our observation cover essentially all stars in the CoRoT-fields.
As usual, a fibre bundle was placed onto a relatively bright star for guiding purposes. In order to monitor any possible field rotation, typically 6 fibres were place onto stars within the FOB. For the observations we used the AAOmega spectrograph with the 580V grating in the blue arm and the 385R in the red arm. The spectra cover the spectral range from 3740 to 5810 Å in the blue arm, and 5650 to 8770 Å in the red arm. The resolution is R=1300.
Each field was observed for 30 to 45 minutes. In order to avoid any saturation, and in order to make the removal of cosmic rays easy, we split the observing time spend on each field into three or more exposures.
All calibration frames (flat, arcs and bias-frames) were taken as it is common practice with AAOmega: Bias frames in the afternoon before each observing night, flats and arcs before the observations of each field. The sky subtraction is not critical, because we observed only stars brighter than 15.0 mag and the observations were carried out during dark time. Nevertheless, for subtracting the spectrum of the night sky from the stellar spectra, we have to calibrate the relative throughput for each fibre. Because the throughput varies depending on the bending of the fibre, these measurements had to be done after the fibres have been positioned. The throughput of each individual fibre was determined by taking spectra during dawn, or observing fields of blank sky during the night. The spectra were reduced using the 2dfrdr-reduction package which has especially been developed for AAOmega.
How the spectral types are determined
=====================================
![The spectral types are obtained iteratively by fitting the observed spectra to templates by minimizing $\sigma^2$. Shown here is the derived $\sigma^2$ vs. the shift in wavelength. The best match is obtained at the minimum of $\sigma^2$.[]{data-label="Abb:1"}](shift.eps){height=".4\textheight"}
As outlined above, we take a two-step approach. In the first step. we select suitable candidates based on their B$-$V colours. The second step then is the determination of the spectral type using the AAOmega observations.
This is done by deriving which spectrum from a library of template spectra matches best the observed spectrum. We use *The Indo-U.S. Library of Coudé Feed Stellar Spectra* (Valdes et al. 2004) for this purpose. For each template $\sigma^2$ is calculated as the sum of the squared differences between the template and the observed spectrum. In order to match each template to the observed spectrum, we first shift the spectrum to the correct position in wavelength, adjust the flux, and remove the extinction. This process is done iteratively, always by varying each of these parameters and then minimizing $\sigma^2$. In this way we automatically determine the radial velocity and the extinction $A_{{\rm V}}$ (Binney & Merrifield 1998; Figs. 1, 2) for each star. Since the library includes templates of different luminosity classes, we can also determine the luminosity class for the star (Fig. 3).
![Same as Fig. 1 but for the flux and the reddening.[]{data-label="Abb:2"}](box.eps){height=".4\textheight"}
The accuracy of the method
==========================
Figure 4 shows an observed spectrum together with the best matching template. The red line is the observed spectrum of the star after correcting it for extinction, radial velocity, and removing also the off-set in flux. The green line is the template spectrum of an A5V star. The templates matches the observed spectrum extremely well.
![Shown is the differences between the templates and the observed spectrum expressed in $\sigma^2$ for templates of different spectral types and luminosity classes. The minimum of $\sigma^2$ is achieved for A2V.[]{data-label="Abb:3"}](lumi.eps){height=".36\textheight"}
![Observed spectrum (red line) and the best matching template (green line) for an A5-star.[]{data-label="Abb:4"}](fieldIIIcBlue.064.a_n.txt.eps){width="\textwidth"}
The question however is, how accurate the method for determining the spectral types is. For 178 stars, we have obtained several spectra. This data thus allows a thorough test how accurate our method is. In these cases, we derive the spectral type of the same star several times using different spectra. The differences of the spectral types derived for the same star thus gives us the error of the measurements. The result is shown in Fig. 5. A value of 1.0 means that the difference of the spectral type of two determinations of the same star is one subclass. We find that our error is $1.13\pm0.08$ subclasses. As can be seen in Fig. 5, there are a few outliers. These are largely due to the fact that a few individual spectra were affected by instrumental problems. As shown in Fig. 3, we can also reliable distinguish between dwarfs and giants. Since the difference between dwarfs and sub-giants is rather small, distinguishing these is less certain.
It turns out that the main limitation of the determination of the spectral types is not the quality of the spectra, or the analysis of the data, but the quality of the templates taken from the literature. The libraries published by different authors gave slightly different results (Le Borgne et al. 2000, and Jacoby et al. 1984).
![The error in subclasses, derived from the difference of the spectral types obtained for stars of which several spectra were taken.[]{data-label="Abb:5"}](histo.eps){height=".4\textheight"}
![Distribution of spectral types in our sample.[]{data-label="Abb:6"}](types.eps){height=".4\textheight"}
The spectral types derived
==========================
As already mentioned above, we selected stars according to their B$-$V colours. As a selection criteria, we used B$-$V$= -0.16^{\rm m}$ to $0.42^{\rm m}$, corresponding to an un-reddened B5V to F5V stars (Binney & Merrifield 1998). In total 805 stars were observed with AAOmega which matched this criterion. After the detailed analysis of the spectra we identified 562 A-stars in IRa01, LRa01, and LRa02. Thus, $\sim70\%$ of the stars in the range between B$-$V$= -0.16^{\rm m}$ and $0.42^{\rm m}$ are A-stars. Figure 6 shows the distribution of spectral types. An interesting feature is that stars with spectral types of A3V and A4V are missing. This does not mean that stars of a certain temperature do not exist but it means that almost no stars match the A3V and A4V templates from the literature. This is caused by the slightly odd definition of the sub-classes in this spectral regime. The absence of these types of stars is already well know.
After identifying 562 A-stars in IRa01, LRa01, and LRa02, we now intend to analyze the CoRoT light-curves of these stars in detail in order to detect the shallow transits of planets orbiting these stars.
We are grateful to the user support group of AAT for all their help and assistance for preparing and carrying out the observations. We would like to particularly thank Rob Sharp, Fred Watson and Quentin Parker. The authors thank DLR and the German BMBF for the support under grants 50 OW 0204, and 50 OW 0603.
[\[9\]]{}
Binney J. & Merrifield M. 1998, *Galactic Astronomy*, Princeton University Press.
Deleuil, M., et al. 2006, ESA Special Publication, 1306, 341.
Guenther, E.W., et al. 2010, *these proceedings*.
Jacoby, G. H., Hunter, D. A., & Christian, C. A. 1984, ApJS, 56, 257.
Le Borgne, J.-F., et al. 2003, A&A, 402, 433,
Saunders, W., et al. 2004, SPIE, 5492, 389.
Smith, G. A., et al. 2004, SPIE, 5492, 410.
Valdes, F., et al. 2004, *The Indo-U.S. Library of Coudé Feed Stellar Spectra*, available from .
|
---
abstract: 'A 4600 Hz pulsed synchrotron is considered as a means of accelerating cool muons with superconducting RF cavities from 4 to 20 GeV/c for a neutrino factory. Eddy current losses are held to less than a megawatt by the low machine duty cycle plus 100 micron thick grain oriented silicon steel laminations and 250 micron diameter copper wires. Combined function magnets with 20 T/m gradients alternating within single magnets form the lattice. Muon survival is 83%.'
author:
- 'D. J. Summers$^*$, A. A. Garren$^{\dag}$, J. S. Berg$^{\P}$ and R. B. Palmer$^{\P}$'
title: A Pulsed Synchrotron for Muon Acceleration at a Neutrino Factory
---
[ address=[$^*$Dept. of Physics and Astronomy, University of Mississippi–Oxford, University, MS 38677\
$^{\dag}$Dept. of Physics, University of California, Los Angeles, CA 90095\
$^{\P}$Brookhaven National Laboratory, Upton, NY 11973 ]{}]{}
Historically synchrotrons have provided economical particle acceleration. Here we consider a pulsed muon synchrotron [@nufact02] for a neutrino factory [@factory]. The accelerated muons are stored in a racetrack to produce neutrino beams ($\mu^- \to e^- \, {\overline{\nu}}_e \, \nu_{\mu}$ and $\mu^+ \to e^+ \, \nu_e \, {\overline{\nu}}_{\mu}$). Neutrino oscillations have been observed at experiments [@homestake] such as Homestake, Super–Kamiokande, SNO, and KamLAND. Further exploration using a neutrino factory could reveal CP violation in the lepton sector [@oscillation].
This synchrotron must accelerate muons from 4 to 20 GeV/c with moderate decay loss ($\tau_{\mu^{\pm}}$ = 2.2 $\mu$S), using magnet power supplies with reasonable voltages. To reduce voltage, magnet gaps are minimized to store less magnetic energy. Cool muons [@cool] with low beam emittance allow this. Acceleration to 4 GeV/c might feature fixed field dogbone arcs [@dogbone; @berg] to minimize muon decay loss. Fast ramping synchrotrons [@dogbone; @snowmass] might also accelerate very cool muons to higher energies for a $\mu^+ \, \mu^-$ collider [@collider].
{width="145mm"}
We form arcs with sequences of combined function cells within continuous long magnets, whose poles are alternately shaped to give focusing gradients of each sign. A cell has been simulated using SYNCH [@synch]. Gradients alternate from positive 20 T/m gradient (2.24 m long), to zero gradient (.4 m long) to negative 20 T/m gradient (2.24 m) to zero gradient (0.4 m), etc. See Fig. 1 and Table 1. It is proposed to use 5 such arc cells to form an arc segment. These segments are alternated with straight sections containing RF. The phase advance through one arc segment is 5 x 72$^0$ = 360$^0$. This being so, dispersion suppression between straights and arcs can be omitted. There are 18 arc segments and 18 straight sections, forming 18 superperiods in the ring. Straight sections (22 m) without dispersion are used for superconducting RF, and, in two longer straights (44 m), the injection and extraction. To assure sufficiently low magnetic fields at the cavities, relatively long field free regions are desirable. A straight consisting of two half cells would allow a central gap of 10 m between quadrupoles, and two smaller gaps at the ends. Details are given in Table 3. Matching between the arcs and straights is not yet designed. The total circumference of the ring including combined functions magnets and straight sections adds up to 917 m ($18 \times 26.5 \, + \, 16 \times 22 \, + \, 2 \times 44$).
[**TABLE 1.**]{} Combined function magnet cell parameters. 5 cells/arc. 18 arcs form the ring.
----------------------------------- ----------------
Cell length 5.28 m
Combined Dipole length 2.24 m
Combined Dipole B$_{\rm central}$ 0.9 T
Combined Dipole Gradient 20.2 T/m
Pure Dipole Length 0.4 m
Pure Dipole B 1.8 T
Momentum 20 GeV/c
Phase advance/cell 72$^0$
beta max 8.1 m
Dispersion max 0.392 m
Norm. Trans. Acceptance 4 $\pi$ mm rad
----------------------------------- ----------------
[**TABLE 2.**]{} Superconducting RF.
=0.3mm
--------------------------- -------------------- --
Frequency 201 MHz
Gap .75 m
Gradient 15 MV/m
Stored Energy 900 J
Muons per train $5 \times 10^{12}$
Orbits (4 to 20 GeV/c) 12
No. of RF Cavities 160
RF Total 1800 MV
$\Delta$U$_{\hbox{beam}}$ 110 J
Energy Loading .082
Voltage Drop .041
Acceleration Time 37 $\mu$S
Muon Survival .83
--------------------------- -------------------- --
The superconducting RF (see Fig. 1 and Table 2 and note that 11 MV/m has been achieved so far [@hartill]) must be distributed around the ring to avoid large differences between the beam momentum (which rises in steps at each RF section) and the magnetic field (which rises continuously). The amount of RF is a tradeoff between cost and muon survival. Time dilation permits extra orbits with little muon decay if the RF sags.
[**TABLE 3.**]{} Straight section lattice parameters.
=1.8mm
-------------------- ------------------
$\phi$ 77$^{\,0}$
L$_{\rm cell}$/2 11 m
L$_{\rm quad}$ 1 m
dB/dx 7.54 T/m
a 5.8 cm
$\beta_{\rm max}$ 36.6 m
$\sigma_{\rm max}$ 1.95 cm
B$_{\rm pole}$ 0.44 T
U$_{\rm mag}$/quad $\approx$ 3000 J
-------------------- ------------------
[**TABLE 4.**]{} Permeability ($B/\mu_0H$). Grain oriented silicon (3% Si) steel has a far higher permeability parallel ($\parallel$) to than perpendicular ($\perp$) to its rolling direction [@armco]. Grain oriented silicon steel permits high fields with little energy ($B^2/2\mu$) stored in the yoke.
=1.4mm
Material 1.0 T 1.5 T 1.8 T
------------------------------ ------- ------- -------
1008 Steel 3000 2000 200
Grain Oriented ($\parallel$) 40000 30000 3000
Grain Oriented ($\perp$) 4000 1000
The muons accelerate from 4 to 20 GeV. If they are extracted at 95% of full field they will be injected at 19% of full field. For acceleration with a plain sine wave, injection occurs at 11$^0$ and extraction occurs at 72$^0$. So the phase must change by 61$^0$ in 37 $\mu$S. Thus the sine wave goes through 360$^0$ in 218 $\mu$sec, giving 4600 Hz.
Estimate the energy stored in each 26.5 m long combined function magnet. The gap is about .14 m wide and has an average height of h = .06 m. Assume an average field of 1.1 Tesla. The permeability constant, $\mu_0$, is $4\pi\times
10^{-7}$. $W = {B^2 / {2{\mu_0}}}[\hbox{Volume}] =$ 110000 Joules. Next given one turn (N=1), an LC circuit capacitor, and a 4600 Hz frequency; estimate current, inductance, capacitance, and voltage.
$$\begin{aligned}
B = {{\mu_0\,NI}\over{h}} \, \rightarrow\,
I = {{Bh}\over{\mu_0\,N}} = 52 \, \hbox{kA;} \quad &
W = {1\over{2}}\,L\,I^2 \, \rightarrow\,
L = {\displaystyle{2\,W}\over\displaystyle{{\rule{0pt}{13pt}}I^2}} =
80\,\mu\hbox{H} \\
f = {1\over{2\pi}}\sqrt{1\over{LC}} \rightarrow
C = {1\over{L\,(2\pi f)^2}} = 15 \mu\hbox{F;} & \
W = {1\over{2}}\,C\,V^2 \rightarrow
V = \sqrt{\displaystyle{2W}\over\displaystyle{C}}
= 120\,\hbox{kV}\end{aligned}$$
The stack of SCRs driving each coil might be center tapped to halve the 120 kV. Nine equally spaced 6 cm coil slots could be created in the top and bottom of each yoke using 6 cm of taller laminations to cut the voltage by ten, while leaving the pole faces continuous. 6 kV is easier to insulate than 120 kV. It will be useful to shield [@nufact02] and/or chamfer [@school] magnet ends to avoid large eddy currents where the field lines typically do not follow laminations. Neutrino horn power supplies are of interest.
Calculate the resistive energy loss in the copper coils. There are two 5cm square copper conductors each 5300cm long. R = ${5300 \ (1.8\,\mu\Omega\hbox{-cm}) \, / \, {(2) \, (5^2)}} =
190\,\mu\Omega.$ So, $P = I^2R\int_0^{2\pi}\!\cos^2(\theta)\,d\theta = \hbox{260\,000 w/magnet}.$ Eighteen magnets give a total loss of 4680 kW. But the neutrino factory runs at 30 Hz. Thirty half cycles of 109 $\mu$sec per second gives a duty factor of 300 and a total $I^2R$ loss of 16 kW. Muons are orbited in opposite directions on alternate cycles. If this proves too cumbersome, the duty cycle factor could be lowered to 150. See if .25 mm (30 gauge) wire is usable. The skin depth [@lorrain], $\delta$, of copper at 4600 Hz is $(\rho \, / \, \pi \, f \,\mu_0)^{1/2}$ = $(1.8\times{10^{-8}} \, / \, \pi \, 4600 \, \mu_0)^{1/2}$ = 0.97 mm.
Now calculate the dissipation due to eddy currents [@sasaki] in a $w$ = .25 mm wide conductor, which consists of transposed strands to reduce this loss [@school; @sasaki]. To get an idea, take the maximum B-field during a cycle to be that generated by a 0.025m radius conductor carrying 26 kA. The eddy current loss in a conductor made of square wires .25 mm wide (Litz wire [@mws]) with a perpendicular magnetic field is as follows. $B = {{\mu_0\,I}/{2\pi r}} = 0.2$ Tesla.
$$P = \hbox{[Volume]}{{(2\pi\,f\,B\,w)^2}\over{24\rho}}
= [2 \ .05^2 \ 53]\, {{(2\pi \ 4600 \ .2 \ .00025)^2} \over
{(24)\,1.8\times{10^{-8}}}} = 1400 \
\hbox {kW}$$
Multiply by 18 magnets and divide by a duty factor of 300 to get an eddy current loss in the copper of 85 kW. Stainless steel water cooling tubes will dissipate a similar amount of power [@dogbone]. Alloy titanium cooling tubes would dissipate half as much.
Grain oriented silicon steel is chosen for the yoke due to its high permeability at high field at noted in Table 4. The skin depth [@lorrain], $\delta$, of a lamination is $(\rho \, / \, \pi \, f \,\mu)^{1/2}$ = $(47\times{10^{-8}} \, / \, \pi \, 4600 \, 1000 \, \mu_0)^{1/2}$ = 160 $\mu$m. $\rho$ is resistivity. Take $\mu = 1000 \mu_0$ as a limit on magnetic saturation and hence energy storage in the yoke. Next estimate the fraction of the yoke inductance that remains after eddy currents shield the laminations [@lucent]. The lamination thickness, $t$, is 100 $\mu$m [@arnold]. L/L$_0$ = $(\delta/t) \, (\sinh(t/\delta) + \sin(t/\delta)) \, / \,
(\cosh(t/\delta) + \cos(t/\delta))$ = 0.995. So it appears that magnetic fields can penetrate 100 $\mu$m thick laminations at 4600 Hz. Thicker 175 $\mu$m laminations [@armco] would be half as costly and can achieve a bit higher packing fraction. L/L$_0$($t$ = 175 $\mu$m) = 0.956.
Do the eddy current losses [@sasaki] in the 100 $\mu$m thick iron laminations. Use equation 3 with a quarter meter square area, a 26.5 m length, and an average field of 1.1 Tesla. P = $[(26.5) \, \, (.5^2)]\,
{{(2\pi \ 2600 \ 1.1 \ .0001)^2} /
[{(24)\,47\times{10^{-8}}}}]$ = 5900 kW. Multiply by 18 magnets and divide by a duty factor of 300 to get an eddy current loss in the iron laminations of 350 kW or 700 watts/m of magnet. So the iron will need some cooling. The ring only ramps 30 times per second, so the $\int{\bf{H}}{\cdot}d\,{\bf{B}}$ hysteresis losses will be low, even more so because of the low coercive force (H$_c$ = 0.1 Oersteds) of grain oriented silicon steel. This value of H$_c$ is eight times less than 1008 low carbon steel.
The low duty cycle of the neutrino factory leads to eddy current losses of less than a megawatt in a 4600 Hz, 917 m circumference ring. Gradients are switched within dipoles to minimize eddy current losses in ends. Muon survival is 83%. This work was supported by the U. S. DOE and NSF. Many thanks to K. Bourkland, S. Bracker [@lasker], C. Jensen, S. Kahn, H. Pfeffer, G. Rees, Y. Zhao, and M. Zisman.
[10]{}
D.Summers [*etal.,*]{} J.Phys.[**[G29]{}**]{} (2003) 1727; PAC, hep-ex/0305070; S.Berg, A.Garren, R.Palmer, G.Rees, D.Summers, Y.Zhao, www-mucool.fnal.gov/mcnotes/public/pdf/muc0259/muc0259.pdf. A. Blondel [*et al.,*]{} Nucl. Instrum. Meth. [**A451**]{} (2000) 102; R. Palmer [*et al.,*]{} [*ibid.,*]{} 265;\
D. Neuffer, IEEE Trans. NS–[**[28]{}**]{} (1981) 2034; D. Cline, D. Neuffer, AIP Conf. Proc. [**68**]{} (1980) 846;\
D. Ayres [*et al*]{}, physics/9911009; N. Holtkamp, D. Finley, [*et al*]{}, “A feasibility study of a neutrino source based on a muon storage ring,” Fermilab-Pub-00-108-E; S. Ozaki, R. Palmer, M. Zisman, J. Gallardo, [*et al*]{}, “Feasibility study II of a muon based neutrino source,” (2001) BNL-52623. R.Davis [*et al.*]{} (Homestake), PRL [**20**]{} (1968) 1205; B.Cleveland [*et al.*]{}, Astrophys.J. [**496**]{} (1998) 505;\
Y. Fukuda [*et al.*]{} (Super-K), Phys. Rev. Lett. [**81**]{} (1998) 1562; Q.Ahmad [*et al.*]{} (SNO), Phys. Rev. Lett. [**89**]{} (2002) 011301; 011302; (2001) 071301; S.Ahmed [*et al.*]{} (SNO), nucl-ex/0309004;\
H. H. Chen, Phys. Rev. Lett. [**55**]{} (1985) 1534; K. Eguchi [*et al.*]{} (KamLAND), [*ibid.*]{} [**90**]{} (2003) 021802. V. Barger [*et al.,*]{} Phys. Rev. Lett. [**45**]{} (1980) 2084; Phys. Rev. [**D62**]{} (2000) 073002; 013004;\
S.Geer, Phys.Rev. [**D57**]{} (1998) 6989; S.Bilenky [*etal., ibid.*]{}[**[D58]{}**]{} (1998) 033001; J.Burguet-Castell [*et al.,*]{} hep-ph/0207080; C.Albright [*et al.,*]{} hep-ex/0008064; K.Kodama [*et al.,*]{} hep-ex/0012035;\
A. De Rujula [*et al.,*]{} Nucl. Phys. [**B547**]{} (1999) 21; A. Romanino, Nucl. Phys. [**B574**]{} (2000) 675;\
A. Cervera [*et al.,*]{} Nucl. Phys. [**B579**]{} (2000) 17; M. Koike and J. Sato, Phys. Rev. [**D61**]{} (2000) 073012.
A.Skrinsky, V.Parkhomchuk, Sov.J.Part.Nucl.[**[12]{}**]{}(1981)223; D.Neuffer, Part.Accel. [**14**]{} (1983) 75;\
R. Fernow, J. Gallardo, Phys. Rev. [**E52**]{} (1995) 1039; M. Alsharo’a [*et al.,*]{} PRSTAB [**6**]{} (2003) 081001;\
G.Penn and J.Wurtele, Phys.Rev.Lett. [**[85]{}**]{} (2000) 764; C.Wang and K.Kim, [*ibid.*]{}[**[88]{}**]{} (2002) 184801;\
J. Norem [*et al.,*]{} PRSTAB [**6**]{} (2003) 072001; D. Li [*et al.,*]{} J. Phys. [**[G29]{}**]{} (2003) 1683;\
R.Palmer, “Ring coolers,” J. Phys. [**G29**]{} (2003) 1577; S.Berg, R.Fernow and R.Palmer, [*ibid.,*]{} 1657;\
D.M.Kaplan [*et al.,*]{} Nucl. Instrum. Meth. [**[A503]{}**]{} (2003) 392; D.M.Kaplan, physics/0306135;\
R.P.Johnson [*et al.,*]{} AIP Conf.Proc. [**671**]{} (2003) 328; Y. Derbenev and R.Johnson, “Six-dimensional muon beam cooling in a continuous, homogeneous, hydrogen absorber,” COOL’03, Mt. Fuji, Japan. D. J. Summers, Snowmass 2001, hep-ex/0208010. S. Berg [*et al.,*]{}, PAC’01, http://accelconf.web.cern.ch/AccelConf/p01/PAPERS/RPPH044.PDF.
D. Summers, D. Neuffer, Q. S. Shu and E. Willen, PAC97, Vancouver, physics/0109002;\
D. Summers, Snowmass’96, physics/0108001; Bull. Am. Phys. Soc. [**39**]{} (1994) 1818.
D.Cline, Nucl.Instr.Meth. [**A350**]{} (1994) 24; D.Neuffer, [*ibid.,*]{} 27; AIP Conf.Proc. [**156**]{} (1987) 201;\
V. Barger [*et al.,*]{} Phys.Rev.Lett.[**[75]{}**]{} (1995) 1462; hep-ph/9803480; S.Choi, J.Lee, hep-ph/9909315;\
C. M. Ankenbrandt [*et al.,*]{} Phys. Rev. ST Accel. Beams [**2**]{} (1999) 081001; R. Palmer [*et al.,*]{} Nucl. Phys. Proc. Suppl. [**51A**]{} (1996) 61; R. Raja and A. Tollestrup, Phys. Rev. [**D58**]{} (1998) 013005.
A. Garren [*et al.,*]{} AIP Conf. Proc. [**297**]{} (1994) 403. R.L.Geng, P.Barnes, D.Hartill, H.Padamse, J.Sears, S.Calatroni, E.Chiaveri, R.Losito and H.Preis, “First RF test at 4.2K of a 200MHz superconducting Nb–Cu cavity,” TPAB049, PAC’03.
http://www.aksteel.com/markets/electrical[\_]{}steels.asp; R. Bozorth, “Ferromagnetism,” (1950) 90–1.
N. Marks, “Conventional Magnets – I and II,” Jyväskylä Accel. School, CERN 94-01, [**II**]{}, 867–911. P. Lorrain, D. Corson, and F. Lorrain, “Electromagnetic fields and waves,” 3rd ed. (1988) 537–42.
H. Sasaki, “Magnets for fast–cycling synchrotrons,” Indore Conf. Synchrotron Rad., KEK 91-216.
MWS Wire Industries, Westlake Village, CA 91362, http://www.mwswire.com/litzmain.htm. K. L. Scott, Proc. Inst. Radio Eng. [**18**]{} (1930) 1750–64. Arnold Engineering, 300 North West St., Marengo, IL 60152, http://www.grouparnold.com.
B. M. Lasker, S. B. Bracker and W. E. Kunkel, Publ. Astron. Soc. Pac. [**85**]{} (1973) 109.
|
---
abstract: |
Despite their very low surface gravities, asteroids exhibit a number of different geological processes involving granular matter. Understanding the response of this granular material subject to external forces in microgravity conditions is vital to the design of a successful asteroid sub-surface sampling mechanism, and in the interpretation of the fascinating geology on an asteroid.
We have designed and flown a Taylor-Couette shear cell to investigate granular flow due to rotational shear forces under the conditions of parabolic flight microgravity. The experiments occur under weak compression. First, we present the technical details of the experimental design with particular emphasis on how the equipment has been specifically designed for the parabolic flight environment. Then, we investigate how a steady state granular flow induced by rotational shear forces differs in varying gravitational environments. We find that the effect of constant shearing on the granular material, in a direction perpendicular to the effective acceleration, does not seem to be strongly influenced by gravity. This means that shear bands can form in the presence of a weak gravitational field just as on Earth.
author:
- 'N. Murdoch'
- 'B. Rozitis'
- 'S. F. Green'
- 'T-L de Lophem'
- 'P. Michel'
- 'W. Losert'
bibliography:
- 'Murdochetal2012\_GranularMatter.bib'
date: 'Published online in Granular Matter: 22 February 2013'
title: Granular Shear Flow in Varying Gravitational Environments
---
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Introduction {#s:intro}
============
Asteroids are covered by granular material that can range in size from a few microns (dust) or few hundreds of microns (sand) to a few centimeters or meters (gravels, pebbles, boulders). This superficial layer extends to variable depths and essentially results from impact processes via excavation, fragmentation and ejection of material. Granular materials (regolith) on the surfaces of planets and moons in the Solar System may be also produced via other geological processes such as volcanic activity, erosion and transport, and are extremely common at the surface of all solid bodies. Therefore, the dynamics of granular materials are involved in planetary evolution but appear to also be critical for the design and/or operations of landers, sampling devices and rovers.
Our experiment was designed to study the dynamics of a granular material subject to shear forces in a microgravity environment using a Taylor-Couette shear cell. Couette flow, a term named after the French physicist Maurice Couette, was first used in the field of fluid dynamics [@couette1888]. It refers to the laminar flow (a streamlined flow in parallel layers within which there are no cross currents or eddies) of a fluid between two parallel plates, one of which is moving relative to the other. In this simple Couette geometry the two plates cannot extend infinitely in the flow direction, and so, in order to study shear-driven flows, Sir Geoffrey Taylor created a Couette shear cell using rotating co-axial cylinders [@taylor23]. The circular Couette shear cell, also known as the Taylor-Couette shear cell, has been used in countless experiments in fluid dynamics and, more recently, in studies of granular material on the ground.
Although a similar experimental set-up can be used to study both fluids and granular materials the two media react very differently to shear stresses. Neglecting possible edge effects and effects due to the curvature of the shear cell, the Couette (angular velocity) profile of a Newtonian fluid (a fluid for which there is a linear relationship between the stress and the strain) is a decreasing function across the entire shear cell width. However, in a similar experiment, a granular material develops a shear band; a narrow zone of large relative particle motion bounded with essentially rigid regions. Almost all of the energy input into the granular system by the inner cylinder is dissipated by friction within this narrow region producing large velocity gradients. Shear bands mark areas of flow, material failure, and energy dissipation, and as such they are important in many geophysical processes [[e.g., ]{} @chemenda12; @jiang95; @gapais89].
Various different configurations of the Taylor-Couette shear cell have been used to study the effect of shear stresses on granular materials experimentally. For example, [@bocquet01] shear a granular material in a Taylor-Couette cell with a rotating inner cylinder and stationary outer cylinder. The inner cylinder is connected to the motor via a spring that allows either stick-slip motion or steady shearing depending on the parameters. By performing force measurements they determined that the shear force acting on the moving cylinder is independent of shearing velocity. The dynamics of individual particles were also investigated by measuring the mean velocity of particles on the top-surface. The authors concluded that the normalised velocity profiles are independent of shearing velocity and of the type of motion of the inner cylinder ([i.e., ]{}stick-slip or continuous sliding). [@tardos98] shear a fine, dry powder in a Couette cell. They measured the torque generated on the rough, inner cylinder and found that, in a column of granular material undergoing continuous shearing, normal and shear stresses increase linearly with depth. [@khosropour97] investigate the size segregation of a binary mixture of spherical glass particles in a Taylor-Couette geometry, where the cylinders are made of smooth glass and the flow is generated by the shearing motion of the inner cylinder. The trajectories of 1, 2, and 3 mm glass particles, placed at the bottom of the cell, were followed as they moved through a 1 mm medium. The authors observed that the larger particles rose to the top and remained on the surface.
Nothing is known regarding the same process in the absence of gravity or in the presence of a weak gravity field. To reduce the ambient gravitational acceleration during our experiments we use the microgravity environment available on board a parabolic flight (Novespace A300 Zero-G aircraft). During each parabola of a parabolic flight there are three distinct phases: a $\sim$20 second $\sim$1.8 $g$ ($g$ being the gravitational acceleration at the surface of the Earth) injection phase as the plane accelerates upwards, a $\sim$22 second microgravity phase as the plane flies on a parabolic trajectory (during this period the pilot carefully adjusts the thrust of the aircraft to compensate for the air drag so that there is no lift), and finally, a $\sim$20 second $\sim$1.8 $g$ recovery phase as the plane pulls out of the parabola. Between the end of one parabola and the start of the next there is a 2 minute 1 $g$ rest period. This means the starts of the parabolas are 3 minutes apart. After each set of 5 parabolas it is standard procedure to take a longer 1 $g$ rest of 4-8 minutes. Our experiments were performed with the company Novespace (in France). During one flight there are 31 parabolas and each flight campaign normally consists of 3 flights. This means that there are 93 parabolas in one flight campaign giving approximately 30 minutes of (disconnected) microgravity in total.
We perform constant shear rate experiments with a Taylor-Couette shear cell to investigate how a steady state granular flow, induced by rotational shear forces, differs in varying gravitational environments. The experiments, which occur under weak compression, also allow us to investigate if the formation of shear bands differs between the two gravitational regimes. All of these experiments were performed in the framework of the competitive ESA program ‘Fly your Thesis’.
In [Section \[s:hardware\]]{} we present the technical details of the hardware developed for our experiment with particular emphasis on how the equipment has been designed specifically for the parabolic flight environment. In [Section \[s:procedures\] and \[s:pp\]]{} we present the experimental procedures and we report on some tests that were performed prior to the microgravity flight campaign. Then, finally, in [Section \[s:results\]]{} we present the first results examining the constant shear rate experiments performed with our shear cell on the ground and during the parabolic flights.
Experimental design {#s:hardware}
===================
The AstEx (ASTeroid EXperiment) experiment flew in the European Space Agency (ESA) 51st Microgravity Research Campaign in November 2009 as part of ESAÕs ‘Fly your Thesis’ programme [@callens11]. Our experiment uses a Taylor-Couette shear cell (modified for the parabolic flight environment) that is housed, along with the mechanical and electrical components, inside an experimental rack. This section presents the technical details of the hardware developed for the AstEx experiment with particular emphasis on how the equipment has been designed specifically for the parabolic flight environment.
The AstEx shear cell {#s:MM_cell}
--------------------
Our experiment uses the simplest Taylor-Couette geometry as shown in [Fig. \[fig:Taylor\_Couette\]]{}. There are two concentric cylinders made from cast Acrylic tubes; the outer cylinder has an inner radius of 195 mm and the inner cylinder has an outer radius of 100 mm. Both the inner and outer cylinders are 200 mm in height. The outer cylinder is fixed and its inner surface is rough with a layer of particles, the outer surface of the inner cylinder is also rough but it is free to rotate, and the floor between the two cylinders is smooth and fixed in place. The gap between the two cylinders is filled to a height of 100 mm with spherical soda lime glass beads (grain diameter, d $=$ 3 or 4 mm; density, $\rho$ $=$ 2.55 g cm$^{-3}$) upon which the rotating inner cylinder applies shear stresses. [Figure \[fig:astex\_cell\]]{} is a photograph of a single shear cell.
\[FIGURE \[fig:Taylor\_Couette\] GOES HERE\]
\[FIGURE \[fig:astex\_cell\] GOES HERE\]
We confine the granular material by exerting a very low positive force on the top surface of the beads using a *pressure plate*: a sprung loaded movable, transparent disk ([Fig. \[fig:internal\_workings\]]{}). This ensures all sidewalls can sustain forces to contain the particles. The microgravity environment on a parabolic flight is not perfect; there are small fluctuations that Novespace aim to maintain within the limits of 0 $\pm$ 0.05 $g$ ([Fig. \[fig:gravity\]]{}). This means that, at times, the experiment will experience small amounts of negative $g$, which will lead to dilation of the granular material. The mass of beads contained in the shear cell, assuming an initial packing fraction ($\psi$) of 0.6 and a filling height of 100 mm is 13.47 kg. As the magnitude of the maximum negative acceleration possible is 0.05 $g$, the force required to compensate for this acceleration, and prevent dilation of the granular material, is 6.6 N. This force is distributed between 3 springs; each spring provides a force of 2.2 N at the normal filling height ($\psi$ = 0.6), a force of 0 N at the minimum filling height ($\psi$ = 0.645) and a maximum force of 4.4 N at the maximum filling height [$\psi$ = 0.555; @onoda90]. This is equivalent to pressure variations of between 0 and 149 Pa, assuming that the pressure is equally distributed over the entire area of the plate.
\[FIGURE \[fig:internal\_workings\] GOES HERE\]
\[FIGURE \[fig:gravity\] GOES HERE\]
The AstEx experimental rack
---------------------------
[Figure \[fig:rack\]]{} shows how our Taylor-Couette shear cell is mounted inside the A300 Zero-G aircraft for testing during the parabolic flights. The experiment rack consists of two parts: a test compartment, and a laptop work station. The test compartment is where the experiments take place and it contains one shear cell. Situated next to the test compartment is the laptop work station where two laptops are mounted to allow two experimenters to control the various components of the experimental hardware, to perform the experiments and to collect and store data. The laptops were specially requested without freefall sensors (a standard feature in most laptop hard drives). The entire experiment rack is 1006 $\times$ 1250 $\times$ 750 mm in size, and has a total mass of $\sim$170 kg.
\[FIGURE \[fig:rack\] GOES HERE\]
The shear cells were designed so that they could be easily exchanged between flights when the plane is on the ground. This allows different shear cells to be flown during the parabolic flight campaign. The shear cells are mounted on a polycarbonate base before being placed into position in the experiment rack ([Fig. \[fig:mounting\]]{}). Only one shear cell can be attached to the experiment rack at one time. The polycarbonate base features two handles used to shake the shear cell between experiments. The inner cylinder has a steel shaft running through the centre and is attached to the outer cylinder by two bearings contained inside a housing unit. This steel shaft attaches to the driving shaft and transmits the torque produced by the motor to the inner cylinder. A Watt Drive HU50C 64K4 inline helical geared three phase motor is used, controlled by a Moeller DF51-322-025 three phase inverter.
To minimise the effects of aircraft vibrations we attempt to isolate the shear cell from the aircraft. This is done by mounting silent blocks (a type of vibration isolator made of rubber) between the two strut profiles on which the shear cell is resting and the rest of the support structure frame ([Fig. \[fig:mounting\]]{}).
\[FIGURE \[fig:mounting\] GOES HERE\]
Data collection {#s:data_collection}
---------------
Two high-speed cameras (Matrix Vision Blue Fox 120aG) image the top and bottom layers of glass beads in the shear cell at $\sim$60 frames sec$^{-1}$ (fps) so that the particles do not move more than 1/10 $d$ between consecutive frames (where $d$ is the particle diameter). The cameras, which each have a resolution of 640 $\times$ 480 pixels, are mounted to the experiment rack in the test compartment and image the glass beads through camera viewports built into the shear cells (see [Figs. \[fig:internal\_workings\] and \[fig:rack\]]{}). Four energy-saving reflector lamps are mounted next to the cameras (two for each camera) to illuminate the glass beads. Several thousand images were taken with each camera during each experimental run and saved to the laptop hard-disks within the time between parabolas ([i.e., ]{}in less than two minutes) using our own customised data acquisition software.
Experimental Procedures {#s:procedures}
=======================
During each parabola the same sequence of events is followed. Approximately 5 seconds before the start of the injection phase of the parabola the high-speed cameras start recording. As soon as the microgravity phase starts the motor is started. The motor continues to run during the $\sim$1.8 $g$ recovery phase. Finally, when the 1 $g$ rest phase starts the motor and high-speed cameras are stopped. During the 2 minute rest period the shear cell is shaken by hand to create reasonably consistent initial arrangement of glass beads and to attempt to remove any contact networks and possible memory effects from the granular material. This procedure is repeated for each parabola; however, during the longer 4-8 minute rests the motor rotation speed was also adjusted. During the parabolic flight campaign, experiments were performed with the inner cylinder rotating at 0.025, 0.05 and 0.1 rad sec$^{-1}$.
After the flights, particle tracking was performed using an adaptation of a subpixel-accuracy particle detection and tracking algorithm [@crocker96], which locates particles with an accuracy of approximately $1/10$ pixel. The raw tracking data was smoothed over time using a local regression weighted linear least squares fit (loess model) and then, using the particle positions in each frame, the average angular velocities of every particle can be computed. The pixel scale for the images of the top surface was determined using beads that are glued to the top surface of the confining pressure plate (these can be seen in [Fig. \[fig:StackedImages\]]{} below). The real separations were measured, centre to centre, with vernier callipers. These beads were then identified in the images and their centres were located allowing the pixel scale to be calculated. For the bottom surface the pixel scale was calculated by choosing several particles in a line from within a crystallised region. The distance between the centres of all of these particles was combined with the known particle radius to calculate the pixel scale. In the few laboratory-based runs that were also performed with no pressure plate (see [Section \[s:pp\]]{}) the pixel scale on the top surface was calculated using bead diameters.
In order to compare experiments at different inner cylinder rotational velocities and with different bead sizes, the velocities are normalised. The mean angular velocity of several experiments of the same type, $\overline{V_{\theta}}$, is plotted in its normalised form,
$$\overline{V^*_{\theta}} = \frac{\overline{V_{\theta}}}{\omega}
\label{e:VstarTheta}$$
where $\omega$ is the inner cylinder angular velocity.
Experimental Tests {#s:pp}
==================
As described above, the AstEx experiment had to be modified for the parabolic flight environment. As a result of the modifications to the hardware and experimental set-up, there are a few factors that may influence the measured particle dynamics, namely the presence of the pressure plate. Experiments were performed on the ground with, and without, the pressure plate to determine if the pressure plate had any influence on the particle velocities. [Figure \[fig:PPInfluence4mm4mHz\]]{} shows the normalised mean angular velocity profiles ([i.e., ]{}$\overline{V^*_{\theta}(r)}$) of the 4 mm particles on the top surface of an experiment with the pressure plate and without the pressure plate with an inner cylinder angular velocity of 0.025 rad s$^{-1}$. The velocity profiles with and without the pressure plate are very similar, although, there are some irregularities in the velocity profiles with the pressure plate. These irregularities seem to occur at similar locations in all experiments and, therefore, may be caused by an inhomegenity in the pressure plate. Part of the pressure plate possibly exerts more pressure on the top layer of particles, or part of the pressure plate is perhaps rougher than the rest. The irregularities also become very slightly more pronounced as the angular velocity increases, which means that the pressure plate has a larger effect on the particles at larger rotation rates.
[Figure \[fig:PPInfluence3mm4mHz\]]{} shows the normalised mean angular velocity profiles of the 3 mm particles on the top surface of an experiment with the pressure plate and without the pressure plate again with an inner cylinder angular velocity of 0.025 rad s$^{-1}$. Contrary to the 4 mm particle results, the pressure plate appears to have a very strong influence on the mean angular velocity profiles of the 3 mm particles. This may be because the filling heights are slightly different in the two shear cells; the mass of beads to be added to the shear cells was calculated assuming a random packing fraction of $\sim$0.6 and a desired filling height of 100 mm. It is possible that the 3 mm particles have a lower packing fraction than the 4 mm beads and the filling height is, therefore, slightly higher. This would mean that the pressure plate is exerting a slightly larger pressure on the 3 mm beads.
However, another possible explanation is that the rate at which the particles are decelerated by the pressure plate depends on their radius. Assuming we have one layer of mono-disperse particles (particle radius, $r$, and density, $\rho$), covering a given surface area, $A_s$, and with a 2d packing fraction, $\phi_{2d}$, then the total number of particles in the layer, $N_p$, is given by,
$$N_p = \frac{A_s \phi_{2d}}{\pi r^2}.
\label{e:NpIn2dLayer}$$
If the total force exerted on the particles by the pressure plate is, $F_N$, then the normal force experienced by each particle ($F_{Np}$) on the surface layer is,
$$F_{Np} = \frac{F_N}{N_p} = \frac{\pi F_N r^2 }{A_s \phi_{2d}}.
\label{e:NpIn2dLayer}$$
The frictional force experienced by each particle ($F_{Sp}$) is related to the normal force experienced by each particle via the following relation:
$$F_{Sp} = \mu_s F_{Np}$$
where $\mu_s$ is the coefficient of static friction. The resulting deceleration ($a_p$) of each particle in the surface layer due to the pressure plate is given by:
$$a_p = \frac{F_{Sp}}{M_p} = \frac{3 \mu_s F_N}{4 \phi_{2d} \rho r}
\label{e:Decel}$$
where the mass of each particle ($M_p$) is,
$$M_p = \frac{4 \pi r^3 \rho}{3}.
\label{e:Mp}$$
Therefore, with constant total normal force, particle density, surface area and packing fraction, the deceleration of each particle should vary inversely with $r$. This means that the smaller (3 mm) particles should be decelerated by the pressure plate more than larger (4 mm) particles. Our experimental results do indeed show that the 3 mm particles have a lower velocity in the presence of the pressure plate, whereas the 4 mm particles do not ([Fig. \[fig:PPInfluence\]]{}). However, from these two data sets it is not possible to conclude with certainty that this mechanism is responsible.
\[FIGURE \[fig:PPInfluence\] GOES HERE\]
Steady state granular flow in varying gravitational environments {#s:results}
================================================================
On the top surface of our ground-based experiments ([Fig. \[fig:Cam2\_4mm\_MeanAngVelProfs1g\]]{}) we observe that the mean particle angular velocity ($\overline{V^*_{\theta}}$) decays exponentially with distance from the shearing surface, the shear band is approximately 6-7 particle diameters wide, and the mean normalised angular velocity profiles at different inner cylinder angular velocities are identical to within the error bars. These results are in agreement with those of many previous ground-based experiments of granular shear with similar experimental set-ups: the angular velocity of particles decreases quickly over a few particle diameters away from the shearing wall and the angular velocity profile, normalised by the shear rate, is independent of the shear rate [[e.g., ]{} @behringer99; @mueth00; @losert00; @bocquet01]. Our microgravity experiments ([Fig. \[fig:Cam2\_4mm\_MeanAngVelProfs0g\]]{}) display the same trends, therefore, we show for the first time, that the same is also true for granular shear in microgravity.
\[FIGURE \[fig:Cam2\_4mm\_MeanAngVelProfs\] GOES HERE\]
Comparing the shape of the top surface angular velocity profiles obtained on the ground and in microgravity shows that the mean normalised angular velocity profiles on the top surface are identical in both microgravity and on the ground, except for the small difference in magnitude near the inner cylinder ([Fig. \[fig:Cam2MeanAngVelProfsALL\]]{}). This difference can be explained by considering the shear cell set-up. The outer wall of the inner cylinder is coated with beads to make it rough. These beads are glued onto the wall of the inner cylinder but, due to the presence of the pressure plate, cannot extend above the height of the top layer of beads in the shear cell. Therefore, as the granular material dilates in the microgravity environment, the particles on the top surface nearest the inner cylinder will be in contact with the smooth surface of the inner cylinder rather than the rough surface of the glued-on beads. This will result in a reduced particle velocity very close to the inner cylinder in microgravity compared to the velocity on the ground.
The same trends occur on the bottom surface of the shear cell on the ground and in microgravity ([Fig. \[fig:Cam1MeanAngVelProfsALL\]]{}), however, the shear band is much narrower (3-4 particle diameters wide) and the angular velocity profile is very steep. It is possible that the difference in the angular velocity profiles results from a difference in the packing fraction between the free top surface and the crystallised ([i.e., ]{}hexagonally packed) bottom surface. Indeed, this same effect was found by [@daniels06] who reported that when a granular material is in the disordered state the shear band extends to several particles, but while in the crystallised state, the shear is localised almost entirely to the first layer of particles. This hypothesis gains strength when stacked images of the top and bottom surfaces are considered in which several thousand images have been super-imposed ([Fig. \[fig:StackedImages\]]{}). The shear band near to the moving inner cylinder can be seen on the right of both images but it is much smaller in the crystallised granular material on the bottom surface than on the top surface. As a narrow shear band is observed in microgravity as well as on the ground, this would imply that the crystallisation continues to infuence the particle dynamics in microgravity.
It is also worth noting that the magnitudes of the angular velocity profiles at the bottom surface ([Fig. \[fig:Cam1MeanAngVelProfsALL\]]{}) are much lower than at the top surface ([Fig. \[fig:Cam2MeanAngVelProfsALL\]]{}). This may be linked to the crystallisation of the bottom surface but it is perhaps more likely to be because, in 1 $g$, the particle motion on the bottom surface is resticted due to the weight of the particles above (recall there are $\sim$13.5 kg of beads in the shear cell; see [Section \[s:MM\_cell\]]{}). The same difference in magnitude of $\overline{V^*_{\theta}}$ on the top and bottom surfaces does not exist for the 3 mm beads because the motion of the top surface is also inhibited, albeit via a different mechanism.
Although not shown here, the 3 mm particles follow all of the same trends mentioned above, however, the magnitudes of the mean particle velocites on the top surface are smaller due to the presence of the pressure plate (see [Section \[s:pp\]]{}).
\[FIGURE \[fig:4mmALLAngProfs\] GOES HERE\]
\[FIGURE \[fig:StackedImages\] GOES HERE\]
Conclusions
===========
We have designed and flown a Taylor-Couette shear cell to investigate granular flow due to rotational shear forces in varying gravitational environments. The technical details of the experiment have been presented along with our first experimental results. The experiments occur under weak compression. From our experiments and analysis of steady state granular shear in varying gravitational environments, we have shown that:
- [The angular velocity of particles decreases quickly over a few particle diameters away from the shearing wall for experiments performed both on the ground and in microgravity.]{}
- [The normalised angular velocity profiles of constant shear rate experiments performed both on the ground and in microgravity are independent of shear rate.]{}
- [The normalised angular velocity profiles of constant shear rate experiments performed both on the ground and in microgravity are almost identical (except for a few differences caused by the experiment set-up).]{}
- [A higher packing density of particles (such as at the bottom surface) reduces both the width of the shear band and the maximum angular velocity of the particles in experiments performed both on the ground and in microgravity.]{}
Taking all of the above into account we conclude that the effect of constant shearing on a granular material in a direction perpendicular to the effective acceleration does not seem to be strongly influenced by gravity. This means that shear bands can form in the presence of a weak gravity field just as on Earth.
These are the first experimental results reporting the response of granular material to shear forces in a microgravity environment. Therefore, although most asteroids have a gravitational acceleration at their surface that is even lower than the levels of gravity obtained in this experiment, these results may be of interest for interpreting images of asteroid surfaces and in the design of future asteroid space missions.
However, we note that shear depends on many factors, for example, wall friction, the effective pressure forcing particles against the side boundaries, the velocity difference between the particles and the walls, the grain characteristics [@knight97; @mair02; @luding08]. There are many more experiments, or indeed numerical simulations, which could be performed to investigate the influence of these factors in varying gravitational environments.
Further parabolic flights experiments have been performed with our Taylor Couette shear cell investigating granular convection and shear reversal in varying gravitational environments. These experiments are currently being analysed and the results will be presented in future papers.
Acknowledgments {#acknowledgments .unnumbered}
===============
Thanks to The Open University, Thales Alenia Space, the UK Science and Technology Facilities Council, the Royal Astronomical Society and the French National Programme for Planetology for providing financial support. Thank you also to ESA ÔFly your ThesisÕ for giving us the opportunity to be part of the 51st ESA microgravity research campaign, and for the financial support. Finally, thanks to the workshop of the Planetary and Space Sciences Research Institite at The Open University for constructing our experimental hardware. This study benefited from discussions with the International Team (\#202) sponsored by the International Space Science Institute (ISSI), Bern, Switzerland.
FIGURES {#figures .unnumbered}
=======
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abstract: 'We define and study opfibrations of ${\mathbf{V}}$-enriched categories when ${\mathbf{V}}$ is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary category $B$, there is an equivalence of 2-categories between ${\mathbf{V}}$-enriched opfibrations over the free ${\mathbf{V}}$-category on $B$, and pseudofunctors from $B$ to the 2-category of ${\mathbf{V}}$-categories. This generalizes the classical (${{\mathsf{Set}}}$-enriched) Grothendieck correspondence.'
author:
- Jonathan Beardsley and Liang Ze Wong
bibliography:
- 'references.bib'
title: The Enriched Grothendieck Construction
---
Introduction
============
The Grothendieck construction and its inverse relate stacks on a Grothendieck site to fibrations, or fibered categories, over that site. More generally, for any category $B$, there is an equivalence between pseudofunctors $B^{op} \to {{\mathsf{Cat}}}$ and fibrations over $B$ [@johnstone1 Theorem B1.3.6]; by duality, there is also an equivalence between pseudofunctors $B \to {{\mathsf{Cat}}}$ and *op*fibrations over $B$. $${{\mathsf{opFib}}}(B) \cong {{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathsf{Cat}}}).$$ We generalize this equivalence to categories enriched over a suitable monoidal category ${\mathbf{V}}$. Our main result is Theorem \[mainthm\]:
Let ${\mathbf{V}}$ be a monoidal category satisfying the assumptions in §\[sec:grcon\], and let $B_{\mathbf{V}}$ be the free ${\mathbf{V}}$-category on a category $B$. There is a $2$-equivalence $${{\mathsf{opFib}}}(B_{\mathbf{V}}) \cong {{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}).$$
Our paper is structured in the following manner: In §\[sec:prelims\], we briefly recall some notions from enriched category theory and $2$-category theory that will be used in the rest of the paper, leaving the details to Appendix \[sec:appendix\].
In §\[sec:opfibinvgrcon\] we define ${\mathbf{V}}$-enriched opfibrations and show that such opfibrations $p \colon {\mathcal{E}}\to {\mathcal{B}}$ give rise to pseudofunctors ${\mathcal{B}}_0 \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$, where ${\mathcal{B}}_0$ is the underlying category of the ${\mathbf{V}}$-category ${\mathcal{B}}$. This gives the *inverse Grothendieck construction* $$I \colon {{\mathsf{opFib}}}({\mathcal{B}}) \to {{\mathsf{Fun}}^\mathsf{ps}}({\mathcal{B}}_0, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}).$$
In §\[sec:grcon\] we show that, under suitable assumptions on ${\mathbf{V}}$, pseudofunctors $B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ give rise to opfibrations over $B_{\mathbf{V}}$, the free ${\mathbf{V}}$-category on $B$. This gives the *Grothendieck construction* $$Gr \colon {{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}) \to {{\mathsf{opFib}}}(B_{\mathbf{V}}).$$ Our hypotheses on ${\mathbf{V}}$ ensure that $(B_{\mathbf{V}})_0 \cong B$, so that it makes sense to ask if our constructions are mutual inverses when ${\mathcal{B}}= B_{\mathbf{V}}$. In §\[sec:grcorr\], we answer this in the affirmative, yielding our main result.
We make use of standard notions and techniques from enriched category theory, and as far as possible, try to relate our constructions back to the classical Grothendieck construction (i.e. the ${{\mathsf{Set}}}$-enriched case). In fact, one of the features of the construction given here is that a young mathematician well versed in the definitions of enriched and 2-category theory could understand all of our proofs.
Motivation and relation to other work
-------------------------------------
The original motivation for this work goes back to the PhD thesis of the first author (some of which is described in [@beardsrelative]). One of the goals of that thesis was to describe certain *coalgebraic* structures (e.g. bialgebras and comodules) in quasicategories, which are one of several models for $(\infty,1)$-categories. The first author found that the $(\infty,1)$-categorical Grothendieck construction described in [@htt] and expanded upon in [@ha] were not rigid enough for these purposes. Recalling from [@bergner] that we may think of simplicially enriched categories as a model for $(\infty,1)$-categories, we may then think of the enriched Grothendieck construction given in this paper, when ${\mathbf{V}}={{\mathsf{sSet}}}$, as a rigidified version of Lurie’s quasicategorical Grothendieck construction over an ordinary category. This perspective is developed further in [@beardsley2018operadic].
Our inverse Grothendieck construction (§\[sec:opfibinvgrcon\]) is an instance of the $2$-functor ${{\mathsf{opFib}}}_{\mathcal{K}}({\mathcal{B}}) \to {{\mathsf{Fun}}^\mathsf{ps}}({\mathcal{B}}_0, {\mathcal{K}})$ that arises from an opfibration ${\mathcal{E}}\to {\mathcal{B}}$ in any $2$-category ${\mathcal{K}}$ with finite $2$-limits. This $2$-functor is due to Ross Street, but precise references are hard to find (see [@riehl2017comprehension §6] for the analogous construction in an $\infty$-cosmos, from which the reader may distill the original $2$-categorical construction). A reader who is familiar with these results may safely skip this section.
An anonymous referee has pointed out that the Grothendieck construction (§\[sec:grcon\]) and the Grothendieck correspondence (§\[sec:grcorr\]) factor through a simpler correspondence between pseudofunctors $B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ and opfibered $B$-*parametrized* ${\mathbf{V}}$-categories. In other words, letting ${\mathbf{V}}_B \textsf{-}{{\mathsf{opFib}}}$ be the category of opfibered $B$-parametrized ${\mathbf{V}}$-categories, there is a $2$-equivalence $${\mathbf{V}}_B\textsf{-}{{\mathsf{opFib}}}\cong {{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}).$$ This result holds for arbitrary monoidal categories ${\mathbf{V}}$, and is seen to be a special case of [@kelly2002categories §7.6]. A similar result for opfibered $B$-*graded* ${\mathbf{V}}$-categories may be found in [@lowen2008hochschild §2.4], where ${\mathbf{V}}$ is required to have coproducts which are preserved by $\otimes$.
However, these opfibered $B$-parametrized or $B$-graded categories are not *functors* in ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$. In light of these observations, our contribution in this work may alternatively be interpreted thus: we identify properties of ${\mathbf{V}}$ under which opfibered $B$-parametrized ${\mathbf{V}}$-categories may be internalized as actual opfibrations over $B_{\mathbf{V}}$ in ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$, so that we have an equivalence $${{\mathsf{opFib}}}(B_{\mathbf{V}}) \cong {\mathbf{V}}_B\textsf{-} {{\mathsf{opFib}}}\cong {{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}).$$
We note that there are other works dealing with various aspects of the Grothendieck construction for categories enriched over specific ${\mathbf{V}}$. Lurie has essentially defined what an opfibration of ${{\mathsf{sSet}}}$-categories ought to be in [@htt], although the results regarding the $\infty$-categorical Grothendieck correspondence are formulated in (marked) simplicial sets. When ${\mathbf{V}}= {{\mathsf{Cat}}}$, Hermida [@hermida1999some], Bakovic [@bakovicgrothendieck] and Buckley [@buckley2014fibred] give a fully enriched Grothendieck construction and Grothendieck correspondence. By constrast, our Grothendieck construction only allows for pseudofunctors from an *ordinary* category $B$ into ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$, which correspond to opfibrations over the *free* ${\mathbf{V}}$-category $B_{\mathbf{V}}$. This is unavoidable at our level of generality, since ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ is not necessarily ${\mathbf{V}}$-enriched[^1], so that it does not make sense to talk about ${\mathbf{V}}$-functors ${\mathcal{B}}\to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ from an arbitrary ${\mathbf{V}}$-category ${\mathcal{B}}$ to ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$. In addition, although we do define opfibrations over an arbitrary ${\mathbf{V}}$-enriched base ${\mathcal{B}}$, one anonymous referee has pointed out that in the case when ${\mathbf{V}}= {{\mathsf{sSet}}}$ or ${{\mathsf{Cat}}}$, our definition agrees with those given in [@htt 2.4.1.10, 2.4.2.1] and [@buckley2014fibred 2.1.6, 3.1.5] only when ${\mathcal{B}}= B_{\mathbf{V}}$. Nevertheless, we believe this work is an important step towards a fully enriched Grothendieck correspondence relating opfibrations over ${\mathcal{B}}$ and ${\mathbf{V}}$-functors ${\mathcal{B}}\to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$, under additional assumptions on ${\mathbf{V}}$ such that ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ is ${\mathbf{V}}$-enriched.
Finally, we note that there is already a preprint of Tamaki [@tamaki] which discusses an enriched Grothendieck construction. However, the definition of an opfibration given there is equivalent to the classical definition (when ${\mathbf{V}}= {{\mathsf{Set}}}$) only when the base category $B$ is a groupoid. While we have taken some inspiration from [@tamaki], our work is significantly different.
Acknowledgements
----------------
The authors thank the two anonymous referees for their very detailed reviews and helpful suggestions, which have greatly simplified and improved this paper. We would also like to thank James Zhang for his support and mentorship during the completion of this paper.
Preliminaries {#sec:prelims}
=============
We begin by recalling a few notions from enriched category theory and $2$-category theory that will be used in this paper. Throughout, we work over a monoidal category $({\mathbf{V}}, \otimes, {\mathbf{1}})$.
Properties of ${\mathbf{V}}$ {#sec:Vprop}
----------------------------
We describe a few additional properties of ${\mathbf{V}}$ that we will later require.
1. If ${\mathbf{V}}$ has coproducts, we say that *$\otimes$ preserves coproducts (in both variables)* if there is a canonical isomorphism $$\bigg(\coprod_{i \in I} A_i\bigg) \otimes \bigg(\coprod_{j \in J} B_j\bigg) \cong \coprod_{i \in I} \coprod_{j \in J} A_i \otimes B_j.$$
2. If ${\mathbf{V}}$ has pullbacks and coproducts, we say that ${\mathbf{V}}$ is *extensive* if pullbacks interact well with coproducts in the following sense:
1. *Pullbacks preserve coproduct injections*: For any set $I$ and family of maps $f_i \colon Y_i \to X_i$ in ${\mathbf{V}}$, the following square is a pullback: $$\begin{tikzcd}
Y_i \ar[d, "f_i"'] \ar[r, hookrightarrow] & \coprod_{i \in I} Y_i \ar[d, "\coprod_i f_i"]
\\
X_i \ar[r, hookrightarrow] & \coprod_{i \in I} X_i
\end{tikzcd}$$
2. *Pullbacks preserve coproduct decompositions*: For any set $I$ and family of maps $f_i \colon X_i \to Z$ and $g \colon Y \to Z$ in ${\mathbf{V}}$, we have a canonical isomorphism $$Y \times_Z \left(\coprod_i X_i \right) \cong \coprod_i \left( Y \times_Z X_i \right),$$ where these fibered products are given by the following pullback diagrams: $$\begin{aligned}
\begin{tikzcd}
Y \times_Z \left(\coprod_i X_i \right) \ar[r] \ar[d] \arrow[dr, phantom, "\lrcorner", very near start] & Y\ar[d, "g"]
\\
\coprod_{i \in I} X_i \ar[r, "\coprod_i f_i"] & Z
\end{tikzcd}
\end{aligned}
\quad \quad
\begin{aligned}
\begin{tikzcd}
Y \times_Z X_i \arrow[dr, phantom, "\lrcorner", very near start] \ar[r] \ar[d] & Y\ar[d, "g"]
\\
X_i \ar[r, "f_i"] & Z
\end{tikzcd}
\end{aligned}$$
3. If the monoidal product $\otimes$ is the cartesian product $\times$, we say that ${\mathbf{V}}$ is *cartesian*. This implies that the monoidal unit ${\mathbf{1}}$ is terminal. If only this last condition holds, we say that ${\mathbf{V}}$ is *semi*cartesian.
4. Finally, if ${\mathbf{V}}$ is extensive, we say that the monoidal unit ${\mathbf{1}}$ is *connected* if the representable functor $${\mathbf{V}}({\mathbf{1}}, -) \colon {\mathbf{V}}\to {{\mathsf{Set}}}$$ preserves all coproducts. If ${\mathbf{V}}$ is also semicartesian, then ${\mathbf{V}}({\mathbf{1}}, {\mathbf{1}}) \cong \{*\}$, so for any set $X$ we have a canonical isomorphism $$\label{eq:V-prop4}
{\mathbf{V}}\bigg({\mathbf{1}}, \coprod_{x \in X} {\mathbf{1}}\bigg) \cong \coprod_{x \in X} \{*\} \cong X.$$ This last isomorphism is equivalent to the left adjoint of ${\mathbf{V}}({\mathbf{1}},-)$ (defined later in (\[eq:set-tensor\])) being fully faithful.
Our Properties 2(i) and 2(ii) are respectively (e1) and (e$2'$) in the characterization of extensivity from [@centazzo97sheaf §4.2]. Another way of stating Property 2(ii) is that *coproducts are universal*. Note that the definition we have used is sometimes called *infinitary* extensive.
The above assumptions hold for the categories of sets, simplicial sets, topological spaces or categories, equipped with the cartesian monoidal product $\times$. They also hold for any locally cartesian closed category with disjoint coproducts and connected unit.
An example where the above assumptions do *not* hold is $({{\mathsf{Vect}}_k}, \otimes_k, k)$.
In §\[sec:invgrcon\], we will only require that ${\mathbf{V}}$ has pullbacks, while in §\[sec:grcon\], we will require all the above but with *semi*cartesian instead of cartesian ${\mathbf{V}}$, and with only the isomorphism (\[eq:V-prop4\]) of Property 4.
Underlying categories and free ${\mathbf{V}}$-categories {#sec:underfree-prelim}
--------------------------------------------------------
The $2$-category of ${\mathbf{V}}$-categories, ${\mathbf{V}}$-functors and ${\mathbf{V}}$-natural transformations (defined in Appendix \[sec:appendix\]) will be denoted ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$.
Throughout this paper, ${\mathcal{B}}$ will denote a ${\mathbf{V}}$-category with hom-objects ${\mathcal{B}}(b,c) \in {\mathbf{V}}$, while $B$ will denote an ordinary category with hom-sets $B(b,c)$. Let ${\mathbbm{1}}$ denote the ${\mathbf{V}}$-category with a single object $*$ and $${\mathbbm{1}}(*,*) := {\mathbf{1}}.$$
When ${\mathbf{V}}$ has coproducts, the representable functor ${\mathbf{V}}({\mathbf{1}}, -)$ has a left adjoint $(-) \cdot {\mathbf{1}}\colon {{\mathsf{Set}}}\to {\mathbf{V}}$ sending $X$ to $$\label{eq:set-tensor}
X \cdot {\mathbf{1}}:= \coprod_{x \in X} {\mathbf{1}}.$$ Further, if coproducts in ${\mathbf{V}}$ are preserved by $\otimes$, this adjunction between ${{\mathsf{Set}}}$ and ${\mathbf{V}}$ induces an adjunction: $$\label{eq:CatVCat}
\begin{tikzcd}[column sep = large]
{{\mathsf{Cat}}}\ar[r, bend left = 20, shift left, "(-)_{\mathbf{V}}", ""{name = L}, start anchor = east, end anchor = west]
&
{{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}\ar[l, bend left = 20, shift left, "(-)_0", ""{name = R}, start anchor = west, end anchor = east] \ar[from = L, to = R, symbol = \dashv]
\end{tikzcd}$$ In detail, the *underlying category* of ${\mathcal{B}}\in {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ is the functor category $${\mathcal{B}}_0 := {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}({\mathbbm{1}}, {\mathcal{B}})$$ whose objects are ${\mathbf{V}}$-functors $b \colon {\mathbbm{1}}\to {\mathcal{B}}$ and morphisms are ${\mathbf{V}}$-natural transformations $f \colon b {\Rightarrow}c$. Equivalently, ${\mathcal{B}}_0$ is the category with the same objects as ${\mathcal{B}}$ and morphisms $f \colon {\mathbf{1}}\to {\mathcal{B}}(b,c)$.
The *free ${\mathbf{V}}$-category* on $B \in {{\mathsf{Cat}}}$ is the ${\mathbf{V}}$-category $B_{\mathbf{V}}$ with the same objects as $B$ and hom-objects $$B_{\mathbf{V}}(b,c) := B(b,c) \cdot {\mathbf{1}}= \coprod_{f \in B(b,c)}\!\!\! {\mathbf{1}}.$$
\[rem:BV0\] Let $\iota$ and $\sigma$ denote the unit and counit of the adjunction $(-)_{\mathbf{V}}\dashv (-)_0$. Their components are $$\begin{aligned}
\iota_B &\colon B \to (B_{\mathbf{V}})_0, \\
\sigma_{\mathcal{B}}&\colon ({\mathcal{B}}_0)_{\mathbf{V}}\to {\mathcal{B}}.
\end{aligned}$$ If the canonical isomorphism (\[eq:V-prop4\]) from Property 4 holds, then $\iota_B$ is an isomorphism of categories $$B \cong (B_{\mathbf{V}})_0.$$ We elaborate on this further in Remark \[rem:identifyBBV0\].
Some examples in the one-object case might be instructive. A one-object category may be identified with a monoid $M$, while a one-object ${\mathbf{V}}$-category may be identified with a monoid ${\mathcal{M}}$ in ${\mathbf{V}}$.
When ${\mathbf{V}}= {{\mathsf{Top}}}$, the free topological monoid $M_{\mathbf{V}}$ on a monoid $M$ is the same monoid given the discrete topology, while the underlying category ${\mathcal{M}}_0$ of a topological monoid ${\mathcal{M}}$ is the same monoid forgetting its topology.
In this case, $M = (M_{\mathbf{V}})_0$, so $\iota_M$ is the identity. The map $\sigma_{\mathcal{M}}$ is also the identity on the underlying sets, but its domain has the discrete topology.
When ${\mathbf{V}}= {{\mathsf{Vect}}_k}$, the $k$-algebra $M_{\mathbf{V}}$ is the monoid-algebra $k[M]$, while the monoid ${\mathcal{M}}_0$ is the $k$-algebra ${\mathcal{M}}$ treated simply as a monoid (forgetting its $k$-linear structure).
Unlike for ${{\mathsf{Top}}}$, in this case $M \neq k[M]$, but $\iota_M \colon M\hookrightarrow k[M]$ is the inclusion of $M$ as a basis. The map $\sigma_{\mathcal{M}}\colon k[{\mathcal{M}}] \to {\mathcal{M}}$ sends formal linear combinations of objects in ${\mathcal{M}}$ to their actual sum in ${\mathcal{M}}$.
Pseudofunctors, natural transformations, modifications {#sec:pseudoFTM}
------------------------------------------------------
We will also be working with *pseudofunctors* $F \colon B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ out of an ordinary category $B$. These are ‘functors’ that are only associative and unital up to coherent isomorphism. More precisely, $F$ consists of the following data,
1. for each $b \in B$, a ${\mathbf{V}}$-category $F_b$;
2. for each $f \colon b \to c$, a ${\mathbf{V}}$-functor $F_f \colon F_b \to F_b$;
3. for each $b \in B$, a natural isomorphism $\xi(b) \colon F_{1_b} \cong 1_{F_b}$ (or simply $\xi$) with components: $$\xi_x \colon {\mathbf{1}}\to F_b(F_{1_b} x, x)$$
4. for each $b \xrightarrow{f} c \xrightarrow{g} d$ in $B$, a natural isomorphism $\theta(f,g) \colon F_{gf} \cong F_g F_f$ (or simply $\theta$) with components: $$\theta_x \colon {\mathbf{1}}\to F_d(F_{g f} x , F_g F_f x)$$
satisfying the following relations: $$\label{eq:pseudo-unit-left}
\begin{aligned}
\begin{tikzcd}
& F_b \ar[dr, "F_f"] &
\\
F_b \ar[ur, ""{name = D, above}, bend right = 20] \ar[rr, bend right = 10, "F_f"',""{name = L, above}] \ar[ur, bend left = 40, equals, ""{name = C, below}] \ar[from = C, to = D, Leftarrow, "\xi"] & & F_d \ar[from = 1-2, to = L, Leftarrow, "\theta", pos=0.6, shorten <=0.8em, shorten >=0.5em]
\end{tikzcd}
\end{aligned}
\quad = \quad
\begin{aligned}
\begin{tikzcd}
& \phantom{F_d} &
\\
F_b \ar[rr, bend left=20, "F_f", ""{name = U, below}] \ar[rr, bend right=20, "F_f"', ""{name = L, above}] & & F_d \ar[from = U, to = L, Leftarrow, "=", pos=0.6, shorten <=0.2em, shorten >=0.2em]
\end{tikzcd}
\end{aligned}$$ $$\label{eq:pseudo-unit-right}
\begin{aligned}
\begin{tikzcd}
& F_d \ar[dr, ""{name = E, above}, bend right = 20] \ar[dr, bend left = 40, equals, ""{name = D, below}] \ar[from = D, to = E, Leftarrow, "\xi"] &
\\
F_b \ar[ur, "F_f"] \ar[rr, bend right=10, "F_f"',""{name = L, above}] & & F_d \ar[from = 1-2, to = L, Leftarrow, "\theta"', pos=0.6, shorten <=0.8em, shorten >=0.5em, ]
\end{tikzcd}
\end{aligned}
\quad = \quad
\begin{aligned}
\begin{tikzcd}
& \phantom{F_d} &
\\
F_b \ar[rr, bend left=20, "F_f", ""{name = U, below}] \ar[rr, bend right=20, "F_f"', ""{name = L, above}] & & F_d \ar[from = U, to = L, Leftarrow, "=", pos = 0.6, shorten <=0.2em, shorten >=0.2em]
\end{tikzcd}
\end{aligned}$$ $$\label{eq:pseudo-assoc}
\begin{aligned}
\begin{tikzcd}
&[-15pt] F_b \ar[r, "F_g"] \ar[drr,, ""{name = L, left}, bend right = 10] & F_d \ar[dr, "F_h"] \ar[to = L, Leftarrow, "\theta"', pos = 0.9] &[-15pt]
\\
F_b \ar[ur, "F_f"] \ar[rrr, "F_{hgf}"', bend right = 10, ""{name = B, left}, pos=0.4 ] & \phantom{F_b} & & F_e \ar[from =1-3, to = B, Leftarrow, "\theta"', pos=0.7, shorten <=1.8em, shorten >=0.5em]
\end{tikzcd}
\end{aligned}
\quad = \quad
\begin{aligned}
\begin{tikzcd}
&[-15pt] F_b \ar[r, "F_g"] & F_d \ar[dr, "F_h"] &[-15pt]
\\
F_b \ar[ur, "F_f"] \ar[rrr, "F_{hgf}"', bend right = 10, ""{name = B,right}, pos = 0.6] \ar[urr, , ""{name = L, right}, bend right = 10] & & & F_e \ar[from =1-2, to= L, Leftarrow, "\theta", pos = 0.9] \ar[from = 1-2, to = B, Leftarrow, "\theta", pos=0.7, shorten <=1.8em, shorten >=0.5em]
\end{tikzcd}
\end{aligned}$$
A *pseudonatural transformation* (or simply a *transformation*) $\alpha \colon F {\Rightarrow}G$ between pseudofunctors consists of $1$-cells $\alpha_b \colon F_b \to G_b$ for each $b \in B$ and natural isomorphisms $$\label{eq:trans}
\begin{tikzcd}
F_b \ar[r, "F_f"]\ar[d, "\alpha_b", swap] & F_b \ar[d, "\alpha_b"]\\
G_b \ar[r, "G_f", swap] \ar[ur, Rightarrow,"\alpha_f", "\cong"', start anchor={north east}, end anchor={south west}, shorten <=0.7em, shorten >=0.7em]& G_b
\end{tikzcd}$$ for each $f \colon b \to c$ in $B$, satisfying further coherence rules given in [@leinsterbicats §1.2].
Finally, a *modification* $\Gamma\colon \alpha{\Rrightarrow}\beta$ between transformations consists of natural transformations $\Gamma_b \colon \alpha_b {\Rightarrow}\beta_b$ for each $b \in B$ satisfying: $$\label{eq:mod}
\begin{aligned}
\begin{tikzcd}[column sep = huge, row sep = huge]
F_b \ar[r, "F_f"]\ar[d, "\alpha_b" description, bend right = 40] & F_b \ar[d, "\alpha_b" description, bend right = 40, ""{name = L}] \ar[d, "\beta_b" description, bend left = 40, ""{name = R, left}] \ar[from = L, to = R, Rightarrow, "\Gamma_b"]\\
G_b \ar[r, "G_f", swap] \ar[ur, Rightarrow,"\alpha_f" near start, start anchor={north east}, end anchor={south west}, shorten <=0.5em, shorten >=3.5em, "\cong"' near start]& G_b
\end{tikzcd}
\end{aligned}
\quad = \quad
\begin{aligned}
\begin{tikzcd}[column sep = huge, row sep = huge]
F_b \ar[r, "F_f"]\ar[d, "\alpha_b" description, bend right = 40, ""{name = L}] \ar[d, "\beta_b" description, bend left = 40, ""{name = R, left}] \ar[from = L, to = R, Rightarrow, "\Gamma_b"] & F_b \ar[d, "\beta_b" description, bend left = 40] \\
G_b \ar[r, "G_f", swap] \ar[ur, Rightarrow,"\beta_f" near end, start anchor={north east}, end anchor={south west}, shorten <=3.5em, shorten >=0.5em, "\cong"' near end]& G_b
\end{tikzcd}
\end{aligned}$$
Let ${{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}})$ denote the $2$-category of pseudofunctors, transformations and modifications from $B$ to ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$.
Opfibrations and the Inverse Grothendieck Construction {#sec:opfibinvgrcon}
======================================================
In this section, we develop the theory of opfibrations in the enriched setting. We define opfibrations over a base ${\mathcal{B}}$, and the 2-category ${{\mathsf{opFib}}}({\mathcal{B}})$ that they form. The inverse Grothendieck construction is then a 2-functor from ${{\mathsf{opFib}}}({\mathcal{B}})$ to the category of pseudofunctors ${{\mathsf{Fun}}^\mathsf{ps}}({\mathcal{B}}_0, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}})$.
Throughout we will assume that ${\mathbf{V}}$, and hence ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$, has pullbacks.
Opfibrations and opfibered functors
-----------------------------------
\[def:opcartesian\] Let $p \colon {\mathcal{E}}\to {\mathcal{B}}$ be a ${\mathbf{V}}$-functor. A map $\chi \colon {\mathbf{1}}\to {\mathcal{E}}(e, e')$ is **$p$-opcartesian** if the following square is a pullback in ${\mathbf{V}}$ for all $d \in {\mathcal{E}}$: $$\label{eq:opcartesian}
\begin{tikzcd}[column sep = large, ]
{\mathcal{E}}(e', d) \ar[d, "p"'] \ar[r, "-\circ \chi"] & {\mathcal{E}}(e,d) \ar[d, "p"]
\\
{\mathcal{B}}(pe', pd) \ar[r, "-\circ p\chi"] & {\mathcal{B}}(pe,pd)
\end{tikzcd}$$
\[def:opfib\] An **opfibration** is a ${\mathbf{V}}$-functor $p\colon {\mathcal{E}}\to {\mathcal{B}}$ along with, for every $e \in {\mathcal{E}}$, $b \in {\mathcal{B}}$ and $f \colon {\mathbf{1}}\to {\mathcal{B}}(pe, b)$, an object ${{f}_!}e \in {\mathcal{E}}$ over $b$ and a $p$-opcartesian map $\chi(f,e) \colon {\mathbf{1}}\to {\mathcal{E}}(e, {{f}_!}e)$ over $f$.
The map $\chi(f,e)$ will be called a **chosen** $p$-opcartesian lift of $f$.
We note a simple but important lemma:
\[lem:iso-to-chosen\] Let $p \colon {\mathcal{E}}\to {\mathcal{B}}$ be an opfibration. A map $\chi \colon {\mathbf{1}}\to {\mathcal{E}}(e,e')$ is $p$-opcartesian if and only if it is isomorphic to the chosen $p$-opcartesian lift $\chi(p\chi, e) \colon {\mathbf{1}}\to {\mathcal{E}}(e, {{(p\chi)}_!} e)$ via a unique isomorphism $$\label{eq:eps-chi}
\varepsilon_\chi \colon {\mathbf{1}}\to {\mathcal{E}}({{(p\chi)}_!}e, e')$$ lying over $1_{e'}$.
This follows from the uniqueness (up to unique isomorphism) of the pullback in Definition \[def:opcartesian\].
An **opfibered functor** from $p\colon {\mathcal{E}}\to {\mathcal{B}}$ to $q \colon {\mathcal{F}}\to {\mathcal{B}}$ is a functor $k \colon {\mathcal{E}}\to {\mathcal{F}}$ that satisfies $qk = p$ and sends $p$-opcartesian maps to $q$-opcartesian maps.
\[lem:opfibered-cocart\] Let $p \colon {\mathcal{E}}\to {\mathcal{B}}$ and $q \colon {\mathcal{F}}\to {\mathcal{B}}$ be opfibrations over ${\mathcal{B}}$, and let $k \colon {\mathcal{E}}\to {\mathcal{F}}$ be such that $qk = p$. Then $k$ is opfibered if and only if it sends chosen $p$-opcartesian maps to (not necessarily chosen) $q$-opcartesian maps.
If $k$ is opfibered, it certainly sends chosen $p$-opcartesian maps to $q$-opcartesian ones.
Conversely, suppose $k$ sends chosen $p$-opcartesian maps to $q$-opcartesian ones. Lemma \[lem:iso-to-chosen\] then shows that $k$ is opfibered: any $p$-opcartesian map $\chi \colon {\mathbf{1}}\to {\mathcal{E}}(e,e')$ is isomorphic to the chosen $p$-opcartesian map $\chi(p\chi, e)$. Since functors preserve isomorphisms, $k\chi$ is isomorphic to the $q$-opcartesian map $k \chi(p\chi, e)$, hence is also $q$-opcartesian.
Let ${{\mathsf{opFib}}}({\mathcal{B}})$ denote the $2$-category whose objects are opfibrations over ${\mathcal{B}}$, morphisms are opfibered functors, and 2-morphisms are natural transformations over ${\mathcal{B}}$.
Properties of opcartesian maps
------------------------------
In this section, we record some results concerning opcartesian maps and the unique maps that their universal properties induce.
Let $p \colon {\mathcal{E}}\to {\mathcal{B}}$ be an opfibration. By the universal property of $p$-opcartesian maps, a pair of maps $\varphi \colon {\mathbf{1}}\to {\mathcal{E}}(e,d)$ and $g \colon {\mathbf{1}}\to {\mathcal{B}}(b, pd)$ such that $p\varphi = gf$ induces a unique $\widetilde{g}$: $$\begin{tikzcd}[sep = large]
{\mathbf{1}}\ar[drr, "\forall \,\varphi", bend left = 13] \ar[ddr, "\forall \, g"', bend right] \ar[dr, dashed, "\exists !\,\widetilde{g}" description]
\\
& {\mathcal{E}}({{f}_!}e, d) \ar[r, "{-\circ \chi(f,e)}"] \ar[d, "p"'] \ar[dr,phantom, "\lrcorner", very near start] & {\mathcal{E}}(e,d) \ar[d, "p"]
\\
& {\mathcal{B}}(b, pd) \ar[r, "-\circ f"] & {\mathcal{B}}(pe,pd)
\end{tikzcd}$$ When ${\mathbf{V}}= {{\mathsf{Set}}}$, the preceding discussion yields the universal property: for every $\varphi \colon e \to d$ and $g \colon b \to pd$ such that $p\varphi = gf$, there exists a unique $\widetilde{g} \colon {{f}_!} e \to d$ such that $p \widetilde{g} = g$ and $\widetilde{g}\chi(f,e) = \varphi$: $$\label{eq:opcart-classical}
\begin{tikzcd}[column sep = large, row sep =large]
e \ar[r, "{\chi(f,e)}"] \ar[d, dotted] & {{f}_!}e \ar[dr, dashed, "\exists !\, \widetilde{g}"] \ar[d, dotted] &
\\
pe \ar[r, "f"] \ar[drr, bend right = 10, "gf"'] & b \ar[dr, "\forall\, g"] & d \ar[d, dotted] \ar[from = 1-1, bend right = 10, crossing over, "\forall\, \varphi", near start]
\\
& & pd
\end{tikzcd}$$ Here, the dotted arrows represent $p$, and indicate which objects and arrows of ${\mathcal{E}}$ lie over which objects and arrows of ${\mathcal{B}}$.
\[lem:g-tilde-opcart\] $\varphi$ is $p$-opcartesian if and only if $\widetilde{g}$ is.
Let $d' \in {\mathcal{E}}$. The various maps involved fit into the following diagram: $$\label{eq:g-tilde-opcart}
\begin{tikzcd}[sep = large]
{\mathcal{E}}(d, d') \ar[r, "-\circ \widetilde{g}"] \ar[rr, bend left, "-\circ \varphi"] \ar[d, "p"]
&
{\mathcal{E}}({{f}_!}e, d') \ar[r, "-\circ {\chi(f,e)}"] \ar[d, "p"] \ar[dr,phantom, "\lrcorner", very near start]
&
{\mathcal{E}}(e,d') \ar[d, "p"]
\\
{\mathcal{B}}(pd,pd') \ar[r, "-\circ g"]
&
{\mathcal{B}}(b, pd') \ar[r, "- \circ f"]
&
{\mathcal{B}}(pe, pd')
\end{tikzcd}$$ Since $\chi(f,e)$ is $p$-opcartesian, the square on the right is a pullback. By the pasting law for pullbacks, the outer square is a pullback ($\varphi$ is $p$-opcartesian) if and only if the square on the left is a pullback ($\widetilde{g}$ is $p$-opcartesian).
\[lem:opcart-compose\] A composite of $p$-opcartesian maps is $p$-opcartesian.
This follows from arguments similar to the previous proof. The relevant diagram is (\[eq:g-tilde-opcart\]), with $\tilde{g}$ taken to be $p$-opcartesian and $\chi(f,e)$ replaced by an arbitrary $p$-opcartesian map.
\[lem:counit-iso\] For each $e \in {\mathcal{E}}$, the lift $\chi(1_{pe},e)$ is an isomorphism. Thus $$e \cong {{(1_{pe})}_!} e.$$
It is easy to see that $1_e$ is $p$-opcartesian. By Lemma \[lem:iso-to-chosen\], there is a unique $\varepsilon_{(1_e)}$ which is a left inverse to $\chi(1_{pe},e)$: $$\label{eq:eps-e}
\varepsilon_{(1_e)} \circ \chi(1_{pe},e) = 1_e.$$ Since $\varepsilon_{(1_e)}$ is an isomorphism, it is a right inverse as well, so $\chi(1_{pe},e)$ is an isomorphism.
Fibers and transport {#sec:fibers}
--------------------
In this section, we define and study the fibers of an opfibration, and show that arrows in the base ${\mathcal{B}}_0$ induce transport functors between fibers. We begin with a more general definition of fibers of any functor.
Let $p\colon {\mathcal{E}}\to {\mathcal{B}}$ be a ${\mathbf{V}}$-functor. For each $b \in {\mathcal{B}}$, treated as a functor $b \colon {\mathbbm{1}}\to {\mathcal{B}}$, the **fiber** of $p$ over $b$ is the category ${\mathcal{E}}_b$ given by the pullback: $$\begin{tikzcd}
{\mathcal{E}}_b \ar[dr,phantom, "\lrcorner", very near start] \ar[r, hookrightarrow] \ar[d] & {\mathcal{E}}\ar[d, "p"]
\\
{\mathbbm{1}}\ar[r, "b"'] & {\mathcal{B}}\end{tikzcd}$$
The objects of ${\mathcal{E}}_b$ are $\left\{e \in {\mathcal{E}}\, | \, pe = b \right\}$, while the morphisms are given by the pullback: $$\label{eq:fiber-homs}
\begin{tikzcd}
{\mathcal{E}}_b(e,e') \ar[r] \ar[d] \arrow[dr, phantom, "\lrcorner", very near start] & {\mathcal{E}}(e,e') \ar[d , "p"]
\\
{\mathbf{1}}\ar[r, "1_b"'] & {\mathcal{B}}(b,b)
\end{tikzcd}.$$
We may think of ${\mathcal{E}}_b$ as the subcategory of ${\mathcal{E}}$ consisting of objects in the pre-image of $b$ and morphisms in the pre-image of $1_b$.
\[prop:transport\] Let $p \colon {\mathcal{E}}\to {\mathcal{B}}$ be a opfibration. For every $f \in {\mathcal{B}}_0(b,b')$, the assignment $e \mapsto {{f}_!}e$ extends to a functor $${{f}_!} \colon {\mathcal{E}}_b \to {\mathcal{E}}_{b'},$$ called the .
On morphisms, take $({{f}_!})_{e,e'} \colon {\mathcal{E}}_b(e,e') \to {\mathcal{E}}_{b'}({{f}_!}e,{{f}_!}e')$ to be the unique map induced by the commuting diagram $$\begin{tikzcd}[sep = large]
{\mathcal{E}}_b(e,e') \ar[r, hookrightarrow] \ar[d] \arrow[dr, phantom, "\lrcorner", very near start] & {\mathcal{E}}(e,e') \ar[d, "p"] \ar[r, "\chi{(f, e')}\circ -"] & {\mathcal{E}}(e, {{f}_!}e') \ar[d, "p"]
\\
{\mathbf{1}}\ar[r, "1_b"] & {\mathcal{B}}(b,b) \ar[r, "f \circ -"] & {\mathcal{B}}(b, b')
\end{tikzcd}$$ and the universal property of the composite pullback: $$\begin{tikzcd}[sep = large]
{\mathcal{E}}_{b'}({{f}_!}e,{{f}_!}e') \ar[r, hookrightarrow] \ar[d] \arrow[dr, phantom, "\lrcorner", very near start] & {\mathcal{E}}({{f}_!}e,{{f}_!}e') \ar[d, "p"] \ar[r, "-\circ \chi{(f, e)}"] \arrow[dr, phantom, "\lrcorner", very near start] & {\mathcal{E}}(e, {{f}_!}e') \ar[d, "p"]
\\
{\mathbf{1}}\ar[r, "1_{b'}"] & {\mathcal{B}}(b',b') \ar[r, "-\circ f"] & {\mathcal{B}}(b, b')
\end{tikzcd}$$ Functoriality of ${{f}_!}$ follows from the uniqueness of each $({{f}_!})_{e,e'}$. We leave it to the reader to check the details.
We may attempt to define a functor $$\begin{aligned}
{\mathcal{E}}_\bullet \colon {\mathcal{B}}_0 &\xrightarrow{?} {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}\nonumber \\
b &\mapsto {\mathcal{E}}_b \label{eq:Fpseudo} \\
(b \xrightarrow{f} b') &\mapsto ({\mathcal{E}}_{b} \xrightarrow{{{f}_!}} {\mathcal{E}}_{b'}). \nonumber
\end{aligned}$$ Unfortunately, ${\mathcal{E}}_\bullet$ fails to be a functor, as it only preserves identities and composites *up to isomorphism*, i.e. ${\mathcal{E}}_\bullet$ is a *pseudo*functor. This is the subject of the next subsection.
The Inverse Grothendieck Construction $I$ {#sec:invgrcon}
-----------------------------------------
The map ${\mathcal{E}}_\bullet \colon {\mathcal{B}}_0 \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ can be given the structure of a pseudofunctor.
We need to supply natural isomorphisms $\xi$ and $\theta$ as per §\[sec:pseudoFTM\].
For each $b \in {\mathcal{B}}_0$ and $e \in {\mathcal{E}}_b$, take $\xi_e$ to be the isomorphism $\varepsilon_{(1_e)}$ from Lemma \[lem:counit-iso\]. For $f \in {\mathcal{B}}_0(b,c)$ and $g \in {\mathcal{B}}_0(c,d)$, we need an isomorphism $$\theta_e \colon {\mathbf{1}}\to {\mathcal{E}}_{d}( {{gf}_!}e, {{g}_!}{{f}_!} e).$$ By Lemma \[lem:opcart-compose\], the composite $
\chi(g,{{f}_!}e) \, \chi(f,e)
$ from $e$ to ${{g}_!}{{f}_!} e$ is $p$-opcartesian. We may thus take $\theta_e$ to be the unique isomorphism $
\varepsilon_{\chi(g,{{f}_!}e) \chi(f,e)}
$ from Lemma \[lem:iso-to-chosen\] between $\chi(gf,e)$ and $\chi(g, {{f}_!}e)\, \chi(f,e)$.
The uniqueness of these components may be used to show that they satisfy equations (\[eq:pseudo-unit-left\], \[eq:pseudo-unit-right\], \[eq:pseudo-assoc\]), so that we do indeed obtain a pseudofunctor.
The inverse Grothendieck construction that sends an opfibration $p \colon {\mathcal{E}}\to {\mathcal{B}}$ to the pseudofunctor ${\mathcal{E}}_\bullet$ extends to a $2$-functor $${I}\colon {{\mathsf{opFib}}}({\mathcal{B}}) \to {{\mathsf{Fun}}^\mathsf{ps}}({\mathcal{B}}_0,{{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}).$$
We have seen above what ${I}$ does to opfibrations i.e. the 0-cells of ${{\mathsf{opFib}}}({\mathcal{B}})$. We need to define what ${I}$ does to 1-cells and 2-cells. Let $p \colon {\mathcal{E}}\to {\mathcal{B}}$ and $q \colon {\mathcal{F}}\to {\mathcal{B}}$ be opfibrations over ${\mathcal{B}}$, and let ${\mathcal{E}}_\bullet, {\mathcal{F}}_\bullet \colon {\mathcal{B}}_0 \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ be the corresponding pseudofunctors.
Suppose we have an opfibered functor $k \colon {\mathcal{E}}\to {\mathcal{F}}$. Pulling this back along each $b$ yields functors $k_b \colon {\mathcal{E}}_b \to {\mathcal{F}}_b$ sending $e$ to $ke$. We seek a natural isomorphism of the form $$\label{eq:alpha-f}
\begin{tikzcd}
{\mathcal{E}}_b \ar[d, "k_b"'] \ar[r, "{{f}_!}"] & {\mathcal{E}}_{b'} \ar[d, "k_{b'}"]
\\
{\mathcal{F}}_b \ar[r, "{{f}_!}"']
\ar[ur, Rightarrow, shorten >=1em, shorten <=1em, "\cong"', "\alpha_f"] & {\mathcal{F}}_{b'}
\end{tikzcd}$$ with components $(\alpha_f)_e \colon {\mathbf{1}}\to {\mathcal{F}}_{b'}({{f}_!} \, k e, k \, {{f}_!}e )$. Since $k$ is opfibered, the map $k \chi(f,e)$ is $q$-opcartesian. By Lemma \[lem:iso-to-chosen\], we may take $(\alpha_f)_e$ to be the unique isomorphism $\varepsilon_{k \chi(f,e)}$ between $\chi(f, ke)$ and $k \chi(f,e)$.
The functors $k_b$ thus form the 1-components of a pseudonatural transformation which we denote $\kappa \colon {\mathcal{E}}_\bullet {\Rightarrow}{\mathcal{F}}_\bullet$, while the 2-components are given by the natural isomorphism above.
Given another opfibered functor $h \colon {\mathcal{E}}\to {\mathcal{F}}$ which induces $\lambda \colon {\mathcal{E}}_\bullet {\Rightarrow}{\mathcal{F}}_\bullet$, and a ${\mathbf{V}}$-natural transformation $\gamma \colon h {\Rightarrow}k$ over ${\mathcal{B}}$, we may once again pull all these back along $b$ to obtain a ${\mathbf{V}}$-natural transformation $$\begin{tikzcd}
{\mathcal{E}}_b \ar[r, bend left, start anchor = north east, end anchor = north west, "h_b", ""{name = U, below}] \ar[r, bend right, start anchor = south east, end anchor = south west, "k_b"', ""{name=D, above}] & {\mathcal{F}}_b \ar[from = U, to = D, Rightarrow, "\gamma_b"]
\end{tikzcd}$$ The 2-cells $\gamma_b$ then form the data of a modification $\Gamma \colon \lambda \Rrightarrow \kappa$.
We leave it to the reader to check that the various coherence conditions are satisfied, and that this indeed yields a $2$-functor.
Note that while the inverse Grothendieck construction takes opfibrations over an arbitrary enriched ${\mathbf{V}}$-category ${\mathcal{B}}$, it only returns pseudofunctors from an unenriched ${\mathcal{B}}_0$. It is thus generally not possible to recover an opfibration $p \colon {\mathcal{E}}\to {\mathcal{B}}$ over an arbitrary base ${\mathcal{B}}$ from its corresponding pseudofunctor ${\mathcal{E}}_\bullet \colon {\mathcal{B}}_0 \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$. The next section describes the best we can do.
The Grothendieck Construction {#sec:grcon}
=============================
We now describe an opfibration $p \colon GrF \to B_{\mathbf{V}}$ associated to a pseudofunctor $F \colon B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$.
Assumptions {#sec:assumptions}
-----------
For what follows, we require the following assumptions described in §\[sec:Vprop\]:
1. ${\mathbf{V}}$ has coproducts, and $\otimes$ preserves coproducts in both variables;
2. ${\mathbf{V}}$ has pullbacks and is extensive (pullbacks preserve coproduct injections and decompositions);
3. ${\mathbf{V}}$ is semi-cartesian (${\mathbf{1}}$ is terminal);
4. For $X$ a set, we have a canonical isomorphism $${\mathbf{V}}\bigg({\mathbf{1}}, \coprod_{x \in X} {\mathbf{1}}\bigg) \cong X,$$ so that $B \cong (B_{\mathbf{V}})_0$ (this holds if ${\mathbf{1}}$ is connected).
We briefly sketch where these properties will be used: Coproducts are required for the formation of $B_{\mathbf{V}}$ and $GrF$, and we require $\otimes$ to commute with coproducts to define composition in these categories. While ${\mathbf{1}}$ need not be terminal to obtain $GrF$, we do need it to obtain a functor $p \colon GrF \to B_{\mathbf{V}}$. Pullbacks are needed in the very definition of an opfibration. Finally, extensivity and the last condition are required in order for $p$ to be an opfibration (see Remark \[rem:assumption4used\]).
\[rem:identifyBBV0\] Property 4 implies that the unit of the adjunction (\[eq:CatVCat\]) is an isomorphism of categories: $$B \cong (B_{\mathbf{V}})_0.$$ We simplify matters by assuming that this isomorphism is in fact *equality*, which we justify in the following manner:
Recall that elements of $(B_{\mathbf{V}})_0$ are ${\mathbf{V}}$-maps $${\mathbf{1}}\to B_{\mathbf{V}}(b,c) = \coprod_{f \in B(b,c)} {\mathbf{1}}.$$ The set of such maps contains the inclusions ${\mathbf{1}}_g \hookrightarrow B_{\mathbf{V}}(b,c)$, where ${\mathbf{1}}_g$ denotes the copy of ${\mathbf{1}}$ corresponding to $g \in B(b,c)$. Assumption 4 then says that these inclusions account for *all* ${\mathbf{V}}$-maps ${\mathbf{1}}\to B_{\mathbf{V}}(b,c)$. By abuse of notation, we may *identify* elements $g \in B(b,c)$ with maps $${\mathbf{1}}= {\mathbf{1}}_g \hookrightarrow B_{\mathbf{V}}(b,c),$$ which we also call $g$. Under this identification, we then have $B = (B_{\mathbf{V}})_0$.
The category $Gr F$
-------------------
\[defi:gc\] Let $B$ be an ordinary (i.e. ${{\mathsf{Set}}}$-enriched) category treated as a $2$-category, and let $F\colon B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ be a pseudofunctor. The **Grothendieck construction of $F$** is the ${\mathbf{V}}$-category $Gr F$ with objects and morphisms $$\begin{aligned}
Ob(GrF) &:= \;\; \coprod_{b\in B} \;\; Ob(F_b) \times \{b\}, \\
GrF\big( (x,b), (y,c) \big) & :=\coprod_{f\colon b\to c} F_{c}(F_f x,y).\end{aligned}$$ Identity morphisms are given by $$\label{eq:grid}
1_{(x,b)} := \xi_x \colon {\mathbf{1}}\to F_b(F_{1_b}x,x) \subset \coprod_{f \colon b \to b} F_b(F_f x, x) = GrF\big( (x,b), (x,b)\big)$$ while composition is induced by the composite $$\begin{tikzcd}[column sep = large]
F_b(F_f x,y)\otimes F_d(F_g y,z) \ar[r, "F_g\otimes 1"] \ar[d, dotted] & F_d(F_g F_f x,F_g y)\otimes F_d(F_g y,z) \ar[d, "\cong", "(-\circ \theta_x)\otimes 1"']
\\
F_d(F_{gf}x,z) & F_d(F_{gf}x,F_g y)\otimes F_d(F_g y,z) \ar[l, "\circ"]
\end{tikzcd}$$ where $b \xrightarrow{f} c \xrightarrow{g} d$. This extends to a functor out of $GrF\big( (x,b), (y,c) \big) \otimes GrF\big( (y,c), (z,d)\big)$ because $\otimes$ preserves coproducts.
We will see in what follows that the $GrF$ admits an opfibration to the free ${\mathbf{V}}$-category $B_{\mathbf{V}}$ on $B$. If instead we had $F \colon B^{op} \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$, we could similarly define the ${\mathbf{V}}$-category $Gr^\vee F$ where $$\begin{aligned}
Ob(Gr^\vee F) &:= \;\; \coprod_{b\in B} \;\; Ob(F_b) \times \{b\}, \\
Gr^\vee F\big( (x,b), (y,c) \big) & :=\coprod_{f\colon b\to c} F_b(x, F_f y),\end{aligned}$$ and show that $Gr^\vee F$ admits a *fibration* to $B_{\mathbf{V}}$. The properties of $Gr^\vee F$ are formally dual to $Gr F$.
We next produce the functor that we want to show is an opfibration.
Let $F \colon B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ be a pseudofunctor, and $GrF$ its Grothendieck construction. There is a ${\mathbf{V}}$-functor $p \colon Gr F\to B_{\mathbf{V}}$.
We will describe the functor but leave checking of the necessary coherences to the reader. On objects, $p$ simply projects down to $B$, sending $(x,b)$ to $b$. On morphisms, we need to give a ${\mathbf{V}}$-morphism $$GrF\big( (x,b), (y,c)\big) = \coprod_{f \colon b \to c} F_b(F_f x,y) \to \coprod_{f\colon b\to c}{\mathbf{1}}= B_{\mathbf{V}}(b,c).$$ Since ${\mathbf{V}}$ is semi-cartesian, each $F_b(F_f x,y)$ has a unique map to ${\mathbf{1}}$. Taking the coproduct of these maps over $B(b,c)$, we obtain the desired morphism.
The Grothendieck construction $Gr$
----------------------------------
\[prop:grisopfib\] The functor $p \colon Gr F \to B_{\mathbf{V}}$ is an opfibration.
For all $(x,b) \in Gr F$ and $f \colon {\mathbf{1}}\to B_{\mathbf{V}}(b,c)$, we need ${{f}_!} (x,b) \in Gr F$ and a $p$-opcartesian lift $\chi\big(f, (x,b)\big) \colon {\mathbf{1}}\to Gr F\big( (x,b), {{f}_!} (x,b) \big)$.
Since we have assumed that $(B_{\mathbf{V}})_0 = B$, such an $f$ is precisely a map $f \colon b \to c$ in $B$. We may thus let ${{f}_!}(x,b) := (F_f x, c)$ and take $\chi\big(f, (x,b)\big)$ to be the identity of $F_f x$: $$\label{eq:Gr-chi}
\chi\big(f, (x,b)\big) \colon {\mathbf{1}}\xrightarrow{1_{F_f x}} F_b(F_f x, F_f x ) \hookrightarrow Gr F\big( (x,b), {{f}_!}(x,b) \big).$$ For every $(y,d) \in Gr F$, we need the following diagram to be a pullback, where we have written $\chi$ for $\chi\big( f, (x,b) \big)$: $$\begin{tikzcd}[column sep = large, ]
Gr F \big( {{f}_!}(x,b), (y,d) \big) \ar[d, "p"'] \ar[r, "-\circ \chi"] & Gr F\big( (x,b) , (y,d)\big) \ar[d, "p"]
\\
B_{\mathbf{V}}(c, d) \ar[r, "-\circ f"] & B_{\mathbf{V}}(b,d)
\end{tikzcd}$$ But by definition of the various objects involved and the pseudofunctoriality of $F$, the diagram above is equivalent to the diagram below, which is a pullback diagram since ${\mathbf{V}}$ is extensive: $$\begin{tikzcd}[column sep = large, ]
\coprod\limits_{g \colon c \to d} F_d(F_{gf} x, y)
\ar[d, "p"'] \ar[r, ] &
\coprod\limits_{h \colon b \to d} F_d(F_h x, y)
\ar[d, "p"]
\\
\coprod\limits_{g \colon c \to d} {\mathbf{1}}\ar[r, "-\circ f"] & \coprod\limits_{h \colon b \to d} {\mathbf{1}}\end{tikzcd}$$ In more detail, Property 2(i) ensures that the following is a pullback, where the horizontal arrows are given by re-indexing along $f$ (i.e. precomposition with $f$): $$\begin{tikzcd}[column sep = large, ]
F_d(F_{gf} x, y)
\ar[d, "p"'] \ar[r, hookrightarrow] &
\coprod\limits_{h \colon b \to d} F_d(F_h x, y)
\ar[d, "p"]
\\
{\mathbf{1}}_g \ar[r, hookrightarrow, "- \circ f"] & \coprod\limits_{h \colon b \to d} {\mathbf{1}}\end{tikzcd}$$ Property 2(ii) then implies that the diagram above it is also a pullback.
\[rem:assumption4used\] Note that we only have functors $F_f$ for $f \in (B_{\mathbf{V}})_0$ arising from actual maps in $B$. This is the reason for requiring $(B_{\mathbf{V}})_0 = B$ .
The construction $F\mapsto GrF$ extends to a 2-functor $$Gr \colon {{\mathsf{Fun}}^\mathsf{ps}}(B,{{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}})\to {{\mathsf{opFib}}}(B_{\mathbf{V}}).$$
From Proposition \[prop:grisopfib\] we have that the construction takes pseudofunctors to opfibrations. It remains to show that it takes pseudonatural transformations to opfibered functors, and modifications to natural transformations of opfibered functors.
Let $\alpha\colon F\Rightarrow G$ be a pseudonatural transformation between pseudofunctors $F,G \colon B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$. In particular, for every $b\in B$ we have a ${\mathbf{V}}$-functor $\alpha_b\colon F_b\to G_b$ such that for any $f\colon b\to c$ in $B$ we have an *invertible* 2-cell $\alpha_f \colon G_f\alpha_{b} \cong \alpha_b F_f$ with components $$\label{eq:alphaGFiso}
(\alpha_{f})_x \colon {\mathbf{1}}\to G_b(G_f \alpha_b x , \alpha_b F_f x).$$ Define an opfibered functor $a \colon GrF \to Gr G$ as follows: on objects, $$a (x,b) := (\alpha_b x, b).$$ On morphisms, take the composite: $$\label{eq:a-on-maps}
\begin{tikzcd}[row sep = small]
GrF\big( (x,b), (y,c)\big) \ar[rr, dotted, "a_{(x,b),(y,c)}"] \ar[d, equals] & & GrG\big( (\alpha_b x, b), (\alpha_b y, c) \big) \ar[d, equals]
\\
\coprod\limits_{f\colon b\to c} F_b(F_f x,y) \ar[r, "\alpha_b"]
&
\coprod\limits_{f\colon b\to c} G_b(\alpha_bF_f x,\alpha_b y) \ar[r, "-\circ (\alpha_{f})_x" , "\cong"']
&
\coprod\limits_{f\colon b\to c} G_b(G_f \alpha_b x,\alpha_b y)
\end{tikzcd}$$ Pseudofunctoriality of $F$ and $G$, along with pseudonaturality of $\alpha$, immediately indicate that these data assemble into a ${\mathbf{V}}$-functor $GrF\to GrG$ and that this ${\mathbf{V}}$-functor is compatible with the opfibrations $p\colon GrF\to B_{\mathbf{V}}$ and $q\colon GrG\to B_{\mathbf{V}}$ (i.e. $qa =p$). However, we must check that $a$ is actually opfibered.
By Proposition \[lem:opfibered-cocart\], it suffices to show that $a$ sends chosen $p$-opcartesian lifts to $q$-opcartesian maps. Recall from (\[eq:Gr-chi\]) that the chosen $p$-opcartesian lifts are induced by the identity maps $1_{F_f x}$ for $(x,b) \in Gr F$ and $f \colon b \to c$ in $B$. Since $\alpha_b$ is a ${\mathbf{V}}$-functor, these are sent to identity maps $1_{\alpha_b F_f x}$, which are precisely the chosen $q$-opcartesian lifts in $Gr G$. By Lemma \[lem:iso-to-chosen\], the composite of $1_{\alpha_b F_f x}$ with the isomorphism $(\alpha_{f})_x$ remains $q$-opcartesian, as desired.
Now it remains to show that $Gr$ takes modifications to natural transformations of opfibered functors. Let $\Gamma\colon\alpha{\Rrightarrow}\beta \colon F {\Rightarrow}G$ be a modification of pseudonatural transformations. This includes the data of natural transformations $\alpha_b {\Rightarrow}\beta_b \colon F_b \to G_b$ with components ${\mathbf{1}}\to G_b(\alpha_b x, \beta_b x)$ for each $b \in B$ and $x \in F_b$. Composing with the isomorphism $\alpha_b x \cong G_{1_b} \alpha_b x$, we obtain maps $${\mathbf{1}}\to G_b(G_{1_b} \alpha_b x, \beta_b x) \hookrightarrow Gr G\big((\alpha_bx,b),(\beta_bx,b) \big)$$ which are precisely the $(x,b)$-components of the desired natural transformation $Gr\Gamma \colon Gr\,\alpha{\Rightarrow}Gr\,\beta$.
The Grothendieck Correspondence {#sec:grcorr}
===============================
In this section we show that the Grothendieck construction of §\[sec:grcon\] and the inverse Grothendieck construction of §\[sec:invgrcon\] do behave as inverses when the base category ${\mathcal{B}}$ is of the form $B_{\mathbf{V}}$. Throughout, we make the same assumptions as §\[sec:grcon\], including the identification $B = (B_{\mathbf{V}})_0$ from Remark \[rem:identifyBBV0\].
$Gr \circ I$
------------
We first prove some properties of opfibrations over an arbitrary ${\mathcal{B}}$, before specializing to opfibrations over $B_{\mathbf{V}}$.
Let $p \colon {\mathcal{E}}\to {\mathcal{B}}$ be an opfibration, and let $q \colon Gr {\mathcal{E}}_\bullet \to ({\mathcal{B}}_0)_{\mathbf{V}}$ be the opfibration that results from applying $I$ and $Gr$ to $p$. The objects of $Gr {\mathcal{E}}_\bullet$ are pairs $(e, pe)$ where $e \in {\mathcal{E}}$, while the morphisms are $$Gr {\mathcal{E}}_\bullet\big( (e,pe), (e', pe') \big) = \coprod_{f \in {\mathcal{B}}_0(pe, pe')} {\mathcal{E}}_{pe'}({{f}_!}e, e').$$
For each $f \in {\mathcal{B}}_0(pe, pe')$, we have $${\mathcal{E}}_{pe'}({{f}_!}e,e') \cong {\mathcal{E}}_f(e,e')$$ where ${\mathcal{E}}_f(e,e')$ is defined to be the pullback: $$\label{eq:}
\begin{tikzcd}
{\mathcal{E}}_f(e,e') \arrow[dr, phantom, "\lrcorner", very near start] \ar[r] \ar[d] & {\mathcal{E}}(e,e') \ar[d, "p" ]
\\
{\mathbf{1}}\ar[r, "f"] & {\mathcal{B}}(pe,pe')
\end{tikzcd}$$
By Definition \[def:opfib\] and (\[eq:fiber-homs\]), we have a composite of pullbacks: $$\label{eq:GrI-Id-components}
\begin{tikzcd}[row sep = large]
{\mathcal{E}}_{pe'}({{f}_!}e, e') \ar[d] \ar[r] \arrow[dr, phantom, "\lrcorner", very near start]
&
{\mathcal{E}}({{f}_!}e,e') \ar[d, "p"'] \ar[r, "-\circ \chi{(f,e)}"] \arrow[dr, phantom, "\lrcorner", very near start]
&
{\mathcal{E}}(e,e') \ar[d, "p"]
\\
{\mathbf{1}}\ar[r, "1_{pe'}"] & {\mathcal{B}}(pe',pe') \ar[r, "-\circ f"] & {\mathcal{B}}(pe, pe')
\end{tikzcd}$$ But the outer cospan is also the defining cospan for the pullback ${\mathcal{E}}_f(e,e')$, hence these two pullbacks are isomorphic.
Let $\epsilon_p \colon Gr {\mathcal{E}}_\bullet \to {\mathcal{E}}$ denote the functor that sends $(e,pe)$ to $e$, and whose action on morphisms is induced by the upper horizontal composite in (\[eq:GrI-Id-components\]): $$Gr {\mathcal{E}}_\bullet \big( (e,pe), (e', pe') \big) = \coprod_{f \in {\mathcal{B}}_0(pe, pe')} {\mathcal{E}}_{pe'}({{f}_!}e, e')
\xrightarrow{\quad -\circ {\chi(f,e)} \quad} {\mathcal{E}}(e,e').$$
\[lem:GrI-pullback\] The functor $\epsilon_p \colon Gr {\mathcal{E}}_\bullet \to {\mathcal{E}}$ fits into the pullback $$\label{eq:GrI-pullback}
\begin{tikzcd}
Gr {\mathcal{E}}_\bullet \ar[dr, phantom, "\lrcorner", very near start] \ar[d, "q"'] \ar[r, "\epsilon_p"] & {\mathcal{E}}\ar[d, "p"]
\\
({\mathcal{B}}_0)_{\mathbf{V}}\ar[r, "\sigma_{\mathcal{B}}"] & {\mathcal{B}}\end{tikzcd}$$ where $\sigma$ is the counit of the adjunction (\[eq:CatVCat\]).
Note that $({\mathcal{B}}_0)_{\mathbf{V}}$ and ${\mathcal{B}}$ have the same objects and $\sigma_{\mathcal{B}}$ is the identity on objects. Morphisms of $({\mathcal{B}}_0)_{\mathbf{V}}$ are given by $$({\mathcal{B}}_0)_{\mathbf{V}}(b,b') = \coprod_{f \colon {\mathbf{1}}\to {\mathcal{B}}(b,b')} {\mathbf{1}},$$ and $\sigma_{\mathcal{B}}$ is the coproduct of the individual maps $f \colon {\mathbf{1}}\to {\mathcal{B}}(b,b')$, i.e. $$\sigma_{\mathcal{B}}= \coprod_{f \colon {\mathbf{1}}\to {\mathcal{B}}(b,b')} f.$$ The outer square in (\[eq:GrI-Id-components\]) then shows that the square in (\[eq:GrI-pullback\]) commutes.
To see that this square is a pullback, we first note that another pullback of $p$ along $\sigma_{\mathcal{B}}$ is given by the category ${\mathcal{P}}$ with the same objects as ${\mathcal{E}}$ and morphisms fitting into the pullback: $$\begin{tikzcd}
{\mathcal{P}}(e,e') \ar[dr, phantom, "\lrcorner", very near start] \ar[d] \ar[r] & {\mathcal{E}}(e,e') \ar[d, "p"]
\\
\coprod\limits_{f \colon {\mathbf{1}}\to {\mathcal{B}}(pe,pe')} {\mathbf{1}}\ar[r, "\coprod f"] & {\mathcal{B}}(pe, pe')
\end{tikzcd}$$ Since pullbacks preserve coproduct decompositions, we obtain $$\begin{aligned}
{\mathcal{P}}(e,e') &\cong \coprod_{f \colon {\mathbf{1}}\to {\mathcal{B}}(pe,pe')} {\mathcal{E}}_f(e, e') \\
&\cong \coprod_{f \colon {\mathbf{1}}\to {\mathcal{B}}(pe,pe')} {\mathcal{E}}_{pe'}({{f}_!}e,e') \\
&= Gr {\mathcal{E}}_\bullet(e,e'),
\end{aligned}$$ where the second isomorphism is given by the previous Lemma. So $Gr$ is isomorphic to ${\mathcal{P}}$, and is thus also a pullback.
Let $p' \colon {\mathcal{E}}' \to {\mathcal{B}}$ be another opfibration, and let $q' = GrI(p')$. For each opfibered functor $k \colon {\mathcal{E}}\to {\mathcal{E}}'$ from $p$ to $p'$, there is an opfibered functor $GrI(k) \colon Gr {\mathcal{E}}_\bullet \to Gr {\mathcal{E}}_\bullet'$ from $q$ to $q'$. On objects, $GrI(k)$ sends $(e, pe)$ to $(ke, pe)$, while the action on morphisms is induced by the composite $${\mathcal{E}}_{pe'}({{f}_!}e, e') \xrightarrow{\quad k \quad} {\mathcal{E}}'_{pe'}(k {{f}_!}e, ke') \cong {\mathcal{E}}'_{pe'}({{f}_!} ke, ke'),$$ where the isomorphism comes from (\[eq:alpha-f\]), and satisfies: $$\label{eq:chi-k}
\begin{tikzcd}
{\mathcal{E}}'_{pe'}(k{{f}_!}e, ke') \ar[r, "\cong"] \ar[d, "- \circ k \chi{(f,e)}"'] & {\mathcal{E}}'_{pe'}({{f}_!} ke, ke') \ar[d, "-\circ \chi{(f,ke)}"]
\\
{\mathcal{E}}'(ke, ke') \ar[r, equals] & {\mathcal{E}}'(ke, ke')
\end{tikzcd}$$ Let $k' \colon {\mathcal{E}}\to {\mathcal{E}}'$ be another opfibered functor and let $\gamma \colon k {\Rightarrow}k'$ be a natural transformation over ${\mathcal{B}}$. Since $\gamma$ lies over ${\mathcal{B}}$, its components factor as $$\gamma_e \colon {\mathbf{1}}\to {\mathcal{E}}'_{pe}(ke,k'e) \hookrightarrow {\mathcal{E}}'(ke, k'e).$$ The components $GrI(\gamma)_{(e, pe)}$ are then given by the composite $${\mathbf{1}}\xrightarrow{\gamma_{e}} {\mathcal{E}}'_{pe}(ke,k'e) \cong {\mathcal{E}}'_{pe}({{1}_!}ke, k'e) \hookrightarrow \coprod_{f \in {\mathcal{B}}_0(pe,pe)} {\mathcal{E}}'_{pe}({{f}_!}ke, k'e).$$
\[lem:GrI-2-natural\] The following diagram commutes for all opfibered functors $k, k'$ and natural transformations $\gamma$ over ${\mathcal{B}}$: $$\begin{tikzcd}[row sep = large, column sep = huge]
Gr {\mathcal{E}}_\bullet \ar[d, "\epsilon_p"'] \ar[r, "Gr I (k)", ""{name = U,below}, bend left = 17] \ar[r, "Gr I (k')"', ""{name = D, above}, bend right = 17] & Gr {\mathcal{E}}_\bullet' \ar[d, "\epsilon_{p'}"] \ar[from = U, to = D, Rightarrow, "\,Gr I(\gamma)", shift right = 3]
\\
{\mathcal{E}}\ar[r, "k", bend left = 17, ""{name = UU, below}] \ar[r, "k'"', bend right=17, ""{name = DD, above}] & {\mathcal{E}}' \ar[from = UU, to = DD, Rightarrow, "\;\gamma", shift right = 2]
\end{tikzcd}$$
We first verify that $\epsilon_{p'}\circ GrI(k) = k \circ \epsilon_p$ for any $k$. On objects, both composites send $(e, pe)$ to $ke$. By (\[eq:chi-k\]) and the functoriality of $k$, the following diagram commutes, $$\begin{tikzcd}
{\mathcal{E}}_{pe'}({{f}_!}e, e') \ar[r, "k"] \ar[d, "- \circ \chi{(f,e)}"'] & {\mathcal{E}}'_{pe'}(k{{f}_!}e, ke') \ar[r, "\cong"] \ar[d, "- \circ k \chi{(f,e)}"] & {\mathcal{E}}'_{pe'}({{f}_!} ke, ke') \ar[d, "-\circ \chi{(f,ke)}"]
\\
{\mathcal{E}}(e,e') \ar[r, "k"] & {\mathcal{E}}'(ke, ke') \ar[r, equals] & {\mathcal{E}}'(ke, ke')
\end{tikzcd}$$ which implies that the corresponding diagram on morphisms commutes.
Next, we verify that $\epsilon_{p'} \circ GrI(\gamma) = \gamma \circ \epsilon_p$ for any $\gamma$. By Definition \[def:whiskering\], $(\gamma \circ \epsilon_p)_{(e,pe)} = \gamma_{\epsilon_p(e,pe)} = \gamma_e$, while $\left(\epsilon_{p'} \circ GrI(\gamma)\right)_{(e,pe)}$ is the composite $${\mathbf{1}}\xrightarrow{\gamma_{e}} {\mathcal{E}}'_{pe}(ke,k'e) \cong {\mathcal{E}}'_{pe}({{1}_!}ke, k'e) \cong {\mathcal{E}}'_{pe}(ke,k'e) \hookrightarrow {\mathcal{E}}'(ke, k'e),$$ which is again just $\gamma_e$.
We now specialize to the case ${\mathcal{B}}= B_{\mathbf{V}}$. Recall from Remark \[rem:identifyBBV0\] that we have $(B_{\mathbf{V}})_0 = B$, so that $Gr$ and $I$ fit into the diagram: $$\begin{tikzcd}
{{\mathsf{opFib}}}(B_{\mathbf{V}}) \ar[r, bend left, "I"] & {{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}) \ar[l, bend left, "Gr"]
\end{tikzcd}$$
\[prop:IGr\] The maps $\epsilon_p \colon Gr {\mathcal{E}}_\bullet \to {\mathcal{E}}$ (for opfibrations $p \colon {\mathcal{E}}\to B_{\mathbf{V}}$) are the components of a $2$-natural isomorphism $$\epsilon \colon Gr \circ I \Rightarrow 1_{{{\mathsf{opFib}}}(B_{\mathbf{V}})}.$$
Lemma \[lem:GrI-2-natural\] is precisely the statement of $2$-naturality of $\epsilon$, while Lemma \[lem:GrI-pullback\] implies that $\epsilon$ is an isomorphism.
In detail, setting ${\mathcal{B}}=B_{\mathbf{V}}$ for some category $B$, we have $$\left((B_{\mathbf{V}})_0 \right)_{\mathbf{V}}= B_{\mathbf{V}}$$ so that $\sigma_{B_{\mathbf{V}}}$ is $1_{B_{\mathbf{V}}}$. By Lemma \[lem:GrI-pullback\], each $q\colon Gr {\mathcal{E}}_\bullet \to B_{\mathbf{V}}$ is a pullback of $p \colon {\mathcal{E}}\to B_{\mathbf{V}}$ along an identity, hence is isomorphic to $p$ via $\epsilon_p$.
As a consequence of the isomorphism $Gr {\mathcal{E}}_\bullet \cong {\mathcal{E}}$ when ${\mathcal{E}}$ is opfibered over $B_{\mathbf{V}}$, we obtain a coproduct decomposition $${\mathcal{E}}(e,e') \cong \coprod_{f \in B(pe,pe')} {\mathcal{E}}_f(e,e').$$ In fact, as long as pullbacks preserve coproduct decompositions, *any* functor into a free ${\mathbf{V}}$-category $p \colon {\mathcal{E}}\to B_{\mathbf{V}}$ yields such a coproduct decomposition simply by pulling $p$ back along $1_{B_{\mathbf{V}}}$. This does not require $p$ to be an opfibration, nor $(B_{\mathbf{V}})_0 = B$.
The results above also answer the question: when can we recover our original opfibration $p \colon {\mathcal{E}}\to {\mathcal{B}}$ from its pseudofunctor ${\mathcal{E}}_\bullet$? In other words, when can hope for an equivalence $${{\mathsf{opFib}}}({\mathcal{B}}) \cong {{\mathsf{Fun}}^\mathsf{ps}}({\mathcal{B}}_0, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}})?$$ One sees that this can happen only if the counit $\sigma_{\mathcal{B}}\colon ({\mathcal{B}}_0)_{\mathbf{V}}\to {\mathcal{B}}$ of the adjunction $(-)_{\mathbf{V}}\dashv (-)_0$ is an equivalence. But since we have also assumed that the unit $\iota_B$ is an equivalence, this means that ${\mathbf{V}}$ is equivalent to ${{\mathsf{Set}}}$.
$I \circ Gr$
------------
\[prop:GrI\] There is a $2$-natural isomorphism $$\eta \colon 1_{{{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}})} \Rightarrow I \circ Gr.$$
Let $F\colon B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ be a pseudofunctor and let $G$ be the pseudofunctor $(Gr F)_\bullet$ obtained by applying $Gr$ and $I$ to $F$. We need a $2$-natural family of isomorphisms (i.e. invertible pseudonatural transformations) $\eta_F \colon F \Rightarrow G$.
For a fixed $F$, we will write $\eta$ instead of $\eta_F$ for brevity. We first produce functors $\eta_b \colon F_b \to G_b$ for each $b \in B$, and show that each $\eta_b$ is an isomorphism of categories. Objects of $G_b$ are of the form $(x, b)$ where $x \in F_b$, so we may take the action of $\eta_b$ on objects to be $x \mapsto (x,b)$. The hom-objects $G_b\big( (x,b), (y,b)\big)$ are precisely the $f = 1_b$ part of $$GrF\big( (x,b), (y,b)\big) = \coprod_{f \colon b \to b} F_b(F_f x, y).$$ We then take $\eta_b$ on morphisms to be the composite $$\begin{tikzcd}
F_b(x,y) \ar[r, "\cong"', "- \circ \xi_x"] & F_b(F_{1_b}x,y) = G_b\big((x,b),(y,b)\big).
\end{tikzcd}$$ This yields an isomorphism of categories $\eta_b \colon F_b \cong G_b$.
Next, for each $f \colon b \to c$, we need an invertible $\eta_f$: $$\begin{tikzcd}
F_b \ar[r, "F_f"]\ar[d, "\eta_b", swap] & F_b \ar[d, "\eta_b"]\\
G_b \ar[r, "G_f", swap] \ar[ur, Rightarrow,"\eta_f", "\cong"', start anchor={north east}, end anchor={south west}, shorten <=0.7em, shorten >=0.7em]& G_b
\end{tikzcd}$$ In fact, we may take $\eta_f$ to be the identity. To see this, first note that $G_f$ is given by $
G_f (x,b) = (F_f x,c)
$ on objects and $$\begin{tikzcd}[row sep = small]
G_b\big((x,b),(y,b)\big)\ar[d, equals] \ar[rr, dotted, "G_f"] & & G_b\big((F_f x,c),( F_f y ,c) \big) \ar[d, equals]
\\
F_b(F_{1_b}x, y) \ar[r, "F_f"] & F_b(F_f F_{1_b}x, F_f y) \ar[r, "\cong"] & F_b(F_{1_b} F_f x, F_f y)
\end{tikzcd}$$ on morphisms. Expressing both $\eta_b F_f$ and $G_f \eta_b$ in terms of $\xi$ and $\theta$, we see that the relevant diagram is given by $$\label{eq:alphaf}
\begin{tikzcd}
{\mathbf{1}}\ar[rr, "\xi_{F_f x}"] \ar[d, "\xi_x"'] & & F_b(F_{1_b} F_f x, F_f x)
\\
F_b(F_{1_b}x,x) \ar[r, "F_f"] & F_b(F_f F_{1_b}x,F_f x) \ar[r, "\theta({f, 1_b})^{-1}", "\cong"'] & F_b(F_f x, F_f x) \ar[u, "\theta({1_b, f})"', "\cong"]
\end{tikzcd}$$ tensored with $F_b(x,y) \xrightarrow{F_f} F_b(F_fx,F_f y)$, then applying composition in $F_b$. The arrow $\xi_{F_f x}$ gives rise to $\eta_b F_f$, while the composite $
\theta(1_b, f) \theta(f, 1_b)^{-1} F_f \xi_x
$ gives rise to $G_f \eta_b$, so that $\eta_b F_f = G_f \eta_b$ if (\[eq:alphaf\]) commutes. This is the case, because the following diagram commutes (both composites being equal to $1_{F_f x}$ by (\[eq:pseudo-unit-left\]) and (\[eq:pseudo-unit-right\])) $$\begin{tikzcd}
{\mathbf{1}}\ar[rr, "\xi_{F_f x}"] \ar[d, "\xi_x"'] & & F_b(F_{1_b} F_f x, F_f x) \ar[d, "\theta({1_b, f})^{-1}", "\cong"']
\\
F_b(F_{1_b}x,x) \ar[r, "F_f"] & F_b(F_f F_{1_b}x,F_f x) \ar[r, "\theta({f, 1_b})^{-1}", "\cong"'] & F_b(F_f x, F_f x)
\end{tikzcd}$$ and $\theta(1_b,f)$ and $\theta(1_b, f)^{-1}$ are mutual inverses.
We thus have invertible pseudonatural transformations $\eta_F \colon F {\Rightarrow}G$ for each $F \in {{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}})$. It remains to be shown that these are part of a $2$-natural transformation $\eta \colon 1 {\Rightarrow}I \circ Gr$. Let $F'\colon B \to {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ be another pseudofunctor, and $G' = (Gr F')_\bullet$. For pseudonatural transformations $\alpha, \alpha' \colon F {\Rightarrow}F'$ and modification $\Gamma \colon \alpha {\Rrightarrow}\alpha'$ we need the following diagram to commute: $$\begin{tikzcd}[row sep = large, column sep = huge]
F \ar[d, "\eta_F"', Rightarrow] \ar[r,"\alpha", Rightarrow, bend left= 20, ""{name = U, below}] \ar[r,"\alpha'"', Rightarrow, bend right=20, ""{name = D, above}] & F' \ar[d, "\eta_{F'}", Rightarrow] \ar[from = U, to = D, symbol = {\Rrightarrow}, "\;\Gamma"]
\\
G \ar[r, "I Gr(\alpha)", Rightarrow, bend left = 20, ""{name = UU, below}] \ar[r, "IGr(\alpha')"', Rightarrow, bend right = 20, ""{name = DD, above}] \ar[from = UU, to = DD, symbol = {\Rrightarrow}, "\,IGr(\Gamma)", shift right = 4] & G'
\end{tikzcd}$$
We first verify that $\eta_{F'} \cdot \alpha = IGr(\alpha)\cdot \eta_F$ for any $\alpha$. Fixing $b \in B$, the appropriate diagram on objects (of $F_b, F'_b$ etc.) obviously commutes, while on morphisms we need the following diagram to commute: $$\begin{tikzcd}[sep = large]
F_b(x,y) \ar[rr, "\alpha_b"] \ar[d, "- \circ \xi_x"'] & & F'_b(\alpha_b x, \alpha_b y) \ar[d, "- \circ \xi'_{\alpha_b x}"]
\\
F_b(F_{1_b}x, y) \ar[r, "\alpha_b"] & F'_b(\alpha_b F'_{1_b} x, \alpha_b y) \ar[r, "- \circ (\alpha_{1_b})_x"] & F'_b(F'_{1_b} \alpha_b x, \alpha_b y)
\end{tikzcd}$$ But this is precisely the second axiom in [@leinsterbicats §1.2] that $\alpha$ satisfies. Similarly, for each $f \colon b\to c$ in $B$, the relevant diagram involving $\alpha_f$ also commutes because of the first axiom in [@leinsterbicats §1.2] that $\alpha$ satisfies.
Next, we verify that $\eta_{F'} \cdot \Gamma = IGr(\Gamma) \cdot \eta_F$ for any $\Gamma$. The components of $\Gamma$ are natural transformations $\Gamma_b \colon \alpha_b {\Rightarrow}\alpha'_b$ for each $b \in B$, which in turn have components $\Gamma_{b,x} \colon {\mathbf{1}}\to F'_b(\alpha_b x, \alpha'_b x)$ for each $x \in F_b$. The components $(\eta_{F}\cdot \Gamma)_{b,x}$ are given by the composite $${\mathbf{1}}\xrightarrow{\Gamma_{b,x}} F'_b(\alpha_b x, \alpha'_b x) \cong F'_b(F'_{1_b}\alpha_b x, \alpha'_b x) = G'_b\left( (\alpha_b x, b), (\alpha'_b x, b) \right),$$ but these are exactly the components $IGr(\Gamma)_{b, (x,b)}$. Since $(\eta_F)_b$ sends $x \in F_b$ to $(x,b)$, we have $(IGr(\Gamma) \cdot \eta_F)_{b,x} = IGr(\Gamma)_{b, (x,b)} = (\eta_{F}\cdot \Gamma)_{b,x}$, as desired.
Putting Propositions \[prop:IGr\] and \[prop:GrI\] together, we obtain:
\[mainthm\] Let ${\mathbf{V}}$ satisfy the assumptions in §\[sec:grcon\], and let $B$ be a category. There is a $2$-equivalence $${{\mathsf{opFib}}}(B_{\mathbf{V}}) \cong {{\mathsf{Fun}}^\mathsf{ps}}(B, {{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}).$$
We have proved an enriched version of the Grothendieck correspondence when ${\mathbf{V}}$ satisfies the assumptions in §\[sec:grcon\], which yields the classical result by Grothendieck when ${\mathbf{V}}= {{\mathsf{Set}}}$.
However, this is somewhat unsatisfactory for reasons we have mentioned in the Introduction. First, requiring that ${\mathbf{1}}$ is terminal and $B \cong (B_{\mathbf{V}})_0$ seems rather restrictive, ruling out examples such as ${\mathbf{V}}= {{\mathsf{Vect}}_k}$. Next, even when these conditions apply, such as when ${\mathbf{V}}= {{\mathsf{Top}}}, {{\mathsf{sSet}}}$ or ${{\mathsf{Cat}}}$, this result really only considers opfibrations over a ‘discrete’ base $B_{\mathbf{V}}$. Subsequent work will involve removing various assumptions on ${\mathbf{V}}$ and retaining more ${\mathbf{V}}$-structure from an opfibration over a non-discrete base.
Appendix: Enriched categories and ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ {#sec:appendix}
============================================================================
We recall some basic information about (small) enriched categories (all of which can be found in [@riehlhct Ch. 3], or [@kellyenriched]). Throughout, ${\mathbf{V}}$ will denote a locally small monoidal category with monoidal product $\otimes \colon {\mathbf{V}}\times {\mathbf{V}}\to {\mathbf{V}}$ and monoidal unit ${\mathbf{1}}$.
A *${\mathbf{V}}$-category* ${\mathcal{C}}$ is the data of:
1. A set of objects, which we will denote by ${\mathcal{C}}$ or $Ob({\mathcal{C}})$, where the former is an obvious abuse of notation.
2. For every pair of objects $c,d$ in ${\mathcal{C}}$, an object ${\mathcal{C}}(c,d)$ of ${\mathbf{V}}$.
3. For every object $c$ in ${\mathcal{C}}$ a morphism $1_c \colon {\mathbf{1}}\to{\mathcal{C}}(c,c)$ in ${\mathbf{V}}$.
4. For each triple of objects $c, d, e$ in ${\mathcal{C}}$, a morphism in ${\mathbf{V}}$, $$\circ_{c,d,e}\colon {\mathcal{C}}(d,e)\otimes{\mathcal{C}}(c,d)\to {\mathcal{C}}(c,e).$$ We will omit subscripts on $\circ$ when it is clear from context.
All of which causes the following diagrams to commute in ${\mathbf{V}}$:
$$\begin{aligned}
\begin{tikzcd}
& {\mathcal{C}}(d,d)\otimes {\mathcal{C}}(c,d)\arrow[d, "\circ"]
\\
{\mathbf{1}}\otimes{\mathcal{C}}(c,d)\arrow[ur, "1_d \otimes 1"]\arrow[r, "\cong"'] & {\mathcal{C}}(c,d)
\end{tikzcd}
\end{aligned}
\quad
\begin{aligned}
\begin{tikzcd}
{\mathcal{C}}(c,d)\otimes {\mathcal{C}}(c,c)\arrow[d, "\circ"'] &
\\
{\mathcal{C}}(c,d) & {\mathcal{C}}(c,d)\otimes{\mathbf{1}}\arrow[ul, "1 \otimes 1_c"']\arrow[l, "\cong"]
\end{tikzcd}
\end{aligned}$$ $$\begin{tikzcd}
{\mathcal{C}}(d,e)\otimes{\mathcal{C}}(c,d)\otimes{\mathcal{C}}(b,c)\arrow[r, "1\otimes\circ"]\arrow[d, "\circ\otimes1"'] & {\mathcal{C}}(d,e)\otimes{\mathcal{C}}(b,d)\arrow[d, "\circ"]\\ {\mathcal{C}}(c,e)\otimes {\mathcal{C}}(b,c)\ar[r, "\circ"] & {\mathcal{C}}(b,e)
\end{tikzcd}$$
Each ${\mathbf{V}}$-morphism $f \colon {\mathbf{1}}\to {\mathcal{C}}(b,c)$ (i.e. a map in the underlying category ${\mathcal{C}}_0$) induces *pre-* and *post-composition* ${\mathbf{V}}$-morphisms: $$\begin{tikzcd}
-\circ f \colon {\mathcal{C}}(c,d) \cong {\mathcal{C}}(c,d)\otimes {\mathbf{1}}\ar[r, "1\otimes f"] & {\mathcal{C}}(c,d) \otimes {\mathcal{C}}(b,c) \ar[r, "\circ"] & {\mathcal{C}}(b,d)
\\
f \circ - \colon {\mathcal{C}}(a,b) \cong {\mathbf{1}}\otimes {\mathcal{C}}(a,b) \ar[r, "f \otimes 1"] & {\mathcal{C}}(b,c) \otimes {\mathcal{C}}(a,b) \ar[r, "\circ"] & {\mathcal{C}}(a,c)
\end{tikzcd}$$ We say that $f$ is an *isomorphism* if the above composites are ${\mathbf{V}}$-isomorphisms for all $a,d \in {\mathbf{V}}$, and that $b$ and $c$ are *isomorphic*. This is equivalent to ${\mathcal{C}}(-,b)$ and ${\mathcal{C}}(-,c)$ being isomorphic functors ${\mathcal{C}}_0 \to {\mathbf{V}}$, with ${\mathcal{C}}(-,f) = f\circ -$ the natural isomorphism between them.
\[def:Vfunctor\] A functor of ${\mathbf{V}}$-categories, or *${\mathbf{V}}$-functor*, $F\colon {\mathcal{C}}\to {\mathcal{D}}$ consists of a function $F\colon Ob({\mathcal{C}})\to Ob({\mathcal{D}})$, and for all $c,d \in {\mathcal{C}}$ a ${\mathbf{V}}$-morphism $$F_{c,d}\colon {\mathcal{C}}(c,d)\to {\mathcal{D}}(Fc,Fd)$$ such that the following diagrams commute in ${\mathbf{V}}$: $$\begin{aligned}
\begin{tikzcd}
& {\mathcal{C}}(c,c) \ar[dd, "F_{c,c}"]
\\
{\mathbf{1}}\ar[ur, "1_b"] \ar[dr, "1_{Fc}"'] &
\\
& {\mathcal{D}}(Fc,Fc)
\end{tikzcd}
\end{aligned}
\qquad
\begin{aligned}
\begin{tikzcd}
{\mathcal{C}}(d,e)\otimes {\mathcal{C}}(c,d) \ar[r, "\circ"] \ar[dd, "F_{d,e} \otimes F_{c,d}"'] & {\mathcal{C}}(c,e) \ar[dd, "F_{c,e}"]
\\
& \phantom{{\mathbf{1}}}
\\
{\mathcal{D}}(Fd, Fe) \otimes {\mathcal{D}}(Fc,Fd) \ar[r,"\circ"] & {\mathcal{D}}(Fc,Fe)
\end{tikzcd}
\end{aligned}$$
When it is clear from context, we may omit the subscripts in $F_{c,d}$, and use $F$ for the functor ${\mathcal{C}}\to {\mathcal{D}}$, the function $Ob({\mathcal{C}}) \to Ob({\mathcal{D}})$ and the ${\mathbf{V}}$-morphism ${\mathcal{C}}(c,d) \to {\mathcal{D}}(Fc,Fd)$.
Let $F,G\colon {\mathcal{C}}\to {\mathcal{D}}$ be ${\mathbf{V}}$-functors. A *natural transformation* of ${\mathbf{V}}$-functors $\alpha\colon F\Rightarrow G$ is a family of ${\mathbf{V}}$-morphisms $\alpha_b\colon {\mathbf{1}}\to {\mathcal{D}}(Fc,Gc)$ for each $c \in {\mathcal{C}}$ such that the following diagram commutes in ${\mathbf{V}}$: $$\begin{tikzcd}[sep = large]
{\mathcal{C}}(c,d) \ar[r, "F"] \ar[d, "G"'] & {\mathcal{D}}(Fc,Fd) \ar[d, "\alpha_d \circ -"]
\\
{\mathcal{D}}(Gc, Gd) \ar[r, "- \circ \alpha_b"] & {\mathcal{D}}(Fc, Gd)
\end{tikzcd}$$
\[def:whiskering\] Given ${\mathbf{V}}$-functors $F,G,H,K$, and a natural transformation $\alpha$ fitting into the following diagram, $$\begin{tikzcd}
{\mathcal{B}}\ar[r, "H"]
&
{\mathcal{C}}\ar[r, bend left, "F", ""{name = U, below}] \ar[r, bend right, "G"', ""{name = D}] \ar[from = U, to = D, Rightarrow, "\alpha"]
&
{\mathcal{D}}\ar[r, "K"]
&
{\mathcal{E}}\end{tikzcd}$$ let $K \circ \alpha \circ H \colon KFH {\Rightarrow}KGH$ (or simply $K \alpha H$) denote the natural transformation whose components for each $b \in {\mathcal{B}}$ are given by the composite $$\begin{tikzcd}
(K \alpha H)_b \colon
{\mathbf{1}}\ar[r, "\alpha_{Hb}"]
&
{\mathcal{D}}(FHb, GHb) \ar[r, "K"]
&
{\mathcal{E}}(KFHb, KGHb)
\end{tikzcd}
.$$ This process is known as **whiskering**, and $K \alpha H$ is the **whiskered composite** of $K$, $\alpha$ and $H$. If $K$ (resp. $H$) is the identity, we write $\alpha H$ (resp. $K \alpha$) for the corresponding whiskered composite.
Let $\alpha\colon F {\Rightarrow}G$ and $\beta\colon G {\Rightarrow}H$ be natural transformations, where $F,G,H\colon {\mathcal{C}}\to {\mathcal{D}}$. Their **vertical composite** is denoted $\beta \cdot \alpha\colon F {\Rightarrow}H$, and has components $(\beta \cdot \alpha)_b$ given by $${\mathbf{1}}\cong {\mathbf{1}}\otimes {\mathbf{1}}\xrightarrow{\beta_b \otimes \alpha_b} {\mathcal{D}}(Gc,Hc) \otimes {\mathcal{D}}(Fc,Gc) \xrightarrow{\circ} {\mathcal{D}}(Fc,Hc).$$
Given $\alpha \colon F {\Rightarrow}G$ and $\beta \colon J {\Rightarrow}K$ as follows $$\begin{tikzcd}
{\mathcal{B}}\ar[r, bend left, "F", ""{name = U1, below}] \ar[r, bend right, "G"', ""{name = D1}] \ar[from = U1, to = D1, Rightarrow, "\alpha"]
&
{\mathcal{C}}\ar[r, bend left, "J", ""{name = U2, below}] \ar[r, bend right, "K"', ""{name = D2}] \ar[from = U2, to = D2, Rightarrow, "\beta"]
&
{\mathcal{D}}\end{tikzcd}$$ their **horizontal composite** $\beta \circ \alpha \colon JF {\Rightarrow}KG$, or simply $\beta \alpha$, is the composite $(\beta G) \cdot (J\alpha)$, or equivalently, the composite $(K \alpha) \cdot (\beta F) $.
Let ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ denote the strict 2-category of ${\mathbf{V}}$-categories, ${\mathbf{V}}$-functors, and natural transformations.
[^1]: Even if ${\mathbf{V}}$ is complete and symmetric monoidal closed, ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$ is only enriched over ${{\mathbf{V}}\textsf{-}{{\mathsf{Cat}}}}$, not ${\mathbf{V}}$.
|
---
abstract: 'We study integrability –in the sense of admitting recursion operators– of two nonlinear equations which are known to possess compacton solutions: the $K(m,n)$ equation introduced by Rosenau and Hyman $$D_t(u) + D_x(u^m) + D_x^3(u^n) = 0 \; ,$$ and the $CSS$ equation introduced by Coooper, Shepard, and Sodano, $$D_t(u) + u^{l-2}D_x(u) + \alpha p D_x (u^{p-1} u_x^2) + 2\alpha D_x^2(u^p u_x) = 0 \; .$$ We obtain a full classification of [*integrable $K(m,n)$ and $CSS$ equations*]{}; we present their recursion operators, and we prove that all of them are related (via nonlocal transformations) to the Korteweg-de Vries equation. As an application, we construct isochronous hierarchies of equations associated to the integrable cases of $CSS$.'
author:
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R Hernández Heredero$^1$, M Euler$^2$, N Euler$^2$ and E G Reyes$^3$\
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date: 'April 2, 2019'
title: 'Compacton equations and integrability: the Rosenau-Hyman and Cooper-Shepard-Sodano equations'
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Introduction
============
We begin by quoting Rosenau [@Ro]: ‘’We define a [*compact wave*]{} as a robust solitary wave with [*compact support*]{} beyond which it vanishes identically. We then define a [*compacton*]{} as a compact wave that preserves its shape after interacting with other compacton’’. Rosenau and Hyman found examples of compactons while studying generalizations of the Korteweg-de Vries equation for which the dispersion term is nonlinear. Their model equation is the so-called $K(m,n)$ equation $$\label{rhintro}
u_t + (u^m)_x + (u^n)_{xxx} = 0 \: ,$$ and an example of a compacton bearing equation within the family is $K(2,2)$. In this case, the function $u(x,t) = (4 c/3) \cos^2((x-c t)/4)$ for $|x-c t| \leq 2 \pi$ and $u(x,t) = 0$ otherwise, is a compacton solution. Further works on compactons are [@LD; @LOR] and the comprehensive review [@RZ].
It turns out that solutions to equations within the $K(m,n)$ can exhibit very complex behaviors; we refer the reader to [@AW; @AW2; @ASWY], and to the papers [@LOR; @RZ; @ZR] authored by Rosenau and his coworkers, for general discussions. Here, we just mention one example: in [@RH] the authors present four local conservation laws of $K(2,2)$, and credit P.J. Olver with the observation that no further local conservation laws seem to exist[^1]. This (non)existence of conservation laws has an important analytic implication, see [@ZR]: initially nonnegative, smooth and compactly supported solutions to $K(m,n)$ lose their smoothness within a finite time.
We wonder if this complex behavior has to do with (lack of) integrability. In this work we present a detailed study of the integrability properties of $K(m,n)$. We find that, module a rather general space of allowable transformations, the only integrable equations belonging to the $K(m,n)$ family are the KdV and modified KdV equations, and that integrable equations within the $K(m,n)$ family cannot have compacton solutions. In particular, we recover the observation in [@HHR; @Vo] that $K(2,2)$ is not integrable.
In order to obtain this result we classify all integrable $K(m,n)$ equations using the theory of formal symmetries (to be summarized in Section 2). The power of this approach has been amply demonstrated by the classification results for evolution equations and systems of equations due to researchers such as Shabat, Fokas, Svinolupov, Sokolov, Mikhailov and others (see [@MSS; @Fok; @SviSok; @HSSv; @Her1; @Her2]), and also by the important papers [@SW; @SW1] on the classification of integrable scalar evolution equations satisfying an homogeneity condition.
Since our search for integrable compacton bearing equations within the $K(m,n)$ class does not yield examples, we also investigate a related family, the Cooper-Shepard-Sodano family of equations $$u_{t}+u^{l-2}u_{x}-\alpha pD_{x}\left(u^{p-1}u_{x}^{2}\right)+2\alpha D_{x}^{2}\left(u^{p}u_{x}\right)=0, \qquad\alpha\neq0 \; , \label{cssintro}$$ introduced in [@CSS]. We quote from this paper: “These equations have the same terms as the equations considered by Rosenau and Hyman, but the relative weights of the terms are quite different leading to the possibility that the integrability properties might be different”. The authors of [@CSS] then proceed to show that their family of equations indeed admits compacton bearing equations. One such equation is with $l=3$, $p=2$. This family of equations is further studied in [@KC; @DK].
Encouraging properties of (\[cssintro\]) are the facts that it admits a Hamiltonian formulation, and that it possesses three physically interesting conservation laws: area, mass and energy. Regretfully, we prove herein that they are not integrable in general. Using formal symmetries once more, we obtain [*six*]{} integrable equations within the (\[cssintro\]) family. None of them can support compacton solutions.
Since the existence of compacton solutions [*is*]{} a rather extraordinary occurrence in the nonlinear world, we believe that our results are not only important by themselves, but also because they seem to express certain rigidity in our present algebraic/geometric/analytic approach to integrability. In other words, $K(2,2)$ say, must be “special”, and so far we have not been able to uncover the deeper source of its special character.
Our paper is organized as follows. We review the theory of formal symmetries and integrability in Section 2 after, essentially, [@MSS], and in Section 3 we use this theory to classify integrable $K(m,n)$ equations. We note that a previous classification has appeared in [@HHR]. One integrable case was missing therein and we single it out here. Fortunately, the missing case does not alter the conclusion in [@HHR] that the only $K(m,n)$ integrable case are (essentially, module a class of allowable transformations specified in Section 3) the KdV and mKdV equations. The present classification also differs from the one appearing in [@HHR] in that here we explain in detail how to connect our integrable cases to KdV (or, to the linear equation) and because in Section 5 we exhibit explicit recursion operators for all our integrable $K(m,n)$ equations. In Section 4 we study integrability of the Cooper-Shepard-Sodano family and again we are able to explain how to connect its integrable cases to KdV (or, to the linear equation), and to exhibit recursion operators. Finally in Section 6 we present an application of our results: we construct integrable isochronous equations, after [@Calogero-2005; @Calogero-Mariani-2005; @Calogero-Euler-Euler], starting from the equations in our classification of integrable CSS equations, explain how to obtain their point symmetries, and present their corresponding recursion operators.
Formal symmetries and integrability
===================================
The formal symmetry approach to integrability [@MSS; @MikSok] begins with the observation that standard (systems of) partial differential equations which are integrable (for instance, in the sense of Calogero, see [@Ca]) usually admit an infinite set of (generalized) symmetries of arbitrarily large differential order. A.B. Shabat and his collaborators, see for instance [@MSS; @MikSok], realized that it is possible to weaken the notion of a (generalized) symmetry to the notion of a [*formal*]{} symmetry —to be defined precisely below— and that this new concept provides a computationally efficient tool for defining integrability and classifying integrable equations. We recall from [@MSS; @O] that $G = (G^\alpha)$ is a symmetry of a system of partial differential equations of the form $\Delta_a(x^i,u^\alpha,u^\alpha_{x^i},\dots) =0$, if $$\label{genSym}
\Delta_*(G)=0 \;$$ whenever $u^\alpha(x^i)$ is a solution to $\Delta_a=0$, where $\Delta_*$ is the formal linearization of the system $\Delta_a=0$, that is, $\Delta_*=\left( \sum_{L}\frac{\partial\Delta_a}{\partial u_L^\alpha}D_L \right)$. If the system $\Delta_a=0$ consists of just one scalar evolution equation, $$\label{de}
\Delta=u_t-F \; ,$$ then equation becomes $D_t G=F_*(G)$ or, equivalently, $$\label{evolSym}
D_tG = D_\tau F \; ,$$ where $${D}_{\tau} = \sum_{\# K \geq 0} {D}_{K}(G) \,\frac{\partial\ }{\partial u_{K}} \; .$$
[*Note: Here and henceforth we use standard notation from the geometric theory of differential equations as presented in [[@O]]{}, see also [[@MSS]]{}.*]{}
Following [@MSS; @MikSok], we apply a second linearization to formula . We obtain, using some formulae appearing in [@MSS], $$\label{dtaulocal2}
(D_tG)_*=(D_\tau F)_*\quad\Leftrightarrow\quad D_t(G_*)+G_*\circ F_*=D_\tau(F_*)+F_*\circ G_* \; ,$$ in which $D_t ( \sum_{L} a_L D_L ) = \sum_{L} D_t( a_L) D_L$ if $G_\ast = \sum_{L} a_L D_L$, and the last equality holding on solutions to . The expression $D_\tau(F_*)$ is defined analogously. We interpret our symmetry condition (\[dtaulocal2\]) using commutators: $$\label{symOp}
D_t(G_*)-[F_*,G_*]=D_\tau(F_*) \; .$$
Let us consider the degree of the operators appearing in . The degree of $F_*$ as a differential operator —let us denote it by $\deg(F_*)$— is the differential order of $F$, i.e. the order of the differential equation and thus, it is fixed. The degree of the left hand side of depends on $G$: $D_t(G_*)-[F_*,G_*]$ is a differential operator generically of degree $\deg(G_*$) plus $\deg(F_*)$ minus 1, much higher than that of the operator in the right hand side, of degree $\deg(F_*)$, if there are high order symmetries $G$. Thus, it is not clear at all that non-trivial solutions to should exist: the existence of (generalized) symmetries $G$ of arbitrarily high differential order must impose extremely strong constraints on the function $F$.
Following [@MSS; @MikSok], and partially motivated by the theory of recursion operators, see [@O], we define formal symmetries using the left hand side of Equation :
\[fs\] Let $u_t=F$ be an evolution equation with $F$ a function of two independent variables $x$, $t$, one dependent variable $u$ and a finite number of derivatives of $u$ with respect to $x$. A formal symmetry of rank $k$ of this partial differential equation is a formal pseudo-differential operator $$\label{pdo}
\Lambda=l_rD^r+l_{r-1}D^{r-1}+\cdots+l_0+l_{-1}D^{-1}+l_{-2}D^{-2}+\cdots,\qquad D=D_x$$ with $l_i$ being functions of $t$, $x$, $u$ and finite numbers of $x$-derivatives of $u$, that satisfies the equation $$\label{formSym}
D_t(\Lambda)=[F_*,\Lambda]$$ whenever $u$ is a solution to $u_t =F$, up to a pseudo-differential operator of degree $r+\deg(F_*)-k$. A formal symmetry of infinite rank is a pseudo-differential operator such that holds identically whenever $u$ is a solution to $u_t =F$.
Note that if $G$ is a symmetry of order $p$ of $u_t=F$, then implies that $G_*$ is a formal symmetry of rank $p$. We also remark that a formal symmetry of infinite rank is a recursion operator, see [@O]. Thus, it generates, in principle, an infinite number of generalized symmetries of the equation at hand. For example, see [@DS], it can be proven that application of a [*quasilocal*]{} recursion operator (in the sense of [@DS Section 1]) to a given symmetry yields a (generalized) symmetry, and so such an operator could indeed generate an infinite chain of (generalized) symmetries.
The main technical point behind Definition \[fs\] is that the space of solutions of equation is much richer and structured than that of equation or even . For example, powers and roots of formal symmetries (computed using the standard theory of formal pseudo-differential operators, see [@MSS; @O]) are also formal symmetries. In fact, this observation was one of the original motivations for the use of formal pseudo-differential operators in Definition \[fs\], because the $r$th root of a differential operator is usually a [*pseudo*]{}-differential operator.
Now we explain why Definition \[fs\] restricts the function $F$. A theorem due to M. Adler, see [@MAdler], states that the residue (the coefficient of $D^{-1}$) of a commutator of formal pseudo-differential operators is always a total derivative. If we apply this result to different powers $\Lambda^{i/r}$ of a generic formal symmetry [^2] $\Lambda$ of rank $k$ inserted into , we obtain $$D_t (\operatorname{residue} (\Lambda^{i/r})) =
\operatorname{residue}{D_t(\Lambda^{i/r})}= \operatorname{residue}[F_*,\Lambda^{i/r}]=D\sigma_i$$ for some differential functions $\sigma_i$, i.e. a sequence of conservation laws $$\label{eq:ccl}
D_t\rho_i\doteq D_x\sigma_i,\qquad i=-1,1,\ldots,$$ which are, together with the special case $D_t\rho_0=D_t(l_r/l_{r-1})=D_x\sigma_0$, the so called *canonical conservation laws*. The symbol $\doteq$ means that equations must hold on solutions of , i.e. all derivatives with respect to $t$ must be substituted using the equation and its differential consequences.
As observed in [@MSS], the [*canonical densities*]{} $\rho_i$ and conserved fluxes $\sigma_i$ are differential functions which can be recursively written in terms of the right hand side $F$ of the equation and its derivatives. The fact that the left hand side of must be a total derivative with respect to $x$ for all $i =-1,1,2\cdots$, produces obstructions that are necessary conditions for the existence of (generalized/formal) symmetries $G$, i.e. for integrability.
For example, for evolution equations of third order, $$\label{eq:eq3}
u_t=F(x,u,u_x,u_{xx},u_{xxx}) \; ,$$ the first canonical density is $\rho_{-1}=\left(\partial F \big/\partial u_{xxx}\right)^{-1/3}$, see [@MSS]. Therefore, a first integrability condition is requiring $D_t\rho_{-1}$ to be the total derivative of a local function $\sigma_{-1}$. The second canonical density imposes further differential restrictions on $F$, and so forth. Usually, after a small number of steps our family of equations either fails to satisfy the integrability conditions, or the right hand side $F$ becomes so specific that we are able to produce a formal symmetry of infinite rank and therefore, in principle, a sequence of generalized symmetries of $u_t = F$. If $u_t =F$ represents a family of equations, this procedure allows us to find all integrable cases in the family. We are led to the following precise definition of integrability, after [@MSS; @O]:
\[int\] A system of evolution equations is integrable if and only if it possesses a formal symmetry of infinite rank.
Let us we write down the first five canonical densities for a third order equation following [@MSS]. We will use them in the next section to study the integrability of the Rosenau-Hyman and Cooper-Shepard-Sodano equations:
\[p1\] Let $u_t=F(x,u,u_x,u_{xx},u_{xxx})$ be an arbitrary third order evolution equation. The first five canonical conserved densities can be written explicitly as
$$\begin{aligned}
\rho_{-1}&=\left({\frac{\partialF}{\partialu_{xxx}}}\right)^{-1/3},\label{ic0}\\
\rho_0&=\rho_{-1}^3{\frac{\partialF}{\partialu_{xx}}},\\
\rho_1&=D_x\left(2\rho_{-1}^{-2}u_{x}+\rho_{-1}^2{\frac{\partialF}{\partialu_{xx}}}\right)+
\rho_{-1}^{-3}(D_x\rho_{-1})^2+{\frac{1}{3}}\rho_{-1}^5\left({\frac{\partialF}{\partialu_{xx}}}\right)^2\nonumber\\
&\qquad {}+\rho_{-1}(D_x\rho_{-1}){\frac{\partialF}{\partialu_{xx}}}
-\rho_{-1}^2{\frac{\partialF}{\partialu_{x}}}+\rho_{-1}\sigma_{-1},\\
\rho_2&={}-{\frac{1}{3}}(D_x^2\rho_{-1}){\frac{\partialF}{\partialu_{xx}}}-(D_x\rho_{-1}){\frac{\partialF}{\partialu_{x}}}+
\rho_{-1}{\frac{\partialF}{\partialu}}+\rho_{-1}^{-1}(D_x\rho_{-1})^2{\frac{\partialF}{\partialu_{xx}}}\nonumber \\
&\qquad {}-{\frac{1}{3}}\rho_{-1}^4{\frac{\partialF}{\partialu_{x}}}{\frac{\partialF}{\partialu_{xx}}}
+{\frac{1}{3}}\rho_{-1}^3(D_x\rho_{-1})\left({\frac{\partialF}{\partialu_{xx}}}\right)^2\nonumber \\
&\qquad {}+{\frac{2}{27}}\rho_{-1}^7\left({\frac{\partialF}{\partialu_{xx}}}\right)^3+{\frac{1}{3}}\rho_{-1}\sigma_{0},
\\
\rho_3&=\rho_{-1}\sigma_{1}-\rho_{1}\sigma_{-1}.\label{ic4}\end{aligned}$$
The condition of existence of a formal symmetry of infinite rank is strictly [*weaker*]{} than the condition of existence of an infinite number of generalized symmetries. Indeed, the $2$ component system $$\left \{
{\setlength\arraycolsep{2pt}
\begin{array}{rl}
u_t =& u_{xxxx} + v^2 \\
v_t =& v_{xxxx} \; ,
\end{array}}
\right.$$ considered by Bakirov in [@Bakirov], possesses exactly [*one*]{} generalized symmetry, as proved by Beukers, Sanders and Wang [@BSW]. On the other hand, it does possess a recursion operator [@Bilge].
Now, a very important remark, see [@MSS] and also [@O], is that the use of transformations between equations is a very convenient way to proceed when seeking classifications. In this paper we deal with integrable equations of the type $$\label{eq:equfu}
u_t=f(u)u_{xxx}+g(u,u_x,u_{xx}) \; ,$$ see [@RH; @CSS]. The general strategy we use to classify these equations consists in performing a sequence of convenient point transformations and differential substitutions that preserve integrability, and then apply and compute the integrability conditions associated to –. Our general procedure is as follows:
First, if $f'(u)\neq0$ the point transformation $u\to f(u)^{-1/3}$ converts Equation into another one of the form $$\label{equdu3}
u_t=D_x\left[\frac{u_{xx}}{u^3}+f_1(u,u_x)\right]+f_2(u,u_x,u_{xx}) \; ,$$ where $f_2(u,u_x,u_{xx})$ [*is not*]{} a total $x$-derivative. This form is very convenient because it follows from (\[ic0\]) that the integrability condition $D_t\rho_{-1}=D_x\sigma_{-1}$ is equivalent to requiring that $f_2=0$. This condition greatly restricts the form of the equation. Once $f_2=0$, the equation admits a potentiation $u\to u_x$ that brings it into the form $u_t=u_{xxx}/u_x^3+f_1(u_x,u_{xx})$. A subsequent hodograph transformation $x\to u$, $u\to x$ simplifies it to one of the form $$u_t=u_{xxx}+h(u_x,u_{xx}).$$ This equation can be “antipotentiated" ($u_x\to u$) to get $$u_t=u_{xxx}+D_xh(u,u_x).$$ Our integrability conditions imply that the integrable cases of this equation are all of the form $$\label{equdx}
u_t=u_{xxx}+D_x\left[h_2(u) u_{x}^2+\left(a+bu\right)u_x+h_0(u)\right].$$ If $h_2(u)\neq0$ a further point transformation $\int\!\exp\left[\frac23\int\!{h_2(u)\,du}\right]\,du\to u$ transforms this equation into another one of the form $$\label{equcdff}
u_t=u_{xxx}+(a+bu)\,u_{xx}+f_1(u)\,u_x^3+f_2(u)\,u_x^2+f_3(u)\,u_x.$$ We will show that all the integrable cases of the RH and CSS equations can be written in the form , as linear equations, KdV or mKdV, or the Calogero-Degasperis-Fokas (CDF) equation (see below; the CDF is Miura-transformable to KdV). Thus, if we “pullback" the recursion operator of KdV (or, the linear equation) by the foregoing transformations, we can construct recursion operators of the original equations, and therefore we obtain an explicit proof of integrability in terms of Definition \[int\]. In actual fact, we seldom perform this pullback operation explicitly. Once we know that a given equation is integrable, it is usually straightforward to compute its recursion operator from first principles, as in [@PNN].
Now we carry out this plan.
Integrability of the Rosenau-Hyman equation
===========================================
We consider the compacton equation of Rosenau and Hyman (see [@RH]) $$\label{eq:eqRH}
D_t(u)+D_x(u^m)+D_x^3(u^n)=0,\qquad n\neq0 \; .$$ As we informed in Section 2, $D_t$ and $D_x$ are total derivatives with respect to independent variables $t$ and $x$. For simplicity, we use the subindex notation $u_t=D_tu$, $u_x=D_xu$, $u_{xx}=D_x^2$ but we prefer to use the total derivative notation when applied to a more complicated differential function, e.g. $D_x(u^m)=mu^{m-1}u_x$.
The case $n=1$ i.e. $$u_t+mu^{m-1}u_x+u_{xxx}=0 \; ,$$ is well-known, see [@MSS Section 4.1]: the only integrable cases are $m=0,1,2,3$, i.e. the linear equation, the KdV and the modified KdV equations. We write them as (using the point transformation $x\to -x$)
$$\begin{aligned}
u_t&=u_{xxx}+\alpha\, u_x+\beta,\quad\alpha,\beta\in\mathbf{C}, \label{34} \\
u_t&=u_{xxx}+2\,u u_x, \label{35} \\
u_t&=u_{xxx}+3\,u^2u_x. \label{36}\end{aligned}$$
If $n\neq1$, the point transformation $x\to-x$, $t\to t/n$, $u\to u^{3/(1-n)}$ changes into equations of the form , namely: $$\label{eq:eq1}
u_{t}=D_{x}\left[\frac{u_{xx}}{u^3}
-\frac{m (n-1) }{n (3 m-n-2)}u^{\frac{2-3 m+n}{-1+n}}
-\frac{9 n}{2 (n-1)}\frac{u_{x}^2}{u^4}
\right]
+\frac{(n+2) (2 n+1) u_{x}^3}{(n-1)^2 u^5}$$ if $3m-n-2\neq0$, and $$\label{eq:eq2}
u_{t}=D_{x}\left[\frac{u_{xx}}{u^3}-\frac{9 n}{2 (n-1)} \frac{u_{x}^2}{u^4}
+\frac{(n+2)}{3 n}\log{u}\right]
+\frac{(n+2) (2 n+1) u_{x}^3}{(n-1)^2 u^5}$$ if $3m-n-2=0$.
The first integrability condition $D_t\rho_{-1} = D_x \sigma_{-1}$ or, equivalently, $u_t\in\operatorname{Im}D_x$ implies that either $n=-2$ or $n=-1/2$ in both cases, because the last term in and must be zero. The composition of a potentiation, a hodograph and an antipotentiation yields the following equations of the form : $$\label{eq:eq1c}
u_t=u_{xxx}+D_x\left[\frac32\frac{n+2}{n-1}\frac{u_{x}^2}{u} +
\frac{m (n-1)}{n (3 m-n-2)}\,u^{\frac{3 (m-1)}{n-1}}\right],$$
$$\label{eq:eq2c}
u_t=u_{xxx}+D_x\left[\frac32 \left( \frac{n+2}{n-1} \right) \frac{u_{x}^2}{u}+\frac{n+2}{3 n}\,u \log{u}\right].$$
If $n=-2$, Equation becomes $$u_t=u_{xxx}-\frac{m-1}{2} u^{-m}u_{x}$$ whose integrable cases are again $m=1,0,-1,-2$, corresponding to linear equations, KdV and mKdV. On the other hand, if $n=-2$, Equation becomes a linear equation included in case .
If $n=-1/2$, Equation can be written in the form , this is, $$u_t=u_{xxx}-\frac12u_{x}^3-\frac{4 m (m-1)}{2 m-1}\, {\rm e}^{(1-2 m)u} u_{x} \; ,$$ after a point transformation $u\to{\rm e}^{u}$. This family of equations satisfies the first two integrability conditions. The third integrability condition is $D_t\rho_1\in\operatorname{Im}D_x$, and the canonical conserved density $\rho_1$ satisfies $$D_t(\rho_1)\sim -2 (m-1) m (2 m-3) (2 m+1) e^{u-2 m u} u_{x}^3 \; ,$$ in which the symbol $\sim$ denotes equality except for the addition of a total $x$-derivative. Thus, integrability can be achieved only in the cases $m=-1/2$, $0$, $1$, $3/2$[^3] which are all subcases of the Calogero-Degasperis-Fokas (CDF) equation $$\label{cdf}
u_t=u_{xxx}-\frac12u_x^3+\left(\alpha{\rm e}^{2u}+\beta{\rm e}^{-2u}+\gamma\right)u_x\; .$$ Finally, when $n=-1/2$ Equation becomes $$u_t=u_{xxx}-3\frac{u_{x} u_{xx}}{u}+\frac32\frac{u_{x}^3}{u^2}-(1+\log{u})u_{x}$$ and for it $$D_t(\rho_1)\sim-2\frac{u_x^3}{u^3}$$ so this case is not integrable.
We note that the CDF equation can be related to the KdV equation through the Miura transformation $$\frac32 u_{xx}-\frac34 u_{x}^2 -\sqrt{6 \beta } {\rm e}^{-u} u_{x}-\frac{1}{2} \alpha e^{2 u}-
\frac{1}{2} \beta e^{-2 u}-\frac{\gamma }{2} \to u\; .$$
We summarize the integrable cases of the Rosenau-Hyman family in the following theorem. We make the point transformation $x\to-x$, $t\to t/n$ and write $$\label{eqRH1}
u_t=\tfrac{1}{n}D_x(u^m)+\tfrac{1}{n}D_x^3(u^n)\; ,\qquad n\neq0$$ instead of . This transformation is invertible and does not affect integrability.
The integrable cases of the Rosenau-Hyman family are
1. $n=1$, $m=0,1,2,3$, corresponding to Equations , and , namely,
$$\begin{aligned}
u_t&=u_{xxx}+\alpha\, u_x+\beta,\quad\alpha,\beta\in\mathbf{C}, \label{341} \\
u_t&=u_{xxx}+2\,u u_x, \label{351} \\
u_t&=u_{xxx}+3\,u^2u_x \; . \label{361}\end{aligned}$$
2. $n=-2$, $m=-2,-1,0,1$, corresponding to Equations
$$\begin{aligned}
u_t&=D_x\left[D_x\left(\frac{u_x}{u^{3}}\right)-\frac{1}{2u^2}\right] \; , \label{eq:eqm2m2} \\
u_t&=D_x\left[D_x\left(\frac{u_x}{u^{3}}\right)-\frac{1}{2u}\right] \; , \label{39} \\
u_t&=-\frac12D_x^3\left[ u^{-2} \right] \; ,\label{395} \\
u_t&=D_x\left[D_x\left(\frac{u_x}{u^{3}}\right)-\frac{1}{2}u\right]\label{310}\end{aligned}$$
respectively.
3. $n=-{\tfrac{1}{2}}$, $m={\tfrac{3}{2}},1,0,-{\tfrac{1}{2}}$, corresponding to Equations
$$\begin{aligned}
u_t&=D_x\left[D_x\left(\frac{u_x}{u^{3/2}}\right)-2u^{3/2}\right], \label{312} \\
u_t&=D_x\left[D_x\left(\frac{u_x}{u^{3/2}}\right)-2u\right], \label{313}\\
u_t&=D_x^2\left[ \frac{u_x}{u^{3/2}}\right], \label{314}\\
u_t&=D_x\left[D_x\left(\frac{u_x}{u^{3/2}}\right)-2\frac{1}{u^{1/2}}\right]\; \label{315}\end{aligned}$$
respectively.
All these equations are related to the linear equation or to the KdV equation through differential substitutions.
It is clear that the equations appearing above cannot admit solutions with compact support, let alone compactons. As explained in Section 1, this theorem extends and enriches the discussion on $K(m,n)$ appearing in [@HHR]. We mention that J. Vodová classified conservation laws of the $K(m,m)$ equations and observed that $K(-2,-2)$, $K(-1/2,-1/2)$ are integrable; integrability of $K(-2,-2)$, $K(-1/2,-1/2)$, and $K(-1,-2)$, is also observed in the later review [@RZ].
Integrability of Cooper-Shepard-Sodano
======================================
In this section we study equations of the form $$u_{t}+u^{l-2}u_{x}-\alpha pD_{x}\left(u^{p-1}u_{x}^{2}\right)+2\alpha D_{x}^{2}\left(u^{p}u_{x}\right)=0,
\qquad\alpha\neq0.\label{eq:eqRHE}$$ We consider the case $p=0$ first. Equation (\[eq:eqRHE\]) becomes $u_{t}+u^{l-2}u_{x}+2\alpha u_{xxx}=0$, which is integrable if and only if $l=2,3,4$, i.e. in the linear, KdV and mKdV case, as observed in [@MSS]. Let us now consider $p\neq0$. We use the same strategy as with the Rosenau-Hyman case: first we apply the change $t\to -t$, $u\to (2\alpha u^3)^{-1/p}$ to obtain $$\label{css1}
u_{t}
= D_{x}\left[\frac{u_{xx}}{u^3}
-\frac{3 (2 p+3)}{2 p}\frac{u_{x}^2}{u^4}
-\frac{p (2 \alpha )^{\frac{2-l}{p}} }{3 l-p-6}u^{\frac{6-3 l+p}{p}}
\right]+\frac{(p+3) (p+6) u_{x}^3}{2 p^2 u^5}$$ if $3l-p-6\neq0$, and $$\label{css2}
u_{t}=D_{x}\left[\frac{u_{xx}}{u^3}
-\frac{3 (2 p+3) }{2 p}\frac{u_{x}^2}{u^4}
+\frac{1}{\sqrt[3]{2 \alpha}}\log{u}\right]
+\frac{(p+3) (p+6) u_{x}^3}{2 p^2 u^5}$$ if $3l-p-6=0$.
Thus, the first integrability condition of Proposition \[p1\] fixes the values $p=-3$ or $p=-6$. If $p=-3$, a combination of potentiation, hodograph, antipotentiation and exponential point transformation $u\to{\rm e}^u$ , with a scaling to absorb the constant $\alpha$, change (\[css1\]) and (\[css2\]) into the equations $$u_{t}=u_{xxx}-\frac{u_{x}^3}{2}+\frac{(l-2) e^{u-l u} u_{x}}{l-1}
\quad\text{ and }\quad
u_{t}=u_{xxx}-\frac{u_{x}^3}{2}+u u_{x}\;$$ respectively. On the other hand, if $p=-6$, the same combination of transformations, using $u\to u^2$ instead of the exponential, changes (\[css1\]) and (\[css2\]) into $$u_{t}=u_{xxx}+\frac{(l-2)}{l}\frac{u_{x}}{u^l}\quad\text{ and }\quad
u_{t}=u_{xxx}+\log (u)u_{x}\;$$ respectively.
Let us assume that $3l-p-6\neq0$. The integrability condition in $\rho_1$ implies that if $p=-3$ we must have $l=-1,2,3$, and if $p=-6$, then $l=-2,-1,2$ and we recover again the linear, KdV, mKdV and CDF equations. When $3l-p-6=0$, we have the CSS equations $$u_{t}=\frac{a}{u^{6}}u_{xxx}-12\frac{a}{u^{7}}u_{x}u_{xx}+21\frac{a}{u^{8}}u_{x}^{3}+\frac{u_{x}}{u^{2}}$$ ($p = -6$ and $l=0$) and $$u_{t}=\frac{a}{u^{3}}u_{xxx}-6\frac{a}{u^{4}}u_{x}u_{xx}+6\frac{a}{u^{5}}u_{x}^{3}+\frac{u_{x}}{u}\; ,$$ ($p=-3$ and $l=1$), in which $a = 2\alpha$. The integrability condition for $\rho_{2}$ implies that both equations are not integrable.
Summarizing, we have the following theorem.
The integrable equations of family are
1. $p=0,$ $l=2,3,4$, corresponding to the linear equation, KdV equation, mKdV equation;
2. $p=-6$, $l=-2,-1,2$, corresponding to the Equations
$$\begin{gathered}
u_{t}=\frac{a}{u^{6}}u_{xxx}-12\frac{a}{u^{7}}u_{x}u_{xx}+21\frac{a}{u^{8}}u_{x}^{3}+\frac{u_{x}}{u^{4}},\label{eq:eq6m2}\\
u_{t}=\frac{a}{u^{6}}u_{xxx}-12\frac{a}{u^{7}}u_{x}u_{xx}+21\frac{a}{u^{8}}u_{x}^{3}+\frac{u_{x}}{u^{3}},\label{eq:eq6m1}\\
u_{t}=\frac{a}{u^{6}}u_{xxx}-12\frac{a}{u^{7}}u_{x}u_{xx}+21\frac{a}{u^{8}}u_{x}^{3}+u_{x}.\label{eq:eq6p2}\end{gathered}$$
respectively.
3. $p=-3,$ $l=-1,2,3$, corresponding to the Equations
$$\begin{gathered}
u_{t}=\frac{a}{u^{3}}u_{xxx}-6\frac{a}{u^{4}}u_{x}u_{xx}+6\frac{a}{u^{5}}u_{x}^{3}+\frac{u_{x}}{u^{3}},\label{eq:eq3m1}\\
u_{t}=\frac{a}{u^{3}}u_{xxx}-6\frac{a}{u^{4}}u_{x}u_{xx}+6\frac{a}{u^{5}}u_{x}^{3}+u_{x},\label{eq:eq3p2}\\
u_{t}=\frac{a}{u^{3}}u_{xxx}-6\frac{a}{u^{4}}u_{x}u_{xx}+6\frac{a}{u^{5}}u_{x}^{3}+uu_{x}.\label{eq:eq3p3}\end{gathered}$$
respectively.
All these equations are related to the linear equation or to the KdV equation through differential substitutions.
As in Section 3, this theorem implies that no integrable CSS equation admits solutions with compact support.
Integrability and recursion operators
=====================================
In this section we construct explicit recursion operators for the equations appearing in the above theorems using the work [@PNN]. First of all, we note that these equations are all in [@MSS]. We have (when we write (4.x.xx) we are referring to the corresponding equation in [@MSS]):
1. Equation is a special case of (4.1.27), namely, $$\label{eq:eqm2m21}
u_t=D_x\left(\frac{u_{xx}}{u^3}-3\frac{u_x^2}{u^4}+\frac{1}{2u^2}\right) =-9\,{\frac {u_{{x}}u_{{xx}}}{{u}^{4}}}
+12\,{\frac {{u_{{x}}}^{3}}{{u}^{5}}}-{\frac {u_{{x}}}{{u}^{3}}}+{\frac {u_{{xxx}}}{{u}^{3}}}
\; .$$
2. Equation (\[39\]) is equivalent to a subcase of (4.1.25) $$\label{391}
u_t=D_x\!\left(\frac{u_{xx}}{u^3}{-}3\frac{u_x^2}{u^4}{-}\frac{3}{2u}{+}cu\right)= -9\,{\frac {u_{{x}}u_{{xx}}}{{u}^{4}}}
+12\,{\frac {{u_{{x}}}^{3}}{{u}^{5}}}+{\frac {3 u_{{x}}}{2 {u}^{2}}}+ c u_x + {\frac {u_{{xxx}}}{{u}^{3}}}\; .$$
3. Equations (\[395\]) and (\[310\]) are equivalent to subcases of (4.1.34)
$$\begin{gathered}
\label{3951}
u_t=D_x\left(\frac{u_{xx}}{u^3}-3\frac{u_x^2}{u^4}+c_1\frac{u_x}{u^2}+c_2u\right)
\\
=
-9\,{\frac {u_{{x}}u_{{xx}}}{{u}^{4}}}
+12\,{\frac {{u_{{x}}}^{3}}{{u}^{5}}}-2c_1{\frac {u_{{x}}^2}{{u}^{3}}}+ c_2 u_x +c_1\frac{u_{xx}}{u^2}
+ {\frac {u_{{xxx}}}{{u}^{3}}} \; .\end{gathered}$$
4. Equations (\[312\])-(\[315\]) are all equivalent to subcases of (4.1.30) $$u_t=D_x\!\left[\frac{u_{xx}}{u^3}-\frac32\frac{u_x^2}{u^4}-\frac32\frac{\lambda u_x^2}{u^3(\lambda u+1)}+
c_1\frac{(\lambda u+1)^3}{u^2}+c_2\frac{u^2}{\lambda u+1}+c_3u\right]$$ with $\lambda=0$, that is, $$\label{2171}
u_t=-6\,{\frac {u_{x}u_{{xx}}}{{u}^{4}}}
+6\,{\frac{{u_{{x}}}^{3}}{{u}^{5}}}
-2 {\it c_1}\,{\frac {u_{{x}}}{{u}^{3}}}+2{\it c_2}\,u u_{x}
+{\it c_3} u_{{x}}+{\frac {u_{{xxx}}}{{u}^{3}}} \; ,$$ after applying the point transformation $u\to u^2$.
Now, Equations , and are special cases (for the values of the constant numbers $c_1,c_2$ relevant to us) of Equation (81) in [@PNN], namely, $$\label{class-0}
u_t = \frac{u_{xxx}}{u^3} - 9 \frac{u_x u_{xx}}{u^4} + 12 \frac{u_x^3}{u^5}
+ \frac{2}{3}\lambda_1 \frac{u_x}{u^3} + \frac{1}{2} \lambda_2 \frac{u_x}{u^2} - c u_x \; ,$$ while Equation (\[2171\]) is Equation (85) in [@PNN], see below. The recursion operator for Equation (\[class-0\]) is as follows:
$$\begin{gathered}
R[u]=u^{-2}D_x^2
-5u^{-3}u_xD_x
-4u^{-3}u_{xx}+12u^{-4}u_x^2
+\frac{2\lambda_1}{3}u^{-2}
+\frac{2\lambda_2}{3}u^{-1}\\
-u_tD_x^{-1}\circ 1
+u_xD_x^{-1}\circ \left(\frac{\lambda_2}{6}u^{-2}-c\right).\end{gathered}$$
Acting $R[u]$ on the $t$-translation symmetry $\displaystyle{u_t{\frac{{\partial}\ }{{\partial}u}}}$, yields a corresponding symmetry-integrable hierarchy of order $2m+3$, namely
$$\begin{gathered}
u_{t}=R^m[u]\circ \left( \frac{u_{xxx}}{u^3} - 9 \frac{u_x u_{xx}}{u^4} + 12 \frac{u_x^3}{u^5}
+ \frac{2}{3}\lambda_1 \frac{u_x}{u^3} + \frac{1}{2} \lambda_2 \frac{u_x}{u^2} - c u_x \right),\\[2mm]
m=0,1,2,\ldots \; ,\end{gathered}$$
and we note that for the $x$-translation symmetry we obtain $$R[u]\circ u_x=0.$$ \
Let us now consider the integrable cases of the CSS equations. We see that Equations $(\ref{eq:eq6m2})$, $(\ref{eq:eq6m1})$ and $(\ref{eq:eq6p2})$ are special cases of Equation (90) in [@PNN], namely
$$\label{class-1}
u_t=\frac{au_{xxx}}{u^6}-12\frac{au_xu_{xx}}{u^7}+21\frac{au_x^3}{u^8}
+\beta_1 \frac{u_x}{u^4}+\beta_2 \frac{u_x}{u^3}+\beta_3 u_x,$$
where $a$, $\beta_1$, $\beta_2$ and $\beta_3$ are arbitrary constants and $a\neq 0$. This equation admits the following recursion operator:
$$\begin{gathered}
R_1[u]=\frac{1}{u^4}D_x^2
-\frac{6u_x}{u^5}D_x
-\frac{6u_{xx}}{u^5}
+\frac{22u_x^2}{u^6}
+\frac{2\beta_2}{a} \frac{1}{u}
+\frac{4\beta_1}{3a} \frac{1}{u^2}
-\frac{2}{a}u_tD_x^{-1}\circ u
\\
+u_xD_x^{-1}\circ
\left(
-\frac{2u_{xx}}{u^6}
+6\frac{u_x^2}{u^7}
+\frac{\beta_2}{a} \frac{1}{u^2}
+\frac{2\beta_1}{3a} \frac{1}{u^3}
+\frac{2\beta_3}{a}u\right).\label{R1}\end{gathered}$$
Acting $R_1[u]$ on the $t$-translation symmetry $\displaystyle{u_t{\frac{{\partial}\ }{{\partial}u}}}$, we obtain a corresponding symmetry-integrable hierarchy of order $2m+3$, namely
$$\begin{gathered}
\label{hierarchy-1}
u_{t}=R_1^m[u]\circ \left(\frac{au_{xxx}}{u^6}-12\frac{au_xu_{xx}}{u^7}+21\frac{au_x^3}{u^8}
+\beta_1 \frac{u_x}{u^4}+\beta_2 \frac{u_x}{u^3}+\beta_3 u_x\right),\\[2mm]
m=0,1,2,\ldots {\nonumber}\; ,\end{gathered}$$
and we note that for the $x$-translation symmetry we obtain $$R_1[u]\circ u_x=0.$$ \
On the other hand, Equations , and are special cases of Equation (85) in [@PNN], namely $$\label{class-2}
u_t=\frac{au_{xxx}}{u^3}
-6\frac{au_xu_{xx}}{u^4}
+6\frac{au_x^3} {u^5}
+\beta_1\frac{u_x}{u^3}
+\beta_2 uu_x
+\beta_3u_x,$$ where $a$, $\beta_1$, $\beta_2$ and $\beta_3$ are arbitrary constants and $a\neq 0$. This equation admits the following recursion operator:
$$\begin{gathered}
\label{R2}
R_2[u]=\frac{1}{u^2}D_x^2
-\frac{3u_x}{u^3} D_x
-3\frac{u_{xx}}{u^3}
+6\frac{u_x^2}{u^4}
+\frac{\beta_2}{3a} u^2
+\frac{\beta_1}{a} \frac{1}{u^2}\\[0.2cm]
-\frac{1}{a} u_t\,D_x^{-1}\circ 1
+\frac{4\beta_2}{3a} u_x\,D_x^{-1}\circ u
+\frac{\beta_3}{a} u_x\,D_x^{-1}\circ 1.\end{gathered}$$
Acting $R_2[u]$ on the $t$-translation symmetry $\displaystyle{u_t{\frac{{\partial}\ }{{\partial}u}}}$, we obtain a symmetry-integrable hierarchy of order $2m+3$, namely
$$\begin{gathered}
\label{hierarchy-2}
u_{t}=R_2^m[u]\circ \left(
\frac{au_{xxx}}{u^3}
-6\frac{au_xu_{xx}}{u^4}
+6\frac{au_x^3} {u^5}
+\beta_1\frac{u_x}{u^3}
+\beta_2 uu_x
+\beta_3u_x
\right),\\[2mm]
m=0,1,2,\ldots{\nonumber}\; ,
\end{gathered}$$
and we note that for the $x$-translation symmetry we obtain $$R_2[u]\circ u_x=0.$$
The isochronous equations for (\[class-1\]) and (\[class-2\])
=============================================================
In this section we construct new integrable evolution equations starting from what we will call the Cooper-Shepard-Sodano model equations (\[class-1\]) and (\[class-2\]). Our new equations are isochronous in the sense of Calogero, see [@Calogero-2005 Chapter 7] and [@Mariani-Calogero-2005; @Calogero-Mariani-2005; @Calogero-Euler-Euler]: they are autonomous evolution PDEs which depend on a positive parameter $\omega$ and possess [*many*]{} solutions which are time-periodic with period $T=2\pi/\omega$. For completeness, we also explain how to obtain the Lie point symmetries of our equations and present their recursion operators.
Isochronous equations
---------------------
Following [@Mariani-Calogero-2005; @Calogero-Mariani-2005; @Calogero-2005; @Calogero-Euler-Euler], we introduce a new dependent variable $v(r,s)$, where $r$ and $s$ are new independent variables, as follows:
$$\begin{gathered}
\label{Iso-transformation-u}
u(x,t)=e^{-i\lambda \omega s} v(r,s)\\[0.2cm]
\label{Iso-transformation-x}
x=re^{i\mu \omega s}\\[0.2cm]
\label{Iso-transformation-t}
t=\frac{1}{i\omega}\left(e^{i\omega s}-1\right).\end{gathered}$$
The prolongations are
$$\begin{gathered}
u_t=e^{-i(\lambda+1)\omega s}\left[v_s-i\lambda \omega\, v-i\mu \omega r\, v_r\right]\\[0.2cm]
u_{nx}=e^{-i(\lambda+n\mu)\omega s}\, v_{nr},\quad n=1,2,3,\dots,\end{gathered}$$
where $$u_{nx}=\frac{{\partial}^n u}{{\partial}x^n},\qquad v_{nr}=\frac{{\partial}^n v}{{\partial}r^n}.$$ With the change of variables – , equation takes the form
$$\begin{gathered}
v_s-i\lambda \omega v-i\mu\omega r v_r=
e^{i(6\lambda-3\mu+1)\omega s}
\left(\frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}\right)\\[0.2cm]
\label{Iso-class-1}
\qquad +\beta_1e^{i(4\lambda -\mu+1)\omega s}\,\left(\frac{v_r}{v^4}\right)
+\beta_2e^{i(3\lambda-\mu+1)\omega s}\,\left(\frac{v_r}{v^3}\right)
+\beta_3 e^{i(-\mu+1)\omega s}\,v_r.\end{gathered}$$
This equation can become autonomous for $a\neq 0$ only if $$\mu=2\lambda+\frac{1}{3},$$ so that (\[Iso-class-1\]) then takes the form
$$\begin{gathered}
v_s-i\lambda \omega v-i\left(2\lambda+\frac{1}{3}\right) \omega r v_r=
\frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}\\[0.2cm]
\label{Iso-subclass-1}
+\beta_1e^{i(2\lambda+\frac{2}{3})\omega s}\,\left(\frac{v_r}{v^4}\right)
+\beta_2e^{i(\lambda+\frac{2}{3})\omega s}\,\left(\frac{v_r}{v^3}\right)
+\beta_3 e^{i(2\lambda+\frac{2}{3})\omega s}\,v_r.\end{gathered}$$
Clearly (\[Iso-subclass-1\]), and therefore (\[Iso-class-1\]), becomes autonomous in the following three cases: \
[**Case 1.1:**]{} $\displaystyle{\lambda=-\frac{1}{3}}$ and $\displaystyle{\mu=-\frac{1}{3}}$ with $\beta_2=\beta_3=0$. Then (\[Iso-class-1\]) becomes $$\label{Iso-class-1.1}
v_s+i\frac{1}{3}\omega v+i\frac{1}{3} \omega r v_r=
\frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}
+\beta_1 \frac{v_r}{v^4}.$$ [**Case 1.2:**]{} $\displaystyle{\lambda=-\frac{2}{3}}$ and $\displaystyle{\mu=-1}$ with $\beta_1=\beta_3=0$. Then (\[Iso-class-1\]) becomes $$\label{Iso-class-1.2}
v_s+i\frac{2}{3} \omega v+i \omega r v_r=
\frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}
+\beta_2\frac{v_r}{v^3}.$$ [**Case 1.3:**]{} $\displaystyle{\lambda=\frac{1}{3}}$ and $\displaystyle{\mu=1}$ with $\beta_1=\beta_2=0$. Then (\[Iso-class-1\]) becomes $$\label{Iso-class-1.3}
v_s-i\frac{1}{3} \omega v-i \omega r v_r=
\frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}+\beta_3 v_r.$$ \
Now we consider (\[class-2\]). With the change of variables (\[Iso-transformation-u\]) – (\[Iso-transformation-t\]), Equation takes the form
$$\begin{gathered}
v_s-i\lambda \omega v-i\mu\omega r v_r=
e^{i(3\lambda-3\mu+1)\omega s}
\left(\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}\right)\\[0.2cm]
\label{Iso-class-2}
\qquad +\beta_1e^{i(3\lambda -\mu+1)\omega s}\,\left(\frac{v_r}{v^3}\right)
+\beta_2e^{i(-\lambda-\mu+1)\omega s}\,vv_r
+\beta_3 e^{i(-\mu+1)\omega s}\,v_r.\end{gathered}$$
This equation can become autonomous for $a\neq 0$ only if $$\mu=\lambda+\frac{1}{3},$$ so that (\[Iso-class-2\]) then takes the form
$$\begin{gathered}
v_s-i\lambda \omega v-i\left(\lambda+\frac{1}{3}\right) \omega r v_r=
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}\\[0.2cm]
\label{Iso-subclass-2}
\qquad +\beta_1e^{i(2\lambda+\frac{2}{3})\omega s}\,\left(\frac{v_r}{v^3}\right)
+\beta_2e^{i(-2\lambda+\frac{2}{3})\omega s}\,vv_r
+\beta_3 e^{i(-\lambda+\frac{2}{3})\omega s}\,v_r.\end{gathered}$$
Clearly (\[Iso-subclass-2\]), and therefore (\[Iso-class-2\]), becomes autonomous in the following three cases:\
[**Case 2.1:**]{} $\displaystyle{\lambda=-\frac{1}{3}}$ and $\displaystyle{\mu=0}$ with $\beta_2=\beta_3=0$. Then (\[Iso-class-2\]) becomes $$\label{Iso-class-2.1}
v_s+i\frac{1}{3} \omega v=
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}
+\beta_1\,\frac{v_r}{v^3}.$$ [**Case 2.2:**]{} $\displaystyle{\lambda=\frac{1}{3}}$ and $\displaystyle{\mu=\frac{2}{3}}$ with $\beta_1=\beta_3=0$. Then (\[Iso-class-2\]) becomes $$\label{Iso-class-2.2}
v_s-i\frac{1}{3} \omega v-i\frac{2}{3} \omega r v_r=
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}
+\beta_2\,vv_r.$$ [**Case 2.3:**]{} $\displaystyle{\lambda=\frac{2}{3}}$ and $\displaystyle{\mu=1}$ with $\beta_1=\beta_2=0$. Then (\[Iso-class-2\]) becomes $$\label{Iso-class-2.3}
v_s-i\frac{2}{3} \omega v-i\omega r v_r=
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}
+\beta_3 \,v_r.$$
Our isochronous equations are (\[Iso-class-1.1\])–(\[Iso-class-1.3\]) and (\[Iso-class-2.1\])–(\[Iso-class-2.3\]).
Symmetries
----------
We list the Lie point symmetries of equation (\[class-1\]), that is $$u_t=\frac{au_{xxx}}{u^6}-12\frac{au_xu_{xx}}{u^7}+21\frac{au_x^3}{u^8}
+\beta_1 \frac{u_x}{u^4}+\beta_2 \frac{u_x}{u^3}+\beta_3 u_x.$$ Besides the obvious $x$-translation, $\displaystyle{Z_x={\frac{{\partial}\ }{{\partial}x}}}$, and $t$-translation symmetry, $\displaystyle{Z_t={\frac{{\partial}\ }{{\partial}t}}}$, Equation also admits the following point symmetries:
- For $\beta_1=\beta_2=\beta_3=0$: $$Z_1=x{\frac{{\partial}\ }{{\partial}x}}+3t{\frac{{\partial}\ }{{\partial}t}},\quad Z_2=u{\frac{{\partial}\ }{{\partial}u}}+6t{\frac{{\partial}\ }{{\partial}t}}.$$
- For $\beta_1=\beta_2=0$ and $\beta_3\neq 0$: $$Z_1=(x-2t\beta_3){\frac{{\partial}\ }{{\partial}x}}+3t{\frac{{\partial}\ }{{\partial}t}},\quad Z_2=u{\frac{{\partial}\ }{{\partial}u}}+6t{\frac{{\partial}\ }{{\partial}t}}-6t\beta_3{\frac{{\partial}\ }{{\partial}x}}.$$
- For $\beta_1=\beta_3=0$ and $\beta_2\neq 0$: $$Z_1=-\frac{2}{3}u{\frac{{\partial}\ }{{\partial}u}}+x{\frac{{\partial}\ }{{\partial}x}}-t{\frac{{\partial}\ }{{\partial}t}}.$$
- For $\beta_2=\beta_3=0$ and $\beta_1\neq 0$: $$Z_1=-u{\frac{{\partial}\ }{{\partial}u}}+x{\frac{{\partial}\ }{{\partial}x}}-3t{\frac{{\partial}\ }{{\partial}t}}.$$
\
We list the Lie point symmetries of equation (\[class-2\]), that is $$u_t=\frac{au_{xxx}}{u^3}
-6\frac{au_xu_{xx}}{u^4}
+6\frac{au_x^3} {u^5}
+\beta_1\frac{u_x}{u^3}
+\beta_2 uu_x
+\beta_3u_x.$$ Besides the obvious $x$-translation symmetry and $t$-translation symmetry, (\[class-2\]) also admits the following point symmetries:
- For $\beta_1=\beta_2=\beta_3=0$: $$Z_1=x{\frac{{\partial}\ }{{\partial}x}}+3t{\frac{{\partial}\ }{{\partial}t}},\quad Z_2=u{\frac{{\partial}\ }{{\partial}u}}+3t{\frac{{\partial}\ }{{\partial}t}},\quad
Z_3=xu{\frac{{\partial}\ }{{\partial}u}}-\frac{1}{2}x^2{\frac{{\partial}\ }{{\partial}x}}.$$
- For $\beta_1=\beta_2=0$ and $\beta_3\neq 0$:
$$\begin{gathered}
Z_1=(x-2t\beta_3){\frac{{\partial}\ }{{\partial}x}}+3t{\frac{{\partial}\ }{{\partial}t}},\quad Z_2=u{\frac{{\partial}\ }{{\partial}u}}+3t{\frac{{\partial}\ }{{\partial}t}}-3t\beta_3{\frac{{\partial}\ }{{\partial}x}}\\[2mm]
Z_3=u(x+\beta_3t){\frac{{\partial}\ }{{\partial}u}}-\frac{1}{2}(x+\beta_3t)^2{\frac{{\partial}\ }{{\partial}x}}.\end{gathered}$$
- For $\beta_1=\beta_3=0$ and $\beta_2\neq 0$: $$Z_1=u{\frac{{\partial}\ }{{\partial}u}}-2x{\frac{{\partial}\ }{{\partial}x}}-3t{\frac{{\partial}\ }{{\partial}t}}.$$
- For $\beta_2=\beta_3=0$ and $\beta_1\neq 0$:
$$\begin{gathered}
Z_1=u{\frac{{\partial}\ }{{\partial}u}}+3t{\frac{{\partial}\ }{{\partial}t}}\\[2mm]
Z_2=u\sin\left(a^{-1/2}\beta_1^{1/2} x\right){\frac{{\partial}\ }{{\partial}u}}+a^{1/2}\beta_1^{-1/2}\cos\left(a^{-1/2}\beta_1^{1/2} x\right){\frac{{\partial}\ }{{\partial}x}}\\[2mm]
Z_3=u\cos\left(a^{-1/2}\beta_1^{1/2}x\right){\frac{{\partial}\ }{{\partial}u}}-a^{1/2}\beta_1^{-1/2}\sin\left(a^{-1/2}\beta_1^{1/2}x\right){\frac{{\partial}\ }{{\partial}x}}.
\end{gathered}$$
We can now obviously map the symmetries of (\[class-1\]) and (\[class-2\]) with the (\[Iso-transformation-u\]) – (\[Iso-transformation-t\]) to symmetries of the isochronous equations (\[Iso-class-1\]) and (\[Iso-class-2\]). For example, in vertical form the $x$-translation symmetry $$u_x{\frac{{\partial}\ }{{\partial}u}},$$ then takes the form $$e^{-i\mu \omega s}v_r{\frac{{\partial}\ }{{\partial}v}},$$ for (\[Iso-class-1\]) and (\[Iso-class-2\]), whereas the $t$-translation symmetry $$u_t{\frac{{\partial}\ }{{\partial}u}},$$ becomes the symmetry $$e^{-i\omega s}\left(v_s-i\lambda \omega v-i\mu\omega rv_r\right){\frac{{\partial}\ }{{\partial}v}}$$ for (\[Iso-class-1\]) and (\[Iso-class-2\]).
The isochronous hierarchies for (\[class-1\]) and (\[class-2\])
---------------------------------------------------------------
For equation (\[class-1\]) we have the hierarchy (\[hierarchy-1\]), namely
$$\begin{gathered}
u_{t}=R_1^m[u]\circ
\left(\frac{au_{xxx}}{u^6}-12\frac{au_xu_{xx}}{u^7}+21\frac{au_x^3}{u^8}
+\beta_1 \frac{u_x}{u^4}+\beta_2 \frac{u_x}{u^3}+\beta_3 u_x,\right)\\[2mm]
m=0,1,2,\ldots\ ,\nonumber \end{gathered}$$
where $R_1[u]$ is given by . Corresponding to the above Case 1.1, Case 1.2 and Case 1.3, the isochronous hierarchies are the following:\
\
[**Case 1.1:**]{} We consider the hierarchy (\[hierarchy-1\]) with $\beta_2=\beta_3=0$. This leads to the following isochronous hierarchy
$$\begin{gathered}
v_{s}+i\left(\frac{1}{2m+3}\right) \omega v+i\left(\frac{1}{2m+3}\right)\omega rv_r\\
\label{Iso-hierarchy-1.1}
\qquad
=R^m_{11}[v]\circ
\left( \frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}
+\beta_1 \frac{v_r}{v^4} \right),\quad m=0,1,2,\ldots\ ,\end{gathered}$$
where
$$\begin{gathered}
R_{11}[v]=
\frac{1}{v^4}D_r^2
-\frac{6v_r}{v^5}D_r
-\frac{6v_{rr}}{v^5}
+\frac{22v_r^2}{v^6}
+\frac{4\beta_1}{3a} \frac{1}{v^2}\\
\qquad -\frac{2}{a}
\left(
\frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}
+\beta_1 \frac{v_r}{v^4}
\right)
D_r^{-1}\circ v \\
\qquad
+v_rD_r^{-1}\circ
\left(
-\frac{2v_{rr}}{v^6}
+6\frac{v_r^2}{v^7}
+\frac{2\beta_1}{3a} \frac{1}{v^3} \right).\end{gathered}$$
The first member of the hierarchy for $m=0$ is the equation (\[Iso-class-1.1\]).\
\
[**Case 1.2:**]{} We consider the hierarchy (\[hierarchy-1\]) with $\beta_1=\beta_3=0$. This leads to the following isochronous hierarchy
$$\begin{gathered}
v_{s}+i\left(\frac{2}{2m+3}\right) \omega v+i\left(\frac{3}{2m+3}\right)\omega rv_r\\
\label{Iso-hierarchy-1.2}
\qquad
=R^m_{12}[v]\circ
\left( \frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}
+\beta_1 \frac{v_r}{v^4} \right),\quad m=0,1,2,\ldots\ ,\end{gathered}$$
where
$$\begin{gathered}
R_{12}[v]=\frac{1}{v^4}D_r^2
-\frac{6v_r}{v^5}D_r
-\frac{6v_{rr}}{v^5}
+\frac{22v_r^2}{v^6}
+\frac{2\beta_2}{a} \frac{1}{v}\\
\qquad
-\frac{2}{a}
\left(
\frac{av_{rrr}}{v^6}
-\frac{12av_rv_{rr}}{v^7}
+\frac{21av_r^3}{v^8}
+\beta_2 \frac{v_r}{v^3}
\right)
D_r^{-1}\circ v \\
\qquad
+v_rD_r^{-1}\circ
\left(
-\frac{2v_{rr}}{v^6}
+6\frac{v_r^2}{v^7}
+\frac{\beta_2}{a} \frac{1}{v^2}
\right).\end{gathered}$$
The first member of the hierarchy for $m=0$ is the equation (\[Iso-class-1.2\]).\
\
[**Case 1.3:**]{} We consider the hierarchy (\[hierarchy-1\]) with $\beta_1=\beta_2=0$. This leads to the following isochronous hierarchy
$$\begin{gathered}
v_{s}+i\lambda \omega v+i\left(2\lambda+\frac{1}{2m+3} \right)\omega rv_r\\
\label{Iso-hierarchy-1.3}
\qquad
=R^m_{13}[v]\circ
\left( \frac{av_{rrr}}{v^6}-\frac{12av_rv_{rr}}{v^7}+\frac{21av_r^3}{v^8}
+\beta_3 v_r \right),\quad m=1,2,\ldots\end{gathered}$$
where $\lambda$ is arbitrary and
$$\begin{gathered}
R_{13}[v]=\frac{1}{v^4}D_r^2
-\frac{6v_r}{v^5}D_r
-\frac{6v_{rr}}{v^5}
+\frac{22v_r^2}{v^6}\\
\qquad
-\frac{2}{a}
\left(
\frac{av_{rrr}}{v^6}
-\frac{12av_rv_{rr}}{v^7}
+\frac{21av_r^3}{v^8}
+\beta_3 v_r
\right)
D_r^{-1}\circ v \\
\label{R13}
\qquad
+v_rD_r^{-1}\circ
\left(
-\frac{2v_{rr}}{v^6}
+6\frac{v_r^2}{v^7}
+\frac{2\beta_3}{a}v_r\right).\end{gathered}$$
Note that equation (\[Iso-class-1.3\]) does [**not**]{} correspond to $m=0$ in . The reason is rather obvious: since $$R^m_{13}[v] \circ \left(\beta_3 v_r\right)=0$$ for all $m=1,2,\ldots$, the $\beta_3$ term disappears in (\[Iso-hierarchy-1.3\]) and there remains only one constraint on $\lambda$ and $\mu$ to assure that the hierarchy does not depend explicitly on $s$, namely $$\mu-2\lambda-\frac{1}{2m+3}=0.$$ \
For the equation (\[class-2\]) we have the hierarchy (\[hierarchy-2\]), namely
$$\begin{gathered}
u_{t}=R_2^m[u]\circ \left(
\frac{au_{xxx}}{u^3}
-6\frac{au_xu_{xx}}{u^4}
+6\frac{au_x^3} {u^5}
+\beta_1\frac{u_x}{u^3}
+\beta_2 uu_x
+\beta_3u_x
\right),\\[2mm]
m=0,1,2,\ldots\ ,{\nonumber}\end{gathered}$$
where $R_2[u]$ is given by . Corresponding to the above Case 2.1, Case 2.2 and Case 2.3, the isochronous hierarchies are the following:\
\
[**Case 2.1:**]{} We consider the hierarchy (\[hierarchy-2\]) with $\beta_2=\beta_3=0$. This leads to the following isochronous hierarchy
$$\begin{gathered}
\label{Iso-hierarchy-2.1}
v_{s}+i\left(\frac{1}{2m+3}\right) \omega v =
R_{21}^m[v]\circ \left( \frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}
+\beta_1\,\frac{v_r}{v^3} \right)\\[2mm]
m=0,1,2,\ldots,{\nonumber}\end{gathered}$$
where
$$\begin{gathered}
R_{21}[v]=\frac{1}{v^2}D_r^2
-\frac{3v_r}{v^3} D_r
-3\frac{v_{rr}}{v^3}
+6\frac{v_r^2}{v^4}
+\frac{\beta_1}{a} \frac{1}{v^2}\\
\qquad
-\frac{1}{a}
\left(
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}
+\beta_1\,\frac{v_r}{v^3}
\right)
\,D_r^{-1}\circ 1.\end{gathered}$$
\
[**Case 2.2:**]{} We consider the hierarchy (\[hierarchy-2\]) with $\beta_1=\beta_3=0$. This leads to the following isochronous hierarchy
$$\begin{gathered}
\begin{split}
v_{s}-i\left(\frac{1}{2m+3}\right) \omega v&-i\left(\frac{2}{2m+3}\right)\omega r v_r
=
\\
&=R_{22}^m[v]\circ \left(
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}+\beta_2 vv_r\right)
\end{split}\label{Iso-hierarchy-2.2}\\[2mm]
m=0,1,2,\ldots,{\nonumber}\end{gathered}$$
where
$$\begin{gathered}
\label{R22}
R_{22}[v]=\frac{1}{v^2}D_r^2
-\frac{3v_r}{v^3} D_r
-3\frac{v_{rr}}{v^3}
+6\frac{v_r^2}{v^4}
+\frac{\beta_2}{3a} v^2\\
\qquad
-\frac{1}{a}
\left(
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}
+\beta_2 vv_r
\right)
\,D_r^{-1}\circ 1
+\frac{4\beta_2}{3a} v_r\,D_r^{-1}\circ v.\end{gathered}$$
\
[**Case 2.3:**]{} We consider the hierarchy (\[hierarchy-2\]) with $\beta_1=\beta_2=0$. This leads to the following isochronous hierarchy
$$\begin{gathered}
\label{Iso-hierarchy-2.3}
\begin{split}
v_{s}-i\lambda \omega v&-i\left(\lambda +\frac{1}{2m+3}\right)\omega r v_r \\
&\qquad\qquad=R_{23}^m[v]\circ \left(
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}+\beta_3 v_r\right)
\end{split}\\[2mm]
m=1,2,3,\ldots,{\nonumber}\end{gathered}$$
where
$$\begin{gathered}
R_{23}[v]=\frac{1}{v^2}D_r^2
-\frac{3v_r}{v^3} D_r
-3\frac{v_{rr}}{v^3}
+6\frac{v_r^2}{v^4}\\
\qquad
-\frac{1}{a}
\left(
\frac{av_{rrr}}{v^3}-\frac{6av_rv_{rr}}{v^4}+\frac{6av_r^3}{v^5}
+\beta_3 v_r
\right)
\,D_r^{-1}\circ 1
+\frac{\beta_3}{a} v_r\,D_r^{-1}\circ 1.\end{gathered}$$
Note that equation (\[Iso-class-2.3\]) does [**not**]{} correspond to $m=0$ in (\[Iso-hierarchy-2.3\]) for the same reason as in Case 1.3. That is, since $$R^m_{23}[v] \circ \left(\beta_3 v_r\right)=0$$ for all $m=1,2,\ldots$, the $\beta_3$ term disappears in (\[Iso-hierarchy-2.3\]) and there remains only one constraint on $\lambda$ and $\mu$ to assure that the hierarchy does not depend explicitly on $s$, namely $$\mu-2\lambda-\frac{1}{2m+3}=0.$$ \
[**Acknowledgements:**]{} E.G.R. has been partially supported by the FONDECYT operating grant \# 1161691. R.H.H. has been partially supported by the “Ministerio de Economía y Competitividad" (MINECO, Spain) under grant MTM2016-79639-P (AEI/FEDER, EU).
[00]{} M. Adler, On the trace functional for formal pseudodifferential operators and the symplectic structure of the KdV type equations, [*Inventiones Math.*]{} 50 (1979), 219–248.
D.M. Ambrose and J.D. Wright, Preservation of support and positivity for solutions of degenerate evolution equations. [*Nonlinearity*]{} 23 (2010) 607–620.
D.M. Ambrose and J.D. Wright, Dispersion versus anti-diffusion:well-posedness in variable coefficient and quasilinear equations of KdV type. [*Indiana Univ. Math. J.*]{} 62 (2013), 1237–1281.
D.M. Ambrose, G. Simpson, J.D. Wright and D.G Yang, Ill-posedness of degenerate dispersive equations. [*Nonlinearity*]{} 25 (2012) 2655–2680.
I.M. Bakirov, On the symmetries of some system of evolution equations. Technical Report, Institute of Mathematics, Russian Academy of Sciences, Ufa, 1991.
F. Beukers, J.A. Sanders, and J.P. Wang, One symmetry does not imply integrability. [*J. Diff. Eq.*]{} 146 (1998) 251–260.
A.H. Bilge, A system with a recursion operator but one higher local symmetry. [*Lie Groups Appl.*]{} 1 (1994) 132–139. (Also available as arXiv:solv-int/9905012).
F. Calogero, Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable? In: ’What Is Integrability?’, E. Zakharov (Ed.), 1-62. Springer-Verlag, Berlin Heidelberg 1991.
F. Calogero, Isochronous Systems, Oxford University Press, Oxford, UK, 2008.
F. Calogero and M. Mariani, A modified Schwarzian Korteweg-de Vries equation in 2+1 dimensions with lots of isochronous solutions, *Phys. Atomic Nuclei* Vol. 68 (2005) 1646–1653; Russian version: Yadernaya Fiz. Vol. 68 (2005) 1710–1717.
F. Calogero, M. Euler and N. Euler, New evolution PDEs with many isochronous solutions, *J. Math. Anbal. Appl.* Vol. 353 (2009) 481–488.
F. Cooper, M. Shepard and P. Sodano, Solitary waves in a class of generalized Korteweg-de Vries equations. [*Phys. Rev. E*]{} 48 (1993), 4027–4032.
D.K. Demskoy and V.V. Sokolov, On recursion operators for elliptic models. [*Nonlinearity*]{} 21 (2008), 1253–1264.
B. Dey and A. Khare, Stability of compacton solutions. [*Physical Review E*]{} 58 (1998), R2741–R2744.
A. S. Fokas, A symmetry approach to exactly solvable evolution equations, [*J. Math. Physics*]{}, 21(6), (1980), 1318–1325.
R. Hernández Heredero, Integrable Quasilinear Equations, [*Teoret. Mat. Fiz.*]{}, 133:2 (2002), 233–246.
R. Hernández Heredero, Classification of Fully Nonlinear Integrable Evolution Equations of Third Order, [*J. Nonlinear Math. Phys.*]{}, 12 (2005), 567–585.
R. Hernández Heredero and E.G. Reyes, Nonlocal symmetries, compacton equations and integrability. *Int. J. Geometric Methods in Modern Physics* Vol. 10, No. 9 (2013) 1350046 (24 pages).
R. Hernández Heredero, V.V. Sokolov, S.I. Svinolupov, Why are there so many integrable equations of third order?, in [*Proceedings of NEEDSâ94, Los Alamos*]{}, ed. E.V. Zhakarov, A.E. Bishop, D.D. Holm (World Scientific, 1994), pp. 42–53.
A. Khare and F. Cooper, One-parameter family of soliton solutions with compact support in a class of generalized Korteweg-de Vries equations. [*Physical Review E*]{} 48 (1993), 4843–4844.
A. Ludu and J. P. Draayer, Patterns on liquid surfaces: cnoidal waves, compactons and scaling, [*Phys. D*]{} 123 (1998), 82–91.
Yi.A Li, P.J. Olver, and P. Rosenau, Non-analytic solutions of nonlinear wave models. In ‘Nonlinear theory of generalized functions (Vienna, 1997)’, 129–145. Chapman & Hall/CRC, Boca Raton, FL, 1999.
M. Mariani and F. Calogero, Isochronous PDEs, *Phys. Atomic Nuclei* Vol. 68 (2005) 899–908; Russian version: Yadernaya Fiz. Vol. 68 (2005) 935–944.
A.V. Mikhailov, A.B. Shabat, V.V. Sokolov, The symmetry approach to classification of integrable equations, in [*What is Integrability?*]{}, ed. E.V. Zhakarov (Springer, 1991), pp. 115–184.
A.V. Mikhailov, V.V. Sokolov. Symmetries of Differential Equations and the Problem of Integrability. Lect. Notes Phys. 767, (2009), 19–88.
P.J. Olver, “Applications of Lie Groups to Differential Equations” (1993). Second Edition, Springer-Verlag, New York.
P. Rosenau and A. Zilburg, Compactons. [*J. Phys. A: Math. Theor.*]{} 51 (2018) 343001 (136pp).
V.V. Sokolov, V.V. Shabat, Classification of integrable evolution equations, *Sov. Sci. Rev. C* **4** (1984) 221–280.
N. Petersson, N. Euler, M. Euler, Recursion operators for a class of integrable third order equations. *Studies in Applied Mathematics* **112** (2004), 201–225.
P. Rosenau, What is $\dots$ a Compacton? [*Notices of the AMS*]{} 52 (2005), 738–739.
P. Rosenau and J.M. Hyman, Compactons: Solitons with finite wavelength. [*Phys. Rev. Lett.*]{} 70 (1993), 564–567.
S.I. Svinolupov, V.V. Sokolov. Weak nonlocalities in evolution equations. [*Mathematical Notes*]{} 48:6 (1990), 1234–1239.
J.A. Sanders, J.-P. Wang, On the integrability of homogeneous scalar evolution equations. [*J. Differential Equations*]{} 147 (1998), 410–434.
J.A. Sanders and J.-P. Wang, On the integrability of non-polynomial scalar evolution equations. [*J. Differential Equations*]{} 166 (2000), 132–150.
J. Vodová, A complete list of conservation laws for non-integrable compacton equations of $K(m,m)$ type. [*Nonlinearity*]{} 26 (2013), 757–762.
A. Zilburg and P. Rosenau, Loss of regularity in the $K(m,n)$ equations. [*Nonlinearity*]{} 31 (2018), 2651–2665.
[^1]: This observation has been proven rigorously by Vodová in 2013, see [@Vo].
[^2]: The foregoing discussion implies that instead of a general formal symmetry $\Lambda$ of degree $r$, we can consider its $r$th root $\Lambda^{1/r}$ of degree 1 without loss of generality, see [@MSS] for details.
[^3]: We note that the case $n=-1/2$, $m=3/2$ was missing in [@HHR].
|
---
abstract: |
**Abstract** Theories based on General Relativity or Quantum Mechanics have taken a leading position in macroscopic and microscopic Physics, but fail when used in the other extremity. Thus, we try to establish a new structure of united theory based on General Relativity by forming certain spacetime property and a new model of particle. This theory transforms the Riemann curvature tensor into spacetime density scalar so that gravitational field can be added to the Quantum Mechanics, and supposes the electromagnetic field in General Relativity to be a kind of spacetime fluid. Also the theory carries on Einstein-Cartan Theory about spacetime torsion to form the energy exchange in order that Hubble’s law and dark mass can be simply explained. On the basis of the definition of signal, measurement and observation, the theory is able to construct a modified Quantum Equation in curved spacetime compatible with General Relativity, and hence unite the General Relativity and Quantum Mechanics in some range and puts forward a new idea of unification.
**Keywords** General Relativity; Field Theory; Quantization; Quantum Theory
author:
- 'Zhang Chang-kai'
title: Construction of Spacetime Field and Quantization of General Relativity
---
Introduction
============
The contradiction between the General Theory of Relativity and Quantum Mechanics was discovered not long after they were developed into a successful theory. Hence, searching for a united theory has become one of the mainstreams of the Theoretical Physics. Nowadays, there are three main approaches to the united theory: the String Theory, the Deformation Quantization and Loop Quantum Gravity.
The String Theory is a popular candidate of the united theory[@STMvol][@STMNS]. This theory supposes the basic element of matter are not the point particles but vibrating strings in a high dimensional space. The newly developed M-Theory supposes that every basic particle and the basic force fields can be united in 11 dimensional space. However, this theory meets difficulties in mathematical tools and experimental verification.
The Loop Quantum Gravity is also a very competitive alternative[@LQGQGR][@QG][@LQGOV]. It is the most successful non-perturbative quantization theory and do not need any extra dimensions. Nevertheless, it only touches upon the quantization of gravity and therefore cannot form a complete united theory.
The Deformation Quantization suggests that any nontrivial associative deformation of an algebra of functions can be explained as a kind of quantization. Based on Poisson Geometry, Group Theory and noncommutative geometry, it mainly develops various of quantization in linear and nonlinear field theories. [@DQPM][@QM]
And there are also efforts on some other threads toward a united theory such as the development of a quantum field theory in curved spacetime[@QFTOPE][@AQFT]. However, these theories have not yet led to a satisfying united theory, either.
We try to construct a new form of a possible united theory (General United Spacetime Field Theory, GUSFT) with the spacetime field and quantized matter distribution. In this theory, the state of a particle system is described by quantized matter distribution, the interaction is described by spacetime field and the evolution is determined as a result.
The present paper is organized as follows. Section mainly concerns the construction of spacetime field and Section focuses on the attempt to quantize the General Relativity. In Section .1, we largely present the definition and formulation of spacetime property, and gain the spacetime formula in the condition of Schwarzschild Solution. In Section .2, we create the 4-velocity of electromagnetic field and therefore reformulate the energy-momentum tensor. In Section .3, we discuss the situations of positive and negative energy exchange based on spacetime torsion, and then illustrate its effect on the evolution of universe. In Section , the definitions of signal, measurement and observation are put forward, on the basis of which the curved Quantum Equation is constructed. And hence, we are able to give the possibility explanation a new meaning. Thus far, we complete the theoretical foundation of GUSFT.
Spacetime Property
==================
This theory is based on the two basic quantities, spacetime density and spacetime fluid, which aims to describe the gravitational field and the electromagnetic field. We try to consider the gravitational field as a kind of density scalar such that it can be used to indicate the curvature of spacetime. And we consider the electromagnetic field as a kind of perfect fluid, satisfying the energy-momentum tensor. Then, through the electromagnetic field can it be combined with the Quantum Mechanics.
All the tensors in this article are written by abstract index notation. All the inner products of Dirac notation do not include the integral.
Spacetime Density
-----------------
According to the General Relativity, the gravitational field is described as the curvature of the spacetime. In Mathematics, the curvature is explained as the change of spacetime geometry. But in Physics, it’s natural to find a physical quantity homologous to it. Due to the effort of the attempt to reflect the effect of dark matter and dark energy, it’s suitable to suppose this quantity is density. In Einstein’s field equation, Riemann curvature tensor takes the leading position. The Riemann tensor represents the curvature degree of spacetime, one of whose significant property is the curve dependence in the parallel-transport of vectors. So we hope it’s possible to express this property by the density.
First we consider the parallel-transport of vectors along a closed path. In consideration of the equivalence principle, it’s necessary to consider the torsion tensor to be infinitely small compared with the curvature tensor. Suppose there is a $2$-dimensional submanifold $S$ in $n$-dimensional manifold $M$, which has the curves without self-intersection parameterized with $t$ and $s$ and can form a closed path with tangent vector $T^a$ and $S^a$. It’s easy to establish a coordinate system compatible with $t$ and $s$, and consequently, the difference of a vector parallel-transported along this closed path will be $$\delta v^{a}={R_{cbd}}^{a}\cdot T^{c}dt\cdot S^{b}ds\cdot v^{d}$$ This equation [@GR] is intended to represent the effect of curvature. We have two curves and each one is of equal importance. Without loss of generality, we take the gauge $$\begin{split}
T^b\nabla_{\!b}S^a=0 \\
S^b\nabla_{\!b}T^a=0
\end{split}$$ Therefore, we construct the density potential as $$\label{eq:original}
T^aS^b\nabla_{\!a}\!\nabla_{\!b}\phi =\varLambda \sqrt{R_{abcd}R^{abcd}}\tag{0}$$ Within, $\varLambda$ is a constant. $\phi$ is the density potential, which is a scalar that can represent the difference of a parallel-transported vector in some degree. It’s easily perceived that $\phi$ can reflect the difference of points as a result of curvature and we should remember the index $a$ and $b$ are limited by the curves. By analogy, it’s natural to define the spacetime density $\rho$ by $$\tag{\ref{eq:original}$'$}\label{eq:origprime}
\rho =-T^a\nabla_{\!a}\phi$$ Solve the equation (0) and ($0'$), and then we’ll have $$\rho=-\varLambda \int \sqrt{R_{abcd}R^{abcd}}\,Tdt$$ This is the computational equation of spacetime density.
Next we put forward two useful theorems.
In the equation of spacetime density (1), if we use the coordinate $x^{\iota }$ as parameter, then the density equation can be simplified as $$\rho=-\varLambda \int \sqrt{R_{abcd}R^{abcd}}\,dx^{\iota }$$ In order to keep the balance of the indexes on both sides of the equation, some abstract index is left out.
There is an equation following for tangent vector and differential parameter $$\begin{split}
T^{a}&=\frac{\partial x^{\iota }}{\partial t}X^{a}\\
dt&=\frac{\partial t}{\partial x^{\iota }}dx^\iota
\end{split}$$
So we have $$Tdt=dx^{\iota }$$ Thus we gain theorem 1.
The coefficient of spacetime density equation satisfies $$\varLambda =\frac{c^{2}}{16\pi G^{2}M}$$
Because equation (1) always holds in any spacetime condition, if we want to solve $\varLambda$, we only need to solve its value in certain condition, and then it can represent the value of $\varLambda$ in all conditions. In General Relativity, the most simple spacetime condition is the static Schwarzschild Solution. So we need to solve $\varLambda$ in this condition.
For Schwarzschild Solution, we have the components of curvature tensor
$$\begin{aligned}
R_{0101}&=-\frac{2M}{r^{3}} & R_{0202}&=\frac{M}{r}\left ( 1\!-\!2M/r \right ) & R_{0303}&=\frac{M}{r}\left ( 1\!-\!2M/r \right )\sin ^{2}\!\theta\\
R_{2323}&=2Mr\sin \!^{2}\theta & R_{1212}&=-\frac{M}{r}\left ( 1\!-\!2M/r \right )^{-1} & R_{1313}&=-\frac{M}{r}\left ( 1\!-\!2M/r \right )^{-1}\sin ^{2}\!\theta\end{aligned}$$
And the metric $$\begin{split}
ds^2=&-\left ( 1\!-\!\frac{2M}{r} \right )dt^{2}+\left ( 1\!-\!\frac{2M}{r} \right )^{-1}\!dr^{2}+r^{2}\left ( d\theta ^{2}+\sin \!^{2} \theta \,d\varphi ^{2}\right )
\end{split}$$
Wherein the coordinate lines of $t$ and $r$ are the only two curves without self-intersection. We gain the spacetime density in this condition $$\rho =\varLambda \frac{2GM}{r^{2}}+C$$ Because the curvature tensor equals zero at infinity, so the spacetime density equals zero at infinity, too. Thus, the constant $C=0$.
Next, integrate the density and we have mass $$M=8\pi\! \varLambda G\!M\!R$$ And we also have the radius of the black hole $$R=\frac{2GM}{c^{2}}$$ Then we can solve the coefficient $\varLambda$.
Using the theorem above, we can solve the spacetime formula in the condition of Schwarzschild Solution $$\rho =\frac{c^{2}}{8\pi Gr^{2}}$$ And find when $r\rightarrow 0$, there is $\rho\rightarrow \infty $, and then we can define this position as the position of matter when there is an observation. Also we discover that the spacetime density in this condition has nothing to do with any intrinsic property of matter but position.
Spacetime Fluid
---------------
In General Relativity, the electromagnetic field is described by energy-momentum tensor. This theory tries to describe the electromagnetic field as a kind of perfect fluid, which makes the theory able to be used microscopically.
In General Relativity, there is the energy-momentum tensor of electromagnetic field $$T_{ab}=\frac{1}{2\mu}\left ( F_{ac}{F_{b}}^{c}+^{\ast }\!\!{F_{ac}}^{\ast }\!{F_{b}}^{c} \right )$$ And the energy-momentum tensor of perfect fluid $$T_{ab}=\left ( \eta +p \right )U_{a}U_{b}+pg_{ab}$$
So if we consider the electromagnetic field as a kind of spacetime fluid, then the energy-momentum tensor of them must be equal. By analogy, we put forward the energy-momentum tensor of spacetime fluid $$T_{ab}=\eta U_aU_b+ph_{ab}-\frac{1}{2}\left |c\eta\varepsilon_{abcd}V^cU^d\right |$$ In the equation, there are two kinds of vector field $$\begin{split}
U_a&=Z_a\vec{l}+c\eta^{-\frac{1}{2}}\varepsilon^{\frac{1}{2}}F_{ab}Z^b\vec{m}\,-\eta^{-\frac{1}{2}}\mu^{-\frac{1}{2}} {}^ \ast\!F_{ac}Z^c\vec{n}\\
V_a&=Z_a\vec{l}+c\eta^{-\frac{1}{2}}\varepsilon^{\frac{1}{2}}F_{ab}Z^b\vec{n}\,+\eta^{-\frac{1}{2}}\mu^{-\frac{1}{2}} {}^ \ast\!F_{ac}Z^c\vec{m}
\end{split}$$ And two kinds of scalar field $$\begin{split}
&\eta= \frac{1}{2}\left (\varepsilon E^2+\mu^{-1}B^2 \right )\\
&p= -\frac{1}{2}\left (\varepsilon E^2+\mu^{-1}B^2 \right )
\end{split}$$ Wherein, $Z^a$ is the 4-velocity of any observer; $U^a$ is the 4-velocity field of spacetime fluid and $V^a$ is the symmetric of it; $\eta$ is the energy density and $p$ is the pressure; $\vec{l}$, $\vec{m}$ and $\vec{n}$ are orthonormalized quantity; $\varepsilon$ is vacuum dielectric constant and $\mu$ is permeability of vacuum; $E$ is electric field strength and $B$ is magnetic field strength; $h_{ab}$ is the induced metric and $\varepsilon_{abcd}$ is the adapted volume element; $c$ is light speed in vacuum.
Then, we can discuss the significance of each quantity in the spacetime fluid. First, it’s easy to verify that the components of the energy-momentum tensor of spacetime fluid are the same as that of the electromagnetic field. We transform the electromagnetic field into a kind of spacetime fluid and construct the 4-velocity by electromagnetic field tensor. The orthonormalized quantity used in the 4-velocity of spacetime fluid has nothing to do with the geometric vector space. Its function is only to insure that (2) will always hold. Thus, the electromagnetic field is described by two vector fields and two scalar fields.
Next, we are going to study further on $\vec{l}$, $\vec{m}$ and $\vec{n}$ . We find that $U^a$ and $V^a$ have an interesting symmetry and it can be easily explained if $\vec{l}$, $\vec{m}$ and $\vec{n}$ have the form of the combination of wave function $e^{\mathbf{i}\theta}$ and spin matrix $\phi$ together with its conjugate $\bar{\phi}$. So make a transformation on $U_a$ $$|U_a\rangle\!=Z_a\phi^\mp \!+\!\eta^{-\frac{1}{2}}\mu^{-\frac{1}{2}}\left ( F_{ab}Z^be^{\mathbf{i}\theta}\phi^\pm\!\!-\!{}^ \ast\!F_{ac}Z^c\mathbf{i}e^{\mathbf{i}\theta}\phi^\pm\right )$$ And see its conjugate $$\langle U_a|\!=Z_a\bar{\phi}^\mp \!+\!\eta^{-\frac{1}{2}}\mu^{-\frac{1}{2}}\left ( F_{ab}Z^be^{-\mathbf{i}\theta}\bar{\phi}^\pm\!\!+\!{}^ \ast\!F_{ac}Z^c\mathbf{i}e^{-\mathbf{i}\theta}\bar{\phi}^\pm\right )$$ Thus, we have $$\langle V_a|=\mathbf{i}\langle U_a|$$ So the energy-momentum tensor can be simplified as $$T_{ab}=\eta \langle U_a|U_b\rangle+ph_{ab}-c\eta\left |{}^{\ast }\mathbf{i} \langle U_a|U_b\rangle\right |$$
From (3) we discover that there is wave inside the electromagnetic field, which may be the origin of the electromagnetic wave. Equation (3) and Equation (4) are the final result of the discussion of spacetime fluid.
Spacetime Torsion and Energy Exchange
-------------------------------------
Einstein used the spacetime without the torsion when developing his General Relativity, which means he only considered the spacetime curvature but not contortion. Then Cartan established the spacetime with torsion and it became the Einstein-Cartan Theory. According to this theory, the distribution of matter will influence the torsion of spacetime. Hence, in the condition of this theory will there be some other phenomena.
In order to express these phenomena, first we consider the definition of torsion tensor $$\left ( \nabla _{\!\!a}\nabla_{\!b} -\nabla_{\!b} \nabla _{\!\!a} \right ) f=-{T^{c}}_{ab}\nabla _{\!\!c}f$$
For ordinary matter field $\mu$, we put $f =\mu$ into the equation, and consider the difference of a periodic transport along a closed path, then we gain $$\delta \mu=\int\!\!\!\!\! \int {T^{c}}_{ba}\nabla _{\!\!c}\,\mu \cdot I^{a}d\iota \cdot N^{b}d\nu$$ Hence, we see the energy of the field in connection with self-spin will dissipate. In this theory, this part of energy will be exchanged with the spacetime. The energy is released or taken in by the spacetime.
For further discussion, it’s necessary to distinguish two kinds of different torsion tensor field: $_{o}{T^{c}}_{ab}$ and $_{i}{T^{c}}_{ab}$. $_{o}{T^{c}}_{ab}$ means the torsion tensor field produced by the matter except the target object, and $_{i}{T^{c}}_{ab}$ means the torsion tensor field produced by the target object itself. Due to the spin of the object, ${T^{c}}_{ab}$ will cause its energy to be absorbed by the spacetime, matter field losing energy, which is called positive energy exchange. And because of $_{i}{T^{c}}_{ab}$ is created by the object itself, it will make the spacetime energy release from spacetime to matter field, the energy of the matter field rise, which is called negative energy exchange.
To construct the equation of energy exchange, we need the theorems below.
The length of space rotation angular velocity vector is invariant to any inertial observer or coordinate system.
In General Relativity, suppose there are world lines of a particle $G$, then for space rotation angular velocity vector is there an equation [@WG] $$g_{ab}\frac{Dw^{b}}{d\tau }=\varepsilon _{abcd}Z^{b}w^{c}\omega ^{d}$$ Within $D/d\tau$ indicates Fermi-Walker derivative, $\varepsilon _{abcd}$ is the volume element adapted with $g_{ab}$, $Z^{b}$ is the tangent vector of the world line, $w^{c}$ is the space vector in any point in any world line of $G$. Change all the indexes and then contract to get the length of $\omega ^{d}$ $$n\frac{Dw^{a}}{d\tau }\frac{Dw_{a}}{d\tau }=\varepsilon^{2}Z^2w^{2}\omega ^{2}$$ Within $Z$, $w$, $\omega$, $\varepsilon$ is the length of $Z^b$, $w^{c}$, $\omega^d$, $\varepsilon _{abcd}$ with contraction. Also there is $$\frac{Dw^{a}}{d\tau }={h^{a}}_{b}Z^{c}\nabla _{c}w^{b}$$ Within ${h^{a}}_{b}$ is the projection map, and without loss of generality, suppose $w^c$ and $\omega^d$ is perpendicular, so we get $$\omega ^{2}=4w^{-2}Z^{c}\nabla _{c}w^{b}Z^{d}\nabla _{d}w_{b}$$ And discover all the quantities in the right side of equation are invariant to any inertial observer and coordinate system.
The angular momentum vector in General Relativity can be formulated by $$J^a=m\omega^a r^\iota r_\iota$$ Within, $m$ is the rest mass, $\omega^a$ is the space rotation angular velocity vector, $r^\iota$ is the displacement vector.
Define the displacement vector and angular momentum by $$\begin{split}
r_\iota &= \int \!\!\sqrt{{\delta_{\iota}}^{c}h_{c\iota}}\\
J_a&=\varepsilon _{abcd}Z^b P^c r^d
\end{split}$$ Wherein, $h_{c\iota}$ is the induced metric and $P^c$ is the 4-momentum.
Then, with the help of $$U_a=\varepsilon _{abcd}Z^br^c\omega^d$$ We can solve the formulation of the angular momentum vector.
Then we can calculate the variation of energy. First consider positive energy exchange. For photons and other particles, there is quantized angular momentum $$J=\sqrt{j\left ( j+1 \right )}\hbar$$ According to the theorem, there is $$J=m\omega r^{2}$$ So we can solve them and gain $$\omega=\frac{\sqrt{j\left ( j+1 \right )}\hbar}{mr^{2}}$$ Hence, there is $$\delta t=T=\frac{2\pi mr^{2}}{\sqrt{j\left ( j+1 \right )}\hbar}$$ So we get the positive energy exchange rate $$\eta =\!\!\int \!\frac{\delta \mu }{\delta t}dr=\!\!\int dr\frac{\sqrt{j\left ( j+1 \right )}\hbar}{2\pi mr^{2}}\int\!\!\!\!\! \int {T^{c}}_{ba}\nabla _{\!\!c}\,\mu \cdot I^{a}d\iota \cdot N^{b}d\nu$$ Next consider the negative energy exchange of ordinary particle and black hole. For ordinary particle, it’s obvious that there is $$\gamma=\int dr\frac{\sqrt{j\left ( j+1 \right )}\hbar}{2\pi mr^{2}}\int\!\!\!\!\! \int {T^{c}}_{ba}\nabla _{\!\!c}\,\mu \cdot I^{a}d\iota \cdot N^{b}d\nu$$ For black hole, because of $$J=Ma$$ So its negative energy exchange is $$\gamma=\int dr \frac{a}{2\pi r^{2}}\int\!\!\!\!\! \int {T^{c}}_{ba}\nabla _{\!\!c}\,\mu \cdot I^{a}d\iota \cdot N^{b}d\nu$$
Next we discuss the result separately. For photons, due to the nonexistence of rest mass, there is no negative energy exchange, but positive energy exchange. As a result, the energy of photons will be dissipated, whose rate depends on total torsion tensor field. For ordinary particles, when $_{o}{T^{c}}_{ab}$ does not appear or is so small that it can be ignored, there is ${T^{c}}_{ab} \approx $ $_{i}{T^{c}}_{ab}$. So the rate of two energy exchange is almost equal, resulting in the invariance of its mass. For the black hole, because there is no quantized angular momentum, there is no positive energy exchange. Supposing a KN black hole with spin, the mass will rise ceaselessly, whose rate depends on the torsion tensor field created by itself and its angular velocity.
Hence, this theory can explain Hubble’s law and parts of the problems of dark matter. For Hubble’s law, due to the loss of the energy of photons produced by stars in galaxies, the spacetime expands, resulting in the distance between galaxies increasing. If we suppose the galaxies homogeneously distribute, it’s easy to see the spacetime expansion and distance is proportional. For some galaxies rotating too fast, it is possible that it is because the black hole absorbs the spacetime energy, the spacetime contracts, causing the galaxies difficult to break up. It’s also possible for some photons released from some matter in the galaxy to run out of energy and become unable to be discovered, forming what is called “dark matter”.
We need to declare that all quantities in connection with spacetime are the spacetime property in the theory, but there is a sort of special quantities called intrinsic spacetime property. Only some of the spacetime quantities such as spacetime curvature, torsion, density and spin belong to the intrinsic property.
Quantization of General Relativity
==================================
This theory finally will try to be combined with Quantum Mechanics. Quantum Mechanics describes phenomena microscopic and quantized, so observation and measurement become very important concepts. So, the theory needs to define observation and measurement.
Signal is a kind of map $\chi\!\!:\!\!V^n\rightarrow V^n$, wherein $V^n$ is the Cartesian product of any vector space $V$ for $n$ times.
Measurement is a set $\mathcal{F}$ of map $f$, satisfy\
$\left (a \right )$ $f\!:\!Q\rightarrow \mathbb{R}^n$ is continuous, wherein $Q$ is the set of certain physical quantities\
$\left (b \right )$ $\forall f,g\in \mathcal{F}$, and $f\!:\!q \mapsto x$, $g\!:\!q \mapsto y$, then $g\circ\! f^{-1}\!:\!x \mapsto y$ is smooth\
$\left (c \right )$ $\exists h$ such that when $f\!:\!q \mapsto x$, $f\!:\!p \mapsto y$ and $h\!:\!p \mapsto q$, $f\circ h\circ\! f^{-1}\!:\!x \mapsto y$ is smooth
Observation is a signal, if\
$\left (a \right )$ It is non spacetime intrinsic property signal\
$\left (b \right )$ It is released by an object called signal emission source\
$\left (c \right )$ It can be measured, and when it becomes any observable quantity $\hat{\alpha }$, the result $a$ satisfy $$\hat{\alpha }\!\left | \varphi \right \rangle=a\left | \varphi \right \rangle$$ Within $\left | \varphi \right \rangle$ indicates some state of a system
From the definition can we see that signal changes the state of an object, which provides us with methods to test the state of a signal emission source by measuring the change of certain physical quantity of the particles in the instrument, which often appears as the interaction of quanta. The definition of measurement guarantees that we can use a series of real numbers to represent the state of the signal emission source, and satisfies the definition of $n$-dimensional differential manifold, which provides the symmetry of unit exchange. Observation is a special signal and the result of measurement is one of its eigenvalues.
According to the definition of observation and measurement above, there are two important points: There is observation without measurement. No matter whether we measure the object, it is the signal source, as long as it is releasing observation, with its wave function collapsed. The observation does not contain spacetime intrinsic property signal, which allow us to measure the spacetime intrinsic property without affecting the wave function in Quantum Theory.
Based on signal, measurement and observation, we try to combine the General Relativity with the Quantum Mechanics separately. First we have to introduce some new models and equations. It’s necessary to leave out some property about vectors in some equation where geometrical vector space and Hilbert space both exist. Then, about the new model of particles is there the quantized matter distribution $$x\left ( r \right )=x\cdot \omega \left ( r \right )=x\cdot \left \langle \varphi\!\!\mid \!\!\varphi \right \rangle$$ Wherein $\omega \left ( r \right )$ is the probability density, $x$ is any quantity in connection with the intrinsic property of a particle, putting a particle into a kind of perfect fluid. One of the most important examples is the mass distribution, after which we can construct the curved Quantum Equation.
Analogized from the wave equation of light, we gain the equation of free particles $$\hbar^2c^2\nabla_{\!\!a} \nabla^a |\varphi\rangle-m^2c^4|\varphi\rangle=0$$ And if we need to consider any gauge field $T^a$, we will have $$\kappa^{2} \sigma^2 \hbar^2c^2\nabla_{\!\!a} \nabla^a |\varphi\rangle-\rho^2 m^2c^4|\varphi\rangle-\delta \kappa^{2}\sigma^2 T_aT^a |\varphi\rangle=0$$ This is the basic curved Quantum Equation. Within, $\nabla_{\!\!a}$ is the curved derivative operator; $m$ is the rest mass of particle; $T^a$ is a possible potential field and $\delta$ is any number; $\kappa$, $\sigma$ and $\rho$ are matrices aimed at the spin.
By defining a new derivative operator as $$\hat\nabla_{\!\!a} =\nabla_{\!\!a}+\!T_{a}$$ And with the help of a gauge condition $$\nabla_{\!\!a}T^{a}=0$$ And combining the matrices together, we can rewrite equation (5) in natural unit as $$\sigma^2 \hat{\nabla}_{\!\!a} \hat{\nabla}^{a} |\varphi\rangle-\rho^2 m^2|\varphi\rangle=0$$
Define the 4-probability-density-current as $$J^a=\mathbf{i}\hbar^2c^2\left (\langle \varphi|\nabla^a|\varphi \rangle- |\varphi\rangle\nabla^a\langle\varphi| \right )$$ Then it’s easy to see (6) leads to the conservation equation below $$\nabla_{\!\!a}J^a=0$$
Then we need the hypotheses below
Arbitrary particle can be expressed by quantized matter distribution.
Evolution of particle states satisfy the curved Quantum Equation.
Interaction between particles are described by spacetime property.
Next we illustrate the rationality of these hypotheses: The General Relativity and Quantum Mechanics should be treated on the merits of each case. When an object has continuous observation, General Relativity functions in one of its neighborhood with continuous observation, and all the contents in Quantum Mechanics embeds into the General Relativity with the wave function collapsed to $\delta$ function, for no matter the observation is created by the object itself or reflected from the instrument, its wave function should collapse in one of its neighborhood with continuous observation. Owing to the non-measurement observation, we can deem that the object has certain position and certain coordinate time and can look upon it as spacetime point in General Relativity without considering the nondeterministic finite automation. When the object has no observation or has only discontinuous observation, Quantum Mechanics functions, and we can construct the spacetime background by quantized matter distribution. Thus, the simultaneous formulas will give all the information of a particle.
Hence, the connection between matter distribution and wave function is established. In the critical point, the variance of wave function indicates the variance of matter distribution, and then affects the spacetime curvature. For instance, suppose there is an electron in a hydrogen atom in ground state with the wave function decreased exponentially. In this condition, the quantized mass distribution functions, and the effect of electrons on spacetime is described by quantized mass distribution. However, if some time a photon comes and gets the electron to have continuous observation, then the wave function of the electron will collapse immediately. The electron becomes a point particle at this time, and the effect on spacetime changes to the effect of point particle mass distribution on spacetime.
Using the conclusion above, we can try to give the probability explanation for Quantum Equation a new meaning. First of all, we suppose there exists a possible energy exchange in the intersection of particles, which is formulated by $$\delta H=\sum_{i,j}^{n}c^{2}\!\!\int \min\left \{ m_i\left \langle \varphi_i|\varphi_i \right \rangle\!,m_j\left \langle \varphi_j|\varphi_j \right \rangle \right \}dv$$ And if $T_i$ is the energy of particle $i$ that may be carried by itself or gain from outside, then the possible energy exchange will be $$\delta E=\sum_{i,j}^{n}c^{2}\!\!\int \min\left \{ m_i\left \langle \varphi_i|\varphi_i \right \rangle\!,m_j\left \langle \varphi_j|\varphi_j \right \rangle \right \}dv\!+\!T_i\!+\!T_j$$ If this quantity at point $p$ equals (or greater than for continuous condition) the energy difference of any two energy levels of one of the particles $i$, then the energy $\delta H$ together with $T_i$ and $T_j$ will be transferred to particle $i$ and at the same time, the wave function of particle $j$ collapses, taking all its energy to point $p$ and releasing the energy equal to $T_i+T_j$.
From (7) we discover that the possibility of finding a particle at a certain point is influenced by the wave function and the possible energy $T_i$. This can be used to explain the point or line that we see in the instrument. If the energy of the instrument particles (for example, the kinetic energy of atoms) is various enough, the possibility of finding a target particle at a certain place will be proportional to the wave function, and hence form the certain image (for example, the interference fringes).
Thus, the probability explanation of Quantum Equation will only hold in the condition that the environment will give various enough $T_i$ for the particles. But this condition is always satisfied since our instrument usually works at the temperature of more than $270K$, which provides at least enough kinetic energy of atoms.
Discussion and Final Remarks
============================
This article advances the GUSFT Theory and aims at establishing a framework of united theory. We mainly present certain spacetime fields and the application of matter distribution in quantization of General Relativity.
The spacetime density provides us with an inspiration that there is mass hidden inside the spacetime. Together with the spacetime torsion, we find that this configuration of mass plays an important part in the evolution of universe.
The spacetime fluid may reflect to the essence of electromagnetic field. We compare the energy-momentum tensor of electromagnetic field and perfect fluid, from which the 4-velocity of electromagnetic field can be formulated. Using the 4-velocity, we rewrite the energy-momentum tensor and discover what may be the origin of electromagnetic wave.
The new approach aimed at the quantization of General Relativity begins with the definition of signal, measurement and observation. The quantized matter distribution is defined as the description of matter in a system so that the theory can overcome the difficulty raised by the uncertainty of matter displacement in Quantum Theory. And thus, the evolution is provided by the simultaneous equations $$\begin{split}
\sigma^2 \hat{\nabla}_{\!\!a} & \hat{\nabla}^{a} |\varphi\rangle-\rho^2 m^2|\varphi\rangle=0 \\
R_{ab} & -\frac{1}{2}Rg_{ab}=\kappa T_{ab}
\end{split}$$
By the discussion above, we are able to point out that the possibility explanation in Quantum Theory is only a statistical effect. The energy of instrument particle may lead to some bizarre phenomena in Quantum Theory.
In summary, this article constructs the GUSFT Theory and combines the General Relativity and Quantum Mechanics to some extent by quantized matter distribution, which provides another inspiration of united theory.
[10]{}
Joseph Polchinski, *String Theory Vol1, Vol2,* [Cambridge University Press]{}, 1998
Elias Kiritsis, *String Theory in a Nutshell,* [Princeton University Press]{}, 2007
Sergiu I. Vacaru, *Loop Quantum Gravity in Ashtekar and Lagrange-Finsler Variables and Fedosov Quantization of General Relativity,* [The ICFAI Univ J Physics]{}, **4**, 2009
Carlo Rovelli, *Quantum Gravity,* [Cambridge University Press]{}, 2003
Hermann Nicolai, Kasper Peeters, and Marija Zamaklar, *Loop Quantum Gravity: an Outside View,* [Class Quant Grav]{}, **22**, R193–R247, 2005
M. Kontsevich, *Deformation Quantization of Poisson Manifolds,* [Lett Math Phys]{}, **66**, 157–216, 2003
S. Twareque Ali, and M. Englis, *Quantization Methods: a Guide for Physicists and Analysts,* [Rev Math Phys]{}, **17**, 391–490, 2005
Stefan Hollands, and Robert M. Wald, *Quantum Field Theory in Curved Spacetime, the Operator Product Expansion, and Dark Energy,* [Gen Rel Grav]{}, **40**, 2051–2059, 2008
Stefan Hollands, and Robert M. Wald, *Axiomatic Quantum Field Theory in Curved Spacetime,* [Commun Math Phys]{}, **293**, 85–125, 2010
Robert M. Wald, *General Relativity,* [The University of Chicago Press]{}, 37–38, 1984
Liang Can-bin, and Zhou Bin, *Beginning of Differential Geometry and General Relativity,* [The Science Press]{}, 202, 2006
|
---
abstract: 'Decomposing models into multiple components is critically important in many applications such as language modeling (LM) as it enables adapting individual components separately and biasing of some components to the user’s personal preferences. Conventionally, contextual and personalized adaptation for language models, are achieved through class-based factorization, which requires class-annotated data, or through biasing to individual phrases which is limited in scale. In this paper, we propose a system that combines *model-defined* components, by learning when to activate the generation process from each individual component, and how to combine probability distributions from each component, directly from unlabeled text data.'
author:
- |
Denis Filimonov\
`denf@amazon.com` Ravi Teja Gadde\
`gadderav@amazon.com` Ariya Rastrow\
`arastrow@amazon.com`
bibliography:
- 'compositional\_nlm.bib'
title: 'Neural Composition: Learning to Generate from Multiple Models'
---
Introduction {#section:intro}
============
Language models are a key component of applications that require generation of coherent natural language text, including machine translation, speech recognition, abstractive text summarization, and many others. For a long time n-gram models [@ueberla1993ngram] dominated the field due to their simplicity, efficiency and scalability. However, recently neural models gained popularity, notably from simple recurrent networks [@mikolov2010rnnlm] to very powerful models including [@gpt2; @yang2019xlnet]. These models often include billions of parameters and they have been shown to do very well at generalizing from vast amounts of data. However, how to *adapt* these models to different users (e.g., personalized contact list in a messaging application), or how to update these models efficiently (for example, when a new movie title is released, which may be important for a ticket booking application) does still remain a challenge. When the number of users is large, or updates are frequent, adapting a large monolithic model becomes impractical and this necessitates the use of composite models in which some components may be updated separately.
For these reasons, class-based models are still widely used in different applications, particularly in automatic speech recognition (ASR) where integrating external knowledge sources and personalized entities in the language model are crucial in achieving accurate transcription: [@contactnames2015; @dynclass2016vasserman; @mcgraw2016personalized; @chen2018endtoend]. Class-based models, however, require annotations in order to learn where these components/classes are used which limits their applicability. Instead of using classes, where content of a class is assumed to be similar in some way, e.g., entities of the same type, [@hall2015ngram; @scheiner2016voicesearch; @he2019e2e] boost scores of individual phrases and n-grams to bias ASR search. Note that this type of biasing can be applied to both WFST-based[^1] and neural models.
[@pundak2018deep] learn a fixed-size representation for every biasing phrase separately. The ASR decoder then uses attention mechanism to interpolate these representations and the result is added to the decoder’s input. As the decoder needs to attend to each individual phrase at every step, scaling this approach to a large number of biasing phrases and entities poses an engineering challenge.
In this paper, we take an approach reminiscent of a class-based model in that we use components (classes) whose elements are expected to be used in similar context.
We call them *model-defined* components because they are defined by their respective models (FST- or neural-based). Unlike class-based models, however, we do not assign any tags to these components. This allows us to do away with one of the main shortcomings of class-based models – the requirement for annotated (manually or automatically) data. The main motivating idea of our method is as follows: given a general generative language model and some components represented as generative LMs, we can learn where these components are *useful*, i.e. where they make better predictions than the general model. Additionally, the proposed model learns, directly from data, how to interpolate different components at each token, which class-based approaches are incapable of due to their explicit factorization into sequence of classes and words. Note that our approach does not require us to assign any semantic tags to components, their meaning is implicit and arises from their content.
The rest of the paper is organized as follows: in Section \[section:model\], we describe the structure of the proposed model, the training procedure is detailed in Section \[section:model-training\]. In Section \[section:experiments\], we present experimental results, and in Section \[section:conclusions\] we conclude and outline future work.
Compositional Language Model {#section:model}
============================
\[section:model-structure\] In this section, we present the structure of the proposed compositional model and describe each of its components in detail. Figure \[fig:model-structure\] outlines the structure of the model. In Sections \[section:components\] through \[section:attention\], we describe every part of the model in detail, and in Section \[section:model-structure-discussion\], we explain the rationale: how these parts work together to achieve our goal.
![Compositional Model Structure. Green color indicates learnable parameters. Component models (in grey) are fixed during the compositional model training.[]{data-label="fig:model-structure"}](./model_structure.pdf){width="0.8\linewidth"}
Components {#section:components}
----------
Component models are language models, $p^{i}(w_t|w_{<t})$ (we use superscript to indicate component-specific items). They are implemented as stateful functions ${\cal F}^{i} : {\cal S}^{i}(w_{<t-1}), w_{t-1} \rightarrow {\cal S}^{i}(w_{<t}), p(w_{t})$, with a special start state ${\cal S}^{i}(\emptyset)$. In this paper, we use LSTM- and FST-based component models.
Each component model is learned independently, the only requirement is shared vocabulary of word (or subword [@tensor2tensor]) tokens. Some components may model entire sentences while others may model only parts of them, e.g. specific entity types.
Default Model
-------------
One component model, component zero, is designated as the *default* model, and it must span the entire sentence. The default model serves as the baseline model which we aim to adapt and improve by combining with other components.
Component Embeddings {#section:component-embeddings}
--------------------
A component embedding, ${\boldsymbol{ce}^{i}}$, is a fixed-size learnable vector associated with each component, and since the components themselves are fixed during the composite model training, this is the only component-specific representation learned by the model.
Context Encoder {#section:context-encoder}
---------------
Context encoder maps the variable length word sequence $w_{<t}$ into a fixed-size vector ${\boldsymbol{ctx}}_{t}$ which serves as an input to activation and attention models described below. In this paper, we use an LSTM network to encode the context $w_{<t}$ with input embedding ${\boldsymbol{e}(w_{<t})}$.
$$\label{eq:context-encoder}
{\boldsymbol{ctx}}_{t} = \mbox{LSTM}({\boldsymbol{e}(w_{<t})})$$
Activation {#section:activation}
----------
Component models that generate only parts of a sentence, such as entities, are ignorant of the context where those entities can be used, and therefore need an explicit binary signal when to generate their first word, i.e., when to output $p^{i}(w|\verb|<s>|)$. In the case of an FST-based component model, the activation signal $act^{i}_{t} = 1$ resets the FST state to its start. When $act^{i}_{t} = 0$ the model generates the next token given its state and $w_{t-1}$. In the case of an LSTM-based component model, we reset the LSTM state to zero values and replace the previous word $w_{t-1}$ with `<s>` (only for that specific component).
In order to generate the activation signal, we define *activation policy* function $\pi_{t}^{i}$ which can be interpreted as probability of activating component $i$ at time $t$:
$$\label{eq:activation}
\pi_{t}^{i} = \sigma(\mbox{PROJ}(\mbox{LSTM}({\boldsymbol{ctx}}_{t}, {\boldsymbol{ce}^{i}}, \boldsymbol{act}_{t-1}^{i}, \log{p^{i}(w_{t-1} = \verb|</s>| | w_{<t-1}, \boldsymbol{act}_{< t}^{i})})))$$
where the comma indicates concatenation, ${\boldsymbol{ctx}}_{t}$ is the context encoding (Eq. \[eq:context-encoder\]), ${\boldsymbol{ce}^{i}}$ is the *component embedding* described in Section \[section:component-embeddings\], ${\boldsymbol{act}}_{t-1}^{i}$ is the binary activation of the component at $t-1$, $\log{p^{i}_{t-1}(w = \verb|</s>|)}$ is the log probability of the component generating `</s>` at $t-1$. Finally, $\mbox{PROJ}$ is a linear projection (with a bias) mapping the LSTM output to scalar input to sigmoid activation function $\sigma$.
For training, we sample binary activations ${\boldsymbol{act}}_{t}^{i}$ from $\pi_{t}^{i}$, and for inference we apply a threshold of 0.5. We describe sampling activations in more detail in Section \[section:lookahead-activation\].
Attention {#section:attention}
---------
The role of the attention is to interpolate the outputs of all components:
$$\label{eq:interpolation}
p(w_{t}| w_{<t}, {\boldsymbol{act}}_{\le t}^{1 \ldots N}) = \sum_{i=0}^{N} \alpha^{i}_{t} \cdot p^{i}(w_{t} | w_{<t}, {\boldsymbol{act}}_{\le t}^{i}),
~~\mathrm{ where } ~\alpha^{i}_{t} = \frac{exp({\boldsymbol{att}}^{i}_{t})}{\sum_{i=0}^{N} exp({\boldsymbol{att}}^{i}_{t})}$$
where
$$\label{eq:attention}
{\boldsymbol{att}}^{i}_{t} = \mbox{PROJ}(\mbox{LSTM}({\boldsymbol{ctx}}_{t}, {\boldsymbol{ce}^{i}}, {\boldsymbol{act}}_{t}^{i}, \log{p^{i}(w_{t} = \verb|</s>| | w_{<t}, {\boldsymbol{act}}_{\le t}^{i} )}))$$
Note that structurally, attention is very similar to activation model in Eq. \[eq:activation\]. The main difference is that at a given time $t$, the activation influences input to the component whereas the attention uses its output. Also note that the activation is computed *independently* for each component (Eq. \[eq:activation\]), while the attention coefficients $\alpha^{i}_{t}$ in Eq. \[eq:interpolation\] are normalized across all components $i$. Figure \[fig:example-sentence\] shows an example of activation and attention output.
Discussion {#section:model-structure-discussion}
----------
Activation and attention networks have similar structure and a related function: they compare the component’s embedding against the sentence context captured by the context encoder (Section \[section:context-encoder\]). When a component generated a limited span, within a sentence, it is important to know where the span *starts* and *ends*. Start is learned by the activation model and it is the responsibility of the attention model to learn where a component ends. However, the attention does not have access to the content of the component, therefore, it needs a signal from the component itself. To that end, we add ${\boldsymbol{act}}_{t}^{i}$ and $\log{p^{i}(w_{t} = \verb|</s>| \ldots )}$ to the attention model’s inputs. The activation model gets the same signals but from the previous step $t-1$. Note that a component predicting `</s>` signals the end of its span but not necessarily the end of sentence which is the same token. The attention model learns to reduce attention to the component once it starts generating `</s>` with high probability until the component is activated again. This way representing the span on a component is agnostic to its internal structure and works well with both neural and FST-based components.
![This figure illustrates the output of activation and attention models on the sentence *“The town contains a village named Sodus and another named Sodus Point .”* (tokenized into subwords). []{data-label="fig:example-sentence"}](./sent4-loc-per-org.pdf){width="0.8\linewidth"}
Figure \[fig:example-sentence\] illustrates how activation and attention models work together. In this example, we use 3 components (apart from the default model), representing location, person, and company entity types. More details about this model are in Section \[section:wikipedia\]. Attention is stacked and sums to 1 (including the default mode which is not shown), but activation is independent for each component. We use the threshold of 0.5 to trigger activation. The bottom graphs shows the difference in token log-likehood between the compositional model and the default model. Note that the “location” activation spikes at positions where a location is plausible. However, the attention model moderates the probability mass assigned to this component and also, where the span of location ends.
Model Training {#section:model-training}
==============
If we consider binary activations to be an input, the rest of the model’s parameters can be learned by minimizing cross-entropy loss. Activations can be viewed as actions in a reinforcement learning framework, and we can utilize policy learning to learn activation probability $\pi^{i}_{t}$ (Eq. \[eq:activation\]). In the rest of this section, we provide details on parameter estimation procedure.
Loss Functions {#section:loss-functions}
--------------
We use a combination of cross entropy (LL) and reinforcement learning (RL) losses[^2],
$$\label{eq:LL}
\textbf{LL:}~ \Delta \theta \propto \sum_{t} \frac{\partial \log p(w_{t})}{\partial \theta}$$
where $\theta$ refers to all parameters of the model except the activation model. The RL loss, on the other hand, aims to maximize expected reward under the activation policy function $E_{\pi^{i}_{t}}[p(w_{t})]$, and the parameter update is as follows (REINFORCE algorithm) [@williams92reinforce]:
$$\label{eq:reinforce}
\textbf{RL:}~ \Delta \zeta \propto - \sum_{t} G_{t} \frac{\partial \log \pi^i_t }{\partial \zeta}$$
where $\pi^{i}_{t}$ is the activation policy (Eq. \[eq:activation\]) with parameters $\zeta$. Here, $G_{t} = \sum_{\tau=t}^{T} R_{\tau}$ is the reward $R_{\tau}$ from time $t$ to the end of the sentence. We use the following reward function:
$$\label{eq:reward}
R_{\tau} = \log p(w_{\tau}) - \log p^{0}(w_{\tau})$$
where $p(w_{\tau})$ and $p^{0}(w_{\tau})$ are the probability of compositional and the default models, respectively. This reward is equivalent to minimizing expected cross-entropy of the compositional model but subtracting the default model reduces variance of the update.
REINFORCE update in Eq. \[eq:reinforce\] relies on activation samples drawn from $\pi^{i}_{t}$. We deviate from this, and draw samples from an interpolated policy $\hat{\pi}^{i}_{t}$ instead:
$$\label{eq:interpolated-policy}
\hat{\pi}^{i}_{t} = \lambda b^{i}_{t} + (1 - \lambda) \pi^{i}_{t}$$
where $b^{i}_{t}$ is a non-parametric behavior activation policy acting as a teacher and $\lambda$ is the interpolation weight which starts from 1 (behavior policy only) and decays by a schedule. In Section \[section:lookahead-activation\], we describe the rationale for using the teacher activation policy and its implementation details. Note that sampling activations from a different policy results in a biased expected cross entropy estimator. The bias, however, decreases as the training progresses, so we do not correct it.[^3]
Lookahead Teacher Activation {#section:lookahead-activation}
----------------------------
There is a subtle but important nuance about learning attention and activation: if the binary activation sequence for a component is too random, the component’s output will not be useful, thus minimizing cross-entropy loss will cause the attention model to ignore that component. Conversely, when the attention to a component is either too random or too small, the reward function (Eq. \[eq:reward\]) becomes indifferent to changes in activation, which prevents the activation model from learning. This is a chicken-and-egg problem: we need a good activation policy to learn attention and we need a good reward function (attention) to learn activation. We resolve this problem by introducing a lookahead teacher activation policy.
Note that during training we know the entire sentence ahead of time, so if we activate a component at a certain time, we can compare the likelihood of generating subsequent words under the component and the default model. Specifically,
$$b^{i}_{t} \propto \frac{p^{i}(w_{t} | act^{i}_{t} = 1)}{p^{0}(w_{t})}$$
Note that this is a non-parametric policy as all components are fixed when we train the composite model. It is possible to extend the lookahead to multiple words in the future, but in our experiments one word was sufficient. However, in our subword models, when the next word consists of multiple subword tokens, we combine the probabilities of all of them.
This teacher policy allows us to bootstrap attention model learning, and once we have a reasonably good attention (and thus reward function), we can start training the activation policy.
Component Model Training {#section:component-training}
------------------------
Component models are trained independently on their respective datasets. We do not make assumptions about component model types, any combination of WFST and neural models is possible. There only two requirements:
1. All models must be share the vocabulary.
2. Components must predict `</s>` with high probability after seeing an unknown input sequence. For neural component models, this can be achieved by adding random input sequences with `</s>` labels, and for WFST models, we add a *dead state* which only generates `</s>` and all unmatched transitions lead to this state.
Experiments {#section:experiments}
===========
In this section, we evaluate performance of our model on two tasks: perplexity on an English Wikipedia dataset, and word error rate (WER) reduction on an ASR 20-best rescoring task using personalized model trained on anonymized transcriptions of interactions with a voice assistant.
Model Parameters
----------------
In both experiments in this section, we use the same structure for our model:
- The default model is an LSTM model comprising of two 300-unit layers, and a skip connection [@deepres2016] over both LSTM layers. The input is 300-dimension subword embedding (${\boldsymbol{e}(w_{<t})}$) learned with the model, and the output is a softmax layer.
- The context encoder is a single layer LSTM with 256 units layer and 0.2 dropout. The input embeddings are shared with the default model and are frozen during the compositional LM training.
- Component embeddings (${\boldsymbol{ce}^{i}}$) have dimensionality of 256.
- The activation model is a 2-layer LSTM with 128 units, the LSTM’s output is projected into a scalar. The input to activation model has dimensionality of $558 = 300+256+1+1$ (see Eq. \[eq:activation\]). We use 0.2 dropout between LSTM layers.
- Finally, the attention model is a single 128-unit LSTM layer with its output projected into a scalar. The attention model’s input has $558$ dimensions (see Eq. \[eq:attention\]). We add 0.2 dropout before and after the LSTM layer.
For component models, we use WFST-based representation which varies in size depending on the component. The WFST is constructed as a union of entities, determinized and minimized under `log semiring` [@mohri1997wfst]. In Wikipedia experiments, each entity is weighted according to its frequency in the training data, and in n-best rescoring models, the distribution is uniform.
Training Procedure {#section:training-procedure}
------------------
We use Adam optimizer [@kingma2014adam] with 0.001 initial learning rate and an exponential decay of 0.7 per 1k updates. We define an epoch as 800 updates. For efficiency, we use truncated backpropagation with a chunk size of 16 and a batch size of 160. Note that due to truncated backpropagation, the reward outside the current chunk will not be added in cumulative reward computation $G_{t}$ (Eq. \[eq:reinforce\]). This introduces some noise but at the same time it reduces the variance of update, especially in long sentences. The impact (reward) of a component’s activation is limited to the span it generates, therefore, as long as the chunk size is significantly larger than the typical component’s output, chunking should not have a negative impact.
First, we do “pre-training” of all parts of the compositional model except activation by using teacher activation only ($\lambda = 1$ in Eq. \[eq:interpolated-policy\], meaning that all activation samples are drawn from the teacher model) and LL loss (Eq. \[eq:LL\]) for 5 epochs. Then we run the main training routine for 20 epochs, alternating between LL and RL losses every 1-3 batches randomly[^4]. At the same time $\lambda$ (Eq. \[eq:interpolated-policy\]) is exponentially decayed at the rate of 0.8 per 1k updates. Once $\lambda$ reaches 0.05, it is set to 0, meaning that all activation samples will come from the activation model.
Wikipedia {#section:wikipedia}
---------
Evaluating the impact of adding components poses a challenge: in order to be useful, the components have to contain information that the default model has not been exposed to. We are not aware of any publicly available datasets that satisfy this property, therefore we simulate this by using an automatically tagged Wikipedia corpus with fine-grained entity tags [@ghaddar2018wiki]. We split the corpus into train/dev/test partitions by document. Component models are then built from entity mentions corresponding to “`/location/city`”, “`/person`”, and “`/organization/company`” entity types using Figer scheme [@liu2014figer] in the entire *train* partition. The default model is built using a random subset of 100,000 training documents. We also filter out stopword mentions (pronouns) as well as entities onger than 3 words (some entities are quite long and contain entire subordinate clauses). The WFST models have 0.7M, 1.2M, and 0.8M arcs, respectively.
The entire Wikipedia corpus contains almost 7M unique words, we segment them into subwords using 30k subword vocabulary. Subwords ending with an underscore indicate end-of-word, and activations are only generated at word boundaries. Figure \[fig:example-sentence\] shows a sentence from this corpus.
We compute perplexity on a subset of 1,000 documents from test partition. This test set contains 30k sentences with 970k subwords. The perplexity of the default model is 64.9 and the compositional model’s perplexity is 64.0 (1.4% reduction). However, we expect our compositional model to make a significant impact over the default model only on a fraction of sentences because most sentences do not contain aforementioned entity types. Besides, the default model should already model frequent entities well enough. In Figure \[fig:wiki-log-likelihood\], we plot log-likelihood of the default model vs. the composite model. We show sentences up to 20 tokens to limit the range.
![Log-likehood (natural log) scatter plot of sentences: compositional model (X axis) vs. the default model (Y axis). Green color indicates sentences with higher likelihood under the compositional model compared to the default, and red color signifies the opposite. The first on the left contains all sentences up to 20 tokens. On the right, we remove sentences whose scores differ by less that 1 to remove the clutter. []{data-label="fig:wiki-log-likelihood"}](./scatterplot-loc-per-org.pdf){width="1.0\linewidth"}
N-best rescoring {#section:rescoring}
----------------
In this section, we evaluate the impact of our model on scoring n-best ASR hypotheses. The test set consists of anonymized transcriptions of interactions with a voice assistant, and each utterances is associated with an anonymized user id, and some user ids have personalized models of “`contact names`” associated with them, used to improve recognition accuracy for “`communication`” domain. We use separate partitions for train/dev/test with disjoined sets of users. Only a small fraction of all interactions belong to “`communications`” domain, and even smaller fraction still involve contact names. Therefore, we present results on “All domains” test set which represents general interactions, and separately, on “Communications” domain test set. Our goal is to improve the performance on interactions that do invoke contact names, while not regressing on the rest of the data.
In Table \[tbl:ppl\], “default component” is trained on a large sample of interactions representative of “all domains”. The “compositional” model is trained on additional 0.8% of data, half of which belongs to the “`communication`” domain.[^5]
The compositional model is trained with a single component (not counting the default) representing “`contact names`” entity. To simplify the training procedure, we used a “unified entity” model built from aggregated contact lists across all users in the training data. However, for evaluation we also report results with personalized entity. We use a 10k subword token vocabulary for all models.
\#Utterances
----------------------------- -------------- --- -------- -------- --
All domains 138,094 - -0.3% 0.1%
Communications 14,943 - -8.9% -22.2%
only w/ contact names 6,574 - -20.6% -43.6%
only w/ personal entities 4,181 - -19.3% -51.8%
: Relative difference in perplexity compared to the default model. Reductions (negative numbers) indicate improvements.
\[tbl:ppl\]
In Table \[tbl:ppl\], we compare perplexity of our models and the default model. On the entire test set, the changes in perplexity are insignificant which indicates that our model does not cause regression on utterances where its component is not used. On subsets of utterances where adding contact names is expected to make a difference, we do observe substantial reductions in perplexity. Note that despite learning the component embedding using unified “`contact names`”, we observe better performance by using personalized entities. This indicates that the component embedding learns a representation of *entity type* rather than its content.
We also evaluate these models on an n-best rescoring task [@rescorenlm2019]. We rescore top 20 hypotheses generated by a hybrid CTC-HMM ASR model [@graves2006ctc], trained on a large amount of anonymized transcriptions, using the default component as the baseline and compare that to rescoring with our proposed compositional model. The results are presented in Table \[tbl:wer\]. Improvements of compositional models over the default component are significant with at least $p<0.0001$.
Oracle
----------------------------- ------- ------- -------- -------- --
All domains -3.0% -2.9% -3.0% -30.3%
Communications -1.3% -2.5% -5.8% -38.8%
only w/ contact names -0.9% -2.3% -6.8% -40.5%
only w/ personal entities -0.8% -2.8% -10.0% -45.8%
: Relative difference in WER compared to 1-best of CTC-HMM model. Reductions (negative numbers) indicate improvements.
\[tbl:wer\]
Conclusions and Future Work {#section:conclusions}
===========================
In this paper, we proposed a novel method how to compose separately trained models, including personalized models, with a general generative language model. We showed that our method is effective at learning the composition directly from data without relying on annotations. While we evaluate our approach on language modeling tasks, we believe our approach can be applied to many sequence-generating applications in natural language processing. In the future, we plan to integrate our model directly into ASR decoder using end-to-end models such as LAS [@chan2015las] and RNN-T [@graves2012rnnt]. We also want to explore other applications, such as machine translation.
Broader Impact {#broader-impact .unnumbered}
==============
We expect our approach to have the most impact on application that aim to adapt models at high cadence to reflect changes in real world (for example, release of new movies, books, etc.) or even to individual requests, taking advantage of available request-specific information such as personalization, location, etc.). Although we focus on models that generate sequences of words, our approach can be extended to any sequential generative models.
[^1]: Weighted finite-state transducers (WFSTs) are widely used in speech recognition to represent language models [@mohri2002wfst].
[^2]: We omit the conditional part of $p(w_{t})$ for simplicity of notation
[^3]: Unbiased estimator using importance sampling [@hurtado1998monte] results in a high update variance which prevents the model from converging.
[^4]: We found that alternating the losses yields slightly better results than their sum.
[^5]: We did evaluate the default component trained on the additional data, but found no significant difference.
|
---
author:
- Joe Zuntz
- Tessa Baker
- 'Pedro G. Ferreira'
- Constantinos Skordis
bibliography:
- 'paper.bib'
title: Ambiguous Tests of General Relativity on Cosmological Scales
---
Introduction {#Intro}
============
The availability of high-precision cosmological data has made it possible to test General Relativity (GR) on cosmological scales. In particular, in the absence of a conclusive theoretical explanation for the dark energy phenomenon, modifications and extensions to that venerable theory remain a viable possibility.
Since the process of comparing a new theory of gravity to cosmological data can be complicated, any test statistic that can be used to quickly reject a theory is a useful one. In this spirit, inspired by the Parameterized Post-Newtonian (PPN) formalism [@Will_living_review_2006; @Will1971; @Thorne_Will; @Will_Nordvedt_1972], a similar approach has been developed, which describes the deviations of a theory from the standard evolution of a perturbed GR universe [@AmendolaKunzSapone; @AminBlandfordWagoner; @ZhangEtAl; @Pogosian_parameterization; @Daniel2010; @Hu_Sawicki; @Daniel_Linder; @Hojjati_Pogosian; @Song2010; @Bertschinger2006; @Zhao2010; @bean; @Caldwell; @skordis; @skordisFerreira; @baker; @isitgr1; @isitgr2]. Unlike the PPN case, however, such a parameterization is not unique; on cosmological scales there is freedom to modify evolution over the whole of cosmic time and over a range of scales. One is therefore led to constrain functions of space and time rather than just a handful of numbers in the weak-field metric. There are then two levels of structure that must be imposed on the parameterized field equations:
Level 1:
: One must choose the way in which free functions are used to extend the field equations. For example, one could use free functions to rescale terms that exist in GR, or allow new terms to be present.
Level 2:
: When implementing the parameterized field equations in numerical codes one is forced to choose a sensible ansatz for the time- and scale-dependence of the free functions. Typically these ansatzes are motivated by the observation that the dark energy-like sector only dominates the energy density of the universe at late cosmological times. See [@MG_report] for an enumeration of such ansatzes.
The choice usually made at level 1 is to introduce two free functions directly into the modified Poisson equation and the ‘slip relation’ (the transverse, traceless component of the field equations), as these are the expressions relevant to observables such as weak lensing of galaxies and measures of structure growth [@bean; @Caldwell] . One of these free functions acts as an effective rescaling of Newton’s gravitational constant, whilst the other is defined as the ratio of the two potentials that describe the perturbed metric in the Conformal Newtonian gauge. We will refer to this approach as ‘phenomenologically-based’. It is *not* the case that particular choices of the two free functions are designed to reproduce the field equations of specific modified gravity theories (except in a handful of special cases). Rather, in the phenomenological approach, the free functions should be considered as indicator flags for non-GR behaviour.
A disadvantage of phenomenological-type parameterizations is their tendency to obscure which regions of theory space they correspond to, because there is no direct mapping between the free functions and the parameters of a specific model. It is therefore difficult to translate constraints on the two free functions into constraints on a given theory of modified gravity.
We have advocated an alternative approach: directly specifying quantities needed to describe a $4\times 4$ tensor of scalar modifications to the linearly perturbed Einstein equations [@skordis; @skordisFerreira; @baker]. We can then derive the corresponding Poisson equation and slip relation, which will contain components of this new tensor.This has been dubbed the ‘Parameterized Post-Friedmann’ approach (PPF), and it leads to a level-1 structure (as classified above) which is different to that of the phenomenological approach.
Arguments can be made in favour of both the phenomenologically-based and PPF approaches. For large classes of theories the two strategies become equivalent at intermediate distance scales where the time derivatives of perturbations can be neglected in comparison with their spatial derivatives. However, this approximation is not applicable on scales comparable to the horizon size.
Whilst such large scales cannot be probed directly with galaxy surveys, they are nonetheless important for accurate calculation of the Integrated Sachs-Wolfe (ISW) effect and large-scale matter power spectrum. These quantities are frequently computed using Einstein-Boltzmann solvers such as [Camb]{} [@camb] or [Class]{} [@class] which evolve perturbations through horizon crossing.
The purpose of this brief report is to illustrate that the choice made for the level-1 structure of a parameterization has a significant influence on the constraints obtained on deviations from GR – a caveat that is commonly forgotten.
We demonstrate the differences between two possible parameterizations by generating some of the simpler perturbative observables - the CMB power spectra, in particular the ISW effect, matter power spectra, and the growth function $f(z)$. In \[ansatzes\] we further show that the choice made for the level-2 structure has an equally important influence on the constraints obtained.
Theory {#theory section}
======
In this section we describe the two parameterizations that we apply in this paper. Our goal here is not to advocate for either of them, but rather to highlight the fact that they lead to different results. Parameterization A described below is derived as a special case of the general PPF form detailed in [@baker_etal_2012]. However, for the purposes of this paper we can ignore its origin and simply regard the resulting field equations as another example of a phenomenological-type parameterization – one with different choices made for the level-1 structure.
Parameterization A
------------------
In [@skordis] a parameterization was proposed by writing the modifications as an additional tensor to the Einstein equations of GR: $$\label{EFE}
G_{\mu\nu}=8\pi G_0 a^2 T^M_{\mu\nu}+ a^2 U_{\mu\nu}$$ where the stress-energy tensor $T^M_{\mu\nu}$ contains all the standard cosmologically-relevant fluids and the tensor $U_{\mu\nu}$ may contain metric, matter and additional field degrees of freedom coming from a theory of gravity. In writing (\[EFE\]) it is assumed that all known matter fields which are part of $T^M_{\mu\nu}$ couple to the same metric $g_{\mu\nu}$, and that $G_{\mu\nu}$ is the Einstein tensor of that same metric. This ensures that the stress-energy tensor of matter obeys its usual conservation equations, and hence $U_{\mu\nu}$ is separately conserved.
The formalism proceeds by parameterizing around the linearly perturbed version of equation (\[EFE\]). To enable a direct comparison with parameterization B below we will specialise to the case of purely metric theories, that is, those for which the action is constructed from curvature invariants (e.g. $f(R)$ gravity) or non-local invariants (e.g. [@DeserWoodard]).
The most general perturbations of $U_{\mu\nu}$ that are allowed in a second-order metric-only theory are as follows [@skordis; @skordisFerreira; @baker; @baker_etal_2012]: $$\begin{aligned}
\label{2nd_order_Us}
-a^2\delta U^0_0 &=& k^2 A_0\hat\Phi\nonumber\\
-a^2 \delta U_i^0 &=&k B_0 \hat\Phi\nonumber \\
a^2\delta U_i^i &=&k^2 C_0 \hat\Phi + k C_1 \dot{\hat\Phi} \nonumber\\
a^2\delta U_j^i &=&D_0 \hat\Phi+\frac{D_1}{k} \dot{\hat\Phi}\end{aligned}$$ where dots denote derivatives with respect to conformal time $\eta$ and $k$ is the Fourier wavenumber. The gauge-invariant metric perturbation $\hat\Phi$ reduces to the curvature perturbation $\Phi$ in the Conformal Newtonian (CN) gauge. In our conventions the CN gauge is defined by: $$ds^2 = -a^2(1 + 2 \Psi) d\eta^2 + a^2 (1 - 2\Phi) d\vec{x}^2$$ Further below we introduce a second gauge-invariant metric perturbation, $\hat{\Psi}$ which reduces to $\Psi$ in the CN gauge.
The coefficients $A_0\ldots D_1$ appearing above are functions of background quantities, $A_0=A_0 (k, \eta)$ etc., and the factors of $k$ are chosen such that $A_0\ldots D_1$ are dimensionless. However, these functions are not all independent. Perturbations of the Bianchi identity $U^{\mu}_{\nu;\mu}=0$ yield a set of additional constraint equations which can be used to reduce the six free functions in equations (\[2nd\_order\_Us\]) down to just two. One of these free functions is defined to be: $$\frac{D_1}{k}=\frac{\tilde g}{\cal H},\quad\;\mathrm{where}\quad\; \tilde{g}=-\frac{1}{2}\left(A_0+3\frac{\cal H}{k}B_0\right)$$ The modified Poisson equation can then be written (in Fourier space): $$\begin{aligned}
-k^2\hat\Phi&=&4\pi \frac{G_0}{1-\tilde{g}} a^2 \rho\Delta \equiv 4\pi G_\mathrm{eff}a^2 \rho\Delta \label{Poisson}\end{aligned}$$ with $G_0$ denoting the canonical value of Newton’s gravitational constant as measured by a Cavendish experiment on the Earth, and where the gauge-invariant comoving density perturbation $\rho\Delta$ is a summation over all conventional cosmologically-relevant fluids. The combination $G_\mathrm{eff}=G_0\,(1-\tilde{g})^{-1}$ plays the role of a modified Newton’s constant.
We choose the second free function to be $D_0$, which we will hereafter relabel as $\zeta=\zeta (k, \eta)$ to distinguish it from its appearance in the more general format of equations (\[2nd\_order\_Us\]). $\zeta$ appears in the ‘slip’ relation between the potentials $\hat\Phi$ and $\hat\Psi$ (which are equal to one another in GR): $$\label{slip}
\hat\Phi-\hat\Psi=8\pi G_0 a^2 (\rho+P)\Sigma+\zeta\hat\Phi+\frac{\tilde g}{\cal H}\dot{\hat\Phi}$$ The anisotropic stress of conventional matter $\Sigma$ is negligible after the radiation-dominated era.
The key point here is that in parameterization A the slip relation and the modified Newton’s constant are not independent, as the function $\tilde{g}$ appears in both. This special case of the PPF formalism does not capture the behaviour of many modified gravity theories, because we have not allowed for additional degrees of freedom to appear [@baker; @baker_etal_2012]. It is, however, directly comparable to a phenomenological-type parameterization. Hence we will regard eqns.(\[Poisson\]) and (\[slip\]) simply as possible alternatives to eqns.(\[phenom\_Poisson\]) and (\[phenom\_slip\]).
Parameterization B
------------------
If one steps back and looks at the structure of the evolution equations for cosmological perturbations, one finds that it is enough to work with only two of the four Einstein field equations: the Newton-Poisson equation and the slip equation. From a practical point of view, the only observable modifications to gravity (at least at the perturbative level) will be modifications to the these equations, which we can write in the following form $$\begin{aligned}
-k^2\Phi-4\pi G_0 a^2\rho\Delta &=&F_1 \label{phenom_Poisson0}\\
(\Psi-\Phi)+8\pi G_0 a^2 (\rho+P)\Sigma&=& F_2\label{phenom_slip0} \end{aligned}$$ where $F_1$ and $F_2$ are arbitrary functions of time [*and*]{} space. A simple ansatz (from the dimensional point of view) is that $F_1=\alpha k^2\Phi$ and $F_2=-\zeta\Phi$. Reorganizing the equation we find that they can be rewritten as $$\begin{aligned}
-k^2\Phi&=&4\pi G_\mathrm{eff} a^2\rho\Delta \label{phenom_Poisson}\\
\Psi&=&-8\pi G_0 a^2 (\rho+P)\Sigma+(1-\zeta)\Phi \label{phenom_slip} \end{aligned}$$ Parameterization B is defined in the CN gauge, so that $\Phi$ and $\Psi$ replace $\hat\Phi$ and $\hat\Psi$ in eqns.(\[Poisson\]) and (\[slip\]). This formulation is equivalent up to small corrections to that used in [@bean]; their anisotropic stress is related to ours by $k^2\Sigma=\sigma$. This parameterization suggests that, once the anisotropic stress has become negligible, a theory of modified gravity could potentially modify one of equations (\[phenom\_Poisson\]) or (\[phenom\_slip\]) whilst leaving the other unchanged. This behaviour does not arise *analytically* from theories below third order [@baker]; but given enough freedom in the functional ansatz the *numerical* behaviour that occurs in any theory can be realised with this parameterization. (We note that a handful of parameterizations that are instead optimized for weak lensing analysis have also been suggested [@Song2010; @Zhao2010; @Daniel_Linder]).
To demonstrate the differences that the alternative parameterization types can generate, we will use the same ansatz for the free functions ${\ensuremath{G_\mathrm{eff}}}$ and $\zeta$ in both parameterizations A and B. Since we are focusing on modifications to gravity associated with dark energy, an appropriate parameterization is an expansion in $\Omega_\Lambda$. We therefore use a polynomial expansion up to third order in $\Omega_\Lambda$.
Spectra
=======
We have modified a version of the Boltzmann code [Camb]{} to use the new equations specified in section \[theory section\]. We use variants which include both parameterizations A and B. In this section we show and analyze CMB and LSS spectra generated by these models. We will show that the two variants, even when set up with equivalently parameterized functions, generate considerably different spectra.
Figure \[linear geff plot\] shows CMB spectra from parameterization A with variations to the linear term. Because we have modified the late-time behaviour of gravity whilst leaving the era before dark energy domination unchanged, the small-scale CMB (which depends mainly on what happens at recombination) is unaffected, but the large-scale features caused by the ISW effect vary. It can be seen that strengthening gravity by a small amount decreases the power spectrum of the ISW effect by counteracting the suppressive effect of $\Lambda$ on structure formation, whereas weakening $G_{\mathrm{eff}}$ increases the ISW effect. This difference is significant because the ISW effect is only weakly present observationally.
The behaviour of the spectra for a simple linear variation in $G_{\mathrm{eff}}$ shows the same general trends for model B as those for model A shown in figure \[linear geff plot\]. The most important difference between the theories, however, appears when the two free functions interact to allow cancellations. This is discussed in detail below.
Examining the plot of linear variations in $G_{\mathrm{eff}}$ alone, one might be tempted to conclude that the perturbative strength of gravity, as measured by the $G_{\mathrm{eff}}$ parameter, is rather well constrained using just CMB data. We wish to emphasise in this paper that this is not so. First, the various ways one can construct a theory mean that direct interpretation of the parameter, except on small scales, is not unique. Secondly and probably more importantly, the linear variation alone masks a richer phenomenology, which we will discuss more in the next section.
![CMB power spectra for varying the linear term of the Taylor expansion in $\Omega_\Lambda$ of the $G_{\mathrm{eff}}$ appearing in the Poisson equation, as defined in equation (\[Poisson\]).[]{data-label="linear geff plot"}](linear_geff_spectra.pdf){width="8cm"}
Figure \[growth function plot\] shows the growth function, $f(z) = \frac{1}{\cal H}\frac{\dot{(\delta_\rho)}}{\delta_\rho}$ scaled by $\sigma_8$, which is measured by, for example, redshift-space distortion experiments. Results from the recent WiggleZ survey [@wigglez] and other data [@Percival2DF; @TegmarkSDSS; @GuzzoVVDS; @SongPercival] are also shown. These results provide weaker constraints than the ISW, but near-future experiments should reach sensitivity levels good enough for precision tests of late-time gravity.
We do not display the matter power spectra because they are very mildly affected by the modifications of gravity, except on very large scales far beyond the reach of galaxy surveys.
![The growth function $\sigma_8(z) f(z)$ for the same models shown in figure \[linear geff plot\].[]{data-label="growth function plot"}](growth_function_plot.pdf){width="8cm"}
Figures \[linear geff plot\] and \[growth function plot\] demonstrate that relatively small modifications of about 20% to the effective Newton’s constant in the Poisson equation can have effects that are easily detectable at large scales in the CMB. But it is the combined effects of [$G_\mathrm{eff}$]{} and $\zeta$ that are of most interest. Figure \[extreme plot\] demonstrates how this combination can cancel even quite extreme individual effects. The curves in this plot have , and . The other cosmological parameters are within normal ranges, though not identical to those in figure \[linear geff plot\]. There is a near-total cancellation of the ISW effect in model A, but model B with the same parameters is completely ruled out. There do exist alternative parameter combinations in model B with the same cancellation, but the fact that they are very different from those in model A is what we want to highlight here.
We are not, of course, advocating these models as actual cosmologies - they simply demonstrate that a vast region of parameter space can be consistent with CMB data. Since other upcoming data sets like lensing and redshift-space distortions also constrain (different) combinations of the potentials there should be equivalent degeneracies present there.
The importance of other data sets in breaking the degeneracy in the CMB is illustrated in figure \[extreme growth function plot\]. That plot shows the growth rates $\sigma_8 f(z)$ for the extreme models described above. The behavior of the extreme case is once again very different between the two parameterizations.
![A demonstration of how extreme changes to the effective gravitational constant can be permitted by the CMB when they can be counteracted by a significant gravitational slip (see text for form of extreme curves). The cancellation is absent for these parameters in model B. This delicate difference underscores the need for a rigorous treatment of modified functions.[]{data-label="extreme plot"}](extreme_mod_plot.pdf){width="8cm"}
![The growth rates $f(z)$ corresponding to the CMB spectra shown in figure \[extreme plot\].[]{data-label="extreme growth function plot"}](extreme_f_plot.pdf){width="8cm"}
Constraints {#constraints section}
===========
Parameter Estimation
--------------------
Now that we have demonstrated that the different parameterization approaches can yield considerably different results for power spectra, we can check how these differences go forward into parameter estimation. Note that at this stage we are *not* attempting to find the tightest possible constraints on these parameterizations using all available data. In particular we are not using data from weak lensing, galaxy-ISW correlations, or growth rates which can provide the strongest constraints.
We are rather intending to demonstrate that the different schemes applied to the same data can produce significantly different constraints, even on quantities we usually regard as rather physical – such as the the effective Newton’s constant controlling the growth of structures under gravity. We hope to show that headline constraints on the two free functions of parameterized approaches should be taken with a degree of caution.
The plots in this section are generated from Monte-Carlo Markov Chains running parameterizations A (equations (\[Poisson\]) and (\[slip\])) and B (equations (\[phenom\_Poisson\]) and (\[phenom\_slip\])). In each case, as in the previous section, the same ansatz is applied to the free functions $G_\mathrm{eff}(z)$ and $\zeta(z)$. The same constraining data is used in each case: the 7-year WMAP CMB data [@wmap7], the SDSS DR7 matter power spectrum [@sdss-dr7], a prior $H_0 = 73.8 \pm 2.4$ [@riess2011], the BBN constraint $\Omega_b h^2 = 0.022 \pm 0.002$ [@bbn], and the Union2 Supernova Ia data [@Amanullah2010]. [^1]. With these data sets the strongest constraining power comes from the ISW effect (or rather lack thereof) in the large-scale CMB temperature power spectrum.
Insufficient ansatzes {#ansatzes}
---------------------
The simplest ambiguity we can consider when constraining the free functions can arise if we use what might be termed an incomplete parameterization – one where the free functions are not free enough to explore the available parameter space. One example of this would be our restriction here to functions only of $z$, with no scale-dependence.
Figure \[linear constraints\] illustrates another example of an incomplete parameterization. In that figure we show how the constraints on the theory depend on the number of terms in the polynomial expansions of ${\ensuremath{G_\mathrm{eff}}}(\Omega_\Lambda)$ and $\zeta(\Omega_\Lambda)$. The larger contours use cubic models for the evolution, and the much smaller ones use only a linear term in the expansion. Both curves are for parameterization A, and the contours are for 68% and 95% probability mass.
Of course we do not claim that the cubic model is general enough to describe the free functions sufficiently - it is the comparison that we highlight. Whenever contours are presented for the free functions of some parameterized model one should bear in mind that adding only a little freedom can change the areas of the error ellipses by a large factor. Hence all constraints of this form should be taken with a pinch of salt.
![Constraints on the free functions, evaluated at redshift zero, in a model restricted to linear evolution in $\Omega_\Lambda$ (filled contours) and one with cubic evolution (line contours). The apparent constraints on the two parameters, for example, are misleadingly small in the more restrictive parameterization.[]{data-label="linear constraints"}](linear_constraints.pdf){width="8cm"}
Variant parameterization
------------------------
The second result of this section is shown in figure \[2d histogram\]; it demonstrates the parameterization-dependence of the constraints, independent of the ansatz. Most clear is that the free functions are significantly *less* correlated in model A. In the model B there is a simple and straightforward impact on the term whose derivative is the ISW source: $$\Phi + \Psi \propto (\zeta-2) {\ensuremath{G_\mathrm{eff}}}\frac{a^2}{k^2} \rho\Delta$$ which yields a multiplicative correlation, whereas case A does not have this simple relation: $$\begin{aligned}
\Phi &+& \Psi \propto {\cal H}^{-1} \left({\ensuremath{G_\mathrm{eff}}}-G_0\right)\frac{a^2}{k^2} \rho\dot{\Delta}\\
&&+\left[ (\zeta-3){\ensuremath{G_\mathrm{eff}}}+ {\cal H}^{-1} \left(1-\frac{G_0}{{\ensuremath{G_\mathrm{eff}}}}\right)\dot{G}_{\mathrm{eff}}+G_0 \right] \frac{a^2}{k^2} \rho\Delta \nonumber
\label{eq isw}\end{aligned}$$
The edge of the curve for parameterization A in figure \[2d histogram\] cuts off rather sharply just below ${\ensuremath{G_\mathrm{eff}}}=1$. This is due to the impact of $G_{\mathrm{eff}}$ and $\zeta$ on the growth of matter density perturbations, and the restrictions that they should obey in order to reproduce the observed amount of structure in the universe today. During the matter-dominated epoch density perturbations on subhorizon scales grow as , where in parameterization A [@baker2]: $$\begin{aligned}
p_{A}&=&\frac{1}{2}\left(1-\frac{3}{2}\frac{{\ensuremath{G_\mathrm{eff}}}}{G_0}\right) +\frac{1}{4}\sqrt{9\left(\frac{{\ensuremath{G_\mathrm{eff}}}}{G_0}\right)^2+12\frac{{\ensuremath{G_\mathrm{eff}}}}{G_0}(3-2\zeta)-20}
\label{p_A}\end{aligned}$$ In parameterization B the exponent is different: $$p_B=-\frac{1}{4}+\frac{1}{4}\sqrt{1+24\frac{{\ensuremath{G_\mathrm{eff}}}}{G_0}(1-\zeta)}
\label{p_B}$$ One can see that in the GR limit ($\zeta=0$, ${\ensuremath{G_\mathrm{eff}}}=G_0$) we recover $p_A=p_B=1$, in agreement with the standard result for an Einstein-de Sitter universe. Equation (\[p\_A\]) implies that in a universe with matter perturbations fail to grow during the matter-dominated epoch for values of ${\ensuremath{G_\mathrm{eff}}}/G_0 < 0.5$. Hence we should expect this region of parameter space to be disfavoured by the matter power spectrum. The analogous boundaries in parameterization B will be different, which may explain the different shapes of the fall-offs of the distributions in figure 7. However, we ought to remember that equations (\[p\_A\]) and (\[p\_B\]) apply only in a pure Einstein-de Sitter setting. The boundaries in parameter space that they imply may not be obeyed rigidly in the real universe, due to growth during the radiation- and $\Lambda$-dominated eras.
For small deviations from GR $p_a\approx p_b$. This is because in EdS universes $\dot\Phi=0$ so the slip equations (\[slip\]) and (\[phenom\_slip\]) are the same. This does not apply when $\Lambda$ becomes important.
It is in the tails of the distributions where the difference between the two parameterizations can be most stark – this will be particularly important when we are trying to decide if GR is threatened by evidence of deviations. The histograms in figure \[G0 zeta0 histogram\] demonstrate this most clearly. In addition, the spectral index is much less constrained in model A than in either GR or model B, because in that model we can more freely cary ${\ensuremath{G_\mathrm{eff}}}$ at low redshift; so the matter power spectrum amplitude is not as constraining.
![Joint constraints on the slip parameter $\zeta$ and $G_\mathrm{eff}$ at $z=0$, for parameterization A (black lines) and B (filled green). In both cases 68% and 95% contours are shown.[]{data-label="2d histogram"}](both.pdf){width="8cm"}
![Likelihoods of the effective gravitational and slip parameters at $z=0$, for parameterization A (thick blue) and B (thin green). These plots are marginalized forms of figure \[2d histogram\]. The distribution means are significantly shifted in each case, and the tails even more so.[]{data-label="G0 zeta0 histogram"}](G0_zeta0_1D_compare.pdf){width="9cm"}
Discussion
==========
The first, and uncontroversial, issue we have noted here is the significant effect that using insufficiently free functions has on the constraints one obtains on modifications to the slip and Poisson relations. We now argue that there are two reasonable ways to get around this problem. One is to embrace the constraints, but to make them correspond to some regime in theory space that we wish to model. This is the approach that we are working towards when developing the parameterization introduced in [@skordis]. The alternative is to evade the constraints as far as is possible by making the functions completely free in both $k$ and $\eta$. This numerically challenging approach is the one taken in [@pca1; @pca2], where the data itself is given the freedom to choose the parameterization using a Principal Component Analysis (PCA) method.
The second argument we have advanced in this brief report is as follows. Different parameterizations – even at the level of where the free functions are placed analytically – can lead to radically different effects on the gravitational potentials $\Phi$ and $\Psi$, and their effects on cosmological observables. We have focused on the the ISW effect, which gives one of the most important constraints on parameterized approaches. In that case significant changes to the ISW plateau can be absent due to cancellations in these theories, even in the case of very large modifications. The details of this cancellation depend on which terms are included in the extension to the Poisson and slip equations. Given that the goal of these parameterized frameworks is to model and detect realistic deviations from GR, it makes sense to consider modifications to the equations that reflect the reasonable regions of theory space as well as possible, as advocated in [@baker]. The differences highlighted in this paper between the parameterizations underline the delicacy of this question.
Numerical approaches with very free functions can generate the same effect by numerically finding the same cancelling solutions, but models with an explicit parameterization will not in general do the same unless they are chosen for that specific purpose.
In the analysis we have undertaken here the differences between the parameterizations are smaller than the error bars on either of them. This is a temporary situation – the data will soon improve to the point where the differences are significant. In the longer term as they improve further they will clearly pick out which part of theory space is correct regardless of which parameterization is used, but in the intermediate phase before then the choice will matter. Furthermore, we have focused on the ISW in the CMB but the arguments follow through to other cosmological observables, such as weak lensing, redshift space distortions and any other probe of the gravitational potentials.
It is often argued that the purpose of these methods is not to model alternative theories, but simply to detect any deviation from GR+$\Lambda$CDM in as simple a way as possible [@Pogosian_parameterization; @Hojjati_Pogosian]. But, as we have shown in this paper, two different parameterizations can lead to different constraints – calling into question the interpretation of any individual detection if not properly put into the context of the class of theories that are being considered.
Furthermore, the best way to maximize the ability of a method to find a signal like this is to model as closely as possible its expected characteristics, as in matched filter methods in signal analysis.
In future work we will be extending parameterization A to model additional field components and modified background evolution terms. We will also be applying it to other data sets like lensing and cross-correlations which react differently to metric potentials. An inclusive approach which carefully includes as many phenomena as possible in a model offers the best chance of detecting any small deviations from GR in the upcoming era of high-precision cosmic structure data.\
*Acknowledgements* We gratefully acknowledge helpful discussions with Tom Zlosnik, Levon Pogosian, Kazuya Koyama and Gongbo Zhao. This work was supported by STFC, the ERC, the Oxford Martin School, and the Beecroft Institute of Particle Astrophysics and Cosmology. CS is supported by a Royal Society University Research fellowship.
[^1]: It is not generally correct to use published supernovae constraints directly when using modified gravitational physics, since the calibration factors applied to them are cosmology-dependent; it is only because we leave our background evolution unchanged from GR that is is possible here.
|
---
abstract: 'Effective assisted living environments must be able to perform inferences on how their occupants interact with one another as well as with surrounding objects. To accomplish this goal using a vision-based automated approach, multiple tasks such as pose estimation, object segmentation and gaze estimation must be addressed. Gaze direction in particular provides some of the strongest indications of how a person interacts with the environment. In this paper, we propose a simple neural network regressor that estimates the gaze direction of individuals in a multi-camera assisted living scenario, relying only on the relative positions of facial keypoints collected from a single pose estimation model. To handle cases of keypoint occlusion, our model exploits a novel confidence gated unit in its input layer. In addition to the gaze direction, our model also outputs an estimation of its own prediction uncertainty. Experimental results on a public benchmark demonstrate that our approach performs on pair with a complex, dataset-specific baseline, while its uncertainty predictions are highly correlated to the actual angular error of corresponding estimations. Finally, experiments on images from a real assisted living environment demonstrate the higher suitability of our model for its final application.'
author:
- |
Philipe A. Dias$^{1}$ Damiano Malafronte$^{2,3}$ Henry Medeiros$^{1}$ Francesca Odone$^{2}$\
[$^{1}$Marquette University (EECE), USA $^{2}$University of Genoa, Italy $^{3}$Italian Institute of Technology(IIT)]{}\
bibliography:
- 'gaze\_estimation.bib'
title: Gaze Estimation for Assisted Living Environments
---
Introduction
============
The number of people aged 60 years or older is expected to nearly double by 2050 [@UN2017World]. The future viability of medical care systems depends upon the adoption of new strategies to minimize the need for costly medical interventions, such as the development of technologies that maximize health status and quality of life in aging populations. Currently, clinicians use evaluation scales that incorporate mobility and Instrumented Activities of Daily Living (IADL) assessments (i.e., a person’s ability to use a tool such as a telephone without assistance) [@pilotto2008development] to determine the health status of elderly patients and to recommend changes of habits. Despite the potential of recent advances in many areas of computer vision, no current technology allows automatic and unobtrusive assessment of mobility and IADL over extended periods of time in long-term care facilities or patients’ homes. Patient activity analysis to date has been limited to simplistic scenarios [@debes2016monitoring], which do not cover a wide range of relatively unconstrained and unpredictable situations.
Vision-based analysis of mobility and characterization of ADLs is rather challenging. As examples in Fig. \[fig:method\_diagram\] and \[fig:apt\_plan\] illustrate, images acquired from assisted living environments cover a wide scene where multiple people can be performing different activities in a varied range of scenarios. Moreover, it encompasses multiple underlying complex tasks including: detection of subjects and objects of interest, identification of body joints for pose estimation, and estimation of the gaze of the subjects in the scene.
![Overview of our apparent gaze estimation approach. The anatomical keypoints of all the persons present in the scene are detected using a pose estimation model [@cao2017realtime]. The facial keypoints of each person are then provided as inputs to a neural network regressor that outputs estimations of their apparent gaze and its confidence on each prediction.[]{data-label="fig:method_diagram"}](diagram_horiz9.eps){width="\linewidth"}
{width="\linewidth"}
In this paper we focus on gaze estimation, which is a critical element to determine how humans interact with the surrounding environment. It has been applied to design human-computer interaction methods [@majaranta2014eye] and to analyze social interactions among multiple individuals [@Varadarajan2018]. For our application, in conjunction with object detection [@dias2017fine], gaze direction could define mutual relationships between objects and their users (e.g. the user is sitting on a chair with a book on his/her lap vs. sitting on a chair reading the book) and classify simple actions (e.g. mopping the floor, getting dressed, reading a book, cooking food, eating/drinking).
The contributions of the present work can be summarized in three main points:
- *we propose an approach that relies solely on the relative positions of facial keypoints to estimate gaze direction*. As shown in Fig. \[fig:method\_diagram\], we extract these features using the off-the-shelf OpenPose model [@cao2017realtime]. From the coordinates and confidence levels of the detected facial keypoints, our regression network estimates the apparent gaze of the corresponding subjects. From the perspective of the overall framework for ADL analysis, leveraging the facial keypoints is beneficial because a single feature extractor module can be used for two required tasks: pose estimation and gaze estimation.
- the complexity of gaze estimation varies according to the scenario, such that the quality of predictions provided by a gaze regressor is expected to vary case-by-case. For this reason, *our model is designed and trained to provide an estimation of its uncertainty for each prediction of gaze direction*. To that end, we leverage concepts used by Bayesian neural networks for estimation of aleatoric uncertainty.
- in cases such as self-occlusion, one or more facial keypoints might not be detected, and OpenPose assigns a confidence of zero to the corresponding feature. To handle the absence of detections, *we introduce the concept of Confidence Gated Units (CGU)* to induce our model to disregard detections for which a zero-confidence level is provided.
Related Work {#sec:relwork}
============
Ambient assisted living applications may benefit from computer vision methods in a variety of scenarios, including safety, well-being assessment, and human-machine interaction [@chaaraoui2012review; @leo2017computer]. Our aim is to monitor the overall health status of a patient by observing his/her behavior, or the way he/she interacts with the environment or with others. Summarized in Section \[sec:facility\] and detailed in [@VISAPP2018; @chessa17], the assisted living environment where our research takes place has been used for studies on automatic assessment of mobility information and frailty [@martini2018data]. Related to our system are the methods presented in [@cao2017realtime; @bathrinarayanan2013evaluation; @zouba2010activity], which propose different smart systems designed to monitor human behavior and way of life incorporating computer vision elements. Estimating the relative pose of subjects is crucial to perform high level tasks such as whole body action recognition and understanding the relationship between a person and the environment. Appearance-based pose estimation systems attempt to infer the positions of the body joints of the subjects present in a scene. Traditional methods relied on models fit to each of the individual subjects found in a given image frame [@zhang2009efficient; @brubaker2006physics]. More recent approaches employ convolutional architectures [@Wei2016ConvolutionalPM; @cao2017realtime] to extract features from the entire scene, therefore making the whole process relatively independent of the number of subjects in the scene.
At a finer level, the analysis of human facial features may provide additional information [@baltruvsaitis2016openface] about well-being. For example, facial expression recognition [@Lopes2017Facial; @Zhang2017Facial] can be used in sentiment analysis [@Jayalekshmi2017Facial]. Facial analysis can also provide information on gaze direction, which is useful to better understand the interaction between a person and his/her surrounding environment [@Varadarajan2018]. Recent contributions in this area attempt to infer the orientation of a person’s head by fitting a 3D face model to estimate both 2D [@Zhang2015Appearance] and 3D gaze information [@zhang2017written]. Other contemporary methods resort to different types of information, which include head detection, head orientation estimation, or contextual information about the surrounding environment [@Murphy2009Headpose]. In the context of human-computer interaction, the work in [@kafka2016eye] employs an end-to-end architecture to track the eyes of a user in real-time using hand-held devices.
However, most works and datasets on inference of head orientation and gaze focus on specific scenarios, such as images containing close-up views of subjects’ heads [@FunesMora_ETRA_2014; @Zhang2015Appearance], with restricted background size and complexity. More similar to our scenario of interest, the GazeFollow dataset introduced in [@recasens2015were] contains more than 120k images of one or more individuals performing a variety of actions in relatively unconstrained scenarios. Together with the dataset, the authors introduce a two-pathway architecture that combines contextual cues with information about the position and appearance of the head of a subject to infer his/her gaze direction. A similar model is introduced in [@chong2018connecting], with applicability extended to scenarios where the subject’s gaze is directed somewhere outside the image.
Gaze estimation is a task with multiple possible levels of difficulty, which vary according to the scenario of observation. Even for humans, it is much easier to tell where someone is looking if a full-view of the subject’s face is possible, while the task becomes much harder when the subject is facing backwards with respect to the observer’s point of view. In modeling terms, this corresponds to heteroscedastic uncertainty, i.e. uncertainty that depends on the inputs to the model, such that some inputs are associated to more noisy outputs than others.
As explained in [@kendall2017uncertainties], conventional deep learning models do not provide estimations of uncertainties for the outputs. Classification models typically employ softmax in their last layer, such that prediction scores are normalized and do not necessarily represent uncertainty. For regression models, usually no information on prediction confidence is provided by the model. Bayesian deep learning approaches are becoming increasingly more popular as a way to understand and estimate uncertainty with deep learning models [@gal2016uncertainty; @kendall2015bayesian; @kendall2018multi]. Under this paradigm, uncertainties are formalized as probability distributions over model parameters and/or outputs. For estimation of heteroscedastic uncertainty in regressors models, outputs can be modeled as corrupted with Gaussian random noise. Then, as we detail in Eq.\[eq:losscos\] of Section \[sec:training\], a customized loss function is sufficient for learning a regressor model that also predicts the variance of this noise as a function of the input [@kendall2017uncertainties], without need for uncertainty labels.
Proposed Approach
=================
Our method estimates a person’s apparent gaze direction according to the relative locations of his/her facial keypoints. As Fig. \[fig:method\_diagram\] indicates, we use OpenPose [@cao2017realtime] to detect the anatomical keypoints of all the persons present in the scene. Of the detected keypoints, we consider only those located in the head (i.e., the nose, eyes, and ears) of each individual.
Let $p_{k,s}^j = [x_{k,s}^j,y_{k,s}^j,c_{k,s}^j]$ represent the horizontal and vertical coordinates of a keypoint $k$ and its corresponding detection confidence value, respectively. The subscript $k\in \{n,e,a\}$ represents the nose, eyes, and ears features, with the subscript $s\in \{l,r,\emptyset \}$ encoding the side of the feature points.
Aiming at a scale-invariant representation, for each person $j$ in the scene we centralize all detected keypoints with respect to the head-centroid $h^j=[x_h^j,y_h^j]$, which is computed as the mean coordinates of all head keypoints detected in the scene. Then, the obtained relative coordinates are normalized based on the distance of the farthest keypoint to the centroid. In this way, for each detected person we form a feature vector $f\in \mathbb{R}^{15}$ by concatenating the relative vectors $\hat{p}_{k,s}^j=[\hat{x}_{k,s}^j,\hat{y}_{k,s}^j,c_{k,s}^j]$ $$f^j = \left[ \hat{p}_{n,\emptyset}^j,\hat{p}_{e,r}^j,\hat{p}_{e,l}^j,\hat{p}_{a,r}^j, \hat{p}_{a,l}^j \right].
\label{eq:feature}$$
Network architecture using Gated units
--------------------------------------
Images acquired from assisted living environments can contain multiple people performing different activities, such that their apparent pose may vary significantly and self-occlusions frequently occur. For example, in lateral-views at least an ear is often occluded, while in back-views nose and eyes tend to be occluded. As consequence, an additional challenge intrinsic to this task is the representation of missing keypoints. In such cases, OpenPose outputs $0$ for both the spatial coordinates $(x,y)_{k,s}^j$ and also the detection confidence value $c_{k,s}^j$. Since the spatial coordinates are centralized with respect to the head-centroid $h^j$ as the $(0,0)$ reference of the input space, a confidence score $c_{k,s}^j = 0$ plays a crucial role in indicating both the reliability and also the absence of a keypoint.
![The proposed Confidence Gated Unit (CGU).[]{data-label="fig:cgu"}](gatedUnit3.eps){width="0.6\linewidth"}
Inspired on the Gated Recurrent Units (GRUs) employed in recurrent neural networks [@cho2014learning], we propose a Confidence Gated Unit (CGU) composed of two internal units: i) a ReLU unit acting on an input feature $q_i$; and ii) a sigmoid unit to emulate the behavior of a gate according to a confidence value $c_i$. As depicted in Figure \[fig:cgu\], we opt for a sigmoid unit without a bias parameter, to avoid potential biases towards models that disregard $c_i$ when trained with unbalanced datasets where the majority of samples are detected with high confidence. Finally, the outputs of both units are then multiplied into an adjusted CGU output $\tilde{q_i}$.
For our application, a CGU is applied to each pair coordinate-confidence $(\hat{x}_{k,s}^j,c_{k,s}^j)$ and $(\hat{y}_{k,s}^j,c_{k,s}^j)$. To properly exploit the full range of the sigmoid function and thus reach output values near $0$ for $c_{k,s}^j=0$, we centralize and standardize the input confidence scores according to the corresponding dataset statistics. In this way, our proposed network for gaze regression has a combination of $10$ CGUs as input layer.
Moreover, the variety of view-points from which a subject might be visible in the scene, occlusions and unusual poses lead to a vast range of scenarios where the difficulty of the gaze estimation varies significantly. Hence, we design a model that incorporates an uncertainty estimation method, which indicates its level of confidence for each prediction of gaze direction. From an application perspective, this additional information would allow us to refine the predictions by choosing between different cameras, models, or time instants.
The gaze direction is approximated by the vector $\tilde{g}^j=\left[\tilde{g}_x,\tilde{g}_y \right]$, which consists of the projection onto the image plane of the unit vector centered at the centroid $h^j$. In terms of architecture design, this corresponds to an output layer with $3$ units: two that regress the $(\tilde{g}_x,\tilde{g}_y)$ vector of gaze direction, and an additional unit that outputs the regression uncertainty $\sigma_{\tilde{g}}$.
Following ablative experiments and weight visualization to identify dead units, we opt for an architecture where the CGU-based input layer is followed by $2$ fully-connected (FC) hidden layers with $10$ units each, and the output layer with $3$ units. Thus, the architecture has a total of $283$ learnable parameters and can be summarized as: (10 CGU, 10 FC, 10 FC, 3 FC).
Training Strategy {#sec:training}
-----------------
While all weights composing fully-connected layers are initialized as in [@He2015], we empirically observed better results when initializing the parameters composing CGU units with *ones*. Since these compose only the input layer, initializing the weights as such does not represent a risk of gradient explosion as no further backpropagation has to be performed. Intuitively, our rationale is that the input coordinate features should not be strongly transformed in this first layer, as at this initial point no information from additional keypoints is accessible. Regarding regularization, we empirically observed better results without regularization in the input and output layers, while a L2 penalty of $10^{-4}$ is applied to parameters of both FC hidden layers.
Regardless of the dataset, we trained our network only on images where at least two facial keypoints are detected. Since we are interested on estimating direction of gaze to verify whether any object of interest is within a person’s field of view, we opt for optimization and evaluations based on angular error. Thus, training was performed using a cosine similarity loss function that is adjusted based on [@kendall2017uncertainties] to allow uncertainty estimation. Let $\mathcal{T}$ be the set of annotated orientation vectors $g$, while $\tilde{g}$ corresponds to the estimated orientation produced by the network and $\sigma_{\tilde{g}}$ represents the model’s uncertainty prediction. Our cost function is then given by $$\label{eq:losscos}
\mathcal{L}_{\text{cos}}(g,\tilde{g}) = \frac{1}{\left|\mathcal{T}\right|}\sum_{g\in\mathcal{T}}\frac{\exp(-\sigma_{\tilde{g}})}{2} \frac{-{g}\cdot{\tilde{g}}}{||{g}||\cdot||{\tilde{g}}||} + \frac{log\:\sigma_{\tilde{g}}}{2}.$$
With this loss function, no additional label is needed for the model to learn to predict its own uncertainty. The $\exp(-\sigma_{\tilde{g}})$ component is a more numerically stable representation of $\frac{1}{\sigma_{\tilde{g}}}$, which encourages the model to output a higher $\sigma_{\tilde{g}}$ when the cosine error is higher. On the other hand, the regularizing component $\log(\sigma_{\tilde{g}})$ helps avoiding an exploding uncertainty prediction.
In terms of model optimization, all experiments were performed using the Adam [@kinga2015method] optimizer with early stopping based on angular error on the corresponding validation sets. Additional parameters such as batch size and learning rate varied according to the dataset. Hence, we describe them in detail in Section \[sec:results\].
Experiments and Results {#sec:results}
=======================
We evaluate our approach on two different datasets. The first is the GazeFollow dataset [@recasens2015were], on which we compare our method against two different baselines. The second dataset, which we refer to as the *MoDiPro* dataset, comprises images acquired from an actual discharge facility as detailed in Section \[sec:facility\].
Evaluation on the GazeFollow dataset
------------------------------------
[**Dataset split and training details.**]{} The publicly available GazeFollow dataset contains more than 120k images, with corresponding annotations of the eye locations and the focus of attention point of specific subjects in the scene. We use the direction vectors connecting these two points to train and evaluate our regressors. In terms of angular distribution, about $53\%$ of the samples composing the GazeFollow training set correspond to subjects whose gaze direction lies within the quadrant $[-90^\circ,0^\circ]$ with respect to the horizontal axis. On the other hand, only $29\%$ of the cases their gaze direction is within the $[-180^\circ,-90^\circ]$ quadrant. To compensate such bias, we augment the number of samples in the later quadrant by mirroring with respect to the vertical-axis a subset of randomly selected samples from the most frequent quadrant. Finally, for training our model we split the training set into two subsets: $90\%$ for *train*, and $10\%$ for validation *val* subset. Training is performed using a learning rate $5\times10^{-3}$, batches of $1024$ samples and early-stopping based on angular error on the *val* subset. The *test* set comprises $4782$ images, with ten different annotations per image. For evaluation, we follow [@recasens2015were] and assess each model by computing the angular error between their predictions and the average annotation vector.
The GazeFollow dataset is structured such that for each image only the gaze from a specific subject must be assessed. For images containing multiple people, this requires identifying which detection provided by OpenPose corresponds to the subject of interest. To that end, we identify which detected subject has an estimated head-centroid that is the closest to the annotated eye-coordinates $E_{GT}$ provided as ground-truth. To avoid mismatches in cases the correct subject is not detected but detections for other subjects on the scene are available, we impose that gaze is estimated only if $E_{GT}$ falls within a radius of $1.5\times \delta$ around the head-centroid, where $\delta$ corresponds to distance between the centroid and its farthest detected facial keypoint.
We compare our method against two baselines. The first, which we refer to as <span style="font-variant:small-caps;">Geom</span>, relies solely on linear geometry to estimate gaze from the relative facial keypoints positions. Comparison against this baseline aims at evaluating if training a network is needed to approximate the regression $f \rightarrow g$, instead of directly approximating it by a set of simple equations. The second baseline is the model introduced together with the GazeFollow dataset in [@recasens2015were], which consists of a deep neural-network that combines a gaze pathway and a saliency pathway that are jointly trained for gaze estimation. We refer to this baseline as <span style="font-variant:small-caps;">GF-model</span>.
[**Comparison against geometry-based baseline.**]{} We refer the reader to our Supplementary Material for a more detailed description of <span style="font-variant:small-caps;">Geom</span>. This baseline is a simplification of the model introduced in [@gee1994determining] for face orientation estimation, which makes minimal assumptions about the facial structure [@gee1994determining] but additionally requires mouth keypoints and pre-defined model ratios. In short, let $\Vec{s}$ represent the facial symmetry axis that is computed as the normal of the eye-axis. We estimate the facial normal $\Vec{n}$ as a vector that is normal to $\Vec{s}$ while intersecting $\Vec{s}$ at the detected nose position. Then, the head pitch $\omega$ is estimated as the angle between the ear-centroid and the eye-centroid, i.e., the average coordinates of eyes and ears detections, respectively. Finally, gaze direction is estimated by rotating $\Vec{n}$ with the estimated pitch $\omega$.
The <span style="font-variant:small-caps;">Geom</span> baseline requires the detection of the nose and at least one eye. Out of the $4782$ images composing the GazeFollow *test* set, <span style="font-variant:small-caps;">Geom</span> is thus restricted to a subset $Set1$ of $4258$ images. As summarized on Tab. \[tab:res\_gf\], results obtained on subset $Set1$ demonstrate that our model $Net$ provide gaze estimations on average $23^\circ$ more accurate than the ones obtained with the simpler baseline. Such a large improvement in performance suggests our network learns a more complex (possibly non-linear) relationship between keypoints and gaze direction. Examples available on Fig. \[fig:example\_gazefollow\] qualitatively illustrate how the predictions provided by our <span style="font-variant:small-caps;">Net</span> model (in green) are significantly better than the ones provided by the baseline <span style="font-variant:small-caps;">Geom</span> (in red).
--------------------------------------------------------------------------- --------------- --------------- ------------
*No. of images* 4258 4671 4782
<span style="font-variant:small-caps;">Geom</span> $42.63^\circ$ - -
<span style="font-variant:small-caps;">Net0</span> $19.52^\circ$ $25.70^\circ$ -
<span style="font-variant:small-caps;">Net</span> $19.41^\circ$ $23.37^\circ$ -
<span style="font-variant:small-caps;">GF-model[@recasens2015were]</span> - - $24^\circ$
--------------------------------------------------------------------------- --------------- --------------- ------------
: Comparison in terms of angular errors between our method and baselines on the GazeFollow test set.[]{data-label="tab:res_gf"}
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[**Comparison against GazeFollow model.**]{} Since our network is trained on images where at least two facial keypoints are detected, we apply the same constraint for evaluation. In the test set, OpenPose detects at least two keypoints for a subset $Set2$ containing $97.7\%$ of the $4782$ images composing the full set.
The results of our evaluation are summarized in Tab. \[tab:res\_gf\], while qualitative examples are provided in Fig. \[fig:example\_gazefollow\]. As reported in [@recasens2015were], gaze predictions provided by the <span style="font-variant:small-caps;">GF-model</span> present an mean angular error of $24^\circ$ on the *test* set. Our <span style="font-variant:small-caps;">Net</span> model provides an mean angular error of $23.37^\circ$ for $97.7\%$ of these images, which strongly indicates that its performance is on pair with <span style="font-variant:small-caps;">GF-model</span> network despite relying solely on the relative position of $5$ facial keypoints to predict gaze. [**Impact of using Confidence Gated Units (CGU).**]{} To verify the benefits of applying our proposed CGU blocks to handle absent keypoint detections, i.e. keypoints with $0$ confidence score, we evaluated the performance of our model with and without feeding the confidence scores as inputs. We refer to the latter case as the <span style="font-variant:small-caps;">Net0</span>, where the CGU blocks composing the input layer are replaced by simple ReLU units initialized in the same way as described in Section \[sec:training\]. Results summarized in Tab. \[tab:res\_gf\] indicate an error decrease of $2.3^\circ$ when providing confidence scores to an input layer composed of CGUs. In addition to experiments summarized in Tab. \[tab:res\_gf\], we also evaluated a model where the CGU units are replaced by simple additional ReLU units to handle confidence scores. For the $1536$ images where OpenPose detects less than $4$ facial keypoints, a significant decrease on angular error is observed when using CGU units: $30.1^\circ$ mean error, in comparison to $30.9^\circ$ provided by the model with solely ReLU based input layer.
[**Quality of uncertainty estimations.**]{} In addition to the overall mean angular error, we also evaluate how accurate are the uncertainty estimations provided by our <span style="font-variant:small-caps;">Net</span> model for its gaze direction predictions. As depicted in Fig. \[fig:cum\_unc\], significantly lower angular errors are observed for gaze predictions accompanied by low uncertainty network predictions. Uncertainties lower than $0.1$ are observed for $80\%$ of the *test* set, a subset for which the gaze estimations provided by our <span style="font-variant:small-caps;">Net</span> model are on average off by only $16.5^\circ$.
Moreover, the high correlation between uncertainty predictions and angular error ($\rho=0.56$) is clearly depicted by the plots provided in Fig. \[fig:polarplots\]. For each sample in these plots, the radial distance corresponds to its predicted uncertainty $\sigma_i$, while the angle corresponds to predicted direction of gaze $\tilde{g}$, i.e $\alpha_i=tan^{-1}(-\tilde{g}_y/\tilde{g}_x)$. For both *train* and *test* sets, the associated colormap shows that lower errors (in dark blue) are observed for predictions with lower uncertainty, with increasingly higher errors (green to red) as the uncertainty increases (farther from the center).
[**Performance according to keypoint occlusions.**]{} Furthermore, the central and the right-most scatter plots in Fig. \[fig:polarplots\] also allow an analysis on how the performance of our model and its uncertainty predictions vary according to specific scenarios. For most cases, the number of detected keypoints ($k$) indicates specific scenarios: $k=2$ is mostly related to back-views, where nose and two other keypoints (both eyes or a pair eye-ear) are missing; $k=3$ and $k=4$ are mostly lateral-views; $k=5$ are frontal-views, where all keypoints are visible. Since images are $2D$ projections from the environment, back- and frontal-views are the ones more affected by the information loss implicit in the image formation process, while for lateral-views estimation of gaze direction tends to be easier.
An analysis of the scatter plots demonstrates that the predictions provided by our model reflect these expected behaviors. For samples with $k=2$ (back-view), both uncertainty predictions and angular error tend to be higher, while for most cases of $k=3$ and $k=4$ the predictions are associated with lower uncertainty and higher angular accuracy. Moreover, the spread of the distribution of predictions for $k=5$ indicates that the model’s uncertainty predictions are not just correlated to the amount of available keypoints and predicted gaze direction, but rather case specific while still highly correlated to angular errors.
Results on the assisted living dataset {#sec:facility}
--------------------------------------
[**Dataset split and training details.**]{} This work is part of a project that focuses on elderly patients with partial autonomy but in need of moderate assistance, possibly in a post-hospitalization stage. Thus, it is critical to evaluate the performance of our gaze estimation model on data from real assisted living environments. To that end, we also evaluate our approach on videos acquired in an assisted living facility in which the patient, after being discharged from the hospital, is hosted for a few days. The facility is a fully-equipped apartment situated in a local hospital, where patients may be monitored by various sensors, including localization systems, RGB-D, and two conventional video cameras, arranged as shown in Fig. \[fig:apt\_plan\].
More specifically, to evaluate the performance of our gaze estimation model we compiled a dataset which we call *MoDiPro*, consisting of 1,060 video frames collected from the two cameras whose positions are indicated in Fig. \[fig:apt\_plan\]. For $CAM1$, $530$ frames were sampled from $46$ different video sequences; for $CAM2$, $530$ frames were sampled from $27$ different video sequences. To limit storage while discarding minimal temporal information, the resolution of the acquired frames was limited to $480\times 270$ pixels, at $25$ fps. In most frames multiple subjects are simultaneously visible, with a total of 22 subjects performing different activities.
As exemplified also in Fig. \[fig:example\_gaze\], cameras $CAM1$ and $CAM2$ cover different parts of the environment. Images acquired with $CAM2$ present significant distortion, which increases the complexity of the task. We randomly split the available sets of images into camera-specific training, validation and test subsets. Since frames composing the same video sequence can be highly correlated, we opt for a stratified strategy where video sequences are sampled. That is, all frames available from a certain video sequence are assigned to either *train*, *val* or *test* subsets. Aiming at an evaluation that covers a wide variety of scenes, the proportions chosen in terms of total number of frames are: $50\%$ for training, $20\%$ for validation, $30\%$ for testing. Fine-tuning experiments are performed using learning rates $1\times10^{-5}$, while $1\times10^{-4}$ is adopted when training models only on *MoDiPro* images. Batches with $64$ samples are used, with early-stopping based on angular error on the *val* subset. Moreover, all results reported on Tab. \[tab:res\_modipro\] and discussed below correspond to average values obtained after train/test on $3$ different random splits.
To assess the cross-view performance of our method, we train our <span style="font-variant:small-caps;">Net</span> model with $7$ different combinations of images from *MoDiPro* and GazeFollow datasets. As summarized in Tab. \[tab:res\_modipro\], models <span style="font-variant:small-caps;">Net</span>\#0-2 are trained in $CAM1$-only, $CAM2$-only, and both *MoDiPro* cameras. <span style="font-variant:small-caps;">Net</span>\#3 corresponds to the model trained only on GazeFollow frames (GF for shortness), while <span style="font-variant:small-caps;">Net</span>\#4-6 are obtained by fine-tuning the pre-trained <span style="font-variant:small-caps;">Net</span>\#3 on three possible sets of *MoDiPro* frames.
[llcccccc]{} & &\
Model & GF & *Cam1* & *Cam2* & *Cam1* & *Cam2* & *Mean*\
<span style="font-variant:small-caps;">Net</span>\#0 & && & $16.16^\circ$ & $39.12^\circ$ & -\
<span style="font-variant:small-caps;">Net</span>\#1& & & & $29.56^\circ$ & $26.37^\circ$ & -\
<span style="font-variant:small-caps;">Net</span>\#2& & & & $18.52^\circ$ & $23.02^\circ$ & $20.94^\circ$\
<span style="font-variant:small-caps;">Net</span>\#3& & & & $27.64^\circ$ & $26.98^\circ$ & $27.31^\circ$\
<span style="font-variant:small-caps;">Net</span>\#4& & & & $16.17^\circ$ & $27.36^\circ$ & -\
<span style="font-variant:small-caps;">Net</span>\#5& & & & $27.56^\circ$ & $24.01^\circ$ & -\
<span style="font-variant:small-caps;">Net</span>\#6& &&& $17.82^\circ$ & $20.15^\circ$ & $19.05^\circ$\
<span style="font-variant:small-caps;">GF-model</span> & & & & $43.49^\circ$ & $60.82^\circ$ & $52.15^\circ$\
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[**Performance according to training/testing subsets.**]{} Cross-view results obtained by <span style="font-variant:small-caps;">Net</span>\#0 on $CAM2$ and <span style="font-variant:small-caps;">Net</span>\#1 on $CAM1$ demonstrate how models trained only on a camera-specific set of images are less robust to image distortions, with significantly higher angular errors for images composing unseen subsets. Trained on both $CAM1$ and $CAM2$, the model <span style="font-variant:small-caps;">Net</span>\#2 demonstrates a more consistent performance across views. In comparison with the camera specific models, a $3^\circ$ lower angular error on $CAM2$ is obtained at cost of only $1.4^\circ$ error increase on $CAM1$.
In addition, error comparisons between models <span style="font-variant:small-caps;">Net</span>\#0-2 and <span style="font-variant:small-caps;">Net</span>\#4-6 demonstrate that pre-training the model on the GF dataset before fine-tuning on *MoDiPro* images leads to consistently lower mean angular errors, with an optimal performance of $17.82^\circ$ for $CAM1$ and $20.15^\circ$ for $CAM2$. This corresponds to an overall average error $1.9^\circ$ lower than the model <span style="font-variant:small-caps;">Net</span>\#2 not pre-trained on GF, while more than $7^\circ$ better than the model <span style="font-variant:small-caps;">Net</span>\#3 trained solely on GF. In terms of camera-specific performance, for $CAM1$ optimal performances with error below $17^\circ$ are obtained when not training on $CAM2$. On the other hand, predictions for $CAM2$ are significantly better when training is performed using additional $CAM1$ and/or GazeFollow images. We hypothesize the distortions characteristic of $CAM2$ images easily lead to overfitting, thus the advantage of training on additional sets of images. As a final remark we may notice that overall <span style="font-variant:small-caps;">Net</span>\#6 provides the best an most stable result across the two views.
[**Comparison against** ]{}**<span style="font-variant:small-caps;">GF-model</span>**. Finally, we compare the predictions provided by our <span style="font-variant:small-caps;">Net</span> models to the ones obtained by the publicly available version of <span style="font-variant:small-caps;">GF-model</span>[^1]. As summarized in Tab. \[tab:res\_modipro\], gaze predictions provided by <span style="font-variant:small-caps;">GF-model</span> on the *MoDiPro* dataset are remarkably worse in terms of angular error than the ones predicted by any of our <span style="font-variant:small-caps;">Net</span>\#0-6 models, including the <span style="font-variant:small-caps;">Net</span>\#3 also trained only on GF images.
Closer inspection of <span style="font-variant:small-caps;">GF-model</span> predictions suggests two disadvantages of this model with respect to ours when predicting gaze on images from real assisted living environments: i) sensitivity to scale; ii) bias towards salient objects. Images composing the GazeFollow typically contain a close-view of the subject of interest, such that only a small surrounding area is covered by the camera-view. In contrast, images from assisted living facilities such as the ones in the *MoDiPro* dataset contain subjects covering a much smaller region of the scene, i.e., they are smaller in terms of pixel area. Our <span style="font-variant:small-caps;">Net</span> model profits from the adopted representation of keypoints, with coordinates centered at the head-centroid and normalized based on the largest distance between centroid and detected keypoints. Moreover, visual inspection of <span style="font-variant:small-caps;">GF-model</span> predictions reveals examples such as the two bottom ones in Fig. \[fig:example\_gaze\]: in the left, while our model correctly indicates that the subjects look at each other, <span style="font-variant:small-caps;">GF-model</span> is misled by the saliency of the TV and possibly the window; in the right, the saliency of the TV again misguides <span style="font-variant:small-caps;">GF-model</span>, while our model properly indicates that the person is looking at the object she is holding.
Runtime Analysis
----------------
Each call to our network requires approximately 0.85 ms on average on a NVIDIA GeForce 970M, with one feedforward execution per person. The overall runtime is thus domined by OpenPose, whose overall runtime is reported in [@cao2017realtime] as taking $77ms$ on COCO images with a NVIDIA GeForce GTX-1080 Ti GPU.
Conclusion
==========
This paper presents an alternative gaze estimation method that exploits solely facial keypoints detected by a pose estimation model. As our end goal is to study the behaviors of individuals in a assisted living environment, exploring a single feature extraction backbone for both pose as well as gaze estimation facilitates the design of a significantly less complex overall model.
Results obtained on the GazeFollow dataset demonstrate that our method estimates gaze with accuracy comparable to a complex task-specific baseline, without relying on any image features except relative position of facial keypoints. In contrast to conventional regression methods, our proposed model also provides estimations of uncertainty of its own predictions, with results demonstrating that the predicted uncertainties are highly correlated to the actual angular error of the corresponding gaze predictions. Moreover, analysis of performance according to the number of detected keypoints indicates that the proposed Confidence Gate Units improve the performance of our model for cases of partial absence of features. Finally, evaluation on frames collected from a real assisted living facility demonstrate the higher suitability of our method for ADL analysis in realistic scenarios, where images cover wider areas and subjects are visible at different scales and poses. In the future, we intend to explore tracking mechanisms to estimate gaze direction in videos without abrupt estimation changes. To identify human-human and human-object interactions, we plan to combine gaze estimations with a semantic segmentation model to be designed specifically for objects of interest for ADL analysis.
[^1]: This version provides $25.8^\circ$ mean angular error on the GazeFollow test set, in comparison to the $24^\circ$ reported in [@recasens2015were]
|
---
abstract: 'When a star comes within a critical distance to a supermassive black hole (SMBH), immense tidal forces disrupt the star, resulting in a stream of debris that falls back onto the SMBH and powers a luminous flare. In this paper, we perform hydrodynamical simulations of the disruption of a main-sequence star by a SMBH to characterize the evolution of the debris stream after a tidal disruption. We demonstrate that this debris stream is confined by self-gravity in the two directions perpendicular to the original direction of the star’s travel, and as a consequence has a negligible surface area and makes almost no contribution to either the continuum or line emission. We therefore propose that any observed emission lines are not the result of photoionization in this unbound debris, but are produced in the region above and below the forming elliptical accretion disk, analogous to the broad-line region (BLR) in steadily-accreting active galactic nuclei. As each line within a BLR is observationally linked to a particular location in the accretion disk, we suggest that the absence of a line indicates that the accretion disk does not yet extend to the distance required to produce that line. This model can be used to understand the spectral properties of the tidal disruption event (TDE) [PS1-10jh]{}, for which lines are observed, but the Balmer series and are not. Using a maximum likelihood analysis, we show that the disruption of a main-sequence star of near-solar composition can reproduce this event.'
author:
- 'James Guillochon, Haik Manukian, and Enrico Ramirez-Ruiz'
bibliography:
- '/Users/james/Dropbox/library.bib'
title: 'PS1-10jh: The Disruption of a Main-Sequence Star of Near-Solar Composition'
---
Introduction {#sec:intro}
============
The tidal disruption of a star by a supermassive black hole (SMBH) splits the star into either two or three ballistically distinct masses. In the event of a full disruption, the star is split into two pieces of nearly-equal mass. One half of the star becomes bound to the black hole after the encounter, and continues along elliptical trajectories with pericenter distances equal to the star’s original pericenter distance. The other half of the star gains orbital energy in the encounter, and is placed on hyperbolic trajectories. For a partial disruption, a third mass in the form of a surviving stellar core emerges from the encounter, with the absolute value of its orbital energy comparable to its own binding energy [@Faber:2005be; @Guillochon:2011be; @MacLeod:2012cd; @Liu:2013er; @Manukian:2013ce].
Determining the fates of these pieces of the star are critical in determining the appearance of the flare that results from the immense gravitational energy that will be released by the accretion disk that eventually forms. Previously, it has been assumed that the unbound material, which was thought to be a wide “fan,” was the primary contributor to the broad emission lines that are produced as the result of a tidal disruption [@Strubbe:2009ek; @Kasen:2010ci; @Clausen:2011fa].
For the tidal disruption event (TDE) [PS1-10jh]{}(@Gezari:2012fk, hereafter ), it was assumed that hydrogen, which is ejected to large distances within the wide debris fan generated by the disruption, can recombine more quickly than the rate at which it is ionized by the central source. This would ensure that the vast majority of the hydrogen is neutral, and thus any ionizing radiation incident upon the fan would produce an emission feature. The absence of any hydrogen emission features was used to derive an upper limit on the amount of hydrogen present, implying that helium is five times more common than hydrogen by mass with the disrupted star.
In this paper, we present three-dimensional hydrodynamical simulations that show that the assumption that this debris fan intercepts a significant fraction of the light is incorrect. As noted by [@Kochanek:1994bn], the width of the stream of unbound material is still controlled by the stream’s self-gravity in the transverse direction, restricting its width to only be a fraction of the star’s original pericenter distance. Through numerical simulations of fully-disruptive encounters with mass ratios $q \equiv M_{\rm h}/M_{\odot} = 10^{3}$ and $10^{6}$, we verify that the transverse containment of the stream’s width does indeed occur. As a result, the stream only grows in the radial direction, and thus the total volume and surface area increase only slightly more steeply than $v_{\rm p}$. Therefore, the emitting volume of this structure is not significant enough to produce bright hydrogen emission lines, even for the disruption of a main-sequence (MS) star composed largely of hydrogen.
But while we find that the area of the unbound debris has been vastly overestimated, we also find that the area occupied by the accretion disk formed from the bound material has been vastly underestimated. Our numerical simulations confirm the prediction that material that returns to pericenter is ballistically launched to very large distances from the black hole, hundreds of times $r_{\rm p}$. Additionally, we find that significant dissipation occurs when this material returns to pericenter. As the debris stream quickly virializes at pericenter and the density of the material is significantly reduced as compared to the star’s original density, self-gravity is suppressed even in the transverse direction. As a result, a fan structure [*is*]{} formed once material returns pericenter. But as this material belongs to the fraction of the original star that is strongly bound to the black hole, the radial extent of this material grows at a rate that is significantly smaller than the unbound fraction.
As the region in which [[H]{}$\alpha$]{}is produced in steady AGN is on the order of a few light days to a few light weeks from the black hole for $L_{5100} \sim 10^{45}$ ergs s$^{-1}$ [@Peterson:2006hr], we show that it is unlikely that the debris ejected by the disruption has traveled the distance necessary to produce an [[H]{}$\alpha$]{}line for [PS1-10jh]{}. Through comparison with the processes responsible for producing the broad line regions (BLRs) of steadily-accreting AGN, we predict that the helium lines that are observed in [PS1-10jh]{}are produced much closer to the black hole [@Korista:1995eh; @Bentz:2010hk], and the debris has sufficient time to reach this distance by the time the first spectrum was observed. Motivated by the results of our hydrodynamical simulations, we model the accretion disk structure and use a Markov-Chain Monte Carlo (MCMC) procedure to determine the combinations of parameters with the highest-likelihood, and we find that the highest-likelihood models that do not include priors on the input parameters involve the disruption of a main-sequence star with mass $4 M_{\odot}$ by a $M_{\rm h} = 2 \times 10^{7} M_{\odot}$ black hole.
In Section \[sec:method\] we describe our method for running hydrodynamical simulations to characterize the behavior of the debris stream after a disruption, and describe the maximum likelihood analysis (MLA) we employed to estimate the parameters of [PS1-10jh]{}. In Section \[sec:debris\] we present a physical interpretation of the results of our hydrodynamical simulations. Bearing these results in mind, we develop our generalized model for the time-dependent, broadband light that would accompany the disruption of a star in Section \[sec:model\]. We then apply this model specifically to [PS1-10jh]{}in Section \[sec:ps1-10jh\]. Finally, we review additional evidence as to why the disruption of a helium-rich star is unlikely to have produced [PS1-10jh]{}in the first place, and look towards the future when TDEs will be regularly observed.
Method {#sec:method}
======
Hydrodynamical Simulations {#subsec:hydrosims}
--------------------------
The black hole at the center of our own galaxy is estimated to be $\simeq 4 \times 10^{6} M_{\odot}$ [@Ghez:2008hf], and is one of the smallest known massive black holes [@Schulze:2010jl]. As the mass of an average main-sequence star is $\sim 0.1 M_{\odot}$ [@Kroupa:1993tm], the majority of stellar tidal disruptions will have $q \gtrsim 10^{6}$. For such disruptions, the timescale of return of the most bound debris is on the order of days to weeks [@Rees:1988ei], with the peak fallback rate occurring approximately one month after the time of the disruption (@Evans:1989jua; @Lodato:2009iba; ).
For hydrodynamical simulations of TDEs, the main limiting factor is the sound-crossing time of the star, which for a solar mass star is approximately one hour. Given an initial stellar model that occupies $100^{3}$ grid cells, each hydrodynamical time-step translates to only one minute of physical time. Thus, the simulation of the tidal disruption of a solar mass star by a $10^{6} M_{\odot}$ black hole that includes the time at which the fallback rate is at a maximum requires $\sim 10^{5}$ time-steps. Additionally, the debris stream resulting from the disruption must be fully resolved in both length and in width. As the stream is self-gravitating (as described in Section \[subsec:stream\]), it mains a very narrow profile, with the aspect ratio of the stream when the first material returns to pericenter being $q^{1/3} (t_{\rm peak}/t_{\rm p})^{1/2} \sim 10^{3}$, where $t_{\rm peak}$ is the time where $\dot{M}$ reaches a maximum, and $t_{\rm p}$ is the pericenter crossing time. If the number of grid cells across the stream is forced to be at least 20, which is necessary to satisfy the @Truelove:1997bj criteria, then $10^{6}$ grid cells would be required to be evolved for $10^{5}$ time-steps.
This means that a complete simulation of the full problem within a single simulation is very computationally expensive. Instead, we run two separate simulations that are each well-equipped to describe the behavior of the debris stream at two different epochs. To determine the fate of the debris liberated from a star during a tidal disruption, we used two similar simulation setups, differing only in the mass ratio $q$. The first simulation sets $q = 10^{6}$ and solely focuses on the evolution of the debris stream as it expands away from pericenter after the star’s initial encounter with the black hole. Because of the computational expense, the return of the debris to pericenter is not followed in this simulation. The second simulation sets $q = 10^{3}$, and follows the return of the debris to pericenter well beyond the peak in the accretion rate . In these encounters, the peak accretion rate is realized only one day after pericenter, and we follow the evolution of the returning debris for a total of $5 \times 10^{5}$ seconds (about one week). Our hydrodynamical simulations were performed in a module written for the [[FLASH]{}]{}adaptive mesh refinement code, the details of which can be found in @Guillochon:2009di [@Guillochon:2011be] and .
The initial conditions of the simulation are similar to those presented in , with the polytropic $\Gamma$ that describes the star’s structure being set to 5/3, and the impact parameter $\beta \equiv r_{\rm p} / r_{\rm t}$ being set to 2. The star is placed on a parabolic trajectory at an initial position that is several times further than $r_{\rm t}$, and is initially resolved by 50 grid cells across its diameter. As realistic equations of state are only sensical at the full-scale of the problem, the hydrodynamics of the gas are treated using a simple adiabatic polytrope $P \propto \rho^{\gamma}$, where $\gamma$ is the adiabatic index. As we do not include any explicit viscosity terms, entropy generation only occurs through dissipation via shocks, the effects of which are captured by simultaneous evolution of the internal energy $\epsilon$. The code utilizes the adaptive-mesh functionality of the [[FLASH]{}]{}software in different ways for the two simulations. In the $q = 10^{3}$ simulation, regions which are less than $10^{-1}$ times as dense as the current peak density are derefined, but maximum refinement is maintained within $4 r_{\rm p}$ at all times. For the $q = 10^{6}$ simulation, each refinement level is assigned to a single decade in $\rho$, using the star’s original central density $\rho_{\rm c}$ as a baseline, with the exception of the first refinement threshold which is set to $\rho = 5 \times 10^{-3} \rho_{\rm c}$.
As both the timescales and the length scales of a $q = 10^{3}$ disruption are different from those of the more typical $q = 10^{6}$ disruption, care must be taken when interpreting the results from these simulations and attempting to scale them up to what would be realized for larger mass ratios. As we will describe in Section \[subsec:dissipative\], the dissipation processes that are observed in the scaled-down simulation are analogous to other dissipation mechanisms that operate for larger values of $q$, given the proper scaling.
Fitting TDE Observations
------------------------
For fitting our models for tidal disruptions to observed events, we have developed the code [[TDEFit]{}]{}, which performs a MLA using an affine-invariant MCMC [@Goodman:2010et; @ForemanMackey:2013io], in which parameter combinations are assigned to individual “walkers” who then exchange positions according to their relative scores. The code is written in Fortran and utilizes the parallel variant of the algorithm presented in @ForemanMackey:2013io. We have designed the software to be flexible in the model parameters it accepts as inputs, any free parameter (either discrete or continuous) can be included in the parameter space exploration by simply listing it and its range of acceptable values within a parameter file. In the same way, both trivial and non-trivial priors can be specified at runtime for single or combinations of input parameters.
As the solutions can often be multi-modal, with small regions of acceptable parameter space separated by large voids of poor parameter space, it can sometimes be difficult to find the deepest global minimum using the vanilla affine-invariant algorithm. To address this issue, we modify the algorithm by performing simulated annealing [SA, @Press:1986vx] on a fraction $F$ of the walkers every $N$ timesteps during a “bake-in” period, where both $F$ and $N$ are adjustable. Each walker that is selected to anneal is used to seed an amoeba whose points are randomly drawn a small distance away from the original walker, these walkers then run through a full SA cycle in which the temperature is gradually reduced until they are unable to improve upon their local solution.
This enables the depths of local minima to be found more quickly, and in tests we have found that this improves the time of convergence to the global solution by orders of magnitude. Additionally, we anneal the ensemble of walkers themselves during the bake-in period, using the temperature schedule proposed in @Hou:2012hd, and periodically compare the scores of walkers to the best so far, removing those that fall below a pre-determined threshold that depends on the current annealing temperature. After the bake-in period, the algorithm reverts to the vanilla affine-invariant MCMC of @Goodman:2010et and run for several autocorrelation times, ensuring that detailed balance is maintained.
As inputs to this method, we use the full functional forms of the fallback rate (parameterized as $dM/dt \equiv {\dot{M}}_{\rm sim}$) presented in (see their Figure 5) for $\gamma = 4/3$ and $\gamma = 5/3$ polytropes. We assume that as the mass ratio $q \gg 1$, the dependence of ${\dot{M}}$ on $M_{\rm h}$, $M_{\ast}$, and $R_{\ast}$ is self-similar, $$\begin{aligned}
{\dot{M}}&= M_{\rm h,6}^{-1/2} M_{\ast,\odot}^{2} R_{\ast,\odot}^{-3/2} \dot{M}_{\rm int}(\beta),\label{eq:mdotpeak}\end{aligned}$$ where $10^{6} M_{\rm h,6} = M_{\rm h}$, $M_{\odot} M_{\ast,\odot} = M_{\ast}$, $R_{\odot} R_{\ast,\odot} = R_{\ast}$ and ${\dot{M}}_{\rm int}$ is an interpolation of ${\dot{M}}_{\rm sim}$, which was determined in directly from ${dM/dE}$ after the debris had relaxed to its final distribution in binding energy $E$ (see Figures 9 and 10 of ). Note that the functions we use as inputs for our calculation is the rate that the stellar debris returns to pericenter, and not the rate of accretion through the inner edge of the accretion disk. We discuss the validity of this assumption in Section \[subsec:dissipative\].
We can eliminate $R_{\ast}$ in Equation \[eq:mdotpeak\] by using known mass-radius relationships (e.g. @Tout:1996waa for MS stars or @Nauenberg:1972dx for white dwarfs). For MS stars, we presume that all objects with $M_{\ast} \leq 0.1 M_{\odot}$ have the radius of a $0.1 M_{\ast}$ star. With these relations, ${\dot{M}}$ is solely a function of $M_{\rm h}$, $M_{\ast}$, and $\beta$.
As the simulations of are only run for specific values of $\beta$, we determine intermediate $\beta$ solutions by rescaling neighboring simulations in $\beta$-space to the same scaled time variable $x \equiv (t - t_{\min}) / (t_{\max} - t_{\min})$, where $t_{\min}$ and $t_{\max}$ are the minimum and maximum times for each ${\dot{M}}_{\rm sim}$ curve, and then interpolating linearly between the two solutions, $$\begin{aligned}
\lfloor\beta\rfloor &= \min \left\{B\in\beta_{\rm sim} \mid B \geq \beta \right\}\\
\lceil\beta\rceil &= \max \left\{B\in\beta_{\rm sim} \mid B \leq \beta \right\}\\
\dot{M}_{\rm int}(\beta, x) &= \dot{M}_{\rm sim}(\lfloor\beta\rfloor, x)\nonumber\\
&+ \frac{\beta - \lfloor\beta\rfloor}{\lceil\beta\rceil - \lfloor\beta\rfloor} \left[\dot{M}_{\rm sim}(\lceil\beta\rceil, x) - \dot{M}_{\rm sim}(\lfloor\beta\rfloor, x)\right],\end{aligned}$$ where $\beta_{\rm sim}$ is the set of all $\beta$ for which a simulation is available, and where $\lfloor\beta\rfloor$ and $\lceil\beta\rceil$ return the values of $\beta_{\rm sim}$ that bracket $\beta$. We find this preserves the overall shape of the ${\dot{M}}$ curves well for values of $\beta$ for which a simulation is not available, as long as the sampling in $\beta_{\rm sim}$ is sufficiently dense to capture the overall trends.
The objective function used within [[TDEFit]{}]{}when comparing our models to the data is the maximum likelihood function, $$\ln {\cal L}_{\rm LC} = \sum_{i=1}^{j} \left[\frac{\left(V_{{\rm obs},i} - V_{{\rm mod},i}\right)^{2}}{\left(\sigma_{{\rm obs},i}^{2} + \sigma_{\rm v}^{2}\right)}+\ln\left(\sigma_{{\rm obs},i} + \sigma_{\rm v}^{2}\right)\right],$$ where $j$ is the number of datapoints, $V_{{\rm obs},i}$ and $V_{{\rm mod},i}$ are respectively the AB magnitude at the $i$th datapoint for the observation and the model, $\sigma_{{\rm obs},i}$ is the measurement error associated with the $i$th datapoint, and $\sigma_{\rm v}$ is the intrinsic variability of the source, assumed to be a constant that scales with black hole mass (see Section \[subsec:modeldesc\]).
Hydrodynamics of post-disruption debris {#sec:debris}
=======================================
Debris stream with self-gravity {#subsec:stream}
-------------------------------
Determining the fate of the various pieces of the star after a disruptive encounter is critical in determining the appearance of the flare that results from the immense gravitational energy that will be released by the accretion disk that eventually forms. Previously, it has been assumed that self-gravity of the disrupted star is unimportant, and therefore the spread in energy imparted to the debris at pericenter leads to a spread in angle as well as semi-major axis [@Strubbe:2009ek; @Kasen:2010ci]. Under this assumption, the unbound debris is a homologously expanding structure, which occupies a constant solid angle and whose volume increases proportional to $v_{\rm p}^{3}$.
![Snapshots from a tidal disruption simulation with , , and $\beta = 1.8$, as compared to a simple model of a tidally-confined debris stream with self-gravity, where we assume that the width of the stream scales as $r^{1/4}$ [@Kochanek:1994bn]. The left panel shows a superposition of the debris stream at different times, with the longest stream depicting the time when the most bound material returns to pericenter at $t = 42$ days. The color along the stream indicates whether it is bound or unbound from the SMBH, with magenta corresponding to bound and cyan corresponding to unbound. The green line shows the original path of the star, and the green circles show the locations of the surviving core corresponding to the eight snapshots shown on the right-hand side of the figure. In each of the right-hand panels, the simple model of the debris stream is shown atop the results from the simulation.[]{data-label="fig:thinstreams"}](thinstreams-eps-converted-to.pdf){width="0.85\linewidth"}
However, @Kochanek:1994bn showed that the stream can be in fact gravitationally confined in the transverse direction by self-gravity and forms a very thin structure (Figure \[fig:thinstreams\]), with a width $\Delta$ and height $H$ that scale as for $\gamma = 5/3$, where $\tilde{r} \equiv r/r_{\rm t}$. For general $\gamma$, $$H^{2} \Lambda^{\frac{2 - \gamma}{\gamma - 1}} \propto {\rm constant}$$ where $\Lambda$ is the mass per unit length [@Ostriker:1964cg], which we define to be $\Lambda = M_{\ast}/2R_{\ast}$ at $t = t_{\rm d}$. Assuming that $\Lambda \propto r^{-1}$ [@Kochanek:1994bn], $$H = R_{\ast} \tilde{r}^{\frac{2 - \gamma}{2 \gamma - 2}}\label{eq:height},$$ where we recover $H \propto \tilde{r}^{1/4}$ for $\gamma = 5/3$.
In the right eight panels of Figure \[fig:thinstreams\], we superimpose the results of our hydrodynamical simulations of the disruption of a star by a black hole with mass ratio $q = 10^{6}$ with this simple prescription. We find excellent agreement between the prediction of @Kochanek:1994bn and our results over the time period in which we ran the simulation. The thinness of the stream is also noted in other hydrodynamical simulations in which the mass ratio is large [@Rosswog:2009gg; @Hayasaki:2013kd].
The surface area of this structure for $\gamma = 5/3$ is $$\begin{aligned}
A_{\rm s} &= \frac{2\pi R_{\ast}^{2} q^{1/3}}{\beta} \int_{1}^{r_{\rm u}/r_{\rm p}} \tilde{r}^{1/4} d\tilde{r}\nonumber\\
&\simeq 1.4 \times 10^{-2} M_{6}^{1/3} \beta^{7/8}\left(\frac{t}{t_{\rm ff}}\right)^{5/4} {\rm AU}^{2},\end{aligned}$$ where $r_{\rm u} \simeq r_{\rm t} \beta^{1/2} (t/t_{\rm ff})$ is the distance to which the most unbound material has traveled [@Strubbe:2009ek], and $t_{\rm ff} \equiv \smash{\pi \sqrt{R_{\ast}^{3}/G M_{\ast}}}$ is the star’s free-fall time. At the peak time of ${\mathord{\sim}}100$ days, [PS1-10jh]{}emits ${\mathord{\sim}}10^{45}$ erg s$^{-1}$ of radiation with an effective temperature of a few $10^{4}$ K, implying a photosphere size of ${\mathord{\sim}}10^{15}$ cm with area ${\mathord{\sim}}10^{5}$ AU$^{2}$. By contrast, the area occupied by the stream is only comparable to this value when $t \approx 10^{5} t_{\rm ff} \approx 10$ yr for $q = 10^{6}$ and $\beta = 1$. @Kasen:2010ci calculated that this component would contribute at most $10^{40}$ ergs s$^{-1}$ of luminosity for the disruption of a solar mass star. However, we believe that this represents an upper limit as the self-gravity of the stream was not included in that work, resulting in an artificially fast rate of recombination.
Because the evolution of the stream is adiabatic, but not incompressible, the stream is resistant to gravitational collapse in both the radial direction perpendicular to the stream, and in the axial direction along the cylinder. Collapse can only occur in the radial direction when $\gamma < 1$, as it becomes energetically favorable to collapse radially [@McKee:2007bd], and can only occur axially for $\gamma > 2$, where the fastest growing mode has a non-zero wavelength and leads to fragmentation [Figure 4 of @Ostriker:1964cg; @Lee:2007em].
In a thin stream, the tidal force applied by the black hole results in the density $\rho$ scaling as $r^{-3}$ [@Kochanek:1994bn]. As the distance $r \propto t^{2/3}$ when $t \rightarrow \infty$ for parabolic orbits, this implies that $\rho \propto t^{-2}$. For cylinders, the time of free-fall $t_{\rm ff}$ is proportional to $\rho^{-1/2}$, the same as it is for spherical collapse [@Chandrasekhar:1961uk], and thus $t_{\rm ff} \propto t$. Therefore, a segment of the stream within which $t_{\rm ff}$ ever becomes greater than $t$ will not experience recollapse at any future time, as the two timescales differ from one another only by a multiplicative constant. It is also evident that self-gravitating cylinders can be bound whereas a self-bound sphere will not, as the Jeans length progressively decreases in size for structures that are initially confined in fewer directions [@Larson:1985to]. This implies that the cylindrical configuration may not remain self-bound if sufficient energy is injected into the star via a particularly deep encounter in which the core itself is violently shocked, which only occurs for $\beta \gtrsim 3$ [@Kobayashi:2004kq; @Guillochon:2009di; @Rosswog:2009gg], approximately 10% of disruption events. Additional energy can also be injected by nuclear burning [@Carter:1982wf; @Rosswog:2008gc], again only for events in which $\beta$ is significantly larger than 1. For most values of $\beta$, the amount of energy injected into the star at pericenter is insufficient to counteract gravitational confinement in the transverse direction at $t = 0$, and thus from the timescale argument given above the unbound stream would forever be confined.
Given this result, and given the computation burden of resolving a structure with such a large aspect ratio for the number of dynamical timescales necessary for the material to begin accreting onto the black hole, we presume that the gravitational confinement continues to hold at larger distances than we are capable of resolving, and show how the profile of the debris stream would appear if the simulation were followed to the point of the material’s return to pericenter in the left panel of Figure \[fig:thinstreams\].
By contrast, the bound material travels a much shorter distance from the black hole before turning around. For the material that remains bound to the hole, it has previously been assumed that the material circularizes quickly after returning to pericenter, resulting in an accretion disk with an outer radius equal to $2 r_{\rm p}$, where $r_{\rm p}$ is the pericenter distance [@Cannizzo:1990hw; @Ulmer:1999jja; @Gezari:2009dn; @Lodato:2010ic; @Strubbe:2011iw]. This is actually a vast underestimate of the distance to which the debris travels, which can be found via Kepler’s third law for the orbital period of a body and dividing by two to get the half-period, and then solving for the semi-major axis $a$, $$r_{\rm o} = 2\left(\frac{G M_{\rm h} t^{2}}{\pi^{2}}\right)^{1/3}\label{eq:ro},$$ where we have made the assumption that $r_{\rm o} \simeq 2 a$, appropriate for the highly elliptical orbits of the bound material (The most-bound material has eccentricity $e = 1 - 2q^{-1/3} = 0.98$ for $q = 10^{6}$). As this material is initially confined by its own gravity, the return of the stream to pericenter mimics a huge $\beta$ encounter, which as we explain in the following section can yield a impressive compression ratio.
{width="0.9\linewidth"}
Dissipative effects within the nozzle {#subsec:dissipative}
-------------------------------------
There are combinations of potentially active mechanisms that can provide the required dissipation for any given event, with each mechanism dominating for particular combinations of $M_{\rm h}$ and $\beta$. To quantify the effect of each of these mechanisms, we define the ratio ${{\cal V}}\equiv (\partial E / \partial t) (T / E)$, where $T$ is the orbital period. ${{\cal V}}$ represents the fraction of gravitational binding energy that is extracted per orbit, with ${{\cal V}}= 1$ indicating a mechanism that fully converts kinetic to internal energy within a single orbit. To have $\dot{M}$ and $L$ trace one another over the duration of a flare, ${{\cal V}}$ must have a value $\gtrsim t_{\rm m} / t_{\rm peak}$, where $t_{\rm peak}$ is the time at which the accretion rate peaks and $t_{\rm m}$ is the time at which the most bound debris returns to pericenter. For partially disruptive encounters, the ratio between these two times is $\sim 3$, but then increases as $\beta^{3}$ for deep encounters in which $t_{\rm m}$ varies more quickly than $t_{\rm peak}$ .
In the following sections we provide a brief description of the dissipation mechanisms that are expected to operate in a TDE. Only the first mechanism (hydrodynamical dissipation) is present within our calculations, as we do not include magnetic fields or the effects of a curved space time. Regardless of the origin of the dissipation, we expect that the dissipation observed in our simulations is likely to be quite analogous to the other dissipative process that operate.
### Hydrodynamical dissipation
As described in Section \[subsec:stream\], self-gravity within the debris stream sets its width and height to be equal to $R_{\ast} \tilde{r}^{1/4}$, and thus when the stream crosses the original tidal radius, its height is approximately equal to the size of the original star. If the return of the material to pericenter behaved in the same way as the original encounter, the maximum collapse velocity $v_{\perp}$ would be equal to the sound speed at $r_{\rm t}$ multiplied by $\beta$, yielding a dissipation per orbit ${{{\cal V}}} = q^{-2/3}/\beta$, equal to 2% for $q = 10^{3}$ and $\beta = 2$ [@Carter:1983tz; @Stone:2013gk]. In our hydrodynamical simulations for $q = 10^{3}$, we find that $\approx 10\%$ of the total kinetic energy of the debris is dissipated upon its return to pericenter through strong compression at the nozzle (Figure \[fig:columndensity\]). This is a factor of a few larger than the expected dissipation.
However, as the star has been stretched tremendously, the sound speed within the stream has dropped by a significant factor, meaning that the distance from the black hole at which the stream’s sound-crossing time is comparable to the orbital time (i.e., where the tidal and pressure forces are approximately in balance) is not the star’s original tidal radius, but is instead somewhat further away.
For two points that are separated by a distance $dr$ within the original star, their new distance $dr^{\prime}$ upon returning to pericenter is related to the difference in binding energy between them, which remains constant after the encounter and is equal to $$\frac{dE}{dr} \simeq \frac{E(r_{\rm p})}{r_{\rm p}} = \frac{G M_\ast}{R_{\ast}^{2}} q^{1/3}\label{eq:deg}$$ As angular momentum is approximately conserved, the two points will cross pericenter at the same location they originally crossed pericenter, but at two different times $t$ separated by a time $dt$ owing to their different orbital energies. Assuming that the star originally had approached on a parabolic orbit, and that all the bound debris are on highly elliptical orbits, the distance from the black hole is $$r' = \left(\frac{9}{2} G M_{\rm h} t^{2}\right)^{1/3},\label{eq:rprime}$$ and thus $$\frac{dr^{\prime}}{dt} = \left(\frac{4 G M_{\rm h}}{3 t}\right)^{1/3}.\label{eq:drprimedt}$$ If we set $t = 0$ to be the time when the first point re-crosses pericenter, the difference in time is simply the difference in orbital period, and thus we can use Kepler’s third law to derive $dE/dt$. Using Equations \[eq:deg\] and \[eq:drprimedt\], we can use the chain rule to determine $dr^{\prime}/dr$, $$\frac{dr^{\prime}}{dr} = \frac{dE}{dr}\frac{dt}{dE}\frac{dr^{\prime}}{dt} = \left(\frac{3\pi}{2}\right)^{2/3} \left(\frac{t}{t_{\rm m}}\right)^{4/3} q^{2/3}.\label{eq:drdrprime}$$ For $q = 10^{3}$, this implies that the material has been stretched by a factor $\gtrsim 10^{2}$ after the most-bound material begins accreting, and for $q = 10^{6}$ this factor is $\gtrsim 10^{4}$. The change in volume is then given by the change in cross-section of the stream multiplied by the change in length given by Equation \[eq:drdrprime\], $$\frac{dV^{\prime}}{dV} = \frac{dr^{\prime}}{dr} \tilde{r}^{\frac{2-\gamma}{\gamma-1}}$$ where we have used Equation \[eq:height\] to estimate the width and height of the stream.
The density of the stream $\rho$ as it returns to pericenter can be approximated by assuming that $dM/dr = \Lambda$, although in reality this distribution can be determined more exactly from the numerical determination of ${\dot{M}}$ by a change of variables from $E$ to $r$. Under this assumption, the change in density is simply related to the change in volume alone. As the tidal radius is proportional to $\rho^{1/3}$, the ratio of the effective tidal radius of the stream $r_{\rm t,s}$ to $r_{\rm t}$ is then $$\begin{aligned}
\frac{r_{\rm t,s}}{r_{\rm t}} &= \left(\frac{dV^{\prime}}{dV}\right)^{1/3}\nonumber\\
&= \left(\frac{3\pi}{2} q \right)^{\frac{2}{3}\frac{\gamma-1}{4\gamma-5}} \left(\frac{t}{t_{\rm m}}\right)^{\frac{4}{3}\frac{\gamma-1}{4\gamma-5}} \equiv \frac{\beta_{\rm s}}{\beta}\label{eq:rtstream}\end{aligned}$$ where we have substituted $r_{\rm t,s}/r_{\rm t}$ for $\tilde{r}$, and where we have presumed that the time until the stream reaches pericenter from $r_{\rm t,s}$ is small compared to the time since disruption.
Under the assumption that the stream expands adiabatically and its pressure is governed by ideal gas pressure, this results in a reduction in the sound speed $c_{\rm s} = \sqrt{dP/d\rho} \propto V^{(1 - \gamma)/2}$, where $\gamma$ is the adiabatic index of the fluid. At $r = r_{\rm t,s}$, the ratio of the sound speed within the stream $c_{\rm s,s}$ to the star’s original sound speed $c_{\rm s,\ast}$ is $$\begin{aligned}
\frac{c_{\rm s,s}}{c_{\rm s,\ast}} &= \left(\frac{dV^{\prime}}{dV}\right)^{\frac{1 - \gamma}{2}}\nonumber\\
&= \left(\frac{3 \pi}{2} q\right)^{\frac{(\gamma-1)^{2}}{5-4\gamma}} \left(\frac{t}{t_{\rm m}}\right)^{\frac{2(\gamma-1)^{2}}{5-4\gamma}}\label{eq:csstream}.\end{aligned}$$
Analogous to the original star, the maximum collapse velocity of the stream $v_{\perp, {\rm s}}$ is equal to the sound speed at $r_{\rm t,s}$ multiplied by the stream’s impact parameter $\beta_{\rm s} \equiv \beta r_{\rm t,s} / r_{\rm t}$, $v_{\perp} = \beta_{\rm s} c_{\rm s,s}$. As the majority of the dissipation comes through the conversion of the kinetic energy of the vertical collapse via shocks, the fractional change in the specific internal energy is $$\begin{aligned}
{{\cal V}}_{\rm H} &= \frac{\beta_{\rm s}^{2} c_{\rm s,s}^{2}}{v_{\rm p}^{2} }\nonumber\\
&= \left(\frac{3\pi}{2} q\right)^{-\frac{2}{3}\frac{(\gamma - 1)(3\gamma-5)}{4\gamma-5}} \left(\frac{t}{t_{\rm m}}\right)^{-\frac{4}{3}\frac{(\gamma - 1)(3\gamma-5)}{4\gamma-5}} \beta q^{-2/3}.\label{eq:vhydro}\end{aligned}$$
As $r_{\rm t} \propto \rho^{1/3}$, and $c_{\rm s} \propto \rho^{-1/3}$ for $\gamma = 5/3$, Equations \[eq:rtstream\] and \[eq:csstream\] are inverses of one another in the adiabatic case, $$\frac{r_{\rm t,s}}{r_{\rm t}} = \frac{c_{\rm s,\ast}}{c_{\rm s,s}} = 60 M_{6}^{4/15} \left(\frac{t}{t_{\rm m}}\right)^{8/15}$$ and thus \[eq:vhydro\] simplifies to ${{\cal V}}_{\rm H} = \beta q^{-2/3}$, identical to the amount of dissipation experienced by the original star. One key difference exists between the original encounter and the stream’s return to pericenter: While the original encounter may only result in the partial shock-heating of the star, even for relatively deep $\beta$ [@Guillochon:2009di], the fact that the collapse of the stream is highly supersonic ($\beta_{\rm s} \sim 60 \beta$) guarantees that shock-heating will occur upon the material’s return to pericenter.
For our $q = 10^{3}$ simulation, the amount of dissipation expected per orbit predicted by Equation \[eq:vhydro\] is $4 \times 10^{-2}$, and for our $q = 10^{6}$ simulation the expected dissipation would only be $4 \times 10^{-4}$. Thus, the conversion of kinetic energy to internal energy via shocks at the nozzle point is inefficient for all but the lowest mass ratios and/or the largest impact parameters, and would be incapable of circularizing material on a timescale that is shorter than the peak timescale of [PS1-10jh]{}(Figure \[fig:dissipation\]). This suggests that a viscous mechanism that involves an unresolved hydrodynamical instability, or a mechanism that is beyond pure hydrodynamics, is responsible for the circularization of the material for this event.
An additional complication that is not addressed here is recombination. As the stream expands, its internal temperature drops below the point at which hydrogen begins to recombine, flooring its temperature to ${\mathord{\sim}}10^{4}$ K until most of the hydrogen is neutral [@Roos:1992cj; @Kochanek:1994bn]. This implies that the ratio between the initial and final sound speeds is somewhat smaller than when assuming adiabaticity holds to arbitrarily-low stream densities, and depends on the initial temperature of the fluid, which is $\sim 10^{4}$ K in the outer layers of the Sun, but ${\mathord{\sim}}10^{7}$ K in its core. This also causes the stream to expand somewhat due to the release of latent heat. However, as the material returns to pericenter, the compression of the material will reionize it. Given these complications, it is unclear if this process would lead to more or less dissipation at the nozzle.
![Fraction of binding energy dissipated at $t = t_{\rm peak}$ for three mechanisms that may contribute to the circularization of material after a tidal disruption, with mint corresponding to hydrodynamical shocks at the nozzle point, light blue corresponding to dissipation through GR precession (presuming $\gamma = 5/3$), and white corresponding to the MRI mechanism. For each mechanism, three contours of ${{\cal V}}$ are shown, with solid corresponding to 100%, dashed corresponding to 10%, and dotted corresponding to 1%. If all three mechanisms operate, the shaded blue regions represent zones in which ${{\cal V}}$ adopts the values specified by the unions of the regions enclosed by the three sets of contours, with the lightest/darkest corresponding to the least/most dissipation. For reference, our two hydrodynamical simulations are shown by the cyan triangle and the red hexagon, and the highest-likelihood fit returned by our MLA (Section \[subsec:generalmodel\]) is shown by the magenta square.[]{data-label="fig:dissipation"}](dissipation-eps-converted-to.pdf){width="\linewidth"}
### Dissipation through General Relativistic Precession
For orbits in which the pericenter is comparable to the Schwarzschild radius $r_{\rm g}$, the orbital trajectory begins to deviate from elliptical due to precession induced by the curved space-time. The precession time in the inner part of the disk is [@Valsecchi:2012gw]: $$\dot{\gamma}_{\rm GR} = \left(\frac{2\pi}{T}\right)^{5/3} \frac{3 G^{2/3}}{c^{2}} \frac{M_{\rm h}^{2/3}}{1-e^{2}},\label{eq:prec}$$ where $T$ and $e$ are respectively the period and eccentricity of the stream. As the debris resulting from a tidal disruption has a range of pericenter distances ($r_{\rm p} \pm R_{\ast}$), there is a gradient in precession times of the returning debris. This precession causes the orbits to cross one another, dissipating energy [@Eracleous:1995fv]. When compared to the standard $\alpha$ viscosity prescription, the timescale of this precession is comparable to the viscous time, $$t_{\rm prec} = 10^{-1.7} M_{6} T_{5}^{-1/2}\left(1+e\right)\left(\frac{r_{\rm p}}{r_{\rm g}}\right)^{2}\;{\rm yr},$$ where $T_{5} \equiv 10^{5} T$ is the local disk temperature. In a tidal disruption, the most-bound material is also the material with the shortest precession time, and it is this timescale that sets the overall rate of dissipation. By setting $T$ and $e$ in Equation \[eq:prec\] to $t_{\rm m} \equiv \sqrt{q/2} \beta^{-3} t_{\rm ff}$ and $e_{\rm m} \equiv 1 - 2 \beta q^{-1/3}$, the period and eccentricity of the most-bound material, and by assuming that precession through an angle $2 \pi$ would lead to complete dissipation, the dissipation due to relativistic precession for the material that corresponds to the peak in the accretion rate is $${{\cal V}}_{\rm GR, peak} = \frac{t_{\rm peak}}{t_{\rm m}} \left(\frac{2\pi}{t_{\rm m}}\right)^{2/3} \frac{3 G^{2/3}}{c^{2}} \frac{M_{\rm h}^{2/3}}{1-e_{\rm m}^{2}}.\label{eq:vgr}$$
In general, the period of the most-bound material tends to smaller values for larger $q$ and $\beta$, resulting in more dissipation, except in the case that $r_{\rm p}$ and $R_{\ast}$ are comparable (Figure \[fig:dissipation\], cyan curves). If none of the other dissipation mechanisms are effective, this means that disruptions by massive black holes, in which $r_{\rm p}$ and $r_{\rm g}$ are closer to one another in value, would be the only cases in which ${\dot{M}}$ and $L$ follow one another closely. As we will show in Section \[sec:ps1-10jh\], the highest-likelihood models of [PS1-10jh]{}seem to be consistent with a relativistic encounter with the central black hole, so it is possible that [PS1-10jh]{}’s identification as a TDE was contingent upon the condition that $r_{\rm p} \sim r_{\rm g}$.
### Super-Keplerian, Compressive MRI
The magnetic field in the bound stellar debris is likely to be amplified by compression and by strong shearing within the nozzle region. The effects of magnetic shearing, the magneto-rotational instability [MRI; @Balbus:1998tw], is expected to lead to the rapid exponential growth of the magnetic field with a characteristic timescale of order the rotational period. This instability has been routinely studied in the context of accretion disks and we argue here that it is likely to operate within the nozzle. However, given that the fluid’s motion is super-Keplerian at pericenter (being on near-parabolic orbits), its exact character is difficult to compare directly to the classical MRI, as the boundary conditions are constantly changing and the material is never in steady-state.
The MRI is present in a weakly magnetized, rotating fluid wherever $${d\Omega^2 \over d \ln r}<0.$$ The ensuing growth of the field is exponential with a characteristic time scale given by $t_{\rm MRI}= 4\pi |d\Omega/d\ln r|^{-1} $ [@Balbus:1998tw]. For a (super-)Keplerian angular velocity distribution $\Omega \propto r^{-3/2}$, this gives $t_{\rm MRI}=(4/3)\Omega^{-1}$. Exponential growth of the field on the timescale $\Omega^{-1}$ by the MRI is likely to dominate over other amplification process such as field compression within the same characteristic time. While a variety of accretion efficiencies are reported in numerical realizations of magnetically-driven accretion disks, which depend on the geometry, dimensionality of the simulation, and included physics, essentially all models find that the strength of the magnetic field amplifies to the point that it is capable of converting fluid motion into internal energy. From global simulations of the MRI, the build-up of the magnetic field strength is confirmed to be exponential, resulting in a time to complete saturation being a constant multiple of the orbital period. In @Stone:1996hk, this constant is found to be 3. Once the magnetic field strength is saturated, the resulting angular momentum transport will be governed by the turbulence and is therefore expected to take place over longer timescales. A simple estimate of the saturation field can be obtained by equating the characteristic mode scale, $\sim v_{\rm A} (d\Omega/d\ln r)^{-1}$, where $v_{\rm A}$ is the Alfvén velocity, to the shearing length scale, $\sim dr/d\ln \Omega$, such that . This saturation field is achieved after turbulence is fully developed, which in numerical simulations takes about a few tens of rotations following the initial exponential growth [@Hawley:1996gh; @Stone:1996hk].
For the Sun, the initial interior magnetic field energy at the base of the convective zone $E_{B,0} \sim 10^{-10} E_{\rm g}$ [@Miesch:2009kb], although larger initial fields are possible in general [@Durney:1993hq]. As the tidal forces stretch the star into a long stream, the volume of the fluid increases by a factor $\beta_{\rm s}^{3}$ (Equation \[eq:rtstream\]) prior to returning to pericenter, reducing the magnetic field strength further. However, when the stream returns to pericenter, it experiences a dramatic decrease in volume by a factor $\smash{\beta_{\rm s}^{2/(\gamma-1)}}$ [@Luminet:1986ch]. Assuming the frozen flux approximation, the new magnetic energy density is $$\begin{aligned}
E_{B} &= E_{B,0} \beta_{\rm s}^{\frac{3\gamma - 5}{\gamma - 1}}\\
&= \left(\frac{3\pi}{2} q \right)^{\frac{2}{3}\frac{3\gamma - 5}{4\gamma - 5}} \left(\frac{t}{t_{\rm m}}\right)^{\frac{4}{3}\frac{3\gamma - 5}{4\gamma - 5}}\end{aligned}$$ For $\gamma = 5/3$, the dependence on $\beta_{\rm s}$ disappears, i.e. the magnetic field strength upon return to pericenter is identical to the star’s initial interior field. Assuming that the magnetic field within the debris has strength $E_{B}$ relative to the local gravitational binding energy $E_{\rm g}$ upon returning to the nozzle, ${{\cal V}}_{\rm MRI}$ adopts a simple form (Figure \[fig:dissipation\], white curves), $${{\cal V}}_{\rm MRI, peak} = \frac{E_{B}}{E_{\rm g}} \exp \left[\frac{t_{\rm peak}}{3 t_{\rm m}}\right]\label{eq:vmri}.$$
Is the Debris Disk Dissipative Enough? {#subsec:disenough}
--------------------------------------
In order for the emergent luminosity $L$ to follow the feeding rate ${\dot{M}}$ closely, the dissipation must be effective enough such that material returning to pericenter can circularize on a timescale $t_{\rm c}$ that is at most the time since disruption $t_{\rm d}$.
In the original calculation of @Cannizzo:1990hw, the initial conditions place a fixed amount of a mass a fixed distance from the black hole at $t = 0$. The matter is then allowed to evolve viscously, resulting in the transport of mass inwards, and the transfer of angular momentum outwards. While this initial condition is acceptable for fallback calculations onto a newly-formed neutron star in a long GRB [@Lee:2006kp; @Kumar:2008fa; @Cannizzo:2009gm; @LopezCamara:2009iq; @Milosavljevic:2012ci] and for a compact binary merger [@Lee:2004fn; @Metzger:2008hp; @Lee:2009iy; @Metzger:2009fu], it is likely not a perfect analogue for a tidal disruption of a star originally on a parabolic orbit, as it neglects the continuous injection of energy from the returning stream.
As material circularizes, it must deposit $(\sqrt{2} - 1)^2 v_{\rm K}^2$ of kinetic energy within a few orbital periods, as it has to slow down from a near-escape velocity to the local Keplerian velocity $v_{\rm K}$. If the circularization is rapid, this additional source of heating leads to an $H/R \sim 1$ and consequently rapid accretion. For rapid accretion, the accretion disk mass remains small, on order $\dot{M}(t) t_{\rm c}$. The total mass accreting onto the black hole via the stream at any one time samples a later segment of the fallback curve, offset by a time $t_{\rm c}$, $\dot{M}(t + t_{\rm c}) t_{\rm c}$. At early times, this mass is always larger than the amount of mass in the disk, as $\dot{M}$ is increasing rapidly with time. This is very similar to the argument made below equation 34 in @Kumar:2008fa for the direct fallback phase of a GRB. At late times, this mass a bit less than the mass in the disk, but not considerably so as $t_{\rm c} \ll t_{\rm fb}$.
So long as the returning material in the debris stream has a comparable mass to the mass present in the accretion disk and circularization is rapid, the timescale for accretion can remain short over the full duration of the flare. If however the returning debris is unable to circularize quickly at some point in the flare’s evolution, matter will build up in a disk with $H/R \ll 1$, with the resulting disk mass being somewhat larger than the incoming mass. This would effectively “erase” the incoming $\dot{M}$’s functional form, and instead result in a fallback rate with a much shallower slope, between $-1$ and $-4/3$ [@Cannizzo:1990hw]. Evidence for such a transition may have been seen in Swift-J1644 [@Cannizzo:2011df]. However, the time of the transition is likely to occur at very late times for an encounter with $\beta \sim 1$, as suggested by our most-likely solutions (see Section \[subsec:generalmodel\]). Using Equation 21 from @Cannizzo:2011df we find that this transition would occur at $\sim 10^{3}$ yr for these parameters, well beyond the time at which [PS1-10jh]{}’s flux dropped below that of its host galaxy.
In Figure \[fig:dissipation\] we show the three sources of dissipation that we estimated above. We find that while significant dissipation is expected for large $\beta$ encounters, or for encounters in which $r_{\rm p} \sim r_{\rm g}$ (as is the case for our best-fitting model for [PS1-10jh]{}), that there are many combinations of $\beta$ and $q$ that may not have the required dissipation necessary to ensure the direct mapping in time of ${\dot{M}}$ to $L$. In our $q = 10^{3}$ hydrodynamical simulation, we found somewhat more dissipation than what is expected from a simple analytical calculation, but the resolution at which we resolved the compression at pericenter was only marginally sufficient to resolve the strong shocks that form there.
One potential resolution to this issue is the adiabatic index of the fluid $\gamma$, which in the above calculations we have assumed $ = 5/3$, although the real equation of state within the stream is likely softer due to the influence of recombination. With a softer equation of state, the cancelations that occur for $\gamma = 5/3$ and eliminate the dependence on $\beta_{\rm s}$ for the hydrodynamical (Equation \[eq:vhydro\]) and MRI (Equation \[eq:vmri\]) dissipation mechanisms would no longer apply, yielding both increased compression and magnetic field strengths, and thus additional dissipation.
While the initial dissipation of the stream may indeed come as the combination of the three previously described mechanisms, it is likely that the mechanism responsible for the accretion onto the black hole once the material has been assembled into a disk is the MRI mechanism, as is suspected for steadily-accreting AGN. Given the computational challenge of simultaneously resolving the nozzle region and the full debris stream, it is clear that local high-resolution magnetohydrodynamic simulations are required to determine the true dissipation rate ${{\cal V}}$ at the nozzle.
The Relationship Between Steadily-Accreting AGN and TDE Debris Disks {#sec:steady}
====================================================================
Within the debris structure formed from a tidal disruption, the same mechanisms that operate in steady-state AGN may continue to operate. There are a number of differences between the structure of a debris disk resulting from tidal disruption and the structure of steadily-accreting AGN, but we will argue that similar processes are responsible for the appearance of both structures. In this section, we will make continued reference to the highest-likelihood model of [PS1-10jh]{}, which is determined in Section \[sec:model\].
The Conversion of Mass to Light
-------------------------------
For steadily-accreting AGN, energy is thought to be released by the viscous MRI process at all radii. The amount of energy available at a particular radius depends on the local gravitational potential, and thus the vast majority of the energy emitted by accreting black holes is produced within a few times the Schwarzschild radius $r_{\rm g}$. The temperature profile that results from this release in energy within the accretion disk is given by the well-known expression first presented in @LyndenBell:1969dq, and scales as $r^{-3/4}$, resulting in a sum of blackbodies with a continuum slope $F_{\nu} \propto \nu^{1/3}$ [@Pringle:1972vb; @Gaskell:2008wx]. AGN are divided into two fundamental categories [@Antonucci:2012vl]: Compton-thick (i.e. non-thermal) AGN, which are obscured by ${\mathord{\sim}}10^{24}$ cm$^{2}$ of column and thus making them optically thick to Compton scattering [@Treister:2009bk], and thermal AGN, which have column densities significantly less than this value, enabling the black hole’s emission to be directly observed. However, all thermal AGN show an excess in the blue known as the “big blue bump” [@Shields:1978ce; @Lawrence:2012hn], and the slopes of their continuum $F_{\nu} \propto \nu^{-1}$ [@Gaskell:2009dc]. This is more consistent with the notion that the light emitted from the central parts of the disk is intercepted by intervening gas before it is observed, with nearly one hundred percent of the light emitted by the disk being reprocessed in this way. This implies that a significant fraction of the mass that may eventually be accreted by the black hole is suspended some distance above the disk plane, where it can intercept a large fraction of the outgoing light.
{width="0.9\linewidth"}
For the accretion structure that forms from the debris of a tidal disruption, the dissipation at the nozzle point provides a means for lifting material above and below the orbital plane, resulting in a sheath of material that surrounds the debris and is very optically thick for certain lines of sight (Figure \[fig:columndensity\]). However, as the spread in energy at the nozzle point does not completely virialize the flow, the resulting distribution of matter is flattened, allowing the central regions of the accretion disk to be visible through material that is close to the Compton-thick limit. The time-series presented in the upper six panels of Figure \[fig:columndensity\] show that despite the continually active dissipation process at the nozzle-point, the region directly above the central parts of the accretion disk remain relatively evacuated of gas before, during, and after the time of peak accretion. For the toy simulation presented here, the optical depth to Thomson scattering directly above the black hole and perpendicular to the orbital plane of the debris is $\sim 1$, depending on the electron fraction of debris (assumed to be pure hydrogen in Figure \[fig:columndensity\]). For more massive black holes, the debris is spread over a larger volume, as the tidal radius grows as , and thus . If the dissipation rate were the same independent of black hole mass, it would be expected that the disruption of stars by more massive black holes would yield more lines of sight for which $\tau \sim 1$.
Source of Broad Emission Lines {#subsec:blrsource}
------------------------------
Broad line emission is visible in many AGN, being thought to be produced by gas above and below the disk plane at distances from light hours to light years away from the black hole. For other AGN, this region is not directly observable, which has been attributed to a torus at large radii that can obscure the broad line region for lines of sight that run within a few tens of degrees of the disk plane [i.e. the AGN unification model, @Antonucci:1993fe]. The emission lines produced within this region have been successfully used to measure black hole masses [@Dibai:1977tr; @Peterson:2004ig; @Marziani:2012gj] based on measurements of the time lag in the response of line luminosity to variations in the output of the central engine [see e.g. @Denney:2009gm]. It is still debated whether this material is in the form of an optically-thick disk wind [@Trump:2011bp] or optically-thick clouds [@Celotti:1999tm], but in either case the material that constitutes the BLR is mostly bound to the black hole [@Proga:2008bz; @Pancoast:2012dd].
In a steady-accreting AGN, material accretes from very large distances (), and the emission from this region is often manifest as an IR bump in Type II AGN [@Koratkar:1999di]. At such distances, the ionizing flux originating from the black hole is not sufficient to maintain a large ion fraction within the disk’s emitting layer. The closer one gets to the central black hole, the greater the incident flux of ionizing radiation on the BLR wind/clouds that generate the observed emission lines. The fraction of atoms in an excited state X$_{+}$ relative to the state directly below it X$_{0}$ is approximately [@Osterbrock:2006ul] $$\frac{n_{{\rm X}_{+}}}{n_{{\rm X}_{0}}} \sim \frac{a(\nu_{\rm ion})}{\alpha_{\rm B}({\rm X}_{0}) h \nu_{\rm ion}}\frac{Q({\rm X}_{0})}{4\pi r^{2} n_{\rm e}},$$ where $Q({\rm X}_{0})$ is the flux in photons capable of ionizing the lower state. This expression shows that that as the distance from the central engine increases, the number of atoms in the high state decreases, assuming that the electron density $n_{\rm e}$ decreases with radius more slowly than $r^{-4}$ (as $Q \propto r^{-2})$, and also shows that species with larger ionization potentials will have less atoms in the high state than species with smaller ionization potentials. This leads to a hierarchy of ions in the disk, with those with the highest ionization potential being predominant in the inner regions of the disk. In a steadily-accreting AGN, the flux in ionizing photons is large enough to fully ionize iron [as evidenced by the existence of Fe K lines, @Fabian:2000hr], and given that atoms at large radii are mostly neutral, all ionic species of all elements exist at some distance from the central engine. Reverberation mapping supports this basic photoionization picture, as $R_{\rm BLR} \sim L^{1/2}$ [@Bentz:2010hk; @Bentz:2013hm]. In particular, the optical wave band hosts several lines from the Balmer series of hydrogen and lines from both singly-ionized and neutral helium [@Bentz:2010hk].
This wide range of scales is in stark contrast to the debris disk formed as the result of a tidal disruption, which we schematically illustrate in Figure \[fig:schematic\]. Rather than material spiraling in from parsec scales, material is instead ejected from the nozzle point, which lies at the star’s original point of closest approach, and typically has scales on the order of a few AU. As a result, the debris disk forms from the inside out. The ratio of line strengths in a TDE is dependent upon the number of atoms in the photosphere that are in the particular ionization state associated with each line. For [PS1-10jh]{}, the lack of an [[H]{}$\alpha$]{}emission line was interpreted by as being attributed to a lack of hydrogen atoms. However, the Balmer series requires neutral hydrogen to be present in sufficient quantities to produce a line in excess of the continuum emission. As shown in our $q = 10^{3}$ simulation, material is ejected from the nozzle point at approximately the escape velocity, with the fastest moving material traversing a distance $r_{\rm t} [(t-t_{\rm d})/t_{\rm p}]^{2/3}$. [*This sets an upper limit on the radial extent of the disk*]{}. Therefore, the lack of an observed emission feature may simply be the result of the disk not being large enough to host the region required for that particular feature’s production.
The specifics as to which particular radii contribute the most to the emission strength of each line is complicated to determine, and requires a more-through treatment of the ionization state of the gas as a function of radius, which depends on the geometry of the structure, and the distribution of density and temperature as functions of height and radius. In a tidal disruption, the matter distribution that ensheathes the black hole is established quickly, forming a steady-state structure that is supported by a combination of gas pressure and angular momentum [@Loeb:1997jv]. Accretion then proceeds through the midplane, in which the majority of light is generated within a few $r_{\rm g}$ at X-ray temperatures. These photons are intercepted by the ensheathing material at higher latitudes. @Korista:2004cp determined the equivalent widths of various lines as functions of volume density and ionizing flux, which is not expected to vary much as a function of column density for $10^{23} \leq N \leq 10^{25}$ cm$^{2}$ [@Ruff:2012tq]. In Figure \[fig:korista\] we show a series of density-ionization curves corresponding to our highest-likelihood model for [PS1-10jh]{}over the range of times at which spectra were taken of this event. These are compared to the equivalent widths measured To calculate the density distribution $n({\rm H})$ as a function of $r$, we once again use the chain rule, $$\begin{aligned}
n({\rm H}) &= \frac{X({\rm H})}{4\pi m_{\rm p} r^{2}} \frac{dM}{dE}\frac{dE}{da} \left(\frac{2 a}{r_{\rm p}}\right)^{3/2}\label{eq:nh}\end{aligned}$$ where $X({\rm H})$ is the mass fraction of hydrogen. and presuming that the radial distribution of mass is determined by the distribution of mass with semi-major axis $a$, $dM/da$, set at the time of disruption. Likewise, $dM/dr$ is directly proportional to $dM/da$, with a scaling factor equal to the ratio of time spent at apocenter versus pericenter, , where we have assumed that $1 - e \rightarrow 0$ and thus the apocenter distance $r_{\rm a} \simeq 2 a$.
As a strong dissipation mechanism likely operates at the nozzle point, and this dissipation mechanism is likely to be as dissipative as the commonly invoked MRI mechanism, it stands to reason that the vertical structure of the debris disk formed through the circularization process is similar to that of a steadily-accreting AGN. Therefore, we would expect that the BLR associated with such structures should be similar to the BLR produced by steadily-accreting black holes. Under this assumption, we can use Equation \[eq:nh\] to approximate the number density of hydrogen as a function of radius and to determine the equivalent width of various emission lines using the models that have been generated for steadily accreting AGN [@Korista:2004cp]. Figure \[fig:korista\] shows the density-ionizations curves calculated from Equation \[eq:nh\] as a function of time for our highest-likelihood model of [PS1-10jh]{}, with the purple curve corresponding to the time of the first acquired spectrum of [PS1-10jh]{}at -22 days, and the red curve corresponding to the last recorded spectrum at +358 days.
From Figure \[fig:korista\], it is clear that the equivalent width of [ $\lambda$4686]{}is significantly larger than that of [[H]{}$\alpha$]{}, [[H]{}$\beta$]{}, and [ $\lambda$5876]{}, all three of which are not observed in [PS1-10jh]{}. The figure does suggest that hydrogen and/or singly-ionized helium emission lines may appear at later times when the ionizing flux has decreased, although this may not ever be observable in [PS1-10jh]{}where the flux originating from the TDE has already dropped below that of the host galaxy.
![Contours of the log of the equivalent width of four emission lines as a function of hydrogen–ionizing flux $\Phi({\rm H})$ and hydrogen number density $n({\rm H})$, where the black dashed and solid contours correspond to 0.1 and 1 decade, respectively [Adapted from Figure 1 of @Korista:2004cp], with the smallest contour corresponding to of equivalent width. The colored triangle within each panel indicates the peak equivalent width for each line. The rainbow-colored curves show the profiles of the debris in the $\Phi({\rm H})-n({\rm H})$ plane resulting from the tidal disruption that corresponds to the highest-likelihood fit of [PS1-10jh]{}(Section \[subsec:generalmodel\]). The curves span the full duration of the event, with the solid curves corresponding to the range between the first and last spectrum taken for the event (with purple being -22 and red being +358 days from peak), and the dashed curves corresponding to unobserved epochs before/after the spectral coverage. The black dotted curve shows the conditions at $r_{\rm o}$ as a function of time over the full event duration. Note that [[H]{}$\alpha$]{}, [[H]{}$\beta$]{}, and [ $\lambda$5876]{}would potentially be observable if additional spectra were collected at later times.[]{data-label="fig:korista"}](PS1-10jh-new_korista-line-locations-eps-converted-to.pdf){width="\linewidth"}
In generating this plot, we have made some assumptions that actually would lead to a [*decrease*]{} in the strength of the unobserved lines if we performed a more-detailed calculation. Firstly, the models of @Korista:2004cp presume that a full annulus of locally optimally emitting clouds exists at each radius; this is not the case in an elliptical accretion disks where the inner annuli are closer to full circles than outer annuli [@Eracleous:1995fv]. In fact, it is unlikely that the outer material can circularize at all, given that there is significantly less angular momentum in the disk than the angular momentum required to support a circular orbit at the distance at which these lines would be produced (at $r = 10^{16}$ cm, ${\mathord{\sim}}30$ times more angular momentum would be required to form a circular orbit than what is available at $r_{\rm p}$). Secondly, we have made the assumption that the material that does the reprocessing remains at the distance determined by the energy distribution set at the time of disruption [*at all times*]{} [à la @Loeb:1997jv], when in reality the entire debris structure will shrink onto the black hole due to dissipation at pericenter. It is possible that this shrinkage of the debris could prevent emission features arising from species with lower ionization potentials from ever being observed.
Radiation pressure (which we ignore in this work) may act to push some fraction of the material outwards, which in principle could produce low-energy emission features [@Strubbe:2009ek]. However, our highest-likelihood models predict a peak accretion rate that is sub-Eddington, and thus only a small fraction of the accreted matter is expected to be driven to large distances via radiation. It is unclear whether the amount of mass in this component would be dense enough to produce these features, as the recombination time may be too long.
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A Generalized Model for the Observational Signatures of TDEs {#sec:model}
============================================================
As emphasized in the previous sections, there are many uncertainties relating to how the material circularized when it returns to pericenter, how this returning material radiates its energy when it falls deeper into the black hole’s potential well, and in the ionization state of the gas within the debris superstructure. Using the code [[TDEFit]{}]{}, developed for this paper, we construct a generalized model of the resultant emission from TDEs. In this section, we describe the results of running this fitting procedure, and how [PS1-10jh]{}specifically allows us to evaluate some of the other models that have been proposed for modeling TDEs.
Model description and free parameters {#subsec:modeldesc}
-------------------------------------
Our generalized model for matching TDEs is one in which an accretion disk forms by the disruption of a star of mass $M_{\ast}$ by a black hole of mass $M_{\rm h}$ with impact parameter $\beta$ and offset time $t_{\rm off} \equiv t_{0} - t_{\rm d}$, where $t_{0}$ is the time [PS1-10jh]{}was first detected in Pg (May 10.55, 2010). This disk spreads both inwards and outwards from $r_{\rm p}$, and is ensheathed by a diffuse layer of material that intercepts some fraction of the light. The disk itself is bounded by an inner radius $r_{\rm i}$ and outer radius $r_{\rm o}$, with $r_{\rm i}$ assumed to be set by the viscous evolution of the material, and $r_{\rm o}$ being set by the ballistic ejection of material as it leaves the nozzle region, which scales as , where $t_{\rm m}$ is the time of return of the most-bound material. The fraction of the full annulus $\theta_{\rm f}$ that is covered by the disk varies as a function of time, with $\theta_{\rm f} = 0$ when $r = r_{\rm o}$, and $= 2 \pi$ when $t = t_{\rm visc} \left(r\right)$, assuming its spread in the azimuthal direction is controlled by the local value of the viscosity. The model is shown pictographically in the left panel of Figure \[fig:threed\], with the aforementioned size scales as functions of time being shown in the right panel of the same figure.
The source of this viscosity may be similar to the source of viscosity at the nozzle point (see Section \[subsec:dissipative\]), or it could be the result of the stream-stream collision that occurs when material reaches apocenter [@Kochanek:1994bn; @Kim:1999dw; @RamirezRuiz:2009gw]. For simplicity, we assume that the same viscous process, parameterized by the free parameter ${{\cal V}}$, applies in both regions. We presume that ${{\cal V}}$ is time-independent, resulting in a simple time-shift of ${\dot{M}}_{\rm acc}$ relative to ${\dot{M}}_{\rm fb}$, ${\dot{M}}_{\rm acc}(t/{{\cal V}}) = {\dot{M}}_{\rm fb}$, where ${\dot{M}}_{\rm acc}$ is the accretion rate onto the blackhole and is normalized such that the total mass accreted is equal to the integral over ${\dot{M}}_{\rm fb}$, the input fallback rate. For ${{\cal V}}= 1$, ${\dot{M}}_{\rm acc} = {\dot{M}}_{\rm fb}$, i.e. $t_{\rm visc} \lesssim t_{\rm m}$.
The emergent emission from the disk is calculated using the prescription of @Done:2012eq, which largely follows the original prescription of @Shakura:1973uy, but amends the no-torque boundary condition to include the effects of the black hole’s spin, parameterized by the dimensionless spin parameter $a_{\rm spin}$. However, it is not immediately clear that the elliptical disk component, which is in the process of circularizing and spans from $2 r_{\rm t}$ to $r_{\rm o}$, would be adequately described by such models. For tidal disruption disks, the densities are low enough such that radiation pressure dominates, but high enough to be optically thick. If we presume that all material returns to pericenter cold, but then is heated to some degree by the circularization process at $r_{\rm p}$, the specific internal energy of fluid at pericenter $\epsilon = (1/2) \rho \alpha v_{\rm K}^2$, which yields a temperature $T = [\alpha v_{\rm K}^2/(\rho a_{\rm rad})]^{1/4}$, where $a_{\rm rad}$ is the radiation constant and $\rho$ is the local density. As the scale height $H \propto \alpha$ near Eddington [@Strubbe:2009ek] and $\rho \propto 1/(r^{2} H)$, $T \propto (v_{\rm K}^2/a_{\rm rad} r^{-3})^{1/4}$. This is proportional (modulo a constant) to the temperature of a steadily-accreting thin disk at the same $r$ [@Beloborodov:1999wb]. After this injection of energy, the internal pressure of this fluid is balanced at all radii with tidal pressure along its elliptical trajectory, which scales as $r^{-3}$ when $H/R$ is fixed, and thus the temperature scaling with radius is identical to that of a steadily-accreting disk. As the inner circular accretion disk necessarily exchanges energy with the outer elliptical disk at their interface at $r_{\rm p}$, the temperature at this interface is likely to equilibrate to the inner disk’s temperature at $r_{\rm p}$; we assume that this sets the normalization constant of the elliptical disk equal to the inner disk. Under this assumption, the temperature structure within the disk would follow @Done:2012eq verbatim.
The model of @Done:2012eq accounts for a shift in the emergent disk spectrum arising from variations in opacity, resulting in an effective temperature that can be ${\mathord{\sim}}2.7$ times larger than expected from the fiducial SS model. We do not permit the luminosity $L$ to exceed the Eddington luminosity $L_{\rm Edd} = 4 \pi G M_{\rm h} m_{\rm p} c / \kappa_{\rm t}$, where $\kappa_{\rm t}$ is the Thomson opacity $\kappa_{\rm t} = 0.2 (1 + X(H))$ cm$^{2}$ g$^{-1}$, and set $L = L_{\rm Edd}$ at times where ${\dot{M}}$ exceeds this limit. We also include the inclination of the structure relative to the observer $\phi$ as a free parameter, where $\phi = 0$ is defined to be edge-on, assuming that both the disk’s height and the ensheathing layer scale with ${{\cal V}}$ in the same way, with the emergent emission from both components being reduced by a factor ${{\cal V}}+ (1 - {{\cal V}})\cos \left(\phi\right)$.
Note that we assume the color correction is intrinsic to the disk emission, and is not the same as the reprocessing that occurs due to the diffuse gas that ensheaths the disk and is ejected from the nozzle region directly. The photosphere of this reprocessing layer, whose size is set by a combination of the mass distribution and the absorption process responsible for intercepting the light, is less constrained. For steadily-accreting AGN with thermal emission, the reprocessing layer has temperatures of several $10^{4}$ K [@Koratkar:1999di; @Lawrence:2012hn], and intercepts nearly 100% of the emission from the disk, resulting in an effective photosphere size that can be hundreds of AU in size for SMBHs accreting at the Eddington limit. However, as there are many non-thermal AGN whose spectra are more representative of bare slim-disk models [@Walton:2013ds], it remains unclear how the size of this reprocessing zone and its fractional coverage are set.
In the models of @Strubbe:2009ek [@Strubbe:2011iw], this layer is presumed to arise as the result of ejection via a super-Eddington wind, and scales with this value when ${\dot{M}}$ exceeds ${\dot{M}_{\rm Edd}}$. From our hydrodynamical simulations, we find that the material that forms the reprocessing layer may be deposited by a process that does not require the accretion rate to exceed ${\dot{M}_{\rm Edd}}$, but instead depends on the details of how energy is injected into the material within the nozzle region. The distribution of mass in radius resulting from the ejection from the nozzle maps is directly related to ${dM/dE}$, although it is modified somewhat by the additional spread in energy introduced at the nozzle point. However, this spread in energy is local to mass that return at a particular time $t$, and thus the distribution of mass with radius after leaving the nozzle point will resemble ${dM/dE}$ with an additional “smear” equal to the spread in energy applied at the nozzle. Therefore, we expect that the mass distribution with radius follows the general shape of ${dM/dE}$, and that there will be a density maximum corresponding to the orbital period of the material that constitutes the peak of the accretion. This peak in density that corresponds to the apocenter of the material that determines is clearly seen in our hydrodynamical simulations (Figure \[fig:columndensity\]).
Thus, the size of the reprocessing layer is likely to be dependent on both the instantaneous value of ${\dot{M}}$, which determines the amount of ionization radiation produced by the disk and the rate of instantaneous mass loss from the nozzle region, and on the integrated amount of mass that has been ejected from the nozzle region since $t = t_{\rm d}$. The optical depth $\tau$ is $$\tau = \kappa \int_{0}^{\infty} \frac{dM}{dr} dr\label{eq:tau},$$ where $\kappa$ is an opacity that is at minimum $\kappa_{\rm t}$.
As we described in the derivation of Equation \[eq:nh\], the amount of mass at a particular distance $r$ is related to the amount of mass at a particular binding energy $E$, and thus we can rewrite Equation \[eq:tau\] in terms of $E$, $$\tau = \kappa \int_{E_{\rm m}}^{E_{\rm o}} \frac{dM}{dE} dE\label{eq:tau2}.$$ where $E_{\rm m}$ and $E_{\rm o}$ are the binding energy of the most bound material and the material at apocenter at time $t$ respectively.
For simplicity, we presume that the reprocessing layer intercepts a fixed fraction of the disk’s light, with the fraction of light $C$ intercepted by the reprocessing layer simply scaling with the optical depth $\tau$, $$C = 1 - e^{-\tau},$$ where $\tau$ is a free parameter. We enforce the condition that $C < c(t - t_{\rm d})/R_{\rm ph}$ (where $R_{\rm ph}$ is the size of the photosphere) at all times, otherwise the photon diffusion time would be greater than the time since disruption, and thus $L$ and ${\dot{M}}$ would not be expected to closely trace one another. In reality, $C$ should have a wavelength dependence, but for the purposes of this work we treat the opacity as being “gray,” absorbing all frequencies of light equally.
As the ionization state of the gas (and therefore the opacity) depends on the current luminosity, the size of the photosphere is expected to vary with time. In general, as the Thomson cross-section is significantly smaller than that of bound-free transitions, the photosphere scale is likely to correspond to the first species is not completely ionized, and in the case of [PS1-10jh]{}where emission is observed, we speculate that this species is helium (Figure \[fig:threed\], magenta region). If we assume that $R_{\rm ph} \propto {\dot{M}}^{l}$, $T_{\rm ph} \propto {\dot{M}}^{m}$, and $L \propto {\dot{M}}\propto R_{\rm ph}^{2} T_{\rm ph}^{4}$, then the power law indices of $R_{\rm ph}$ and $T_{\rm ph}$ are simply related, $$2l + 4m = 1.\label{eq:powlaws}$$ If the opening angle of the reprocessing layer is independent of $r$, the flux in ionizing photons intercepted is constant, implying $l = 1/2$ and thus $m = 0$, i.e. $T_{\rm ph}$ is independent of time. However, as we find that the geometry may in reality be somewhat more complicated (Figure \[fig:columndensity\]), we do not assume the intercepting area necessarily scales as ${\dot{M}}$, and instead leave $l$ as a free parameter. For any $l \neq 1/2$, the temperature of the photosphere will evolve with time. We leave $l$ as a free parameter and relate $m$ and $l$ through Equation \[eq:powlaws\]. The size of the photosphere is then defined to be $$\begin{aligned}
R_{\rm ph} &= R_{\rm ph,0} a_{\rm p} \left(\frac{{\dot{M}}}{{\dot{M}_{\rm Edd}}}\right)^{l}\label{eq:rph}\\
a_{\rm p} &= \left[8 G M_{\rm h}\left(\frac{t_{\rm peak} - t_{\rm m}}{\pi}\right)^{2}\right]^{1/3},\end{aligned}$$ where $a_{\rm p}$ is the semi-major axis of the material that accretes at $t = t_{\rm peak}$, and $R_{\rm ph,0}$ is a dimensionless free parameter.
The amount of reddening in the host galaxy is also an unknown quantity that must be fitted to simultaneously with the parameters of the disruption. For extinction in the IR through the UV, we adopt the reddening law fits of @Cardelli:1989dp, in which the amount of reddening is defined by $A(\lambda) = A_{\rm V} [a(\lambda) + b(\lambda)/R_{\rm V}]$, in which $a(\lambda)$ and $b(\lambda)$ are fitted parameters, $R_{\rm V}$ is a fitted parameter that ranges between 2 and 10 [@Goobar:2002ea], and where we take $N_{\rm h} = 1.8 \times 10^{21} A_{\rm V}$ g cm$^{-3}$. For the X-rays, we adopt the cross-sections presented in @Morrison:1983bh. Extinction in the X-rays is particularly sensitive to metallicity and temperature [@Gnat:2012gz], and the uncertainty in the amount expected for a particular event is large given the environment of a galactic center is likely to have super-solar metallicities [@Cunha:2007gp] and a wide range of temperatures and densities [@Quataert:2002jc; @Cuadra:2006ju; @DeColle:2012bq].
An advantage of the MCMC method employed here is that it permits the inclusion of discrete parameters that can only assume particular values. This enables us to simultaneously fit multiple physical models, as long as the continuous parameters are shared between the models. We include two discrete free parameters in this work: ${\cal A}_{\ast}$, which parameterizes the type of object that was disrupted, and ${\cal A}_{\gamma}$, which parameterizes the polytropic model that is assumed. We include two distinct object types, the white dwarf sequence and the main sequence. Within each of these sequences, different mass ranges are characterized by different polytropic $\gamma$; we use the ${\dot{M}}$ functions derived from our hydrodynamical simulations appropriate to each mass range, $$\begin{aligned}
{\dot{M}}=\; &\dot{M}_{4/3}\left(t\right) \left\{
\begin{array}{ll}
{\rm MS}&:
\begin{array}{ll}
&0.3 < M_{\ast}/M_{\odot} < 22\\
\end{array}\\
{\rm WD}&:
\begin{array}{ll}
&M_{\ast}/M_{\odot} > 1.0
\end{array}\\
\end{array}
\right.\\
{\dot{M}}=\; &\dot{M}_{5/3}\left(t\right) \left\{
\begin{array}{ll}
{\rm MS}&:
\begin{array}{ll}
&M_{\ast}/M_{\odot} < 1.0\\
&M_{\ast}/M_{\odot} > 22
\end{array}\\
{\rm WD}&:
\begin{array}{ll}
&M_{\ast}/M_{\odot} < 1.0
\end{array}\\
\end{array}
\right.\end{aligned}$$ where some overlap is permitted in the mass range $0.3 < M_{\ast}/M_{\odot} < 1.0$ to account for the gradual transition between fully radiative and fully convective stars in this range. It was found that the white dwarf sequence, which is only permits very low mass black holes, is excluded to very high confidence for all of the combinations of parameters that were considered, especially when accounting for the measurement of the black hole’s mass presented in , which restricts $M_{\rm} > 2 \times 10^{6} M_{\odot}$. For simplicity, we exclude discussion of the white dwarf channel for the rest of this work.
{width="0.9\linewidth"}
In addition to modifying the emergent disk spectrum, $a_{\rm spin}$ also affects the minimum approach distance of a star on a parabolic trajectory (i.e. the innermost bound circular orbit) $r_{\rm IBCO}$, and the spread in energy across the star at pericenter [@Kesden:2012kv]. For simplicity in this work we only consider prograde encounters ($a_{\rm spin} > 0$), and apply first-order correction factor to the binding energy, $$E^{\prime} = \left(1 - \frac{1}{2} \frac{r_{\rm IBCO}}{r_{\rm p}}\right)^{-1/2} E.$$ In general, retrograde and/or orbits in which the orbit’s inclination is not equal to the black hole’s spin inclination would be expected.
AGN show variability from the radio to the X-ray, with variability on the order of a few tenths of a dex being common [@Webb:2000iv]. While the photometric errors are small for this event, it is clear that the light curve exhibits some intrinsic variability, as may be expected for an accreting black hole. To model this, we add an additional intrinsic spread $\sigma_{\rm v}$ in quadrature with the observational errors associated with each data point. In addition, the variability has been shown to be dependent on the black hole mass [@Uttley:2005jq].
In total, our fitting procedure includes 15 parameters, 13 of which are continuous ($M_{\ast}$, $M_{\rm h}$, $\beta$, $t_{\rm off}$, $a_{\rm spin}$, ${{\cal V}}$, $\phi$, $\tau$, $l$, $R_{\rm ph,0}$, $R_{\rm v}$, $N_{\rm h}$, and $\sigma_{\rm v}$), and 2 of which are discrete (${\cal A}_{\gamma}$ and ${\cal A}_{\ast}$).
Using the existence/absence of emission lines and their properties to constrain TDEs
------------------------------------------------------------------------------------
In addition to using the quality of the fit of the model light curves to the data, we also impose additional constraints depending on which lines do or do not exist in spectra taken at various times (Figure \[fig:peterson\]). To do this, we measure the emergent flux at $\lambda = 5100 \AA$, which is used in steadily-accreting AGN to measure the continuum, and compare $r_{\rm o}$ to the distance implied by the relationship between $\lambda L_{\lambda} (5100 \AA)$ and $R_{\rm BLR}$, as first determined for [[H]{}$\beta$]{}by @Wandel:1999ev. Since then, the relationship between $L$ and $R_{\rm BLR}$ has been more-accurately determined for [[H]{}$\beta$]{}[@Peterson:2004ig; @Bentz:2013hm], and for several other emission lines [@Bentz:2010hk]. For all lines, it is found that $L \propto R_{\rm BLR}^{0.5}$, indicating that the source of ionizing photons is point-like and that the disk maintains a relatively constant scale-height for a wide range of $r$, which gives the natural result that the number of ionizing photons $\Phi \propto r^{-2}$.
Our modification to the likelihood function is simple: If a line exists in a spectrum and $r_{\rm o} < R_{\rm BLR}$, or if a line doesn’t exist and $r_{\rm o} > R_{\rm BLR}$, we reduce the log-likelihood measured from the light curve alone ${\cal L}_{\rm LC}$ by a factor $$\ln {\cal L_{\rm BLR}} = \ln \left[1 - \frac{1}{2}{\rm erfc}\left(\frac{\left|\ln R_{\rm BLR} - \ln r_{\rm o}\right|}{2 \sigma_{\rm BLR}}\right)\right],\label{eq:lblr}$$ where $\sigma_{\rm BLR}$ is the error in the measured $L-R_{\rm BLR}$ relation, which we take from @Bentz:2010hk.
If a line exists, and a velocity for that line has been measured, we can use that additional information to constrain the event further by relating $R_{\rm BLR}$ to the underlying velocity expected at the position. An uncertainty exists in TDE debris disks in that the underlying velocity $v_{\rm BLR}$ can range from Keplerian ($v_{\rm BLR} = v_{\rm K}$) to parabolic ($v_{\rm BLR} = \sqrt{2} v_{\rm K}$), and therefore we cannot constrain the distance implied by $v_{\rm BLR}$ better than a factor of $\sqrt{2}$. Bearing this in mind, our reduction to the log-likelihood assumes the following functional form, $$\ln {\cal L}_{v} =\; \left\{
\begin{array}{ll}
v < v_{\rm BLR}&\frac{1}{2}\left(\frac{v - v_{\rm BLR}}{\sigma_{\rm v}}\right)^{2}\\
v_{\rm BLR} < v < \sqrt{2}v_{\rm BLR}&0\\
v > v_{\rm BLR}&\frac{1}{2}\left(\frac{v - \sqrt{2}v_{\rm BLR}}{\sigma_{\rm v}}\right)^{2}.
\end{array}
\right.$$
We show the results of imposing these constraints for [PS1-10jh]{}in Figure \[fig:peterson\], where we plot the distance to which material has traveled $r_{\rm o}$ versus the luminosity at 5100 Å. The specific constraints we have applied are that [ $\lambda$4686]{}must be produced, and [[H]{}$\alpha$]{}must not be produced, in the four spectra in which the observed light is not dominated by the host galaxy (at -22, +227, +254, and +358 restframe days). We find that $r_{\rm o}$ is sufficiently large to produce [ $\lambda$4686]{}in all four of these spectra, and that [[H]{}$\alpha$]{}would potentially be observable at later times if the host galaxy did not dominate the observed light (the light is already subdominant to the host galaxy at +254 days). [[H]{}$\beta$]{}, which is produced at smaller distances than [[H]{}$\alpha$]{}in steadily-accreting AGN, may potentially be observable in late-time spectra, but its wavelength notably overlaps with the observed broad [ $\lambda$4686]{}feature, and is usually a factor of ${\mathord{\sim}}3$ than [[H]{}$\alpha$]{}in most AGN [@Osterbrock:2006ul]. We also might expect that [[H]{}$\gamma$]{}and/or [ $\lambda$5876]{}may appear at later times, as the distances at which these lines are produced are only slightly larger than the distance at which [ $\lambda$4686]{}is produced [@Bentz:2010hk]. As we had mentioned in Section \[subsec:blrsource\], we may also be overestimating the size of reprocessing region if it accretes onto the black hole quickly, which would tend to predict the existence of more emission features. Given these uncertainties, we do not impose a constraint on the non-existence of [ $\lambda$5876]{}, [[H]{}$\beta$]{}, or [[H]{}$\gamma$]{}in this work; we note that their inclusion would likely restrict the size of the accretion disk further, which would tend to favor lower-mass black holes.
We note that the constraints we are imposing do not consider the specific luminosity of the lines versus the continuum (see Section \[subsec:blrsource\]), which can strongly affect whether a line is identified within a collected spectrum. This means we also cannot consider in detail the effects of the elliptical accretion disk structure resulting from a tidal disruption on the strengths of the observed lines, which would preferentially reduce the strength of lines originating at large distances as the BLR does not occupy a full $2\pi$ in azimuth in the outskirts of the debris structure (Figure \[fig:columndensity\]).
Model Fitting of [PS1-10jh]{}, a Prototypical Tidal Disruption {#sec:ps1-10jh}
==============================================================
Available Data
--------------
For the fitting procedure, we use all of the available data to constrain the event, including four [*Pan-STARRS*]{} bands (Pg, Pr, Pi, Pz), the X-ray upper limits from the [*Chandra*]{} space-based X-ray telescope (cycle 12), and the spectra taken by the Hectospec instrument on the MMT telescope, all of which are taken from . As the data presented in is already corrected for extinction assuming $N_{\rm h} = 7.2 \times 10^{19}$ cm$^{-2}$, we remove this correction before using the data as an input, as we self-consistently determine the extinction in the model fitting process. We assume a redshift $z = 0.1696$ as is determined in from template fitting to the host galaxy.
Bare Disk Models {#subsec:baredisk}
----------------
In steadily-accreting AGN disks, the majority of radiation produced by the disk is thought to be intercepted by intervening gas that reprocesses the original emission from the disk. In a TDE, this layer may take some time to form, or may not form at all, depending on the dissipative processes at work. In this case, the light produced as the result of a TDE would resemble a bare Shakura-Sunyaev [SS, @Shakura:1973uy] or slim-disk [@Abramowicz:1988bz] model, with peak emission that extends well beyond the tens of eV that is characteristic of AGN spectra. For this model, we do not include the additional constraints imposed by the existence/absence of emission lines.
{width="0.333\linewidth"}{width="0.333\linewidth"}{width="0.333\linewidth"}
{width="0.5\linewidth"}{width="0.5\linewidth"}
It has previously been assumed that the size of the disk is controlled by the angular momentum content of the returning material, which is limited to $\sqrt{2 G M_{\rm h} r_{\rm t}}$ (see Section \[subsec:stream\] for references). In this case, the disk resembles a bare disk that is “truncated” at $r = 2r_{\rm t}$. We find that these models are a very poor match to the event (Figure \[fig:fits\], left panel).
Fits that do not truncate the disk and extend to $r_{\rm o}$, which are equivalent to our generalized model for TDE debris disks sans the reprocessing layer (cyan region of Figure \[fig:threed\]), are shown in Figure \[fig:fits\] (middle panel). This model is also a poor match to [PS1-10jh]{}, although the increase in surface area relative to the truncated disk does enable the model to at least reproduce [PS1-10jh]{}’s peak luminosity. In order for the bare disk model to closely match the data, the fitting routine settles upon one of two non-ideal solutions: An SED with peak that centers about the range of wavelengths covered by the observed bands, or an SED in which the bands are all within the Rayleigh-jeans tail. In the first case, the luminosity $L$ can closely follow ${\dot{M}}$, but the color evolves tremendously as the peak of the summed blackbody curves shift into/out of the observed bands. In the latter case, the ratio of fluxes between the observed bands remains constant, but $L$ scales as a much weaker power of ${\dot{M}}$, $L \propto {\dot{M}}^{1/4}$ (Figure \[fig:seds\]).
Fits to Generalized Model With Reprocessing Layer {#subsec:generalmodel}
-------------------------------------------------
From the previous section, we know that bare disk models can either reproduce a constant color, or reproduce a luminosity that follows ${\dot{M}}$, but cannot reproduce both behaviors simultaneously. This suggests that a secondary process is involved that reprocesses a large fraction of the light prior to reaching the observer. In section \[subsec:generalmodel\] we suggested that this mechanism is the absorption of the soft X-ray photons produced primarily at $r \sim r_{\rm g}$ by material deposited at $r \sim a_{\rm p}$.
As can be seen in the right panels of Figures \[fig:fits\] and \[fig:seds\], these models provide excellent fits to the data; the parameters associated with the fits of highest likelihood are shown in Table \[tab:parameters\]. We immediately caution the reader that the reported medians of the probability distributions, and the small spread in distributions of some parameters, should not be taken at face value. In our generalized model, which is only a simplified realization of the true structure of the debris, we have made many assumptions, and the uncertainty in some of these assumptions is likely to be greater than spread of solutions about the highest-likelihood models presented here. That being said, it is encouraging that such a simple model with relatively few free parameters can provide a reasonable fit to the data, and is highly suggestive of the true values of the underlying parameters. Note that the dominance of the reprocessing region over the emission from the disk in the optical/UV is similar to the model of @Armijo:2013bo in which an average temperature is calculated from the disk and used to fit [PS1-10jh]{}as a single, time-evolving blackbody.
[cccll]{} ${{\rm Log}\xspace}_{10} M_{\ast}$ & $M_{\odot}$ & Flat & $-3 {\leq x \leq}2$ & $0.576_{-0.143}^{+0.151}$\
${{\rm Log}\xspace}_{10} M_{\rm h}$ & $M_{\odot}$ & Flat & $4 {\leq x \leq}8.6$ & $7.25_{-0.08}^{+0.09}$\
$\beta$ & & Flat & $0.5 {\leq x \leq}4$ & $1.32_{-0.02}^{+0.02}$\
$t_{\rm off}$ & days & Flat & $-700 {\leq x \leq}700$ & $77_{-8}^{+9}$\
$a_{\rm spin}$ & & Flat & $0 {\leq x \leq}1$ & $0.37_{-0.26}^{+0.34}$\
${{\rm Log}\xspace}_{10} {{\cal V}}$ & & Flat & $-4 {\leq x \leq}0$ & $-0.18_{-0.05}^{+0.05}$\
$\phi$ & radians & Flat & $0 {\leq x \leq}\pi/2$ & $0.40_{-0.29}^{+0.36}$\
${{\rm Log}\xspace}_{10} \tau$ & & Flat & $-6 {\leq x \leq}6$ & $-0.31_{-0.28}^{+0.27}$\
$l$ & & Flat & $0 {\leq x \leq}4$ & $2.7_{-0.3}^{+0.3}$\
${{\rm Log}\xspace}_{10} R_{\rm ph, 0}$ & & Flat & $-4 {\leq x \leq}4$ & $0.17_{-0.19}^{+0.28}$\
$R_{\rm v}$ & & Flat & $2 {\leq x \leq}10$ & $6.5_{-0.4}^{+0.4}$\
${{\rm Log}\xspace}_{10} N_{\rm h}$ & cm$^{-2}$ & Flat & $17 {\leq x \leq}23$ & $21_{-0.03}^{+0.03}$\
$\sigma_{\rm v}$ & & Flat & $0 {\leq x \leq}1$ & $0.05_{-0.007}^{+0.009}$ \[tab:parameters\]
In the generalized model without priors, we find that the disruption is best matched by the complete disruption ($\beta = 1.32$) of a moderate mass star ($M_{\ast} = 4 M_{\odot}$) by a $M_{\rm h} = 2 \times 10^{7} M_{\odot}$ black hole. This combination of parameters is close to the most common sub-Eddington disruption expected [@DeColle:2012bq], but predicts the black hole mass is a factor of a few larger than the black hole mass suggested by . Most of this discrepancy is likely to arise not from improper template fitting of the host galaxy, but rather the large intrinsic scatter in the $M_{\rm h}$-$L$ relation, as black holes of mass $10^{6} \leq M_{\rm h}/M_{\odot} \leq 10^{9}$ have been found for other galaxies of similar magnitude [@Graham:2013fc]. The disruption is predicted to have occurred 42 days prior to the first observation, about 20 days prior to what was originally suggested in .
We find that $a_{\rm spin}$ is only loosely constrained, with the main effects of a larger spin being that deeper-$\beta$ encounters would be permitted (which are disfavored anyway), and an increase in the efficiency of converting mass to light. The inclination $\phi$ is highly degenerate with this parameter, and shows a strong anti-correlation (i.e. more-slowly spinning black holes tend to be more face-on). We find that the preferred models increase ${{\cal V}}$ to as large a value as possible, and likely this result would be altered given a physical model for ${{\cal V}}$ that accounts for all the various dissipation processes (see Section \[subsec:dissipative\]).
If the mass from the disruption were spread evenly in azimuth, its optical depth would be quite large, $\tau \gtrsim \rho \kappa_{\rm t} r_{\rm p} \sim 100$, but the $\tau$ values returned by our fitting routine suggest that $\tau \simeq 0.1$. This suggests that the material reprocessing the outgoing light has significant angular momentum support, and that the particular line of sight through which [PS1-10jh]{}was observed contained only a fraction of the total mass accreted, $\lesssim 10^{-2} {M_{\odot}}$. For the power-law evolution of the reprocessing component (Equation \[eq:rph\]), we find that $l = 2.76$, which indicates that the photosphere of the reprocessing component evolves significantly during the encounter, $m = -1.18$ (Equation \[eq:powlaws\]). The fact that the highest-likelihood models deviate from $m = 0$ indicates that the distribution of matter in radius and height may be non-trivial, or that the ionization state may be changing as a function of time. The photosphere scale parameter $R_{\rm ph,0} = 0.17$ corresponds to a physical size of $9 \times 10^{14}$ cm at peak (approximately 50 times larger than pericenter distance $r_{\rm p} = 1.8 \times 10^{13}$ cm), the time evolution of which is shown in the right panel of Figure \[fig:threed\].
For the extinction in the host galaxy, we find that a column of $N_{\rm h} = 10^{21}$ cm$^{-2}$ is preferred, with a reddening parameter $R_{\rm v} = 6.5$. This value is somewhat higher than what is typically observed within the Milky Way ($R_{\rm v} = 3.1$), and is more representative of “gray” dust in which all wavelengths are absorbed equally. We verified that such a gray opacity is necessary by running a separate MCMC in which we fitted the extinction in each band independently, finding that the extinction in Pg is only 0.17 magnitudes greater Pz. Such values of $R_{\rm v}$ have been observed outside of the Milky Way [see e.g. @Falco:1999ip], and are typical of dense molecular clouds [@Draine:2003di]. Another possibility that our generalized model simply does not produce enough UV photons, necessitating a gray opacity law to compensate.
![Power-law index $n \equiv \partial \ln X / \partial \ln t$ for the highest-likelihood model shown in the right-hand panel of Figures \[fig:fits\] and \[fig:seds\] (corresponding to the generalized model), where $X$ is a placeholder for either the fallback rate ${\dot{M}}$ or the amount of flux incident on the detector in a given filter. The solid black curve shows the fallback rate ${\dot{M}}$, which is assumed in our model to be proportional to the bolometric luminosity, whereas the dashed black curve shows $n = -5/3$, the power-law index expected for the canonical constant-density star. The colored curves show $n$ for each band [PS1-10jh]{}was observed. Note that there is little color evolution at early times through peak (at $n = 0$), but some color evolution at late times.[]{data-label="fig:bandsn"}](PS1-10jh-new-dc-no-priors-most_probable-bandsn-eps-converted-to.pdf){width="\linewidth"}
Lastly, we find that the model requires $\sigma_{\rm v} = 0.05$ magnitudes of intrinsic variability, about double that expected for a steadily-accreting black hole with mass $10^{7} M_{\odot}$ [@Kelly:2011ku], $$\begin{aligned}
\sigma_{\rm v} &= t_{\rm H} \zeta^{2}\nonumber\\
&= 0.0253^{+0.071}_{-0.038} M_{7}^{-0.19 \pm 0.78}\label{eq:sigmav},\end{aligned}$$ where $t_{\rm H}$ is the timescale of the break in the PSD, and $\zeta$ is the square root of the variability amplitude measured at the break. This is surprisingly small given the potentially chaotic nature of the accretion process, and suggests that the accretion process is smooth and regular, with no major changes in global structure over short timescales.
In Figure \[fig:bandsn\] we show the power-law index $n$ for the feeding rate ${\dot{M}}$ and luminosity measured in each of the [*Pan-STARRS*]{} and [*GALEX*]{} bands. It is clear that ${\dot{M}}$ does not asymptotically approach $n = -5/3$ for the most-likely model, as is expected given that the asymptotic $n$ ranges from $-1.4$ to $-2.2$ for MS disruptions [@Guillochon:2013jj]. The individual bands also deviate from $-5/3$ in the asymptotic limit, which again is not surprising given that the flux is a given band depends on the photosphere temperature [@Strubbe:2009ek], which in our models varies as a function of time.
We find that our highest-likelihood models with and without the BLR constraints are very similar to one another. In Figure \[fig:fit-dists\], we present the posteriors of four fundamental parameters ($M_{\rm h}$, $M_{\ast}$, $a_{\rm spin}$, and $\beta$), both with (red) and without (blue) the BLR constraints, and find that difference in the posteriors is on the same order as the scatter about the median. This suggests that the timescale, luminosity, and color of [PS1-10jh]{}are sufficient to constrain most of the physical parameters of an event, whereas the BLR constraints can be used as a sanity check to ensure there is no discrepancy between the existent BLR emission regions and the observed spectra.
Discussion {#sec:discussion}
==========
Arguments against the helium star interpretation
------------------------------------------------
The discovery of a flare with no noticeable hydrogen features certainly hints at the possibility that the disrupted star may have been relatively devoid of hydrogen. Aside from the hypothesis presented in the previous sections, there are other reasons to believe why the helium-rich progenitor scenario might be unlikely.
Firstly, helium-rich stars are rare in the universe. The known candidates are SdB/SdO stars [${\mathord{\sim}}10^{6}$ in the MW, @Han:2003hl], helium WDs [${\mathord{\sim}}10^{7}$ in the MW, @Nelemans:2001ia], and WR stars [${\mathord{\sim}}10^{4}$ in the MW, @vanderHucht:2001gf]. While there is some evidence that the mass function around SMBHs is not well-represented by a canonical IMF [@Bartko:2010hh], it seems unlikely that the numbers of these stars could be increased by the factor of ${\mathord{\sim}}10^{4}$–$10^{7}$ required to plausibly explain why the first well-resolved TDE happened to be a helium-rich star.
A second possibility is that the helium-rich star comes as the result of the previous interaction of a giant star with the SMBH, or potentially through a collision between the giant and a more compact stellar object [@Davies:1991hp]. However, in both of these cases, hydrogen is not completely removed from the star. In fact, even for deep tidal encounters, the core tends to retain an atmospheric mass of hydrogen comparable to its own mass [@MacLeod:2012cd]. Additionally, giant stars do not frequently get deposited into highly-bound orbits in which the core itself is likely to be disrupted, as the densities of their cores are $\gtrsim 10^{3}$ times larger than their envelopes, and their orbital migration into the loss-cone is largely dictated by diffusion [@Wang:2004jy; @MacLeod:2012cd; @MacLeod:2013dh].
We can thus conclude that while helium-rich disruptions will occur occasionally, they will not be the dominant contributor to the rate, and as a result, it is highly unlikely by chance that these disruptions would be among the first to be observed.
![Posterior distributions of $M_{\rm h}$, $M_{\ast}$, $a_{\rm spin}$, and $\beta$ for [PS1-10jh]{}. Within each panel are the probability $P$ scaled to the maximum probability $P_{\max}$, with the red curves showing the posteriors when no BLR constraints and no priors are included, the blue curves showing the posteriors when BLR constraints alone are included, and the green curves showing the posteriors when BLR constraints are combined with priors for $M_{\ast}$ and $\beta$. The BLR constraints alone (red) are only slightly different from posteriors calculated with no constraints (blue), but the inclusion of a prior on $M_{\ast}$ and $\beta$ suggests that the star that was disrupted was lower in mass ($0.5 M_{\odot}$ vs. $4 M_{\odot}$) and that the black hole was rapidly spinning (0.9 vs. 0.3).[]{data-label="fig:fit-dists"}](PS1-10jh-new-most_probable-1d-p-eps-converted-to.pdf){width="\linewidth"}
Inclusion of priors
-------------------
In the previous sections, we did not make any assumptions about the distribution of any of our input parameters, but in reality these parameters are likely to have non-flat priors. For example, the distribution of stars around SMBHs is likely to possess a current mass function (CMF) that is strongly related to the initial mass function (IMF), which would suggest that the most likely stars to be disrupted are those with $M_{\ast} \sim 0.1 M_{\odot}$ [@Kroupa:2001ki], in contrast to our unconstrained fits in which $M_{\ast} \simeq 4 M_{\odot}$. Additionally, we might expect that that grazing encounters (e.g. small $\beta$) should outnumber deep encounters [@Frank:1976tg], and the black hole mass should follow established $M_{\rm h}$-$L$ relations [@Graham:2013fc]. The green posteriors in Figure \[fig:fit-dists\] show how our most-likely parameters change when priors on $M_{\ast}$ [@Kroupa:2001ki] and $\beta$ (${\rm Prob.} \propto \beta^{-2}$) are included in our MLA, yielding a lower mass star with $M_{\ast} \simeq 0.5 M_{\odot}$ that was disrupted by a rapidly-spinning black hole ($a_{\rm spin} \simeq 0.9$). We find that the quality of the fit is only slightly poorer when priors are included, suggesting that a low-mass stellar disruption can fit equally well within the context of our generalized model.
However, each of these priors has a great deal of uncertainty associated with it. In our own galactic center, it is not clear if the distribution of stars is similar to the general IMF observed in the field, especially given the prevalence of short-lived B-stars within several lt-days of the black hole [@Gillessen:2009fn]. These stars may have been deposited through binary disruption [@Ginsburg:2006jg], or through a disk [@Madigan:2011ej], both of which would lead to different distribution in $\beta$ than what would be produced by a steady-state, spherically symmetric cluster around the black hole [à la @Wang:2004jy]. Lastly, while a clear trend has been demonstrated between the luminosity of the host galaxy and the mass of its black hole, there is significant scatter about this trend [@Gultekin:2009hj]. In principle, once a significant number of TDEs have been identified, these distributions can be determined from the collection of fits to all disruptions, which could potentially improve the accuracy of parameter estimations of future events.
How BLRs can help us understand TDEs
------------------------------------
The (non-)existence of various lines in spectra acquired of TDEs can be used to great effect to constrain the parameter space of allowed encounters for any particular event. These features enable one to place a time-dependent size constraint on the size of the debris structure resulting from a tidal disruption, which is directly related to the combination of three parameters: $M_{\rm h}$, $M_{\ast}$, and $\beta$ (Equation \[eq:ro\]). In this paper, we have focused specifically on [ $\lambda$4686]{}and [[H]{}$\alpha$]{}in regards to [PS1-10jh]{}, but our technique could be used in general with other emission lines. [PS1-10jh]{}appears to originate from a moderately-massive SMBH, but disruptions of stars by more or less massive black holes would respectively produce larger or smaller structures from which emission lines could be produced. As an example, the disruption of the same star by a $10^{8} M_{\odot}$ may show [[H]{}$\beta$]{}and [ $\lambda$5876]{}features early, with [[H]{}$\alpha$]{}appearing later, whereas a disruption by a $10^{6} M_{\odot}$ black hole may never show any helium or hydrogen emission lines.
In the optical at $z = 0$, the number of strong emission lines is limited, but many more emission lines are available in the UV and X-ray where metals with larger ionization potentials begin to lose their electrons. These lines, which would be produced nearer to the SMBH, could potentially be used to constrain the size scale at early times, and would provide a spatial map of the accretion disk as it grows. It is critical that TDEs are identified early and followed up spectroscopically to obtain this valuable information.
How TDEs can help us understand BLRs
------------------------------------
The BLR has long been used to measure the masses of black holes from the lag times observed in the response of various emission lines, which are thought to lie at various distances. However, there remains much uncertainty in these models, namely the form of the BLR itself. If the dissipation mechanism within the debris disks resulting from tidal disruptions is similar to the dissipation mechanism that controls angular momentum transport in steadily-accreting AGN, it is reasonable to expect that the two structures should have many similarities in terms of their density and temperature profiles, velocity structures, and in the components of the structure that conspire to produce the emergent light.
In this paper, we have made direct comparisons to BLRs in order to understand the emission features that are observed in a particular event. As we have shown, the dependence between the distance at which a particular emission line is produced and the flux originating from the central engine is even more exaggerated than in the case of steadily-accreting AGN, as some line-emitting regions do not exist at all due to the absence of mass beyond a certain distance. With a larger catalogue of TDEs, we can reverse the arguments presented here to learn more about the structure of the BLR present in TDE debris disks.
Caveats and future directions
-----------------------------
At the time of this writing, [PS1-10jh]{}is the only event that is claimed to be a TDE and also captures the rise, peak, and decay of the flare. By capturing all three phases, and with the addition of spectroscopic information, this event provides significantly more information on the underlying mechanisms than the small number of poorly sampled UV/optical TDEs that only capture the decay phase and may have no spectroscopic data [@Gezari:2006gd; @Gezari:2008iv; @Cappelluti:2009jl; @vanVelzen:2011gz; @Cenko:2012fg]. While the models presented here provide compelling evidence of the similarities between steadily-accreting AGN and luminous flares resulting from the tidal disruptions of stars, there are many aspects that can be improved upon. Some uncertainties in the generalized model presented here, such as details on the viscous processes that govern accretion and how matter light is reprocessed, could potentially be resolved with a more-complete collection of well-sampled TDEs.
It is clear from our purely hydrodynamical simulations that mere gas dynamics is incapable of generating the necessary dissipation for high mass-ratio encounters, as we described in Section \[subsec:disenough\]. This suggests that magnetohydrodynamical simulations that focus on the nozzle region need to be performed to examine the growth of the MRI, which by our estimate may be capable of providing the required dissipation. If this mechanism is incapable of operating, then it is possible that only deeply-penetrating encounters in which $r_{\rm p} \sim r_{\rm g}$ will yield rapidly-rising light curves.
A second critical uncertainty is our treatment of the reprocessing layer, which is inextricably linked to the BLR of TDE debris disks. In this work, we have presumed that this reprocessing layer is spherical, parameterized the amount of light absorbed by an average gray opacity, and have ignored potentially complex radiation transport and line-of-sight effects. It is also unlikely that the BLR relations we compare to here are identical for debris disks resulting from disruption, given their elliptical geometry and different radial mass distributions. While the scaling relations determined for steadily-accreting AGN are likely to be similar to TDE scaling relations, meaningful constraints on individual events can only be obtained by revising these relations to account for the differences.
Given a more-accurate prescription of how the viscous and reprocessing mechanisms operate, [[TDEFit]{}]{}can easily be improved to include these additional aspects of the problem, which can potentially yield accurate estimates of the parameters associated with individual disruption events. With a large library of TDEs, which will likely exist in the LSST era when potentially thousands of TDEs may be detected [@vanVelzen:2011gz], it should be possible to obtain detailed demographics of the stellar clusters that surround SMBHs.
Lessons Learned
---------------
For the readers convenience, we summarize the main findings of this paper below.
1. The unbound material, while ejected at high velocity from pericenter after a disruption, is gravitationally confined in the two directions transverse to its motion. This constricts the debris to a thin stream that presents a negligible surface area as compared to the emitting surface generated by the return of the stream to pericenter, and is unlikely to affect the flare’s appearance.
2. When material returns to pericenter, it is heated via hydrodynamical shocks, but this dissipation is likely insufficient to explain the tight relationship between $L$ and for large-$q$ encounters. Additional dissipation via an MRI-like mechanism or through general relativistic precession is probably required to explain this observed relationship.
3. A disk that is truncated at $2 r_{\rm t}$ fails drastically in explaining the observed flare, and cannot match the observed shape of the light curves without extreme color evolution.
4. The light curve of [PS1-10jh]{}is well modeled by a single blackbody whose temperature evolves weakly in time, and whose size is tens of times larger than $r_{\rm t}$. We speculate that this distance is roughly co-spatial with the distance at which helium is doubly-ionized.
5. The fact that emission lines are observed, but [[H]{}$\alpha$]{}and [[H]{}$\beta$]{}are not, is consistent with the size constraint on the bound debris that is ejected from the nozzle point upon returning to pericenter. In general, the presence or absence of various emission lines can be used as a probe of the size of the elliptical debris disk.
6. When prior information is not included, the parameters for [PS1-10jh]{}of our highest-likelihood fits indicate that a $4 M_{\odot}$ main-sequence star was disrupted by a $2 \times 10^{7} M_{\odot}$ black hole. We find that the inclusion of a reasonable prior on $M_{\ast}$ and $\beta$ yields a lower stellar mass, $0.5 M_{\odot}$. While there is uncertainty in the proper prior to use in a galactic center environment, both the fits with and without priors involve the disruption of a common star by a common SMBH with an impact parameter near the expected average [@DeColle:2012bq], and thus TDEs of the kind we associate with [PS1-10jh]{}are likely to be among the most common sub-Eddington disruption events. However, given that we are analyzing a single event in this paper, we cannot eliminate the possibility that an event of this type was among the first observed due to observational bias. Once more well-sampled TDEs are available, a joint analysis of many events similar to what we perform here is required for a complete understanding of the demographics of tidal disruption.
We have benefited from many useful discussions with J. Arnold, J. Braithwaite, B. Cenko, R. da Silva, K. Denney, C. Dorman, R. Foley, D. Foreman-Mackey, S. Gezari, J. Goodman, J. Halpern, D. Kasen, S. Kulkarni, D. N. C. Lin, M. MacLeod, C. Matzner, C. Miller, M. Pessah, M. Rees, S. Rosswog, A. Socrates, L. Strubbe, J. Trump, and E. Zweibel. We thank the anonymous referee for their constructive comments and suggestions, and C. M. Gaskell for detailed comments. The software used in this work was in part developed by the DOE-supported ASCI/Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago. Computations were performed on the UCSC Pleiades, Hyades, and Laozi computer clusters, and the NASA Pleiades computer cluster. We acknowledge support from the David and Lucille Packard Foundation, NSF grants PHY-0503584 and ST-0847563, and the NASA Earth and Space Science Fellowship (JG).
|
---
date: 'July 31 - August 7, 2003'
title: |
HEGRA Contributions to the\
**28th International Cosmic Ray Conference**\
Tsukuba, Japan
---
|
---
abstract: 'In a missing-data setting, we have a sample in which a vector of explanatory variables $\mathbf{x}_{i}$ is observed for every subject $i$, while scalar outcomes $y_{i}$ are missing by happenstance on some individuals. In this work we propose robust estimates of the distribution of the responses assuming missing at random (MAR) data, under a semiparametric regression model. Our approach allows the consistent estimation of any weakly continuous functional of the response’s distribution. In particular, strongly consistent estimates of any continuous location functional, such as the median or MM functionals, are proposed. A robust fit for the regression model combined with the robust properties of the location functional gives rise to a robust recipe for estimating the location parameter. Robustness is quantified through the breakdown point of the proposed procedure. The asymptotic distribution of the location estimates is also derived.'
author:
- |
Mariela Sued (msued@dm.uba.ar) and Víctor J. Yohai (vyohai@uolsinectis.com.ar)\
University of Buenos Aires and CONICET
date:
title: Robust location estimation with missing data
---
Introduction
============
Suppose we have a sample of a population, such that for every subject $i$ in the sample we observe a vector of explanatory variables $\mathbf{x}_{i}$ while a scalar response $y_{i}$ is missing by happenstance on some individuals. A classical problem is to construct consistent estimators for the mean value of the response based on the observed data. In order to identify the parameter of interest in terms of the distribution of observed data, missing at random (MAR) is assumed.
This hypothesis establishes that the value of the response does not provide additional information, on top of that given by the explanatory variables, to predict whether an individual will present a missing response (see Rubin [@Rubin]). To be more rigorous, let us introduce a binary variable $a_{i}$ such that $a_{i}=1$ whenever the response is observed for subject $i$. In this way, MAR states that$$\mathrm{P}(a_{i}=1|\mathbf{x}_{i},y_{i})=\mathrm{P}(a_{i}=1|\mathbf{x}_{i}).
\label{MAR}$$ Under this condition, if $\mathrm{P}(a_{i}=1|\mathbf{x}_{i})>0$, we have that$$\mathrm{E}[y_{i}]=\mathrm{E}\left[ \frac{a_{i}y_{i}}{\pi(\mathbf{x}_{i})}\right] , \label{IPW}$$ where $\pi(\mathbf{x}_{i})=$$(a_{i}=1|\mathbf{x}_{i})$, and identifiability of $\mathrm{E}[y_{i}]$ holds. One approach to estimate consistently $\mathrm{E}[y_{i}]$, called inverse probability weight (IPW), is based on (\[IPW\]) and requires to estimate the propensity score function $\pi(\mathbf{x)}$. Then, the estimate of $\mathrm{E}[y_{i}]$ can be obtained replacing in (\[IPW\]) $\pi(\mathbf{x}_{i})$ by its estimate and the expectation by its empirical version. MAR also implies that the conditional distribution of the responses given the vector of explanatory variables remains the same, regardless of the fact that the response is also observed: $y_{i}|\mathbf{x}_{i}\sim y_{i}|\mathbf{x}_{i},a_{i}=1$. Then $\mathrm{E}[y_{i}|\mathbf{x}_{i}]=\mathrm{E}[y_{i}|\mathbf{x}_{i},a_{i}=1]$. Since $\mathrm{E}[y_{i}]=\mathrm{E}\left[ \mathrm{E}[y_{i}|\mathbf{x}_{i}]\right]
$, a second approach to estimate $\mathrm{E}[y_{i}]$ is based on a regression model (parametric or nonparametric ) for $\mathrm{E}[y_{i}|\mathbf{x}_{i}]=g(\mathbf{x}_{i})$, which is fitted using only the individuals for whom the response is observed. Then a second estimate for $\mathrm{E}[y_{i}]$ is obtained by averaging $\widehat{g}(\mathbf{x}_{i})$ over the whole sample, where $\widehat{g}$ is an estimate of $g$. There is third approach (doubly protected) that postulates models for $\pi (\mathbf{x)}$ and $g(\mathbf{x)}$ and obtains a consistent estimate of $\mathrm{E}[y_{i}]$ if at least one of the two models is correct. A recent survey and discussion on these three approaches can be found in Kan and Schafer [@KanSchafer] and Robins, Sued, Lei-Gomez and Rotnitzky [@Robins].
As it is well known, the mean is not a robust location parameter, i.e., a small change in the population distribution may have a large effect on this parameter. As a consequence of this, the mean does not admit consistent non-parametric robust estimates, except when strong properties on the distribution are assumed, as for example symmetry. For this reason, to introduce robustness in the present setting, we start by reformulating the statistical object of interest: instead of estimating the mean value of the response, we look for consistent estimates of $T_{L}(F_{0}),$ where $T_{L}$ is a robust location functional and $F_{0}$ is the distribution of $y_{i}$. Bianco, Boente, Gonzalez-Manteiga and Perez-Gonzalez [@Bianco] used this approach to obtain robust and consistent estimates of an M location parameter of the distribution of $y_{i}$. In their treatment they assumed a partially linear model to describe the relationship between $y_{i}$ and $\mathbf{x}_{i}$, and also that the distributions of the response $y_{i}$ and of the regression error under the true model are both symmetric.
In this paper we introduce a new estimate of any continuous location functional assuming that the relation between $y_{i}$ and $\mathbf{x}_{i}$ is given by means of a semiparametric regression model. We show that once the regression model is fitted using robust estimates, we can define a consistent estimate of the distribution function of the response. Then, any parameter of the response distribution defined throughout a weak continuous functional, may be also consistently estimated by evaluating the functional at the estimated distribution function. The consistency of this procedure does not require the symmetry assumptions used by Bianco et al. [@Bianco].
A robust fit for the regression model combined with the robust properties of the location functional to be considered, gives rise to a robust recipe for estimating the location parameter. Robustness is quantified looking at breakdown point of the proposed procedure. In particular our results can be applied when the location functional is the median or an MM location functional.
The proposed procedure may be considered as a robust extension of the second approach described above for estimating $\mathrm{E}[y_{i}]$. We have not found a way to robustify the approaches that use the propensity score $\pi
(\mathbf{x)}$. The main difficulty in such cases is to obtain a consistent procedure avoiding the assignment of very large weights to those observations with very small $\pi(\mathbf{x}_{i}\mathbf{)}$.
This work is organized as follows. In Section 2 we formalize the problem of the robust estimation of a location parameter with missing data. We propose a family of procedures which depend on the location functional to be estimated and also on the robust regression estimate for the parameter of the regression model postulated to describe the relationship between $\mathbf{x}_{i}$ and $y_{i}$. In Section 3 we show that, under some assumptions on the location functional and the regression estimate, the proposed estimates are strongly consistent and asymptotically normal. In Section 4 we study the breakdown point of the proposed estimates. In Section 5 we show that when the location and regression estimates are of MM type, the assumptions that guarantee consistency and asymptotic normality of the proposed estimates are satisfied. In Section 6 we present the results of a Monte Carlo study which shows that the proposed estimates are highly efficient under Gaussian errors and highly robust under outlier contamination. Proofs are presented in the Appendix.
Notation and Preliminaries
==========================
We first introduce some notation. Henceforth $\mathrm{E}_{G}[h(\mathbf{z})]$ and $\mathrm{P}_{G}\left( A\right) $ will respectively denote the expectation of $h(\mathbf{z})$ and the probability that $\mathbf{z}\in A,$ when $\mathbf{z}$ is distributed according to $G$. If $\mathbf{z}$ has distribution $G$ we write $\mathbf{z}\sim G$ or $\mathcal{D}\left(
\mathbf{z}\right) =G.$ Weak convergence of distributions, convergence in probability and convergence in distribution of random variables or vectors are denoted by $G_{n}\rightarrow_{w}G,$ $\mathbf{z}_{n}\rightarrow_{p}\mathbf{z}$ and $\mathbf{z}_{n}\rightarrow_{d}\mathbf{z},$ respectively. By an abuse of notation, we will write $\mathbf{z}_{n}\rightarrow_{d}G$ to denote $\mathcal{D}\left( \mathbf{z}_{n}\right) \rightarrow_{w}G.$ We use $o_{P}(1)$ to denote any sequence that converges to zero in probability. The complement and the indicator of the set $A$ are denoted by $A^{c}$ and $\mathbf{1}_{A},$ respectively. The scalar product of vectors $\mathbf{a,b}\in{\mathbb{R}}^{s}$ is denoted by $\mathbf{a}^{\prime}\mathbf{b}$. $\mathbb{R}_{+}$ denotes the set of positive real numbers.
Along this paper we use the expression *empirical distribution * of $\ $a$\ $sequence on $\ n$ points $\mathbf{z}_{1},\mathbf{z}_{2},...,\mathbf{z}_{n}$ in $\mathbb{R}^{k}$* * to denote the function $F_{n}:\mathbb{R}^{k}\rightarrow\lbrack0,1]$ such that given $\mathbf{z}\in\mathbb{R}^{k},$ $F_{n}(\mathbf{z})$ $=m/n,$ where $m$ is the number of points $\mathbf{z}_{i}$ such that all its coordinates $\ $ are smaller or equal than the corresponding ones of $\mathbf{z}.$
Describing our setting: the data, the problem and the model
-----------------------------------------------------------
Throughout this work, we have a random sample of $\ n$ subjects and for each subject $i$ in the sample, $1\leq i\leq n$, a vector of explanatory variables $\mathbf{x}_{i}$ is always observed, while the response $y_{i}$ is missing on some subjects. Let $a_{i}$ be the indicator of whether $y_{i}$ is observed at subject $i$: $a_{i}=1$ if $y_{i}$ is observed and $a_{i}=0$ if it is not.
We will be concerned with the estimation of a location functional at the distribution of the response. A location functional $T_{L}$, defined on a class of univariate distribution functions $\mathcal{G}$, assigns to each $F\in$ $\mathcal{G}$ a real number $\ T_{L}(F)$ satisfying $T_{L}(F_{ay+b})=aT_{L}(F_{y})+b,$ where $F_{y}$ denotes the distribution of the random variable $y$.
Example of locations functionals are the mean and median. Another important class of location functionals that includes the mean and median and other robust estimates is the class of M location functionals. This class also includes S and MM estimators that will be described in Section 4. We should also mention the class of L location functionals, see e.g. Chapter 2 of Maronna, Martin and Yohai [@MaronnaMartin; @and; @Yohai], but we do not study this class in this work.
A functional $T$ is said to be weakly continuous at $F$ if given a sequence $\{F_{n}\}$ of distribution functions that converges weakly to $F$ $(F_{n}\rightarrow_{w}F$), then $T(F_{n})\rightarrow T(F)$. In order to obtain a consistent estimate of a location parameter defined by means of a weakly continuous functional, it is sufficient to have a sequence of estimates $\widehat{F}_{n}$ such that converges weakly to the distribution of the $y_{i}$’s.
To be more precise, denote by $F_{0}$ the distribution of the outcomes $y_{i}
$. Let $\ T_{L}$ be a weakly continuous location functional at $F_{0}$ . We are interested in estimating $$\mu_{0}=T_{L}(F_{0}).$$
We assume a semiparametric regression model $$y_{i}=g(\mathbf{x}_{i},\mathbf{\beta}_{0})+u_{i},1\leq i\leq n,
\label{modelogral}$$ with $y_{i},u_{i}\in{\mathbb{R}}$, $\mathbf{x}_{i}\in{\mathbb{R}}^{p}$, $u_{i}$ independent of $\mathbf{x}_{i}$, **$\beta$**$_{0}\in
B\subset{\mathbb{R}}^{q}$, $g:{\mathbb{R}}^{p}\times B\rightarrow{\mathbb{R}}$. Furthermore, in order to guarantee the MAR condition, we assume that $u_{i}$ is independent of $(\mathbf{x}_{i},a_{i})$. We denote by $Q_{0}$ and $\mathrm{K}_{0}$ the distributions of $\mathbf{x}_{i}$ and $u_{i}$, respectively.
To identify **$\beta$**$_{0}$, without assuming that either (i) $\mathrm{K}_{0}$ is symmetric around $0$ or (ii) $\mathrm{K}_{0}$ satisfies a centering condition, (as, e.g. , E$_{K_{0}}u=0)$ we assume that $$\mathrm{P}_{Q_{0}}\left( g(\mathbf{x},\mathbf{\beta}_{0})=g(\mathbf{x},\mathbf{\beta})+\alpha\right) <1 \label{IDCOND}$$ for all **$\beta$**$\neq$**$\beta$**$_{0}$, for all $\alpha$. This condition requires that in case there is an intercept, it will be included in the error term $u_{i}$ instead as of a parameter of the regression function $g(\mathbf{x},$**$\beta$**$)$. For linear regression we have $g(\mathbf{x},$**$\beta$**$)=$**$\beta$**$^{\prime}\mathbf{x}$ and then this condition means that the vector $\mathbf{x}_{i}$ is not concentrated on any hyperplane.
The proposal
------------
Recall that $\mathrm{K}_{0}$ denotes de distribution of $u_{i}$ and let $R_{0}$ denotes the distribution of $g(\mathbf{x}_{i},\beta_{0})$. Independence between $\mathbf{x}_{i}$ and $u_{i}$ guarantees that $F_{0}$ is the convolution between $R_{0}$ and $K_{0}$. Then by convoluting consistent estimators $\widehat{R}_{n}$ and $\widehat{K}_{n}$ of each of these distributions, we get a consistent estimator for $F_{0}$.
In order to estimate $R_{0}$ and $K_{0}$ we need to have a robust and strongly consistent estimator $\widehat{\mathbf{\beta}}_{n}$ of **$\beta $**$_{0}$. This estimator may be, for example, an S estimate (see Rousseeuw and Yohai [@RousseeuwYohai]) or an MM-estimate (see Yohai [@Yohai87]). Since $u_{i}$ is independent of $a_{i}$, $\widehat{\mathbf{\beta}}_{n}$ may be obtained by a robust fit of the model using the data for which $\
y_{i}$ is observed: i.e., using the observations $(\mathbf{x}_{i},y_{i})$ with $a_{i}=1$. Let $\widehat{R}_{n}$ be the empirical distribution of $g(\mathbf{x}_{j},\widehat{\mathbf{\beta}}_{n})$, $1\leq j\leq n$ defined by$$\widehat{R}_{n}=\frac{1}{n}\,\sum_{j=1}^{n}\delta_{g(\mathbf{x}_{j},\widehat{\mathbf{\beta}}_{n})}, \label{Rn}$$ where $\delta_{s}$ denotes the point mass distribution at $s$. Let $A=\{i:a_{i}=1\}$ and $m=\#A$. For $\ i\in A$ consider $$\widehat{u}_{i}=y_{i}-g(\mathbf{x}_{i},\widehat{\mathbf{\beta}}_{n}).$$ The estimator $\widehat{K}_{n}$ of $K_{0}$ is defined as the empirical distribution of $\{\widehat{u}_{i}:i\in A\}$: $$\widehat{K}_{n}=\frac{1}{m}\,\sum_{i\in A}\delta_{\widehat{u}_{i}}=\frac
{1}{\sum_{i=1}^{n}a_{i}}\,\sum_{i=1}^{n}a_{i}\delta_{\widehat{u}_{i}}.
\label{Fn}$$
Then, we estimate $F_{0}$ by $\widehat{F}_{n}=\widehat{R}_{n}\ast\widehat
{K}_{n}$, where $\ast$ denotes convolution. Note that $\widehat{R}_{n}\ast\widehat{K}_{n},$ is the empirical distribution of the $nm$ points$$\widehat{y}_{ij}=g(\mathbf{x}_{j},\widehat{\mathbf{\beta}}_{n})+\widehat
{u}_{i},\text{ }1\leq j\leq n,\text{ }i\in A,$$ and therefore we can also express $\widehat{F}_{n}$ as$$\widehat{F}_{n}=\frac{1}{nm}{\displaystyle\sum\limits_{i\in A}}
\sum_{j=1}^{n}\delta_{\widehat{y}_{ij}}=\frac{1}{n\,\sum_{i=1}^{n}a_{i}}{\displaystyle\sum\limits_{i\in A}}
\sum_{j=1}^{n}\delta_{\widehat{y}_{ij}}. \label{Gnsom}$$ Finally, we estimate $\mu_{0}$ by $$\widehat{\mu}_{n}=T_{L}(\widehat{F}_{n}). \label{OUREST}$$
Since we have assumed weak continuity of $T_{L}$ at $F_{0}$, in order to prove that $\widehat{\mu}_{n}$ is a strongly consistent estimate of $\mu_{0}$ we only need to prove that $\widehat{F}_{n}\rightarrow_{w}F_{0}$ a.s. Observe that $$\mathrm{E}_{\widehat{F}_{n}}h(y)=\frac{1}{nm}{\displaystyle\sum\limits_{i\in A}}
\sum_{j=1}^{n}h(\widehat{y}_{ij}).$$ The right hand side of this equation was proposed by Müller [@Muller] to estimate $\mathrm{E}_{F_{0}}h(y).$
Consistency and asymptotic distribution\[results\]
==================================================
Let $(\mathbf{x}_{i},y_{i})$ and $\ u_{i}$ satisfy model (\[modelogral\]), with $u_{i}$ independent of $(\mathbf{x}_{i},a_{i})$. Denote by $\mathrm{G}_{0}$, $\mathrm{Q}_{0}$ and $\mathrm{K}_{0}$ the distributions of $(\mathbf{x}_{i},y_{i})$, $\mathbf{x}_{i}$ and $u_{i}$, respectively, and denote by $\mathrm{G}_{0}^{\ast}$ and $\mathrm{Q}_{0}^{\ast}$ the distributions of $(\mathbf{x}_{i},y_{i})$ and $\mathbf{x}_{i}$ conditioned on $a_{i}=1$, respectively.
The MAR condition implies that under $G_{0}^{\ast}$ model (\[modelogral\]) is still satisfied with $\ \mathbf{x}_{i}^{\ast}$ and $u_{i}^{\ast}$ independent, $\mathbf{x}_{i}^{\ast}$ with distribution $Q_{0}^{\ast}$ and $\ u_{i}^{\ast}$ with distribution $K_{0}.$ We also assume that the regression function $\ g$ satisfies following assumption:
**A0** $g(\mathbf{x},$**$\beta$**$)$ is twice continuously differentiable with respect to **$\beta$** and there exists $\delta>0$ such that$$\mathrm{E}_{Q_{0}}\sup_{\left\Vert \mathbf{\beta}-\mathbf{\beta}_{0}\right\Vert \leq\delta}\left\Vert {\dot{g}}(\mathbf{x}_{1},\mathbf{\beta
})\right\Vert ^{2}<\infty\text{ }\mathrm{and}\ \ \mathrm{E}\,_{Q_{0}}\sup_{\left\Vert \mathbf{\beta}-\mathbf{\beta}_{0}\right\Vert \leq\delta
}\left\Vert {\ddot{g}}(\mathbf{x}_{1},\mathbf{\beta})\right\Vert <\infty,
\label{A0}$$
where $\dot{g}(\mathbf{x},$**$\beta$**$)$ and $\ddot{g}\left(
\mathbf{x},\mathbf{\beta}\right) $ denote the vector of first derivatives and the matrix of second derivatives of $g$ respect to **$\beta$**, respectively.
In order to prove the consistency and the asymptotic normality of $\widehat{\mu}_{n}$ the following assumptions on $\widehat{\mathbf{\beta}}_{n}$ and $T_{L}$ are required.
**A1** $\{\widehat{\mathbf{\beta}}_{n}\}$ is strongly consistent for **$\beta$**$_{0}$.
**A2** The regression estimate $\widehat{\mathbf{\beta}}_{n}$ satisfies $$\sqrt{n}(\widehat{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})=\frac{1}{n^{1/2}}\sum_{i=1}^{n}a_{i}I_{R}(\mathbf{x}_{i},y_{i})+o_{P}(1), \label{TRexpansion}$$ for some function $I_{R}(\mathbf{x},u)$ with $\mathrm{E}a_{i}I_{R}(\mathbf{x_{i}},y_{i})=0$ and finite second moments.
**A3** $T_{L}$ is weakly continuous at $F_{0}$.
**A4** The following expansion holds: $$\sqrt{n}\left( T_{L}(\widehat{F}_{n})-T_{L}(F_{0})\right) =\sqrt
{n}\mathrm{E}_{\widehat{F}_{n}}I_{L}(y)\,+\,o_{P}(1), \label{TLexpansion}$$ for some differentiable function $I_{L}(y)$ with $\mathrm{E}_{F_{0}}I_{L}(y)=0$, $\mathrm{E}_{F_{0}}I_{L}^{2}(y)<\infty$ and $|I_{L}^{\prime}(y)|$ bounded.
It can be shown that when expansion (\[TLexpansion\]) holds, $I_{L}$ is given by the influence function (as defined by Hampel (1974)) of $T_{L}$ at $F_{0}$. When $\widehat{\mathbf{\beta}}_{n}$ is obtained using a regression functional, a similar statement holds.
The following Theorem shows the consistency of $\widehat{\mu}_{n}=T(\widehat{F}_{n}).$
\[lemaconvdebil1\] Let $\widehat{F}_{n}$ be defined as in (\[Gnsom\]) and assume that A1 holds. Then (a) $\{\widehat{F}_{n}\}$ converges weakly to $F_{0}$ a.s., i.e., $$\mathrm{P}(\widehat{F}_{n}\rightarrow_{w}F_{0})=1.$$ (b) Assume also that A3 holds; then $\widehat{\mu}_{n}=T_{L}(\widehat{F}_{n})$ converges a.s. to $\mu_{0}=T_{L}(F_{0})$.
In order to find the asymptotic distribution of $\widehat{\mathbf{\mu}}_{n}\ $, consider $$\begin{aligned}
\eta & =\mathrm{E}a_{i},\mathbf{c}=\mathrm{E}\left[ a_{1}\,I_{L}^{\prime
}(y_{1}-g({\mathbf{\beta}_{0}},\mathbf{x}_{1})+{g}({\mathbf{\beta}_{0}},\mathbf{x}_{2}))\,\{\dot{g}({\mathbf{\beta}_{0}},\mathbf{x}_{2})-\dot
{g}({\mathbf{\beta}_{0}},\mathbf{x}_{1})\}\right] ,\\
e(\mathbf{x}_{i},u_{i},a_{i}) & =\mathrm{E}\left[ a_{i}I_{_{T_{L},F_{0}}}(u_{i}+g(\mathbf{x}_{j},\mathbf{\beta}_{0}))|u_{i},a_{i}\right]
=a_{i}\mathrm{E}\left[ I_{_{T_{L},F_{0}}}(u_{i}+g(\mathbf{x}_{j},\mathbf{\beta}_{0}))|u_{i},a_{i}\right] ,\\
f(\mathbf{x}_{j}\mathbf{)} & \mathbf{=}\mathrm{E}\left[ a_{i}I_{_{T_{L},F_{0}}}(u_{i}+g(\mathbf{x}_{j},\mathbf{\beta}_{0}))|\mathbf{x}_{j}\right] ,\\
\tau^{2} & =\frac{1}{\eta^{2}}\mathrm{E}\left[ \left\{ e(\mathbf{x}_{i},u_{i},a_{i})+f(\mathbf{x}_{i})+a_{i}{\mathbf{c}^{\prime}I_{R}(\mathbf{x}_{i},u_{i})}\right\} ^{2}\right] .\end{aligned}$$ Then, the following Theorem gives the asymptotic normality of the estimate $\widehat{\mathbf{\mu}}_{n}$, defined in (\[OUREST\]).
\[asym\] Assume A0-A4. Then$$n^{1/2}(\widehat{\mu}_{n}-\mu_{0})\rightarrow_{d}N(0,\tau^{2}).
\label{Vnasnor}$$
The median as location parameter
--------------------------------
The median is one of the most popular robust location parameters. However, since this estimate does not satisfy A4, we cannot prove its asymptotic normality using Theorem \[lemaconvdebil1\]. In this section, we will prove consistency and asymptotic distribution for the median of $\widehat{F}_{n}$, defined at (\[Gnsom\]), assuming that A0 holds and that $\{\widehat{\beta }_{n}\}$ satisfy A1 and A2.
The functional $T_{\text{med}}$ is defined by $$T_{\text{med}}(F)=\arg\min_{\mu}\mathrm{E}_{F}|y-\mu|. \label{meddef}$$ When there are more than one value attaining the minimum, the functional is defined by choosing any of them. We have the following result, whose proof needs an extra argument to compensate the absence of differentiability of $I_{T_{\text{med}},F_{0}}(y)$.
\[median\] Assume that $\mu_{0}=T_{\text{med}}(F_{0})$ is well defined and let $\widehat{\mu}_{n}=T_{\text{med}}(\widehat{F}_{n})$. Suppose that $F_{0}$ is continuous and strictly increasing at $\mu_{0}$. Then, (a) under A1 we have $\widehat{\mu}_{n}\rightarrow\mu_{0}$ a.s.(b )Assume A0-A2. Assume also that $F_{0}$ and $K_{0}$ have continuous and bounded densities $f_{0}$ $\ $and $k_{0}$ respectively, and that $f_{0}(\mu_{0})>0$. Then$$n^{1/2}(\widehat{\mu}_{n}-\mu_{0})\rightarrow_{d}N(0,\tau^{2}),
\label{Vnasnor1}$$ where $\tau^{2}$ is as in Theorem \[asym\] with $\mathbf{c}$ replaced by$$\mathbf{c}^{\ast}=\frac{1}{\eta f_{0}(\mu_{0})}\mathrm{E}[a_{1}k_{0}(-g(\mathbf{x}_{2},\mathbf{\beta}_{0})+\mu_{0})\{\dot{g}(\mathbf{x}_{2},\mathbf{\beta}_{0})-\dot{g}(\mathbf{x}_{1},\mathbf{\beta}_{0})\}]\ \label{cstar}$$ and $I_{T_{L},F_{0}}(y)$ replaced by $$I_{T_{\text{med}},F_{0}}(y)=\frac{\text{sign}(y-\mu_{0})}{2f_{0}(\mu_{0})}.$$
Breakdown point \[secBDP\]
==========================
Consider first a dataset of $n$ complete observations $\mathbf{Z=\{z}_{1}\mathbf{,..,z}_{n}\mathbf{\}},$ where $\mathbf{z}_{i}\in\mathbb{R}^{j}$, and let $\widehat{\theta}_{n}(\mathbf{Z})$ be an estimate of a parameter $\mathbf{\theta}\in\mathbb{R}^{k}$ defined on all possible datasets. Donoho and Huber [@Dh] define the finite sample breakdown point (FSBP) of $\widehat{\mathbf{\theta}}_{n}$ at $\ \mathbf{Z}$ by $$\varepsilon^{\ast}(\widehat{\mathbf{\theta}}_{n},\mathbf{Z})=\min\left\{
\frac{s}{n}:\sup_{\mathbf{Z}^{\ast}\in\mathcal{Z}_{s}}\Vert\widehat{\theta
}_{n}(\mathbf{Z}^{\ast})\Vert=\infty\right\} ,$$ where$$\mathcal{Z}_{s}=\{\mathbf{Z}^{\ast}=\{\mathbf{z}_{1}^{\ast},...,\mathbf{z}_{n}^{\ast}\}:{\displaystyle\sum\limits_{i=1}^{n}}
I\{\mathbf{z}_{t}^{\ast}\neq\mathbf{z}_{i}\}\leq s\}.$$ Then $\ \varepsilon^{\ast}$ is the minimum fraction of outliers that is required to take the estimate beyond any bound.
Now, we extend the notion of FSBP to the present setting, where there are missing data, as follows. Let$$\mathbf{W}=\{(\mathbf{x}_{1},y_{1},a_{1}),....(\mathbf{x}_{n},y_{n},a_{n})\}
\label{Wcom}$$ be the set of all observations and missingness indicators, and let $A=\{i:1\leq i\leq n,\,a_{i}=1\}$, $m=\#A.$ Denote by $\mathcal{W}_{st}$ the set of all samples obtained from $\mathbf{W}$ by replacing at most $t$ points by outliers, with at most $s$ of these replacement corresponding to the non missing observations. Then $\mathbf{W}^{\ast}=\{(\mathbf{x}_{1}^{\ast},y_{1}^{\ast},a_{1}),....(\mathbf{x}_{n}^{\ast},y_{n}^{\ast},a_{n})\}$ belongs to $\mathcal{W}_{t,s}$ if $$\sum_{i\in A}I\{(\mathbf{x}_{i}^{\ast},y_{i}^{\ast})\neq(\mathbf{x}_{i},y_{i})\}\ +\sum_{i\in A^{C}}I\{\mathbf{x}_{i}^{\ast}\neq\mathbf{x}_{i}\}\leq
t$$ and $$\sum_{i\in A}I\{(\mathbf{x}_{i}^{\ast},y_{i}^{\ast})\neq(\mathbf{x}_{i},y_{i})\}\ \leq s.$$ Given an estimate $\widehat{\mu}_{n}$ of $\mu_{0},$ we define $$M_{ts}=\sup_{\mathbf{W}^{\ast}\in\mathcal{W}_{t,s}}\left\vert \widehat{\mu
}_{n}(\mathbf{W}^{\ast})\right\vert$$ and $$\kappa(t,s)=\max\left( \frac{t}{n},\frac{s}{m}\right) .$$ Then, we define the finite sample breakdown point (FSBP) of an estimate $\widehat{\mu}_{n}$ at $\mathbf{W}$ $$\varepsilon^{\ast}=\min\{\kappa(t,s):M_{ts}=\infty\}.$$ Then $\ \varepsilon^{\ast}$ is the minimum fraction of outliers in the complete sample or in the set of non missing observations that is required to take the estimate beyond any bound.
In order to get a lower bound for the FSBP of the location estimate $\widehat{\mu}_{n}$ introduced in (\[OUREST\]), we need to define the * uniform asymptotic breakdown point* $\varepsilon_{U}^{\ast}$ of $T_{L}$ as follows:
Given a functional $T_{L}$, its *uniform asymptotic breakdown point* $\ $(UABP) $\varepsilon_{U}^{\ast}( T_{L})$ is defined as the supremum of all $\varepsilon>0$ satisfying the following property: for all $M>0$ there exists $K>0$ depending on $M$ so that $$\mathrm{P}_{F}(|y|\leq M)>1-\varepsilon\text{ implies }|T_{L}(F)|<K.
\label{epsilonun}$$
For any location functional $T_{L}$ we have that $\varepsilon_{U}^{\ast}(T_{L})\leq0.5.$ This is an immediate consequence of the following two facts: (a) $\varepsilon_{A}^{\ast}$ $($ $T_{L},F)$ $\leq0.5$, for all location functionals $T_{L}$ and all $F$, where $\varepsilon_{A}^{\ast}$ $($ $T_{L},F)$ is the asymptotic breakdown point of $T_{L}$ at the distribution $F$, while (b) $\varepsilon_{U}^{\ast}(T_{L})$ $\leq\varepsilon_{A}^{\ast}$ $($ $T_{L},F)$ for all $F$. In the case that $T_{L}$ is the median it is immediate to show that $\varepsilon_{U}^{\ast}=0.5$. In fact, for any $\varepsilon<0.5$, choosing $K=M$ we get that (\[epsilonun\]) holds. This proves that $\varepsilon_{U}^{\ast}\geq0.5$ and therefore $\varepsilon
_{U}^{\ast}=0.5.$ The following Theorem gives a lower bound for the FSBP of the estimate $\widehat{\mu}_{n}$ defined in (\[OUREST\]).
\[BDP\]Let $\mathbf{W}$ be given by (\[Wcom\]) and let $\mathbf{Z}=\{(\mathbf{x}_{i},y_{i}):i\in A\}.$ Suppose that $\widehat{\mathbf{\beta}}_{n}$=$\widetilde{\mathbf{\beta}}_{m}$ $(\mathbf{Z}),$ where $\widetilde
{\mathbf{\beta}}_{m}$ is a regression estimate for samples of size $m.$ Let $\varepsilon_{1}>0$ be the FSBP at $\mathbf{Z}$ of $\widetilde{\mathbf{\beta
}}_{m}$ and call$\ \varepsilon_{2}>0$ the UABP of $T_{L}.$ Then the FSBP $\varepsilon^{\ast}$ of the estimate $\widehat{\mu}_{n}$ at $\ \mathbf{W}$ satisfies $$\varepsilon^{\ast}\geq\varepsilon_{3}=\min(\varepsilon_{1},1-\sqrt
{1-\varepsilon_{2}}).$$
In the next Section we introduce MM estimates of regression and location. The maximum value of $\varepsilon_{1}$ for an MM estimate of regression is $(n-c(G_{n}^{\ast}))/(2n)$, where $c(G)$ is defined be (\[cG\]) (see Martin et al. [@MaronnaMartin; @and; @Yohai]). In Theorem \[locationbound\] we show that maximum value of $\varepsilon_{2}$ for an MM estimate of location is $0.5$. Then, if $\ c(G_{n}^{\ast})/n$ is small, we can have have $\varepsilon_{3}$ close to $1-\sqrt{0.5}=0.293.$ A similar statement holds when we change $T_{L}$ by the median.
MM Regression and Location Functionals\[SMM\]
=============================================
Several robust estimates for the parameters of the regression model (\[modelogral\]) based on complete data $(\mathbf{x}_{1},y_{1}),\dots,(\mathbf{x}_{n},y_{n})$ have been proposed. In this paper we will consider MM estimates. These estimates were introduced by Yohai [@Yohai87] for the linear model while Fasano, Maronna, Sued and Yohai [@Fasano; @et; @al] extended these estimates to the case of nonlinear regression. For linear regression, MM estimates may combine the highest possible breakdown point with an arbitrarily high efficiency in the case of Gaussian errors. It will be convenient to present MM-estimates of $\
$**$\beta$**$_{0}$ in their functional form, i.e., as a functional $\mathbf{T}_{MM,\beta}(G)$ defined on a set of distributions in $\ \mathbb{R}^{p+1}$, taking values in $\mathbb{R}^{q}.$ Given a sample $(\mathbf{x}_{1},y_{1}),\dots,(\mathbf{x}_{n},y_{n})$ the corresponding estimate of **$\beta$**$_{0}$ is given by $\widehat{\mathbf{\beta}}_{MM}=\mathbf{T}_{MM,\beta}(G_{n}),$ where $G_{n}$ is the empirical distribution of the sample. As we explained in the Introduction, we have excluded the intercept in model (\[modelogral\]). However in order to guarantee the consistency of the estimates without requiring symmetric errors it is convenient to estimate an additional parameter which can be naturally interpreted as an intercept or a center of the error distribution. For this purpose put **$\xi$**$=($**$\beta$**$,\alpha)$ with $\alpha\in{\mathbb{R}}$, and define $\underline{g}(\mathbf{x},$[$\mathbf{\xi}$]{}$)=g(\mathbf{x},$**$\beta$**$)+\alpha$.
To define a regression MM functional $\mathbf{T}_{MM}(G)=(\mathbf{T}_{MM,\beta}(G),T_{MM,\alpha}(G))$ two loss functions, $\ \rho_{0}$ $\ $ and $\rho_{1}$ are required$.$ The function $\rho_{0}$ is used to define a dispersion functional $S(G)$ of the error distribution$.$Then $\mathbf{T}_{MM}$ is defined as a regression M functional with loss function $\rho
_{1}$ and scale given by $\ S(G).$
Throughout this work, a *bounded* $\rho$–*function* is a function $\rho\left( t\right) $ that is a continuous nondecreasing function of $|t|,$ such that $\rho(0)=0,$ $\rho\left( \infty\right) =1,$ and $\rho\left(
v\right) <1$ implies that $\rho\left( u\right) <\rho(v)$ for $|u|<|v|.$ We also assume that $\rho_{1}(t)\leq\rho_{0}(t)$ for all $t.$
We start by defining the dispersion functional. For any $\ $distribution $G$ of $(\mathbf{x},y)$ and $\mathbf{\xi}=($**$\beta$**$,\alpha),$ let $S^{\ast}(G,\mathbf{\xi})$ be defined by$$\mathrm{E}_{G}\rho_{0}\left( \frac{y-\underline{g}(\mathbf{x},\mathbf{\xi})}{S^{\ast}(G,\mathbf{\xi})}\right) =\delta, \label{Sestrella}$$ where $\delta\in(0,1)$. Then the dispersion functional $S(G)$ is defined by $$S(G)=\min\limits_{\mathbf{\xi}\in B\times\mathbb{R}}S^{\ast}(G,\mathbf{\xi})
\label{SS}$$ and the MM estimating functional $\mathbf{T}_{\mathrm{MM}}(G)=(\mathbf{T}_{\mathrm{MM},\mathbf{\beta}}(G),T_{\mathrm{MM},\alpha}(G))$ by$$\mathbf{T}_{\mathrm{MM}}(G)=\arg\min\limits_{\mathbf{\xi}\in B\times
{\mathbb{R}}}\mathrm{E}_{G}\left[ \rho_{1}\left( \frac{y-\underline
{g}(\mathbf{x},\mathbf{\xi})}{{S}(G)}\right) \right] . \label{TMM}$$
We can also consider another regression functional $\mathbf{T}_{\mathrm{S}}(G)=(\mathbf{T}_{\mathrm{S},\mathbf{\beta}}(G),T_{\mathrm{S},\alpha}(G))$, called *regression S functional*, as follows: $$\mathbf{T}_{\mathrm{S}}(G)=\arg\min\limits_{\mathbf{\xi}\in B\times
{\mathbb{R}}}\mathrm{E}_{G}\left[ \rho_{0}\left( \frac{y-\underline
{g}(\mathbf{x},\mathbf{\xi})}{{S}(G)}\right) \right] . \label{TSS}$$
In the case of linear regression, the asymptotic breakdown point of both $\mathbf{T}_{MM}$ and $\mathbf{T}_{S}$ is given by $$\varepsilon^{\ast}=\min(\delta,1-\delta-c(G)), \label{BDPa}$$ where$$\label{cG}c(G)=\sup_{\mathbf{\gamma}\neq0,\mathbf{\gamma}\in\mathbb{R}^{p+1}}\mathrm{P}_{G}(\mathbf{\gamma}^{\prime}(\mathbf{x}^{\prime},1)^{\prime}=0).$$
The maximum breakdown point occurs when $\delta=(1-c(G))/2$ and its value is ($1-c(G))/2.$ It can be proved that this is the maximum possible breakdown point for equivariant regression functionals. In the case of nonlinear regression both $\mathbf{T}_{MM}$ and $\mathbf{T}_{S}$ $\ $ have also the same breakdown point but it is not given by a simple closed expression (see Fasano [@FasanoTesis]). Yohai [@Yohai87] showed that MM estimates for linear regression may combine the highest possible breakdown point $(1-c(G))/2$ with a Gaussian efficiency as high as desired. Instead, Hössjer [@Hossjer] showed that this is not possible for S estimates. The maximum asymptotic Gaussian efficiency of an S estimate with $\varepsilon^{\ast}=(1-c(G))/2$ is 0.33.
Let $(\mathbf{x},y)$ and $\ u$ satisfy model (\[modelogral\]). Let $\{\mathrm{G}_{n}^{\ast}\}$ be the sequence of empirical distribution associated with observed pairs $(\mathbf{x}_{i},y_{i})$, i.e., those pairs such that $\ a_{i}=1:$$$G_{n}^{\ast}=\frac{1}{\sum_{i=1}^{n}a_{i}}\,\sum_{i=i}^{n}a_{i}\delta
_{(\mathbf{x}_{i},y_{i})}. \label{pseudoempiricas}$$ Then we can estimate **$\beta$**$_{0}$ by $$\widehat{\mathbf{\beta}}_{n}=\mathbf{T}_{MM,\mathbf{\beta}}(G_{n}^{\ast}).
\label{OURMM}$$
We can also choose as location functional $T_{L}$, whose value at $\mu
_{0}=T_{L}(F_{0})$ we want to estimate, a location MM functional. MM and S location functionals are defined similarly to the regression case. Let $\rho_{1}^{L}$ and $\rho_{0}^{L}$ be bounded$\ \rho$-functions. We start by defining the dispersion functional. For any $\ $distribution $F$ of $y$ and $\mu\in\mathbb{R}\ $ let $S_{L}^{\ast}(F,\mu)$ be defined by$$\mathrm{E}_{F}\rho_{0}^{L}\left( \frac{y-\mu}{S_{L}^{\ast}(F,\mathbf{\xi})}\right) =\delta,$$ where $\delta\in(0,1)$. Then the dispersion functional $S_{L}(F)$ is defined by $$S_{L}(F)=\min\limits_{\mu\in\mathbb{R}}S_{L}^{\ast}(F,\mu)$$ and the MM location functional $T_{\mathrm{MM}}^{L}(F)$ by$$T_{\mathrm{MM}}^{L}(F)=\arg\min\limits_{\mu\in\mathbb{R}}\mathrm{E}_{F}\left[
\rho_{1}^{L}\left( \frac{y-\mu}{{S}_{L}(F)}\right) \right] .
\label{del loc}$$
The S location functional $T_{\mathrm{S}}^{L}(F)$ is defined similarly to the regression S functional. We denote by $\mu_{00}=T_{\mathrm{S}}^{L}(F_{0})$ and $\mu_{01}=T_{\mathrm{MM}}^{L}(F_{0})$, whenever they are well defined. Location MM estimates may also combine high breakdown point with high Gaussian efficiency and their breakdown point is given by $\varepsilon^{\ast
}=\min(\delta,1-\delta).$
For the validity of assumptions A1-A4, the $\rho$-functions used to define the location and regression MM functionals should satisfy assumptions R1 and R2 below.
**R1** For some $m,$ $\rho(u)=1$ iff $|u|\geq m,$ and $\log(1-\rho)$ is concave on $(-m,m)$ .
**R2** $\rho$ is twice continuously differentiable
A family of very popular bounded $\rho-$function satisfying R0,R1 and R2 is Tukey’s bisquare family: $$\rho_{k}^{T}\left( u\right) =1-\left( 1-\left( \frac{u}{k}\right)
^{2}\right) ^{3}I(|u|\leq k)$$
for $\ k>0.$
We denote by $\psi_{0}$, $\psi_{1},\psi_{0}^{L}$ and $\psi_{1}^{L}$ the derivatives of $\rho_{0}$, $\rho_{1},\rho_{0}^{L}$ and $\rho_{1}^{L}$. Put $\alpha_{01}=T_{MM,\alpha}(G_{0}^{\ast}),$ $\alpha_{00}=T_{S,\alpha}(G_{0}^{\ast})$ and $\sigma_{0}=S(G_{0}^{\ast})$
Both regression and location MM and S functionals are studied in detail in Fasano et al. [@Fasano; @et; @al]. There we can find sufficient conditions for weak continuity and Fisher-consistency.Moreover, a weak differentiability notion involving the influence function of the functionals is also developed. This notion allows to obtain asymptotic expansions, like those required in (\[TRexpansion\]) and (\[TLexpansion\]). The following numbers will be used to derive the influence functions of the regression functionals: $$a_{0i}=\mathrm{E}_{G_{0}^{\ast}}\psi_{i}^{\prime}\left( (y-g(\mathbf{x},\mathbf{\beta}_{0})-\alpha_{0i})/\sigma_{0}\right) =\mathrm{E}_{K_{0}}\psi_{i}^{\prime}\left( (u-\alpha_{0i})/\sigma_{0}\right) ,i=0,1,$$$$e_{0i}=\mathrm{E}_{K_{0}}\left[ \psi_{i}^{\prime}\left( (u-\alpha
_{0i})/\sigma_{0}\right) (u-\alpha_{0i})/\sigma_{0}\right] ,i=0,1,$$$$d_{0}=\mathrm{E}_{K_{0}}\left[ \psi_{0}\left( (u-\alpha_{00})/\sigma
_{0}\right) (u-\alpha_{00})/\sigma_{0}\right] \quad\hbox{and}\quad
\mathbf{b}_{0}=\mathrm{E}_{G_{0}^{\ast}}\dot{g}(\mathbf{x},\mathbf{\beta}_{0}).$$ Similarly we define $a_{0i}^{L}$, $\ e_{0i}^{L}$, $d_{0}^{L}$ and $\sigma
_{0}^{L}$ replacing $\psi_{_{i}}$ by $\psi_{i}^{L}$, $K_{0}$ by $F_{0}$, $g(\mathbf{x},$**$\beta$**$_{0})$ by $0$, $\alpha_{0i}$ by $\mu_{0i}$ and $\sigma_{0}$ by $\sigma_{0}^{L}=S_{L}(F_{0})$. We denote by $A_{0}$ the covariance matrix of $\dot{g}(\mathbf{x},$**$\beta$**$_{0})$ under $Q_{0}^{\ast}.$
Theorems \[asslreg\] and \[assloc\] summarize the results for MM functionals of regression and location, respectively.
\[asslreg\]Let $\rho_{0}$ and $\rho_{1}$ be bounded $\rho$-functions satisfying $\ $R1, with $\rho_{1}\leq\rho_{0}$. Assume that $K_{0}$ has a strongly unimodal density and that (\[IDCOND\]) holds replacing $Q_{0}$ by $Q_{0}^{\ast}$. We will consider that either (a) $B$ is compact or (b) $g(\mathbf{x},$**$\beta$**$)=\mathbf{\beta}^{\prime}\mathbf{x}$ and $\delta<1-c(G_{0}^{\ast})$. Then
- $\lim_{n\rightarrow\infty}$ $\mathbf{T}_{\mathrm{MM,}\mathbf{\beta}}(G_{n}^{\ast})=$**$\beta$**$_{0}$ $\ $ a.s. and therefore A1 is satisfied.
- Assume also that $a_{00},$ $a_{01}$ and $d_{0}$ are different from $0$, that A0 holds and that $\rho_{0}$ and $\rho_{1}$ satisfies R2. Then (\[TRexpansion\]) holds with $I_{R}(\mathbf{x},y)=I_{\mathbf{T}_{MM,\mathbf{\beta}},G_{0}^{\ast}}(\mathbf{x,}y\mathbf{)/}\mathrm{E}(a_{1})$ , where $I_{\mathbf{T}_{MM,\mathbf{\beta}},G_{0}^{\ast}}(\mathbf{x,}y\mathbf{)}
$ is the influence function of $\mathbf{T}_{MM,\mathbf{\beta}}$ at $G_{0}^{\ast}.$ Moreover, we have that $$I_{\mathbf{T}_{MM,\mathbf{\beta}},G_{0}^{\ast}}(\mathbf{x,}y\mathbf{)}=\frac{\sigma_{0}}{a_{01}}\psi_{1}\left( \frac{y-\underline{g}(\mathbf{x},(\mathbf{\beta}_{0},\alpha_{01}))}{\sigma_{0}}\right) A_{0}^{-1}(\dot
{g}(\mathbf{x},\mathbf{\beta}_{0})-\mathbf{b}_{0}), \label{ICEX2}$$ and therefore A2 holds.
\[assloc\] Let $\rho_{0}^{L}$ and $\rho_{1}^{L}$ be bounded $\rho
$-functions satisfying $\ $R1, with $\rho_{1}^{L}\leq\rho_{0}^{L}$. Assume that $\ F_{0}$ has a strongly unimodal density. Then
- There is only one value $\mu_{01}=T_{MM}^{L}$ $(F_{0})$ that attains the minimum at (\[del loc\]), $T^{L}_{MM}$ is continuous at $F_{0}$, and so A3 holds. In the case that $F_{0}$ is symmetric around $\nu_{0},$ we have $\mu_{01}=\nu_{0}$.
- Assume also A0, that $\rho_{0}^{L}$ and $\rho_{1}^{L}$ satisfy R2 and that $a_{00}^{L},$ $a_{01}^{L}$ and $d_{0}^{L}$ are different from $0.$ Then (\[TLexpansion\]) holds when $I_{L}(y)$ is the influence function of $T_{MM}^{L}$ at $F_{0}.$ Moreover we have $$I_{L}(y)=\frac{\sigma_{0}^{L}}{a_{01}^{:L}}\psi_{1}^{L}\left( \frac
{y-\mu_{01})}{\sigma_{0}^{L}}\right) -\frac{e_{01}^{L}\ \sigma_{0}^{L}}{a_{01}^{L}d_{0}^{L}}\left( \rho_{0}^{L}\left( \frac{y-\mu_{00}}{\sigma
_{0}^{L}}\right) -\delta\right) , \label{ifloc1}$$ and therefore A4 holds.
- In case that $F_{0}$ is symmetric with respect to $\nu_{0}$ we have $e_{0}=0$ and $$I_{L}(y)=\frac{\sigma_{0}^{L}}{a_{01}^{:L}}\psi_{1}^{L}\left( \frac{y-\nu
_{0})}{\sigma_{0}^{L}}\right) .$$
To end this Section, we state the announced result regarding the uniform bound required for the location functional in order to deduce a lower bound for the FSBD of $\widehat\mu_{n}$, introduced in Section \[secBDP\].
\[locationbound\] Let $T_{MM}^{L}$ be an MM location functional. Then its uniform asymptotic breakdown point is $\varepsilon_{U}^{\ast}=\min
(1-\delta,\delta)$.
Monte Carlo study \[SECMC\]
===========================
In order to assess how the proposed robust method compares to the classical procedure that uses as $\widehat{\mathbf{\beta}}_{n}$ the least squares and as $T_{L}$ the mean functional, we performed a Monte Carlo study. We consider the following model $$y_{i}=3x_{i1}+...+3x_{i5}+u_{i},1\leq i\leq100,$$ where $x_{i1},...,x_{i5}$ are i.i.d. random variables with uniform distribution in the interval $[0,1]$, $u_{i}$ are standardized normal variables ($u_{i}\sim{\mathcal{N}}(0,1)$) and $\beta_{1}=\beta_{2}=...=\beta_{5}=3$. The missingness indicators $a_{i}$ were generated using a logistic model. Let $\mathbf{x}_{i}=(x_{i1},...,x_{i5})$, then $$\log\frac{\mathrm{P}(a_{i}=1|\mathbf{x}_{i})}{1-\mathrm{P}(a_{i}=1|\mathbf{x}_{i})}=0.57(x_{i1}+...+x_{i5}).$$ Using this model and the distribution of the covariables, we have $\mathrm{P}(a_{i}=1)=0.80$.
We study (a) the case with no outlier contamination and (b) the case where 10% of the observations $(\mathbf{x,}$ $y_{i})$’s with $a_{i}=1$ are replaced by $(\mathbf{x}^{\ast},y^{\ast})$, with $\mathbf{x}^{\ast}=(x^{\ast
},...,x^{\ast})$. We take two values for $x^{\ast}$: $1$ and $3$, and for $y^{\ast}$ we take a grid of values over the interval $[8,50]$, with steps of $0.20$. For each case we performed 1000 replications. We consider four functionals $T_{L}$ : (i) the mean (MEAN in Figure 1), (ii) the median (MEDIAN in Figure 1) (iii) an MM location functional with $\rho_{i}^{L}=\rho_{T,k_{i}},$ $k_{0}$=$1.57$, $k_{1}=3.88$ and $\delta=0.5$ $.$ The corresponding location estimate has a Gaussian asymptotic efficiency of 90% (MM90 in Figure 1). (iv) Finally we study an MM location functional defined as in (iii) with constants $k_{0}$=$1.57$, $k_{1}=4.68$ and $\delta=0.5$. This location estimate has a Gaussian asymptotic efficiency of 95% (MM95 in Figure 1). Note that in the case in which there is no outlier contamination, the distribution $F_{0}$ is symmetric with center of symmetry $7.5$, and then $T_{L}(F_{0})=\mathrm{E}(y)=7.5$ in the four cases. When $T_{L}$ is the mean, $\widehat{\mathbf{\beta}}_{n}$ is the least squares (LS) estimate. In the other 3 cases $\widehat{\mathbf{\beta}}_{n}$ is an MM estimate with $\rho
_{i}=\rho_{T,k_{i}},$ $k_{0}$=$1.57$, $k_{1}=3.44$ and $\delta=0.5.$ This estimate has an asymptotic efficiency of 85% in the case of Gaussian errors and breakdown point close to $0.5.$ In Table 1 we show the mean square errors (MSE), and the relative efficiencies of the four estimates when there is no outlier contamination. In Figure 1 we plot the MSE of the four estimates under outlier contamination.
Table 1. MSE and efficiencies without outliers
$$\begin{tabular}
[c]{lllll}Estimates & MEAN & MEDIAN & MM90 & MM95\\
MSE & 0.047 & 0.056 & 0.051 & 0.049\\
Efficiency & 100\% & 83\% & 91\% & 95\%
\end{tabular}
$$
\[ptb\]
[fig1.eps]{}
As expected, when there are no outliers the classical estimate based on the mean is the most efficient, but the estimates based on the MM estimates are highly efficient too. The estimate based on the median is less efficient, but its efficiency is larger than that of the sample median which is 64%. Note that the estimate based on the median is an U-statistics similar to the Hodges–Lehmann estimate, which is also more efficient than the median.
When there are outliers, we observe that the MSE of the estimate based on the mean increases beyond any limit, while for the robust estimates the MSE remains bounded. In the case of $x^{\ast}=1$ the MSE of MM95 is larger than those of MEDIAN and MM90. For $x^{\ast}=3$ the MSE of MEDIAN is larger than those of the other two robust estimates. The MSEs of MM90 and MM95 are practically the same. Based on these results we recommend to use MM90 which has a very good behavior with and without outliers.
Acknowledgements {#acknowledgements .unnumbered}
=================
The authors would like to thank Graciela Boente and Sara van der Geer for valuable discussions and suggestions and also to Damian Scherlis for his careful reading of the first manuscript.
Appendix[\[Apendice\]]{}
========================
The following result plays a crucial role in the proof of Theorem \[lemaconvdebil1\].
\[convdebil\] Let $\{\mathbf{z}_{i}\}$ be a sequence of i.i.d. random vectors taking values in ${\mathbb{R}}^{k}$ and let $h:{\mathbb{R}}^{k}\times{\mathbb{R}}^{q}\rightarrow{\mathbb{R}}$ be a continuous function. Assume that $\widehat{\mathbf{\beta}}_{n}$ is a strongly consistent sequence of estimators of $\beta_{0}\in{\mathbb{R}}^{q}$. Denote by $\widehat{H}_{n}$ the empirical distribution at $h(\mathbf{z}_{i},\widehat{\mathbf{\beta}}_{n})$, $1\leq i\leq n$ and by $H_{0}$ the distribution of $h(\mathbf{z}_{1},\beta_{0})$. Then $\widehat{H}_{n}$ converges weakly to $H_{0}$ a.s., i.e.$$\mathrm{P}(\widehat{H}_{n}\rightarrow_{w}H_{0})=1. \label{convdebilae}$$
Recall that weak convergence is characterized by the following property: $$H_{n}\rightarrow H\hbox{weakly}\Leftrightarrow\int f\,dH_{n}\rightarrow\int
f\,dH,\quad\forall f\in{\mathcal{C}}_{B}({\mathbb{R}}),$$ where ${\mathcal{C}}_{B}({\mathbb{R}})$ denotes the set of continuous bounded functions. Denote by $\tilde{H_{n}}$ the empirical distribution at $h(\mathbf{z}_{i},\beta_{0})$, for $1\leq i\leq n$. By the Glivenko-Cantelli Theorem, $\tilde{H_{n}}$ converges uniformly to $H_{0}$, a.s. and so it also converges weakly a.s. Then, it remains to find a set of probability one where $$\lim\limits_{n\rightarrow\infty}\left\vert \int f\,d\widehat{H}_{n}-\int
f\,d\tilde{H_{n}}\right\vert =0,\text{ }\forall f\in{\mathcal{C}}_{B}({\mathbb{R}}).$$ Observe that $$\int f\,d\widehat{H}_{n}=\frac{1}{n}\sum_{i=1}^{n}f\left( h(\mathbf{z}_{i},\widehat{\beta}_{n})\right) ,\text{ }\int f\,d\tilde{H_{n}}=\frac{1}{n}\sum_{i=1}^{n}f\left( h(\mathbf{z}_{i},\widehat{\beta}_{0})\right) ,$$ and so $$\left\vert \int f\,d\widehat{H}_{n}-\int f\,d\tilde{H_{n}}\right\vert
\leq\frac{1}{n}\sum_{i=1}^{n}\left\vert f\left( h(\mathbf{z}_{i},\widehat{\beta}_{n})\right) -f\left( h(\mathbf{z}_{i},\widehat{\beta}_{0})\right) \right\vert I_{\{|\mathbf{z}_{i}|\leq K\}}+2||f||_{\infty}\frac{1}{n}\sum_{i=1}^{n}I_{\{|\mathbf{z}_{i}|>K\}}.$$
Put $C_{K}=\{(\mathbf{z},\beta):||\mathbf{z}||\leq K,||\beta-\beta_{0}||\leq1\}$. We have that $f\circ h:C_{K}\rightarrow\mathbb{R}$ is uniformly continuous and so, given $\varepsilon>0$, there exists $\delta>0$ such that if $(\mathbf{z}_{i},\beta_{i})\in C_{K}$ and $||(\mathbf{z}_{1},\beta
_{1})-(\mathbf{z}_{2},\beta_{2})||\leq\delta$, then $|f(h(\mathbf{z}_{1},\beta_{1}))-f(h(\mathbf{z}_{1},\beta_{2}))|\leq\varepsilon$. With probability one there exists a random integer $n_{0}$ $\ $ such that $|\widehat
{\mathbf{\beta}}_{n}-$**$\beta$**$_{0}|\leq\delta$ for all $n\geq n_{0}$. Then we get $$\left\vert \int f\,d\widehat{H}_{n}-\int f\,d\tilde{H_{n}}\right\vert
\leq\varepsilon+2||f||_{\infty}\frac{1}{n}\sum_{i=1}^{n}I_{\{|\mathbf{z}_{i}|>K\}},$$ for all $n\geq n_{0}$. Assume also that $$\frac{1}{n}\sum_{i=1}^{n}I_{\{|\mathbf{z}_{i}|>K\}}\rightarrow\mathrm{P}\left( |\mathbf{z}_{1}|>K\right) ,\forall K.$$ Then, with probability one $$\lim\limits_{n\rightarrow\infty}\left\vert \int f\,d\widehat{H}_{n}-\int
f\,d\tilde{H_{n}}\right\vert \leq\varepsilon+2||f||_{\infty}\,\mathrm{P}\left( |\mathbf{z}_{1}|>K\right) ,\forall\varepsilon>0\,,\forall K.$$ To get the desired result, let $\varepsilon\rightarrow0$ and $K\rightarrow
\infty$.
The following results will be used throughout the proofs of the Theorems stated in the previous Sections. We start proving that the convolution preserves weak continuity.
\[convol\] Assume that $K_{n}\rightarrow_{w}K_{0}$ and $\ R_{n}\rightarrow_{w}R_{0}.$ Then $K_{n}\ast R_{n}\rightarrow_{w}K_{0}\ast R_{0}.$
Let $(U,V)$ be independent random variables, both with uniform distribution on $[0,1]$. Given a distribution function $F$, denote by $F^{-1}$ the generalized inverse function of $F$, whose value at $t$ is given by the infimum of the set $\{s:t\leq F(s)\}$. Consider $U_{n}=K_{n}^{-1}(U)$ and $V_{n}=R_{n}^{-1}(V)$. It is known that (i) $U_{n}$ and $V_{n}$ are distributed according $K_{n}$ and $R_{n}$, respectively and (ii) $U_{n}$ and $V_{n}$ converge a.s. to $U_{0}=K_{0}^{-1}(U)$ and $V_{0}=R_{0}^{-1}(V)$, respectively (see Theorem 25.6 (Billingsley (1995)) for details). Then $U_{n}+V_{n}$ converges a.s. to $U_{0}+V_{0}$, and then the convergence holds also in distribution. The independence between $U$ and $V$ implies that $U_{n}+V_{n}\sim K_{n}\ast R_{n}$ while $U_{0}+V_{0}\sim K_{0}\ast
R_{0}$, proving the Lemma.
\[g-c\]Consider $\{(a_{i},\mathbf{z}_{i})\}$ i.i.d. random vectors, with Bernoulli $a_{i}$ and $\mathbf{z}_{i}\in\mathbb{R}^{h}$ Then $$\sup_{z\in{\mathbb{R}}^{h}}\left\vert \frac{1}{n}\sum_{i=1}^{n}a_{i}I_{\{\mathbf{z}_{i}\leq\mathbf{z\}}}-\mathrm{E}\left[ a_{1}I_{\{\mathbf{z}_{1}\leq z\}}\right] \right\vert =0,\;\text{as..}$$
Note that$$a_{i}I_{\{\mathbf{z}_{i}\leq\mathbf{z\}}}=I_{\{\mathbf{z}_{i}\leq\mathbf{z\}}}-I_{\{\mathbf{z}_{i}\leq\mathbf{z,}a_{i}\leq0\mathbf{\}}}. \label{GC0}$$ By the Glivenko-Cantelli Theorem we have$$\sup_{z\in{\mathbb{R}}^{h}}\left\vert \frac{1}{n}\sum_{i=1}^{n}I_{\{\mathbf{z}_{i}\leq\mathbf{z\}}}-\mathrm{E}\left[ I_{\{\mathbf{z}_{1}\leq z\}}\right]
\right\vert =0,\;\text{a.s.} \label{GC1}$$ and$$\sup_{z\in{\mathbb{R}}^{h}}\left\vert \frac{1}{n}\sum_{i=1}^{n}I_{\{\mathbf{z}_{i}\leq\mathbf{z,}a_{i}\leq0\mathbf{\}}}-\mathrm{E}\left[ I_{\{\mathbf{z}_{1}\leq\mathbf{z,}a_{1}\leq0\mathbf{\}}}\right] \right\vert
=0,\;\text{a.s.} \label{GC2}$$ From (\[GC0\]),(\[GC1\]) and (\[GC2\]) we get$$\sup_{z\in{\mathbb{R}}^{h}}\left\vert \frac{1}{n}\sum_{i=1}^{n}a_{i}I_{\{\mathbf{z}_{i}\leq\mathbf{z\}}}-\mathrm{E}\left[ I_{\{\mathbf{z}_{1}\leq
z\}}-I_{\{\mathbf{z}_{1}\leq\mathbf{z,}a_{1}\leq0\mathbf{\}}}\right]
\right\vert .$$ and by applying (\[GC0\]) to $i=1$ the Lemma follows.
The proof of the following Lemma is similar to that of Lemma 4.2 presented by Yohai in [@Yohai85]. It suffices to replace the law of large numbers for i.i.d., variables by the same law for U statistics.
\[victor4.2\] Assume that $\{\mathbf{z}_{i}\}$ are i.i.d. random vectors taking values in ${\mathbb{R}}^{k}$, with common distribution $Q$. Let $f:{\mathbb{R}}^{k}\times{\mathbb{R}}^{k}\times{\mathbb{R}}^{h}\rightarrow
{\mathbb{R}}$ be a continuous function. Assume that for some $\delta>0$ we have that$$\mathrm{E}\sup_{||\lambda-\lambda_{0}||\leq\delta}|f(\mathbf{z}_{1},\mathbf{z}_{2},\lambda)|<\infty$$ $\ $and that $\widehat{\lambda}_{n}\rightarrow\lambda_{0}$ a.s. Then$$\frac{1}{n^{2}}\sum_{j=1}^{n}\sum_{1=1}^{n}f(\mathbf{z}_{i},\mathbf{z}_{j},\widehat{\lambda}_{n})\rightarrow\mathrm{E}f(\mathbf{z}_{1},\mathbf{z}_{2},\lambda_{0})\,\text{\ a.s.} \label{lgnu}$$
<span style="font-variant:small-caps;">Proof of Theorem \[lemaconvdebil1\].</span> According to Lemma \[convol\], it only remains to prove the a.s. weak convergence of $\widehat{R}_{n}$ $\ $ and $\widehat{K}_{n}$ to $\ R_{0}$ and $K_{0}$ respectively. The a.s. weak convergence of $\widehat{R}_{n}$ to $R_{0}$ follows from Lemma \[convdebil\], putting $\mathbf{z}=\mathbf{x}$ and $h(\mathbf{z},\beta)=g(\mathbf{x},\beta)$. Weak convergence of $(\widehat
{K}_{n})_{n\geq1}$ to $K_{0}$ requires an extra argument. If $\mathbf{z}=(\mathbf{x},y)$ and $h(\mathbf{z},\beta)=y-g(\mathbf{x},\beta)$, we get that $$\widehat{K}_{n}(u)=\frac{1}{\sum_{i=1}^{n}a_{i}}\,{\sum_{i=1}^{n}a_{i}\,I_{\{h(\mathbf{z}_{i},\widehat{\mathbf{\beta}}_{n}))\leq u\}}}.$$ By Lemma \[g-c\], we obtain$$\sup_{u\in{\mathbb{R}}}\left\vert \frac{1}{n}\sum_{i=1}^{n}a_{i}I_{\{u_{i}\leq
u\}}-\mathrm{E}\left[ a_{1}I_{\{u_{1}\leq u\}}\right] \right\vert
=0\;\text{a.s.}$$ Since $a_{1}$ and $u_{1}$ are independent, we conclude that $$\sup_{u\in{\mathbb{R}}}\left\vert \frac{1}{\sum_{i=1}^{n}a_{i}}\sum_{i=1}^{n}a_{i}I_{\{u_{i}\leq u\}}-K_{0}(u)\right\vert =0\;\text{a.s.}$$ and $\ $ then $\sum_{i=1}^{n}a_{i}I_{\{u_{i}\leq u\}}/\sum_{i=1}^{n}a_{i}$ converges weakly to $K_{0}$ a.s. An argument similar to the one used in Lemma \[convdebil\] shows that with probability one we have$$\lim\limits_{n\rightarrow\infty}\left\vert \int f\,d\widehat{K}_{n}-\int
f\,d\tilde{K_{n}}\right\vert =0,\quad\forall f\in{\mathcal{C}}_{B}({\mathbb{R}}),$$ proving the a.s. weak convergence of $\widehat{K}_{n}$ to $K_{0}$ . This concludes the proof of part (a) of Theorem \[lemaconvdebil1\]. (b) is an immediate consequence of weak continuity of $T_{L}$. $\square$
<span style="font-variant:small-caps;">Proof of Theorem \[asym\].</span> According to A4, we have that $$\sqrt{n}(\widehat{\mu}_{n}-\mu_{0})\;=\;\sqrt{n}\Big\{T_{L}(\widehat{F}_{n})-T_{L}(F_{0})\Big\}\;=\;\sqrt{n}\mathrm{E}_{\widehat{F}_{n}}I_{L}(y)\,+\,o_{P}(1).$$ Note that $$\mathrm{E}_{\widehat{F}_{n}}I_{L}(y)=\frac{1}{\eta_{n}n^{2}}{\sum_{j=1}^{n}}{\sum_{i=1}^{n}}a_{i}I_{L}(y_{i}-g(\mathbf{x}_{i},\widehat{\mathbf{\beta}}_{n})+g(\mathbf{x}_{j},\widehat{\mathbf{\beta}}_{n})),$$ where $\eta_{n}=\sum_{i=1}^{n}a_{i}/n$. Since $\eta_{n}\rightarrow
\mathrm{E}[a_{i}]=\eta$, to prove Theorem \[asym\], it is enough to show that$$V_{n}\rightarrow_{d}N(0,(\eta\tau)^{2}),$$ where $$V_{n}=\frac{1}{n^{3/2}}{\displaystyle\sum_{j=1}^{n}}
{\displaystyle\sum_{j=1}^{n}}
a_{i}I_{L}(y_{i}-g(\mathbf{x}_{i},\widehat{\mathbf{\beta}}_{n})+g(\mathbf{x}_{j},\widehat{\mathbf{\beta}}_{n})).$$ Performing a Taylor expansion, we can write $$V_{n}=d_{n}+\mathbf{c}_{n}^{\prime}n^{1/2}(\widehat{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0}),$$ where $$d_{n}=\frac{1}{n^{3/2}}{\displaystyle\sum_{i=1}^{n}}
{\displaystyle\sum_{j=1}^{n}}
a_{i}I_{L}(u_{i}+g(\mathbf{\beta}_{0},\mathbf{x}_{j}))$$ and$$\mathbf{c}_{n}=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\ell(a_{i},\mathbf{x}_{i},y_{i},a_{j},\mathbf{x}_{j},y_{j},\beta_{n}^{\ast})$$ with **$\beta$**$_{n}^{\ast}$ between $\widehat{\mathbf{\beta}}_{n}$ and **$\beta$**$_{0}$, and $$\ell(a_{i},\mathbf{x}_{i},y_{i},a_{j},\mathbf{x}_{j},y_{j},\beta)=a_{i}\,I_{L}^{\prime}(y_{i}-g({\mathbf{\beta}},\mathbf{x}_{i})+{g}({\mathbf{\beta}},\mathbf{x}_{j}))\,\{\dot{g}({\mathbf{\beta}},\mathbf{x}_{j})-\dot
{g}({\mathbf{\beta}},\mathbf{x}_{i})\}.$$ By Lemma \[victor4.2\] $$\mathbf{c}_{n}\rightarrow\mathbf{c}=\mathrm{E}\ell(a_{1},\mathbf{x}_{1},y_{1},a_{2},\mathbf{x}_{2},y_{2},\beta_{0})\text{ a.s.} \label{asf1}$$ From the U-statistics projection Theorem we get $$d_{n}=\frac{1}{n^{1/2}}{\displaystyle\sum\limits_{i=1}^{n}}
e(\mathbf{x}_{i},u_{i},a_{i})+f(\mathbf{x}_{i})+o_{P}(1). \label{asf2}$$ Finally, using (\[TRexpansion\]), we get that $$V_{n}=\frac{1}{n^{1/2}}{\displaystyle\sum\limits_{i=1}^{n}}
e(\mathbf{x}_{i},u_{i},a_{i})+f(\mathbf{x}_{i})+a_{i}{\mathbf{c}}^{{\prime}}{I_{R}(\mathbf{x}_{i},y_{i})}+o_{P}(1),$$ and using the Central Limit Theorem we get (\[Vnasnor\]). $\square$
To prove Theorem \[median\] we need an asymptotic expansion for $n^{1/2}(\widehat{\mu}_{n}-\mu_{0})$. Let $\mathbf{z}_{i}=(a_{i},\mathbf{x}_{i},y_{i})$ and consider $$\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\beta,\mu)=a_{i}\text{sign}\left(
g(\mathbf{x}_{j},\beta)+y_{i}-g(\mathbf{x}_{i},\beta)-\mu\right) ,
\label{Psi}$$$$\mathbf{\Lambda}_{1}(\beta,\mu)=\mathrm{E}\Psi(\mathbf{z}_{1},\mathbf{z}_{2},\beta,\mu)\;,\Lambda_{1\mathbf{\beta}}(\mathbf{\beta},\mu\mathbf{)}=\frac{\partial\Lambda_{1}(\mathbf{\beta},\mu)}{\partial\mathbf{\beta}},\text{
}\Lambda_{1\mu}(\mathbf{\beta},\mu\mathbf{)}=\frac{\partial\Lambda
_{1}(\mathbf{\beta},\mu)}{\partial\mu},$$ and $$J_{n}(\beta,\mu)=\frac{1}{n^{3/2}}\sum_{j=1}^{n}\sum_{i=1}^{n}\Psi
_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\beta,\mu).$$ The independence between $a_{1}$ and $(u_{1},\mathbf{x}_{2})$ and the fact that $u_{1}+g(\mathbf{x}_{2},\mathbf{\beta}_{0})$ has distribution $F_{0}$, allow to conclude that $\Lambda_{1}(\beta_{0},\mu_{0})=\mathrm{E}(a_{1})\mathrm{E}_{F_{0}}\text{sign}(y-\mu_{0})=0$. Since $I_{T_{\text{med}},F_{0}}(y)$ is not differentiable we have to use an extra argument to obtain an asymptotic linear expansion for $n^{1/2}(\widehat{\mu}_{n}-\mu_{0})$. To this purpose, the following Lemma is crucial. It is related to a very general linear expansion satisfied by empirical processes based on U-statistics.
\[tayfa\] Suppose the same assumptions as in Theorem \[median\]. Then if $n^{1/2}$ $(\overline{\beta}_{n}-\beta_{0})$ and $n^{1/2}(\overline{\mu
}_{n}-\mu_{0})$ are bounded in probability we have that $$J_{n}(\overline{\beta}_{n},\overline{\mu}_{n})=J_{n}(\beta_{0},\mu_{0})+\sqrt{n}\Lambda_{1\beta}(\mathbf{\beta}_{0},\mu_{0})^{\prime}(\overline
{\mathbf{\beta}}-\mathbf{\beta}_{0})+\sqrt{n}\Lambda_{1\mu}(\mathbf{\beta}_{0},\mu_{0})(\overline{\mu}_{n}-\mu_{0})+o_{p}(1). \label{YYY}$$
The proof of Lemma \[tayfa\] is based on a small number of intermediate results, being Proposition \[Hub67-3\] the most important of them. It may be considered the U-statistics version of Lemma 3 of Huber [@Huber67]. Since we believe that these results can be useful in many other situations, we decided to make a presentation in a general setting. Consider a sequence $\mathbf{z}_{i},i\geq1$ of i.i.d. random vectors of dimension $m$ and let $\ \mathbf{\Psi}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta}):\mathbb{R}^{m}\times\mathbb{R}^{m}\times\Theta
\rightarrow\mathbb{R}^{p}$, where $\Theta\subset\mathbb{R}^{p}.$ Note that here we are resorting to the same notation already adopted for the particular case considered above.
Let $\mathbf{\Lambda}(\mathbf{\theta})=$$\Psi(\mathbf{z}_{1},\mathbf{z}_{2,}\mathbf{\theta})$ and assume that $\mathbf{\
\Lambda }(\mathbf{\theta}_{0})=0$, for some $\theta_{0}\in R^{p}$. Consider $$\mathbf{Z}_{n}(\mathbf{\theta})=\frac{\left\Vert \sum_{j=1}^{n}\sum_{i=1}^{n}[\mathbf{\Psi}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta})-\mathbf{\Lambda}(\theta)-\mathbf{\Psi}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta}_{0})]\right\Vert
}{n^{3/2}+n^{2}||\mathbf{\Lambda
}(\mathbf{\theta})||} \label{Zn}$$ and $$U(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta},d)=\sup_{||\theta^{\ast
}-\mathbf{\theta}||\leq d}\left\Vert \mathbf{\Psi}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta}^{\ast})-\mathbf{\Psi}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})\right\Vert .$$ We need the following assumptions:
C1. For a fixed $\theta,\mathbf{\Psi}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})$ is measurable and $\mathbf{\Psi}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})$ is separable. For the definition of separability, see Huber [@Huber67].
C2. There exist numbers $b>0,$ $c>0$ and $d_{0}>0$ such that $\ $(i) $\Lambda(\mathbf{\theta})$ is continuously differentiable for $|\mathbf{|\theta-\theta}_{0}||\leq d_{0}$ and $\mathbf{\dot{\Lambda}(\theta }_{0})$ is nonsingular, where $\mathbf{\dot{\Lambda}(\theta)}$ is the differential matrix of $\Lambda(\mathbf{\theta}),$ (ii) $\mathrm{E}U(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta},d)\leq bd$ if $||\mathbf{\theta}-\mathbf{\theta}_{0}||+d\leq d_{0}$ and (iii) $\mathrm{E}U^{2}(\mathbf{z}_{1},\mathbf{z}_{2},\theta,d)\leq bd$ if $||\mathbf{\theta }-\mathbf{\theta}_{0}||+d\leq d_{0}.$
C3. $\mathrm{E}||\mathbf{\Psi}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta
}_{0}\mathbf{)||}^{2}<\infty.$
\[Hub67-3\] Suppose that assumptions C1-C3 hold. Then we have $$\sup_{||\mathbf{\theta}-\mathbf{\theta}_{0}||\leq d_{0}}\mathbf{Z}_{n}(\mathbf{\theta})\rightarrow_{p}0.$$
The proof is similar to that of Lemma 3 in Huber [@Huber67]. The only difference is that all the sums of independent variables need to be replaced by U-statistics. Moreover the U-statistics counterparts of $U_{n},V_{n}$ and the right hand side of equation (51) in Huber [@Huber67], must be approximated by sums of independent random variables using the Projection Theorem.$\square$
Let now $\Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\theta):\mathbb{R}^{m}\times\mathbb{R}^{m}\times\Theta\rightarrow\mathbb{R},$ where $\Theta
\subset\mathbb{R}^{p},$ and let $\Lambda_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})=\mathrm{E}\Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})$. Take $\mathbf{\theta}_{0}=(\theta_{01},...,\theta_{0p})$ such that $\Lambda_{1}(\mathbf{\theta}_{0})=0$. Put $$U_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta},d)=\sup_{||\theta^{\ast
}-\mathbf{\theta}||\leq d}\left\vert \Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\theta^{\ast})-\Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta
})\right\vert$$ and$$Z_{n1}(\mathbf{\theta})=\frac{\left\vert
\sum_{j=1}^{n}\sum_{i=1}^{n}[\Psi
_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta})-\Lambda_{1}(\mathbf{\theta
})-\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta}_{0})]\right\vert
}{n^{3/2}+n^{2}|\Lambda_{1}(\mathbf{\theta}_{0})|}.$$ Denote by $\ \ \ \
\mathbf{\dot{\Lambda}}_{1}(\mathbf{\theta})=(\dot{\Lambda
}_{11}(\mathbf{\theta}),...\dot{\Lambda}_{1p}(\mathbf{\theta}))=\partial
\Lambda_{1}(\mathbf{\theta)/\partial}\theta.$
In order to prove a statement analogous to Proposition \[Hub67-3\] for the univariate statistics $Z_{n1}(\mathbf{\theta)}$, the following assumptions will be needed. D1. For a fixed $\theta$, $\Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})$ is measurable and separable.
D2. There exist numbers $b>0,$ $c>0$ and $d_{0}>0$ such that $\ $(i) $\Lambda_{1}(\mathbf{\theta})$ is continuously differentiable for $|\mathbf{|\theta-\theta}_{0}||\leq d_{0}$ and $\dot{\Lambda}_{1}(\mathbf{\theta}_{0})$ $\neq0$ $\mathbf{\ }$ (ii) $\mathrm{E}U_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta},d)\leq bd$ if $||\mathbf{\theta
}-\mathbf{\theta}_{0}||+d\leq d_{0}$ and (iii) $\mathrm{E}U_{1}^{2}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta},d)\leq bd$ if $||\mathbf{\theta }-\mathbf{\theta}_{0}||+d\leq d_{0}.$
D3 $\mathrm{E}\Psi_{1}^{2}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta}_{0})<\infty.$
\[HUB1\] Suppose that assumptions D1-D3 are satisfied$.$ Then$$\sup_{||\theta-\mathbf{\theta}_{0}||\leq d_{0}}Z_{n1}(\mathbf{\theta
})\rightarrow_{p}0. \label{Zn1}$$
Let $\dot{\Lambda}_{1}(\mathbf{\theta})=(\dot{\Lambda}_{11}(\mathbf{\theta
}),...\dot{\Lambda}_{1p}(\mathbf{\theta}))^{\prime}$. Without loss of generality, by D2, we can assume that $\dot{\Lambda}_{11}(\mathbf{\theta })\neq0$. For $2\leq i\leq p$, define $\Psi_{i}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})=\theta_{i}-\theta_{0i}$ and consider $\mathbf{\Psi}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta
)=(}\Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta),}\Psi
_{2}\mathbf{(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta),}...},\Psi
_{p}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta))}^{\prime}$. Doing $\mathbf{\Lambda(\theta
)=}\mathrm{E}\Psi(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta),}$ we have $\mathbf{\Lambda(\theta}_{0}\mathbf{)=0}$ and $$\mathbf{\dot{\Lambda}}(\mathbf{\theta})=\left(
\begin{array}
[c]{cc}\dot{\Lambda}_{11}(\mathbf{\theta}) &
\dot{\Lambda}_{12}(\mathbf{\theta
})...\dot{\Lambda}_{1p}(\mathbf{\theta})\\
0 & I_{p-1}\end{array}
\right) .$$
Then det$(\mathbf{\dot{\Lambda}}(\mathbf{\theta}_{0}))=\dot{\Lambda}_{11}(\mathbf{\theta})\neq0$ and it is easy to check that the remaining assumptions C1-C3 are also satisfied. Let $\mathbf{Z}_{n}(\mathbf{\theta})$ be given by (\[Zn\]), then by Proposition \[Hub67-3\] we get that $\sup_{||\mathbf{\theta}-\mathbf{\theta}_{0}||\leq d_{0}}\mathbf{Z}_{n}(\mathbf{\theta})\rightarrow_{p}0.$ This implies (\[Zn1\]).
\[corlem\] Suppose the same assumptions as in Proposition \[HUB1\] and let $\overline{\mathbf{\theta}}_{n}$ be a sequence of estimates of $\ \mathbf{\theta}_{0}$ such that $n^{1/2}|||\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0}||=O_{p}(1).$ Then$$\frac{1}{n^{3/2}}\sum_{j=1}^{n}\sum_{i=1}^{n}\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\overline{\mathbf{\theta}}_{n})=\frac{1}{n^{3/2}}\sum_{j=1}^{n}\sum_{i=1}^{n}\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta}_{0})+\dot{\Lambda}_{1}(\mathbf{\theta}_{0})^{\prime}(n^{1/2}\
(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0}))+o_{p}(1).
\label{lecorlem}$$
By Proposition \[HUB1\] we have$$Z_{n1}(\overline{\mathbf{\theta}}_{n})=\frac{\left\vert \sum_{j=1}^{n}\sum_{i=1}^{n}[\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\overline{\mathbf{\theta
}}_{n})-\Lambda_{1}(\overline{\mathbf{\theta}}_{n})-\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta}_{0})]\right\vert }{n^{3/2}+n^{2}|\Lambda_{1}(\overline{\mathbf{\theta}}_{n})|}\rightarrow_{p}0. \label{con0}$$ Using the Mean Value Theorem we get$$\begin{aligned}
& \frac{\left\vert \sum_{j=1}^{n}\sum_{i=1}^{n}[\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\overline{\mathbf{\theta}}_{n})-\dot{\Lambda}_{1}(\mathbf{\theta}_{n}^{\ast})^{\prime}(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0})-\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta}_{0})]\right\vert }{n^{3/2}+n^{2}\left\vert \dot{\Lambda}_{1}(\mathbf{\theta}_{n}^{\ast})^{\prime}(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0})\right\vert }\\
& =\frac{\left\vert \sum_{j=1}^{n}\sum_{i=1}^{n}[\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\overline{\mathbf{\theta}}_{n})-\dot{\Lambda}_{1}(\mathbf{\theta}_{n}^{\ast})^{\prime}(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0})-\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta}_{0})]\right\vert }{n^{3/2}(1+\left\vert \dot{\Lambda}_{1}(\mathbf{\theta}_{n}^{\ast})^{\prime}n^{1/2}(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0})\right\vert )},\end{aligned}$$ where $\mathbf{\theta}_{n}^{\ast}\rightarrow_{p}\mathbf{\theta}_{0}.$ Since $\dot{\Lambda}(\mathbf{\theta}_{n}^{\ast})n^{1/2}(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0})$ is bounded in probability, (\[con0\]) implies$$\frac{1}{n^{3/2}}\sum_{j=1}^{n}\sum_{i=1}^{n}[\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\overline{\mathbf{\theta}}_{n})-\dot{\Lambda}_{1}(\mathbf{\theta}_{n}^{\ast})^{\prime}(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0})-\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta}_{0})]\ \rightarrow_{p}0,$$ and so $$\frac{1}{n^{3/2}}\sum_{j=1}^{n}\sum_{i=1}^{n}\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\overline{\mathbf{\theta}}_{n})=\frac{1}{n^{3/2}}\sum_{j=1}^{n}\sum_{i=1}^{n}\Psi_{1}(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\theta}_{0})+\dot{\Lambda}_{1}(\mathbf{\theta}_{n}^{\ast})(n^{1/2}\
(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0}))+o_{p}(1).$$ Finally, using the continuity of $\dot{\Lambda}$ at $\mathbf{\theta}_{0}$, the order of convergence of $\overline{\theta}_{n}$, and the fact that $\mathbf{\theta}_{n}^{\ast}\rightarrow_{p}\mathbf{\theta}_{0}$, we get (\[lecorlem\]).
In the following Proposition we give closed formulas for $\Lambda_{1\mathbf{\beta }}(\mathbf{\beta}_{0},\mu_{0}\mathbf{)}$ and $\Lambda_{1\mu}(\mathbf{\beta }_{0},\mu_{0})$, which are part of the expansion stated in Lemma \[tayfa\].
\[caldev\] We have$$\Lambda_{1\mathbf{\beta}}(\mathbf{\beta}_{0},\mu_{0})=2\mathrm{E}[a_{1}k_{0}(-g(\mathbf{x}_{2},\mathbf{\beta}_{0})+\mu_{0})({g}(\mathbf{x}_{2},\mathbf{\beta}_{0})-{g}(\mathbf{x}_{1},\mathbf{\beta}_{0}))]$$ and$$\Lambda_{1\mu}(\mathbf{\beta}_{0},\mu_{0}\mathbf{)=-}2\eta
f_{0}(\mu_{0}).$$
Let $\Delta($**$x$**$_{i},\mathbf{\beta})=g(\mathbf{x}_{i},\mathbf{\beta})-g(\mathbf{x}_{i},\mathbf{\beta}_{0}).$ Then $$\begin{aligned}
\Lambda_{1}(\mathbf{\beta,\mu}) & =\eta\mathrm{E}(\text{sign}(u_{1}+g(\mathbf{x}_{2},\mathbf{\beta}_{0})+\Delta(\mathbf{x}_{2},\mathbf{\beta
})-\Delta(\mathbf{x}_{1},\mathbf{\beta})-\mu|a_{1}=1)\\
& =\eta\mathrm{E}(\mathrm{E}(\text{sign}(u_{1}+g(\mathbf{x}_{2},\mathbf{\beta}_{0})+\Delta(\mathbf{x}_{2},\mathbf{\beta})-\Delta
(\mathbf{x}_{1},\mathbf{\beta})-\mu|a_{1}=1,\mathbf{x}_{1},\mathbf{x}_{2}\mathbf{))}\\
& =\eta\mathrm{E}((1-2K_{0}(-g(\mathbf{x}_{2},\mathbf{\beta}_{0})-\Delta(\mathbf{x}_{2},\mathbf{\beta})+\Delta(\mathbf{x}_{1},\mathbf{\beta
})+\mu)|a_{1}=1).\end{aligned}$$ Differentiating the last equation we get$$\begin{aligned}
{\Lambda_{1}}_{\mathbf{\beta}}(\mathbf{\beta},\mu) & =-2\eta\mathrm{E}[k_{0}(-g(\mathbf{x}_{2},\mathbf{\beta}_{0})-\Delta(\mathbf{x}_{2},\mathbf{\beta})+\Delta(\mathbf{x}_{1},\mathbf{\beta})+\mu)(-{g}(\mathbf{x}_{2},\mathbf{\beta})+{g}(\mathbf{x}_{1},\mathbf{\beta
}))]|a=1)\nonumber\\
&
=-2\eta\mathrm{E}[k_{0}(-g(\mathbf{x}_{2},\mathbf{\beta}_{0})-\Delta
(\mathbf{x}_{2},\mathbf{\beta})+\Delta(\mathbf{x}_{1},\mathbf{\beta})+\mu)|a=1)\nonumber\\
& =-2\eta\mathrm{E}[k_{0}(-g(\mathbf{x}_{2},\mathbf{\beta}_{0})+\mu_{0})(-{g}(\mathbf{x}_{2},\mathbf{\beta}_{0})+{g}(\mathbf{x}_{1},\mathbf{\beta
}_{0}))]|a_{1}=1)\nonumber\\
& =2\mathrm{E}[a_{1}k_{0}(-g(\mathbf{x}_{2},\mathbf{\beta}_{0})+\mu_{0})({g}(\mathbf{x}_{2},\mathbf{\beta}_{0})-{g}(\mathbf{x}_{1},\mathbf{\beta}_{0}))]\end{aligned}$$ and $$\begin{aligned}
{\Lambda_{1}}_{\mathbf{\mu}}(\mathbf{\beta}_{0},\mu_{0}) & =-2\eta
\mathrm{E}[k_{0}(-g(\mathbf{x}_{2},\mathbf{\beta}_{0})+\mu_{0})]\nonumber\\
& =-2\eta f_{0}(\mu_{0}).\end{aligned}$$
These prove the Proposition.
<span style="font-variant:small-caps;">Proof of Lemma \[tayfa\]:</span> By Proposition \[corlem\], we only need to verify that under the assumptions of Theorem 3, considering $\mathbf{\theta
=}(\mathbf{\beta,}\mu)$, $$\Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})=a_{1} \text{sign}(y_{1}-g(\mathbf{x}_{1},\mathbf{\beta})+g(\mathbf{x}_{2},\mathbf{\beta})-\mu).$$ and $$U_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta},d)=\sup_{||\theta^{\ast
}-\theta||\leq d}|\Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta}^{\ast})-\Psi_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta})|,$$ then, assumptions D1-D3 are satisfied.
Assumptions D1 and D3 follow immediately. Assumption D2(i) follows from Proposition \[caldev\] and the fact that $f_{0}(\mu_{0})>0$.
We now prove D2 (ii) and (iii). Take $d_{0}=\delta$ as in A0, then if we put$$\omega(\mathbf{x})=\sup_{\left\Vert \mathbf{\beta}-\mathbf{\beta}_{0}\right\Vert \leq\delta}\left\Vert
{\dot{g}}(\mathbf{x},\mathbf{\beta })\right\Vert ,$$ we have $\mathrm{E}\omega(\mathbf{x}_{1})<\infty$. To prove D2 (i) and (ii), we have to show that there exist $K_{1}$ and $K_{2}$ such that for all $\mathbf{\theta}$ and $d$ with $||\theta-\theta_{0}||+d\leq d_{0}$, we have $$\mathrm{E}U_{1}^{i}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta},d)\leq
K_{i}d\;,\text{ }i=1,2.\label{doudes}$$ For that purpose, we can write $$\begin{aligned}
U_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\theta,}d) & \leq\sup
_{||\mathbf{\beta}^{\ast}-\mathbf{\beta}||\leq
d,|\mu^{\ast}-\mu|\leq
d}|\text{sign}(u_{1}+g(\mathbf{x}_{1},\mathbf{\beta}_{0})-g(\mathbf{x}_{1},\mathbf{\beta}^{\ast})+g(\mathbf{x}_{2},\mathbf{\beta}^{\ast})-\mu^{\ast
}) \nonumber\\
& -\text{sign}(u_{1}+g(\mathbf{x}_{1},\mathbf{\beta}_{0})-g(\mathbf{x}_{1},\mathbf{\beta})+g(\mathbf{x}_{2},\mathbf{\beta})-\mu)|.\label{HH1}$$
Then if $||\theta-\theta_{0}||+d\leq d_{0},$ $\ $and $||\mathbf{\theta}^{\ast }-\mathbf{\theta}||\leq d$ we get that $||\mathbf{\theta}^{\ast }-\mathbf{\theta}_{0}||\leq d_{0}$, and therefore $||\mathbf{\beta}^{\ast }-\mathbf{\beta}_{0}||\leq d_{0}$ too. Note also that $$\begin{aligned}
& \left\vert
(g(\mathbf{x}_{1},\beta_{0})-g(\mathbf{x}_{1},\mathbf{\beta
}^{\ast})+g(\mathbf{x}_{2},\mathbf{\beta}^{\ast})-\mu^{\ast})-(g(\mathbf{x}_{1},\beta_{0})-g(\mathbf{x}_{1},\mathbf{\beta})+g(\mathbf{x}_{2},\mathbf{\beta})-\mu)\right\vert \nonumber\\
&
\leq(\omega(\mathbf{x}_{1})+\omega(\mathbf{x}_{2}))||\mathbf{\beta}^{\ast
}-\mathbf{\beta}||+|\mu^{\ast}-\mu|\leq(\omega(\mathbf{x}_{1})+\omega
(\mathbf{x}_{2})+1)d. \label{HH2}$$
Let $z=$ $(\omega(\mathbf{x}_{1})+\omega(\mathbf{x}_{2})+1)$, $v=|u|$ and $w=v/z.$ The left hand side of (\[HH1\]) is different from 0 when the two arguments of the sign function have different signs. By (\[HH2\]) this occurs only if $|u_{1}|\leq(\omega(\mathbf{x}_{1})+\omega (\mathbf{x}_{2})+1)d.$ Then we can write $$U_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\beta},\mu\mathbf{,}d)\leq2I(w\leq
d).$$ Observe that $\mathrm{E}z<\infty$ and that the density of $v$ is given by $f_{v}(v)=k_{0}(v)+k_{0}(-v)$, which is bounded by $2\sup
k_{0}$. Then, since the density of $\ w$ is$$f_{w}(w)={\displaystyle\int\limits_{0}^{\infty}}
zf_{v}(wz)dF_{z}\leq2\sup k_{0}{\displaystyle\int\limits_{0}^{\infty}}
zdF_{z}=2\sup k_{0}\mathrm{E}(z)$$ $\ $ we get $$\mathrm{E}U_{1}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\beta},\mu
\mathbf{,}d)\leq2P(w\leq d)\leq4\sup k_{0}\mathrm{E}(z)d$$ and$$\mathrm{E}U_{1}^{2}(\mathbf{z}_{1},\mathbf{z}_{2},\mathbf{\beta},\mu
\mathbf{,}d)\leq4P(w\leq d)\leq8\sup k_{0}\mathrm{E}(z)d,$$ and so (\[doudes\]) holds with $K_{1}=4\sup k_{0}$$(z)$ and $K_{2}=8\sup k_{0}$$(z).$
The expansion obtained in Lemma \[tayfa\] requires that $\sqrt
{n}(\overline{\mathbf{\theta}}_{n}-\mathbf{\theta}_{0})=O_{P}(1)$. The following Lemma shows that $\widehat{\mathbf{\theta}}_{n}=(\widehat
{\mathbf{\beta}}_{n},\widehat{\mu}_{n})$ satisfies this condition.
\[jacot\] Under the assumptions of Theorem \[median\] we have that (a) $n^{1/2}(\widehat{\mu}_{n}-\mu_{0})$ is bounded in probability, (b) $J_{n}(\widehat{\mathbf{\beta}}_{n},\widehat{\mu}_{n})\rightarrow^{p}0.$
Let $$J_{n}^{\ast}(\mathbf{\beta,}\mu)=\frac{1}{n^{3/2}}\sum_{j=1}^{n}\sum_{i=1}^{n}\Psi(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\beta}_{0},\mu_{0}))+n^{1/2}\mathbf{\Lambda_{1}}_{\mathbf{\beta}}(\mathbf{\beta}_{0},\mu
_{0})^{\prime}(\mathbf{\beta}-\mathbf{\beta}_{0})+n^{1/2}\Lambda
_{1\mathbf{\mu}}(\mathbf{\beta}_{0},\mu_{0})(\mu-\mu_{0}).$$ Take $\varepsilon>0$ and let $\widetilde{\mu}_{1n}$ $\ $ and $\widetilde{\mu
}_{2n}$ be defined by $$J_{n}^{\ast}(\widehat{\mathbf{\beta}}_{n}\mathbf{,}\widetilde{\mu}_{1n})=\varepsilon\text{ and }J_{n}^{\ast}(\widehat{\mathbf{\beta}}_{n}\mathbf{,}\widetilde{\mu}_{2n})=-\varepsilon.$$
By Lemma \[caldev\] $\Lambda_{1\mu}(\mathbf{\beta}_{0},\mu
_{0})\neq0,$ and it holds that $$n^{1/2}(\widetilde{\mu}_{1n}-\mu_{0})=-\frac{1}{\Lambda_{1\mu}(\mathbf{\beta
}_{0},\mu_{0})}\left[ \frac{1}{n^{3/2}}\sum_{j=1}^{n}\sum_{i=1}^{n}\Psi(\mathbf{z}_{i},\mathbf{z}_{j},\mathbf{\beta}_{0},\mu_{0})+n^{1/2}\mathbf{\Lambda_{1}}_{\mathbf{\beta}}(\mathbf{\beta}_{0},\mu_{0})^{\prime
}(\widehat{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})-\varepsilon\right] .$$ Both the first and second terms on the right hand side are bounded in probability, the former by the Central Limit Theorem for U-statistics and the later by Assumption A2 and the Central Limit Theorem. Then $n^{1/2}(\widetilde{\mu}_{1n}-\mu_{0})$ is bounded in probability. Thereafter, by Lemma \[tayfa\] we get $$J_{n}(\widehat{\mathbf{\beta}}_{n},\widetilde{\mu}_{1n})=\varepsilon
+o_{p}(1).$$ Similarly we can prove that $n^{1/2}(\widetilde{\mu}_{2n}-\mu_{0})$ is bounded in probability and that $$J_{n}(\widehat{\mathbf{\beta}}_{n},\widetilde{\mu}_{2n})=-\varepsilon
+o_{p}(1).$$ Then since $J(\beta,\mu)$ is nonincreasing in $\mu,$ by a property of the median we get that $$\lim_{n\rightarrow\infty}\mathrm{P}(\widetilde{\mu}_{1n}\leq\widehat{\mu}_{n}\leq\widetilde{\mu}_{2n})=1,$$ and therefore $n^{1/2}(\widehat{\mu}_{n}-\mu_{0})$ is bounded in probability. We also have that $$\mathrm{P}(J_{n}(\widehat{\mathbf{\beta}}_{n},\widetilde{\mu}_{2n})\leq
J_{n}(\widehat{\mathbf{\beta}}_{n},\widehat{\mu}_{n})\leq J_{n}(\widehat
{\mathbf{\beta}}_{n},\widetilde{\mu}_{1n}))\rightarrow1$$ and therefore $$\mathrm{P}(-2\varepsilon\leq J_{n}(\widehat{\mathbf{\beta}}_{n},\widehat{\mu
}_{n})\leq2\varepsilon)\rightarrow1.$$ Since this holds for all $\varepsilon>0,$ part (b) of the Lemma is proved.
<span style="font-variant:small-caps;">Proof of Theorem \[median\].</span>
\(a) To prove this part of the Theorem it suffices to show that $T_{\text{med}}$ is weakly continuous at $F_{0}.$ Take $\varepsilon>0$ and $y_{1\text{
}}$ , $y_{2}$ continuity points of $\ F_{0}$ $\ $ such that $\mu
_{0}-\varepsilon<y_{1}<\mu_{0}<y_{2}<\mu_{0}+\varepsilon$. Since $F_{0}$ is continuous and strictly increasing at $F_{0}$ we have that $F_{0}(\mu
_{0})=0.5$ and there exists $\delta>0$ such that $F_{0}(y_{1})<0.5-\delta
<0.5+\delta<F_{0}(\mu_{2}).$ Suppose that $F_{n}\rightarrow_{w}F_{0},$ then there exists $\ n_{0}$ such that for $n\geq n_{0}$ we have $F_{n}(y_{1})<0.5-\delta$ and $F_{n}(y_{2})>0.5+.\delta.$ This proves that $n\geq
n_{0}$ implies that $\ \mu_{0}-\varepsilon<y_{1}\leq T_{\text{med}}(F_{n})\leq y_{2}<\mu_{0}+\varepsilon.$ (b) Since $n^{1/2}(\widehat{\mu}_{n}-\mu_{0})$ and $n^{1/2}(\widehat
{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})$ are bounded in probability, by Lemma \[tayfa\] we get $$J_{n}(\widehat{\mathbf{\beta}}_{n},\widehat{\mu}_{n})=J_{n}(\mathbf{\beta}_{0},\mu_{0})+\sqrt{n}\Lambda_{1\beta}(\mathbf{\beta}_{0},\mu_{0})(\widehat{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})+\sqrt{n}\Lambda_{1\mu
}(\beta_{0},\mu_{0})(\widehat{\mu}_{n}-\mu_{0})\}+o_{p}(1),$$ and using Lemma \[jacot\] (b) we get $$\begin{aligned}
\sqrt{n}\{\widehat{\mu}_{n}-\mu_{0}\} & =-\frac{1}{\Lambda_{1\mu}(\beta
_{0},\mu_{0})\ }\left\{ \Lambda_{1\mathbf{\beta}}(\mathbf{\beta}_{0},\mu
_{0})^{\prime}n^{1/2}(\widehat{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})+J_{n}(\mathbf{\beta}_{0},\mu_{0})\right\} +o_{p}(1)\\
& =d_{n}^{\ast}+n^{1/2}\mathbf{c}_{n}^{\ast\prime}(\widehat{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})+o_{p}(1),\end{aligned}$$ where $$\mathbf{c}_{n}^{\ast}=\frac{1}{_{n^{2}}}\sum_{j=1}^{n}\sum_{i=1}^{n}a_{i}\,I_{L}^{\prime}(y_{i}-g({\mathbf{\beta}},\mathbf{x}_{i})+{g}({\mathbf{\beta}},\mathbf{x}_{j}))\,\{\dot{g}({\mathbf{\beta}},\mathbf{x}_{j})-\dot{g}({\mathbf{\beta}},\mathbf{x}_{i})\}$$ and$$d_{n}^{\ast}=\frac{1}{n^{3/2}}{\displaystyle\sum_{i=1}^{n}}
{\displaystyle\sum_{j=1}^{n}}
a_{i}I_{T_{med}}(u_{i}+g(\mathbf{\beta}_{0},\mathbf{x}_{j})).$$ Then it suffices to show that $$V_{n}^{\ast}=d_{n}^{\ast}+\mathbf{c}_{n}^{\ast\prime}n^{1/2}(\widehat
{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})\rightarrow N(0,\tau^{2}).$$ The proof of this result is similar to that of $d_{n}+\mathbf{c}_{n}^{\prime}n^{1/2}(\widehat{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})\rightarrow^{d}$N$(0,\tau^{2})$ in Theorem \[asym\]. $\square$ <span style="font-variant:small-caps;">Proof of Theorem \[BDP\]</span>. Let** **$\mathbf{W}$ be as in (\[Wcom\]). We have to show that given $t<n\varepsilon_{3}$ and $s<m\varepsilon_{3},$ there exists $K$ such that for any sample $\mathbf{W}^{\ast}\in\mathcal{W}_{ts},$ we have that $|T_{L}(\widehat{F}_{n}^{\ast})|\leq K,$ where $\ \widehat{F}_{n}^{\ast}$ is the distribution constructed as in (\[Gnsom\]), based on $\mathbf{W}^{\ast}$. According to the definition of $\varepsilon_{U}^{\ast}(T_{L})$, it suffices to show that there exists $M$ such that for any $\mathbf{W}^{\ast}\in\mathcal{W}_{ts}$ we have that the corresponding $\widehat{F}_{n}^{\ast}$ satisfies $$\mathrm{P}_{\widehat{F}_{n}^{\ast}}(|y|\leq M)>1-\varepsilon_{2}. \label{Ihin}$$
Let $$\mathcal{Z}_{s}=\{\mathbf{Z}^{\ast}=\{(\mathbf{x}_{i}^{\ast},y_{i}^{\ast
}):i\in A\}:{\displaystyle\sum_{i\in A}}
I\{(\mathbf{x}_{i}^{\ast},y_{i}^{\ast})\neq(\mathbf{x}_{i},y_{i})\}\leq s\}.$$ Since $s/m<\varepsilon_{1}$ we can find $M_{1}$ such that $$\sup_{\mathbf{Z}^{\ast}\in\mathcal{Z}_{s}}||\widetilde{\mathbf{\beta}}_{m}(\mathbf{Z}^{\ast})||\leq M_{1}, \label{cot0}$$ and then we can find $M$ such that $$\sup_{1\leq j\leq n}\sup_{|\mathbf{|\beta}||\leq M_{1}}|g(\mathbf{x}_{j},\mathbf{\beta}|\leq M/2 \label{cot1}$$ and$$\sup_{i\in A}\sup_{||\beta||\leq M_{1}}|y_{i}-g(\mathbf{x}_{i},\mathbf{\beta
})|\leq M/2. \label{cot2}$$
Given $\ \mathbf{W}^{\ast}\in\mathcal{W}_{t,s}$, if $\widehat{\beta}_{n}^{\ast}=\widetilde{\beta}_{m}(\mathbf{Z}^{\ast})$, with $\mathbf{Z}^{\ast}\in\mathcal{Z}_{s}$. Consider $B=\{j:$ $1\leq j\leq n,\mathbf{x}_{j}=\mathbf{x}_{j}^{\ast}\}$ and $C=\{i\in A:$ $(\mathbf{x}_{j},y_{j})=(\mathbf{x}_{j}^{\ast},y_{j}^{\ast})\}$. Then $\#B>(1-\varepsilon
_{3})n$ and $\#C>(1-\varepsilon_{3})m$. For $1\leq j\leq n,$ $i\in A,$ put $\widehat{y}_{ij}^{\ast}=g(\mathbf{x}_{j}^{\ast},\widehat{\mathbf{\beta}}_{n}^{\ast})+(y_{i}^{\ast}-g(\mathbf{x}_{i}^{\ast},\widehat{\mathbf{\beta}}_{n}^{\ast})).$ Then, when $\ j\in B$ and $\ i\in C$, by (\[cot0\]), (\[cot1\]) and (\[cot2\]), we have that $|\widehat{y}_{ij}^{\ast}|\leq M$ and so $$\#\{(i,j):|\widehat{y}_{ij}^{\ast}|\leq M\}>mn(1-\varepsilon_{3})^{2}\geq(1-\varepsilon_{2})mn.$$ Since there are $mn$ pairs $(i,j)$ subindexing $\widehat{y}_{ij}^{\ast}$, we get that $\mathrm{P}_{\widehat{F}_{n}^{\ast}}(|y|$ $\leq
M)>1-\varepsilon_{2}$ and then (\[Ihin\]) holds. $\square$
<span style="font-variant:small-caps;">Proof of Theorem \[asslreg\]</span>. The proof of this Theorem is essentially based on Theorem 7 of Fasano et. at. [@Fasano; @et; @al]. As is mentioned in Section \[results\], if $(\mathbf{x}_{i},y_{i})$ has distribution $G_{0}^{\ast},$ then (\[modelogral\]) is satisfied with $\mathbf{x}_{i}^{\ast}$ having distribution $Q_{0}^{\ast}$ and $u_{i}^{\ast}$ with distribution $K_{0}.$ Moreover, since by Lemma \[g-c\] $G_{n}^{\ast }\rightarrow_{w}G_{0}^{\ast}$ , by parts (i), (ii) and (iii) of Theorem 7 of [@Fasano; @et; @al] with $G_{0}$ replaced by $G_{0}^{\ast}$, we get part (i) of the present Theorem.
We now prove (ii). We start proving that for any function $d$ such the $_{G_{0}^{\ast}}\left\vert d(\mathbf{x},y)\right\vert <\infty$, we have that $$\mathrm{E}_{G_{n}^{\ast}}d(\mathbf{x},y)\rightarrow\mathrm{E}_{G_{0}^{\ast}}d(\mathbf{x},y)\,\text{\ a.s.} \label{con3}$$ Since$$\mathrm{E}_{G_{n}^{\ast}}d(\mathbf{x},y)=\frac{{\displaystyle\sum\limits_{i=1}^{n}}
a_{i}d(\mathbf{x}_{i},y_{i})}{{\displaystyle\sum\limits_{i=1}^{n}}
a_{i}}\ =\frac{1}{\eta_{n}}\frac{1}{n}{\displaystyle\sum\limits_{i=1}^{n}}
a_{i}d(\mathbf{x}_{i},y_{i})$$
and $\eta_{n}\rightarrow\eta,$ by the Law of Large Numbers we have that $_{G_{n}^{\ast}}d(\mathbf{x},y)\rightarrow$$a_{1}d(\mathbf{x}_{1},y_{1})/\eta$ a.s. Since $\mathrm{E}a_{1}d(\mathbf{x}_{1},y_{1})/\eta$ $=$$_{G_{0}^{\ast}}d(\mathbf{x},y)$ , we obtain (\[con3\]).
Put now $\mathbf{T=(T}_{S},\mathbf{T}_{MM},S)$ and let $I_{\mathbf{T,}G_{0}^{\ast}}(\mathbf{x,}y)$ be its influence function at $G_{0}^{\ast}.$ We now prove that $$n^{1/2}\mathrm{E}_{G_{n}^{\ast}}I_{\mathbf{T,}G_{0}^{\ast}}(\mathbf{x,}y)\rightarrow_{d}H,$$ where $H$ is a multivariate normal distribution. This follows by applying the Central Limit Theorem from$$n^{1/2}\mathrm{E}_{G_{n}^{\ast}}I_{\mathbf{T,}G_{0}^{\ast}}(\mathbf{x,}y)=\frac{1}{\eta_{n}}\frac{1}{n^{1/2}}{\displaystyle\sum\limits_{i=1}^{n}}
a_{i}I_{\mathbf{T,}G_{0}^{\ast}}(\mathbf{x}_{i},y_{i}),$$ the facts that $_{G_{0}^{\ast}}I_{\mathbf{T,}G_{0}^{\ast}}(\mathbf{x},y)=0,$ and the fact that under $G_{0}^{\ast},$ the influence function $I_{\mathbf{T,}G_{0}^{\ast}}(\mathbf{x},y)$ has finite second moments. Then all the conditions required to apply parts (iv) and (v) of Theorem 7 of Fasano et al. [@Fasano; @et; @al] are satisfied. Then$$n^{1/2}(\mathbf{T_{MM,\beta}}(G_{n}^{\ast})-\mathbf{\beta}_{0})=n^{-1/2}\sum_{i=1}^{n}a_{i}\frac{1}{\mathrm{E}[a_{1}]}I_{\mathbf{T_{MM,\mathbf{\beta}}},G_{0}^{\ast}}(\mathbf{x}_{i},y_{i})+o_{P}(1).$$ Finally, using the expression for $I_{\mathbf{T_{MM,\mathbf{\beta}}},G_{0}^{\ast}}$ derived in Fasano et. al. [@Fasano; @et; @al], we obtain part (ii) of the Theorem. Part (iii) is an immediate consequence of the fact that in this case $e_{01}=0.\square$
<span style="font-variant:small-caps;">Proof of Theorem \[assloc\].</span> Part (i) follows from parts (i), (ii) and (iii) of Theorem 8 of Fasano et al.[@Fasano; @et; @al]. Let $\mathbf{T}^{L}(F)$ be the complete functional $$\mathbf{T}^{L}\left( F\right) =\left( {T}_{{S}}^{L}\left( F\right)
,{T}_{{MM}}^{L}\left( F\right) ,S_{L}\left( F\right) \right) .$$ Since $\{\widehat{F}_{n}\}$ is a sequence of random distribution with finite support converging a.s. to $F_{0}$, by part (iv) of Theorem 8 of Fasano et al. [@Fasano; @et; @al] we get that $\mathbf{T}^{L}$ is weakly differentiable at $\{\widehat{F}_{n}\}$ a.s., and so $$\mathbf{T}^{L}(\widehat{F}_{n})-\mathbf{T}^{L}(F_{0})=\mathrm{E}_{\widehat
{F}_{n}}I_{\mathbf{T}^{L},F_{0}}(y\mathbf{)+}o\left( \left\Vert
\mathrm{E}_{\widehat{F}_{n}}I_{\mathbf{T}^{L},F_{0}}(y\mathbf{)}\right\Vert
\right) , \label{expb}$$ where $I_{\mathbf{T}^{L},F_{0}}$ is the influence function of $\mathbf{T}^{L}$ at $F_{0}$.
We prove now that $n^{1/2}\mathrm{E}_{\widehat{F}_{n}}I_{\mathbf{T}^{L},F_{0}}(y\mathbf{)}$ is bounded in probability. Using a Taylor expansion, we get$$\sqrt{n}\,\mathrm{E}_{\widehat{F}_{n}}I_{T^{L},F_{0}}(y)=\frac{1}{\eta_{n}}\left\{ D_{n}+\mathbf{C}_{n}^{\text{'}}n^{1/2}(\widehat{\mathbf{\beta}}_{n}-\mathbf{\beta}_{0})\right\} , \label{TE1}$$ where$$D_{n}={n^{-3/2}}{\displaystyle\sum_{i=1}^{n}}
{\displaystyle\sum_{j=1}^{n}}
a_{i}I_{T^{L},F_{0}}(u_{i}+g(x_{j},\mathbf{\beta}_{0})), \label{TE2}$$ $$\mathbf{C}_{n}=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}h(\mathbf{x}_{i},y_{i},a_{i},\mathbf{x}_{j},\beta_{n}^{\ast}),\ \label{TE3}$$ $\mathbf{\beta}_{n}^{\ast}$ between $\widehat{\mathbf{\beta}}_{n}$ and $\mathbf{\beta}_{0}$ and $$h(\mathbf{x}_{i},y_{i},a_{i},\mathbf{x}_{j},\beta)=a_{i}\,I_{{T^{L},F_{0}}}^{\prime}(y_{i}-g(x_{i},{\mathbf{\beta}})+g(x_{j},{\mathbf{\beta}}))\,\{\dot{g}(x_{j},{\mathbf{\beta}})-\dot{g}(x_{i},{\mathbf{\beta}})\}.$$ Assuming A0, by Lemma \[victor4.2\], we get $$\mathbf{C}_{n}\longrightarrow\mathbf{C}=\mathrm{E}_{G_{0}}a_{i}\,I_{{T^{L},F_{0}}}^{\prime}(y_{i}-g(x_{i},\mathbf{\beta}_{0})+g(x_{j},\mathbf{\beta}_{0}))\,\{\dot{g}(x_{j},\mathbf{\beta}_{0})-\dot{g}(x_{i},\mathbf{\beta}_{0})\},\text{a.s.} \label{TE5}$$ Using (\[TE1\])-(\[TE5\]), the expansion (\[TRexpansion\]) guaranteed by part (ii) of Theorem \[asslreg\], and the fact that by the U-statistics projection Theorem $\{D_{n}\}$ converges to a normal distribution, we conclude [that $\{\sqrt{n}\Vert$$_{\widehat{F}_{n}}I_{T^{L},F_{0}}(y\mathbf{)}\Vert\}$ is bounded in probability.]{} Therefore, from (\[expb\]) we get $$\sqrt{n}\{\mathbf{T}^{L}(\widehat{F}_{n})-\mathbf{T}^{L}(F_{0})\}=\sqrt
{n}\,\mathrm{E}_{\widehat{F}_{n}}I_{\mathbf{T}^{L},F_{0}}(y\mathbf{)+}o_{P}(1).$$ This implies $$\sqrt{n}\{\mathbf{T}_{MM}^{L}(\widehat{F}_{n})-\mathbf{T}^{L}(F_{0})\}=\sqrt{n}\,\mathrm{E}_{\widehat{F}_{n}}I_{\mathbf{T}_{MM}^{L},F_{0}}(y\mathbf{)+}o_{P}(1),$$ and therefore (\[TLexpansion\]) is satisfied with $I_{L}=I_{\mathbf{T}_{MM}^{L},F_{0}}.$ Finally (\[ifloc1\]) follows from formula (44) of Fasano et al. [@Fasano; @et; @al]. Part (ii) follows immediately from $e_{01}^{L}=0.\square$
To prove Theorem \[locationbound\], the following result is required.
\[BDP1\] Given $M$ and $\gamma>0,$ there exists $M^{\ast}$ such that $\mathrm{P}_{F}(|y|\leq M)\geq1-\delta+\gamma$ implies $S^{L}(F)\leq M^{\ast
}.$
It is enough to show that there exists $M^{\ast}$ such that $S^{\ast }_{L}(F,0)\leq M^{\ast}$, where $S^{\ast}_{L}(F,\mu)$ is the location version of the object defined by (\[Sestrella\]) for the regression case.
Let $M^{\ast}$ be such that $\rho_{0}^{L}(M/M^{\ast})<\gamma/2$. Suppose that $S_{L}^{\ast}(F,0)>M^{\ast}$. By definition of $S_{L}^{\ast}(F,0)$, $$\delta=\mathrm{E}_{F}\rho_{0}^{L}\left( {y}/{S_{L}^{\ast}(F,0)}\right)
\ \label{SBP1}$$
On the other hand, let $A=\{|y|\leq M\}$. By hypothesis, $_{F}(A)\geq1-\delta+\gamma$, and so $$\mathrm{E}_{F}\rho_{0}^{L}\left( \frac{y}{S_{L}^{\ast}(F,0)}\right)
\leq\mathrm{E}_{F}\rho_{0}^{L}\left( \frac{y}{M^{\ast}}\right)
\ \ \leq(\gamma/2)\mathrm{P}_{F}(A)+\mathrm{P}_{F}(A^{c})\leq\gamma
/2+\delta-\gamma\leq\delta-\gamma/2,$$ contradicting (\[SBP1\]).
<span style="font-variant:small-caps;">Proof of Theorem \[locationbound\].</span> We will prove that, given $M$ and $\gamma>0,$ there exists $K$ such that $|T_{MM}^{L}(F)|\leq K$, for all $F$ with $\mathrm{P}_{F}(|y|\leq M)\geq\min(1-\delta+\gamma,\delta+\gamma)$. In fact , note that $$\mathrm{E}_{F}\rho^{L}\left( \frac{y-T_{MM}^{L}(F)}{S^{L}(F)}\right)
\leq\mathrm{E}_{F}\rho^{L}\left( \frac{y-T_{S}^{L}(F)}{S^{L}(F)}\right)
\leq\mathrm{E}_{F}\rho_{0}^{L}\left( \frac{y-T_{S}^{L}(F)}{S^{L}(F)}\right)
=\delta. \label{SBDP2}$$
Let $\ M^{\ast}$ be as in Lemma 1 and let $\ a$ so that $\rho^{L}(a/M^{\ast
}))(\delta+\gamma)=\delta+\gamma/2$. Put $K=M+a$ and observe that $|y|\leq M$ and $|T_{MM}^{L}(F)|>K$ imply that $|y-T_{MM}^{L}(F)|>a$. Suppose that $|T_{MM}^{L}(F)|>K$ . Then $$\mathrm{E}_{F}\rho^{L}\left( \frac{y-T_{MM}^{L}(F)}{S^{L}(F)}\right)
\geq\mathrm{E}_{F}\rho^{L}\left( \frac{y-T_{MM}^{L}(F)}{M^{\ast}}\right)
\geq\mathrm{P}_{F}(A)\rho^{L}(a/M^{\ast}\geq\rho^{L}(a/M^{\ast})(\delta
+\gamma)\geq\delta+\gamma/2,$$ contradicting (\[SBDP2\]). $\square$
[99]{}
<span style="font-variant:small-caps;">Bianco, A., Boente, G., González-Manteiga, W. and Pérez-González, A.</span> (2010). Estimation of the marginal location under a partially linear model with missing responses. *Computational Statistics & Data Analysis*. 546–564.
<span style="font-variant:small-caps;">Billingsley, P.</span> (1995) *Probability and Measure*. (3rd. ed.), Wiley, New York.
<span style="font-variant:small-caps;">Billingsley, P.</span> (1999). *Convergence of Probability Measures* (2nd. ed.) Wiley, New York.
<span style="font-variant:small-caps;">Donoho, D.L. and Huber, P.J.</span> (1983), The notion of breakdown point, *A Festschrift for E. L. Lehmann*, P.J. Bickel, K.A. Doksum and J.L. Hodges, (eds.). 157–184, Belmont, CA: Wadsworth.’ <span style="font-variant:small-caps;">Fasano, M.V.</span> (2009). Robust estimation in nonlinear regression. Ph. D. Thesis, University of La Plata. <span style="font-variant:small-caps;">Fasano, M.V., Maronna, R.A., Sued, M. and Yohai, V.J.</span> (2010) Continuity and differentiability of regression M–estimates. Available at http://arxiv.org/abs/1004.4314.
<span style="font-variant:small-caps;">Fernholz, L.T.</span> (1983). Von Mises Calculus for Statistical Functionals. *Lecture Notes in Statistics*. **19** Springer-Verlag, New York.
<span style="font-variant:small-caps;">Hampel, F.R.</span> (1971). A general qualitative definition of robustness. *The Annals of Mathematical Statistics*. **42** 1887–1896.
<span style="font-variant:small-caps;">Hampel, F.R.</span> (1974).The influence curve and its role in robust estimation, *Journal of the American Statistical Association*. **69** 383–393.
<span style="font-variant:small-caps;">Huber, P.J.</span> (1967).The behavior of maximum likelihood estimates under nonstandard conditions. *Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. I: Statistics* 221–233.
<span style="font-variant:small-caps;">Hössjer, O.</span> (1992). On the optimality of S-estimators, *Statistics and Probability Letters*. **14** 413–419.
<span style="font-variant:small-caps;">Kang, J.D.Y, Schafer, J.L.</span> (2007). Demystifying double robustness: a comparison of alternative strategies for estimating a population mean from incomplete data. *Statistical Science.* **22** 523–539.
<span style="font-variant:small-caps;">Maronna, R.A., Martin, R. D. and Yohai, V.J.</span> (2006). *Robust Statistics: Theory and Methods.* Springer, Chichister.
<span style="font-variant:small-caps;">Müller, U.U.</span> (2009). Estimating linear functionals in nonlinear regression with responses missing at random.*Annals of Statistics*. **37** 2245–2277.
<span style="font-variant:small-caps;">Rubin, D.</span> (1976). Inference and missing data. *Biometrika*. **63** 581–592.
<span style="font-variant:small-caps;">Robins,M,. Sued,M., Lei-Gomez, Q., Rotnitzky, A.</span> (2007). Comment: Performance of Double-Robust Estimators When “Inverse Probability” Weights Are Highly Variable. *Statistical Science.* **22** 544–559.
<span style="font-variant:small-caps;">Rousseeuw, P.J and Yohai, V.J.</span> (1984). Robust regression by means of S-estimators, *Robust and Nonlinear Time Series*, J. Franke,W. Hardle and R.D. Martin (eds.), Lectures Notes in Statistics. **26** 256–272, New York: Springer.
<span style="font-variant:small-caps;">van der Vaart, A. & Wellner, J.</span> (1996). *Weak Convergence and Empirical Processes. With Applications to Statistics*. New York: Springer.
<span style="font-variant:small-caps;">Yohai, V.J.</span> (1985) High Breakdown Point and High Efficiency Robust Estimates for Regression. Technical report No. 66. Department of Statistics, University of Washington. .Available at http://www.stat.washington.edu/research/reports/1985/tr066.pdf.
<span style="font-variant:small-caps;">Yohai, V.J.</span> (1987), High breakdown–point and high efficiency estimates for regression, *The Annals of Statistics*. **15** 642–65.
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---
abstract: 'Investigation of near field of QED requires the refuse from an averaging of the Lorentz condition that smooths out some field peculiarities. Instead of it Schwinger decomposition of the 4-potential with the Bogoliubov method of interaction switching in time and in space regions is considered. At such approach near field is describable by the part of covariant Green function of QED, the fast-damping Schwinger function formed by longitudinal and scalar components of $A_{\mu }$ none restricted by light cone. This description reveals possibility of superluminal phenomena within the near field zone as a “nonlocality in the small”. Some specification of Bogoliubov method allows, as examples, descriptions of near fields of point-like charge and at FTIR phenomena. Precisely such possibilities of nonlocal interactions are revealed in the common QED expressions for the Van-der-Waals and Casimir interactions and in the Förster law.'
author:
- 'Mark E. Perel’man'
title: 'Near field in quantum electrodynamics: Green functions, Lorentz condition, “nonlocality in the small”, frustrated total reflection'
---
**Key words:** Lorentz condition, near field, propagators, superluminal, FTIR.
Introduction
============
The usage of 4-component vector $A_{\mu }$ for the field quantization, when for the Maxwell equations in vacuum only two components are needed, induced serious problems at early stages of QED. These problems were initially obviated by Fermi via averaging the Lorentz condition as $\partial A_{\mu }/\partial x_{\mu
}\left\vert 0\right\rangle =0$ \[1\]. This way had led to the indefinite metrics that formally exhausts the problem of exception of superfluous components of $A_{\mu }$ by an artificial their averaging and leaves the theory completely local, although the used procedure has not a direct physical sense, e.g. \[2\]. It excludes also near field from consideration, but at that time its effects were out of common interests.
With the discovery of Aharonov-Bohm effect \[3\] the complexity of real situation could already become more evident: it is impossible to simply cast away two additional components of 4-potential, as the $\mathbf{E}$ and $%
\mathbf{B}$ do not describe all features of electromagnetic fields. But moreover, the development of optics into recent decades show greater complexity of near field than was commonly accepted. The most unexpected peculiarities represent the phenomena of interference in the scope of near field (e.g. \[4, 5\] and references therein) and numerous observations of superluminal signalling (the reviews \[6\]), which also may be related with near fields. All it requires a returning to general problems of near field and to a revision of common representations for description of these new phenomena, at that possibly of nonlocal nature.
But can exist nonlocal solutions of covariant equations for little distances? If to search solutions of the wave equation as $f(x)=f(\mathbf{v}%
t-\mathbf{r})$, then the non-fading solution exists only for $v\leq c$. This peculiarity strongly forbids superluminal movement as an effective nonlocality on arbitrary distances. However, this *equation does not forbid* propagation of faster-than-$c$ “perturbations” on short distances, of order of uncertainty principle distances, at condition of their quick damping.
In our article \[7\] has been established, in the frame of general phenomenological approach, that the transferring of excitation within near field on the distance $\Delta x$ are possible only and only as the instant tunneling, when these distances are inverse relatively energy deficiency to the nearest stable or resonant (quasistable) state $\Delta \hbar \omega $:
$$\Delta x\cdot \Delta \omega \sim \pi c.\ \label{(1)}$$
It defines a “nonlocality in the small” of near field (this “nonlocality” is restrained by the especial form of the energy-time duration uncertainty principle, such additional proof of (1) is given in \[8\]). Thus the superluminal phenomena can occur, for example, in the domain of anomalous dispersion or at the frustrated total internal reflection (FTIR). And it is one of peculiarities of near field that is not evident at usual approach.
Notice, that the common description of field is related to far fields only: as must be especially underlined, the features of locality were established and many times checked for far fields only. But the locality of near fields never was experimentally checked and possibility of their general or at special conditions nonlocality can not be a priori excluded.
With the purpose of investigating such possibilities we shall try to examine the structure of near fields without averaging of the Lorentz condition. Instead of it the Schwinger method of $A_{\mu }$decomposition on transverse, longitudinal and scalar components will be used \[9\]. These procedures (Section 3) will lead to revealing the existence of nonlocal terms within the known decomposition of covariant Green functions (Section 2).
Then the most general procedure of interaction on and off switching, the Bogoliubov method \[10\] for description of QED interactions without the averaged additional condition, will be used (Section 4). It completes the construction of near field in absence averaging procedure and shows its possible nonlocality, manifestations of which quickly disappear with distance.
In the subsequent Sections we try to apply this approach to some real phenomena. It requires special partial specializations of the Bogoliubov method suggested in the Appendix, but contains some arbitrariness’ and therefore subsequent considerations have rather a hypothetical character.
In the Section 5 a near field of point-like charge will be considered with estimation of its characteristic energies at different distances. In the Section 6 on some similar base the phenomena of FTIR are considered, which reveals the dependence of wave numbers values on distances from the surface and corresponding possibility of interference picture formation leading to effects of near field optics.
Then it will be shown that the Van-der-Waals forces, in the form deduced in \[11\], and the Förster law of excitation transfer on short distances are describable in the frame of QED via near-field nonlocal interactions. The revealing of such description means that near-field effects are not so exotic as it can seems and that they are responsible for some known phenomena.
The results of executed examinations, which can be considered as some detalisation of the general results of \[7, 8\], are summed in the Conclusions, where some further perspectives of the offered method are mentioned.
Decomposition of canonical Green functions
==========================================
Let’s consider the covariant Pauli-Jordan function $D(t,\mathbf{r})$ in the variant of Coulomb gauge suggested by Dzjaloshinski and Pitayevski \[11\]
$$D_{ij}(\omega ,\mathbf{r})=(\delta _{ij}+\partial _{i}\partial _{j}/\omega
^{2})D(\omega ,\ \mathbf{r});\qquad D_{i0}=D_{00}=0. \label{2}$$
After differentiation it leads to the representation of Green functions of wave equation (\[12\], cf. \[5\],) as
$$D_{ij}(\omega ,\ \mathbf{r})=\{(\delta _{ij}+e_{i}e_{j})-\frac{i}{\omega r}%
P_{ij}\cot (\omega r)+\frac{1}{(\omega r)^{2}}P_{ij}\}D(\omega ,\mathbf{r}),
\label{3}$$
here and below $c=\hbar =1$, $P_{ij}=\delta _{ij}-3e_{i}e_{j}$. (Note that the decomposition of transfer functions is simpler than decomposition of field strengths \[13\].)
Three terms of (3), which correspond to far, intermediate (or transient) and near fields, are expressed in the $(t,\mathbf{r})$ representation through the Pauli-Jordan function and the generalized singular function $D_{N}(t,\
\mathbf{r})$($\partial _{x}\equiv \partial /\partial x$):
$$D_{ij}(t,\ \mathbf{r})|_{FF}=(\delta _{ij}+e_{i}e_{j})D(t,\mathbf{r});
\label{4}$$
$$D_{ij}(t,\ \mathbf{r})|_{IF}=\frac{1}{4\pi r}P_{ij}\theta
(r^{2}-t^{2})\equiv \frac{1}{r}P_{ij}\ \partial _{t}D_{N}(t,\mathbf{r});
\label{5}$$
$$D_{ij}(t,\ \mathbf{r})|_{NF}=\frac{1}{4\pi r^{2}}P_{ij}\{sgn(t)\theta
(t^{2}-r^{2})+\frac{t}{r}\theta (r^{2}-t^{2})\}\equiv \frac{1}{r^{2}}%
P_{ij}D_{N}(t,\ \mathbf{r}). \label{6}$$
The function $D_{N}(t,r)$, valuable for all subsequent analysis, can be determined immediately by (6) and in more details will be considered below.
This decomposition shows nonlocality of near field, increasing with time, and demonstrates that transitions (5) between near and far fields are concentrated in the space-like domain. It means that near field of any charge can be nonlocal, i.e. it supposes, in particular, the possibility of propagation of quickly relaxing superluminal perturbations within near fields.
Virtual “quanta of near field” and “dressed”, i.e. free photons are clearly distinguishable in the $(\omega ,\mathbf{k})$ representation (here and below $k=|\mathbf{k}|$):$\qquad $$$D_{ij}(\omega ,\mathbf{k})|_{FF}=(\delta _{ij}+e_{i}e_{j})\frac{2}{(2\pi
)^{3}i}\varepsilon (\omega )\delta (\omega ^{2}-k^{2}); \label{4'}$$
$$D_{ij}(\omega ,\ \mathbf{k})|_{IF}=\frac{1}{4\pi 2i\omega k}P_{ij}\theta
(k^{2}-\omega ^{2}); \label{5'}$$
$$D_{ij}(\omega ,\ \mathbf{k})|_{NF}=\frac{1}{8\pi ^{2}i\omega ^{2}k}%
P_{ij}\{|\omega |\theta (\omega ^{2}-k^{2})+k\theta (k^{2}-\omega ^{2})\}.
\label{6'}$$
These expressions clearly show that when quanta of far field are the free photons with light speed $c$ in vacuum, the speed of virtual quanta propagation is not restricted. The transitions between them, described by an intermediate field, are evidently tunnel processes with an energy deficiency relative to momentum.
Note, that for functions $D(\omega ,\ \mathbf{k})$, $D^{(1)}(\omega ,\
\mathbf{k})$ and their combinations the gauge (2) actually coincides with the more widespread Coulomb gauge:
$$D_{ij}(\omega ,\ \mathbf{k})=(\delta _{ij}-k_{i}k_{j}/k^{2})D(\omega ,\
\mathbf{k});\quad D_{00}=-1/k^{2};\quad D_{i0}=0. \label{7}$$
The decomposition similar (3) can be executed, certainly, with singular functions of non-uniform wave equation $D_{c}(\omega ,\mathbf{k})$, etc.
Schwinger decomposition of 4-potential
======================================
For uncovering the origin and physical sense of the function $D_{N}$ some properties of 4-potential construction should be considered without averaging of its 4-divergence over vacuum, i.e. without usage of the Fermi-Lorentz condition.
For this aim the 4-vector $A_{\mu }$ or its frequencies parts in any system of readout must be covariantly decomposed by the Schwinger method \[9\] onto the far field $A_{\mu }^{(f)}$ and two auxiliary fields, longitudinal $%
\Lambda _{||}$ and scalar (temporal) $\Lambda _{0}$:
$$A_{\mu }(x)=A_{\mu }^{(f)}(x)+n_{\mu }(n\partial )\Lambda _{0}(x)-(\partial
_{\mu }+n_{\mu }(n\partial ))\Lambda _{||}(x), \label{8}$$
$n_{\mu }$ is the unit time-like vector, $n_{\mu }^{2}=1$, $(n\partial )\equiv n_{\mu }\partial _{\mu }$. All three fields independently obey the wave equations:
$$\Box A_{\mu }^{(f)}(x)=\Box \Lambda _{0}(x)=\Box \Lambda _{||}(x)=0.
\label{9}$$
Far field $A_{\mu }^{(f)}$ is transverse and satisfies the classical Lorentz condition:
$$n_{\mu }A_{\mu }^{(f)}(x)=\partial _{\mu }A_{\mu }^{(f)}(x)=0. \label{9'}$$
Auxiliary fields are defined via $A_{\mu }$:
$$\partial _{\mu }A_{\mu }=(n\partial )^{2}(\Lambda _{0}-\Lambda _{||});\qquad
n_{\mu }A_{\mu }=-(n\partial )\Lambda _{0}(x); \label{10}$$
$$(\partial _{\mu }+n_{\mu }(n\partial ))A_{\mu }=-(n\partial )^{2}\Lambda
_{||}, \label{11}$$
they are invariant relative coordinates inversion and changing sign at the time reversing. As the commutator of $A_{\mu }^{(f)}$ obeys the Pauli-Jordan function, additional fields form the nonlocal commutators:$\qquad $
$$\lbrack \Lambda _{0}(x),\Lambda _{0}(y)]=-[\Lambda _{||}(x),\Lambda
_{||}(y)]=iD_{N}(x-y), \label{12}$$
Let’s try to establish the reasons of appearance of this function.
The wave equation in vacuum is the operator record of dispersion identity $%
p^{2}\equiv \omega ^{2}-|k|^{2}=0$ and functions, that satisfy this identity, are represented by the Fourier integral:$\qquad $
$$G(x)=\int d^{4}p\ f(p)\ \delta (p^{2})e^{i(px)} \label{13}$$
with arbitrary, non-singular at $p^{2}=0$ function $f(p)$.
The partial solutions of corresponding non-uniform equation are represented by the Fourier integral:
$$\overline{D}(x)=\frac{2}{(2\pi )^{4}}P\int d^{4}p\ \frac{1}{p^{2}}e^{i(px)}.
\label{14}$$
If to demand that (13) and (14) compose one analytical function that corresponds to the Kramers-Kronig dispersion relations and to an opportunity of energy transition from induced oscillations into free waves and back, then $f(p)$ in (13) can consists of step operators $\theta (\pm \omega )$ only. If, however, such fields closely to the source are examining that does not immediately generate far field waves, but can consist from the own or confinement field of source only, this restriction on the form of $f(p)$ in (13) loses its force (therefore the function $D_{N}$ had been appeared at calculation of the electron self-field \[9\]). But it simultaneously means that the acceleration of near field (e.g. at charge acceleration) should lead to occurrence of free, far field waves, i.e.
$$\partial _{t}^{2}D_{N}(x)\rightarrow D(x). \label{15}$$
$\quad $
Hence the response functions of near field should be connected to the corresponding Green functions (13) by the determination (possible constant and linear on $t$ terms are omitted):
$$D_{N}^{(.)}(x)=\frac{1}{2}\int\nolimits_{-\infty }^{\infty }d^{4}y\ n_{\mu
}|x_{\mu }-y_{\mu }|\ D^{(.)}(y), \label{16}$$
or, in the evident time-space form, as$$D_{N}^{(.)}(t,\mathbf{r})=\frac{1}{2}\int\nolimits_{-\infty }^{\infty }d\tau
\ |\tau -t|\ D^{(.)}(\tau ,\mathbf{r}), \label{16'}$$
that just corresponds the function, introduced by Schwinger.
The direct calculation, with the Green functions $D(t,r)$, $D^{(1)}(t,r)$ and $D^{(\pm )}(t,r)$ of wave equation in the right-hand side of (16’), leads to such singular functions of near field:$\qquad $
$$D_{N}(t,\mathbf{r})=\frac{1}{8\pi r}\{|t-r|\ -|t+r|\}\equiv \frac{1}{4\pi }%
\{sgn(t)\theta (t^{2}-r^{2})+\frac{t}{r}\ \theta (r^{2}-t^{2})\}; \label{17}$$
$$D_{N}^{(1)}(t,\mathbf{r})=\frac{1}{4\pi ^{2}r}\{(t+r)\ln (t+r)-(t-r)\ln
(t-r)\}\equiv \frac{1}{4\pi ^{2}}\{\ln (t^{2}-r^{2})-\frac{t}{r}\ln \frac{t\
-\ r}{t\ +\ r}\}; \label{18}$$
$$D_{N}^{(\pm )}(t,r)=\frac{1}{2}\{D_{N}(t,\ \mathbf{r})\mp iD_{N}^{(1)}(t,\
\mathbf{r})\}, \label{19}$$
terms omitted in (16) can be restored for balancing dimensions of arguments of logarithms in (18). Notice that $D_{N}(t,\ \mathbf{r})$ equals zero at $%
t=0$ just as $D(t,\ \mathbf{r})$.
It is remarkable that the temporal change of function (17) occurs completely in the spatial direction:
$$\partial _{t}D_{N}(t,\ \mathbf{r})=\frac{1}{t}\ \mathbf{r\ \nabla }D_{N}(t,\
\mathbf{r})=-\frac{1}{4\pi r}\ \theta (r^{2}-t^{2}), \label{20}$$
which emphasizes the nonlocal character of near field in a concordance with the tunnel character of (5’).
Some features of these functions can be seen more obviously in the mixed $%
(\omega ,\mathbf{r})$-representation:
$$D_{N}(\omega ,\ \mathbf{r})=-\frac{1}{\omega ^{2}}D(\omega ,\ \mathbf{r})=-%
\frac{1}{2\pi i}\frac{\sin (\omega r)}{\omega ^{2}r}; \label{17'}$$
$$D_{N}^{(1)}(\omega ,\mathbf{r})=\ sgn(\omega )D_{N}(\omega ,\ \mathbf{r});
\label{18'}$$
$$D_{N}^{(\pm )}(\omega ,\mathbf{r})=\theta (\pm \omega )D_{N}(\omega ,\
\mathbf{r}). \label{19'}$$
So, in contrast to $D^{(.)}(\omega ,\mathbf{r})$, these functions are singular at $\omega =0$.
Thus, the near zone of electromagnetic field, as can be asserting, is nonlocal, in part at least, and is formed, in accordance with (12), by two (additional) components of vector $A_{\mu }$ or by scalar fields corresponding to their changes. (The opportunity of real replacement of these field components by two scalar fields demands the in-depth analysis and researches.)
But for all that the electric field $\mathbf{E}$, as must be emphasized, remains local:
$$\left\langle T(E_{i}(x)E_{k}(y))\right\rangle =\partial _{x}\partial
_{y}\left\langle T(A_{i}(x)A_{k}(y))\right\rangle \rightarrow D(x-y).
\label{21}$$
A little differently the locality of $\mathbf{E}$ and $\mathbf{H}$ fields can be shown by consideration of commutators:
$$\lbrack E_{i}(x),E_{j}(y)]=[H_{i}(x),H_{j}(y)]=\frac{1}{4\pi i}\{\partial
_{i}\partial _{j}-\delta _{ij}\partial _{t}^{2}\}D_{N}(x-y);$$
$$\lbrack E_{i}(x),H_{j}(y)]=\frac{1}{4\pi i}\partial _{t}\partial
_{j}D_{N}(x-y), \label{22}$$
double differentiation of the function $D_{N}(x)$ repays in $(\omega ,%
\mathbf{r})$ representation the factor $\omega ^{-2}$ responsible for nonlocal effects in (6).
Thus, the nonlocality should be effective, in particular, in such $A_{\mu }$-depending phenomena as the Aharonov-Bohm effect, the Casimir effect, some effects of near field optics, which become apparent at absence of electric and magnetic fields and consequently without the Lorentz force.
On quantum generalization of Lorentz condition
==============================================
The Lorentz-Fermi condition is written in QED as
$$\partial _{\mu }A_{\mu }^{(-)}\left\vert 0\right\rangle \equiv -iP_{\mu
}(0)\ A_{\mu }^{(-)}\left\vert 0\right\rangle =0, \label{23}$$
where $P_{\mu }(0)$ is the linear 4-momentum.
However the 4-moment dependence on a degree of inclusion of interaction (the general adiabatic hypothesis) must be taken into account. In the covariant Stueckelberg-Bogoliubov method \[10\] these features are described by the interaction switching function (FIS) $g(x)\in \lbrack 0,1]$. So, in particular, the 4-momentum depends on a degree of interaction switching as
$$P_{\mu }(g)=P_{\mu }(0)-\int d^{4}x\ H(x)\ \partial _{\mu }g(x), \label{24}$$
$H(x)$ is the Hamiltonian of interaction.
Hence the condition (23) must be generalized as:
$$-iP_{\mu }(g)A_{\mu }^{(-)}\left\vert g\right\rangle \equiv \partial _{\mu
}A_{\mu }^{(-)}\left\vert g\right\rangle -\int d^{4}y\ D^{(-)}(x-y)j_{\nu
}(y;g)\partial _{\nu }g(x)\left\vert g\right\rangle =0. \label{25}$$
The additional relation is executable at the Schwinger decomposition of potential without a vacuum averaging . Therefore the performing of inner part of (25) without averaging in near field can be *assumed*:
$$\partial _{\mu }A_{\mu }^{(-)}=\int d^{4}y\ D^{(-)}(x-y)\ j_{\nu }(y;g)\
\partial _{\nu }g(x). \label{26}$$
In the particular system $n_{\mu }=\delta _{\mu 0}$ potentials of near field are expressed by (11) as
$$A_{0}^{(N)}(x)\equiv A_{0}-A_{0}^{(f)}=-\partial _{t}\Lambda _{0}(x);
\label{27}$$
$$\mathbf{A}^{(N)}(x)\equiv \mathbf{A}-\mathbf{A}^{(f)}=-\mathbf{\nabla }%
\Lambda _{||}(x). \label{28}$$
It shows that near field is non-vortex ($\mathbf{B}^{(N)}\equiv 0$), and its electric component is pure longitudinal:
$$\mathbf{E}^{(N)}(x)\equiv -\partial _{t}\mathbf{A}^{(N)}-\mathbf{\nabla }%
A_{0}^{(N)}=\partial _{t}\mathbf{\nabla }(\Lambda _{0}+\Lambda _{||}).
\label{29}$$
Scalar potential in the Coulomb gauge is equal zero, i.e. the field $\Lambda
_{0}(x)$ is stationary and does not take part in the definition (29). Therefore in such gauge
$$\mathbf{E}^{(N)}(x)\equiv -\partial _{t}\mathbf{\nabla }\Lambda _{||}=\int
d^{4}y[\partial _{t}\mathbf{\nabla }D_{N}^{(-)}(x-y)]\ j_{\mu }(y;g)\
\partial _{\mu }g(y)\equiv \mathbf{K}(x)\otimes _{x}J(x;g), \label{30}$$
where $\mathbf{K}(x)=\partial _{t}\mathbf{\nabla }D_{N}^{(-)}(x)$ and $%
J(x;g)=j_{\mu }(y;g)\partial _{\mu }g(y)$ .
After inserting (20) the relation (30) represents the general expression of near field strength and demonstrates the nonlocality of near field, nonzero at any distance even at $t=0$.
As $D_{N}^{(-)}(x)=f(t,\ r)$, the formula (30) shows that near field is determined by the radial component $E_{r}^{(N)}(x)$ only (the index $r$ is below omitted). So in the $(\omega ,\mathbf{r})$-representation the kernel of these forms is simplified:
$$K(\omega ,\mathbf{r})=\frac{\theta (-\omega )}{2\pi i\omega }\partial _{r}%
\frac{\sin (\omega r)}{r}, \label{31}$$
and
$$K(\omega ,\ \mathbf{k})=\frac{\theta (-\omega )}{(2\pi )^{3}i}\left\{ \frac{4%
}{(k\ -\ i0)^{2}\ -\ \omega ^{2}}-\frac{1}{\omega k}\ln \frac{k\ -\ \omega }{%
k\ +\ \omega }\right\} . \label{32}$$
Via the equation $\mathbf{\nabla E}^{(N)}(x)=4\pi \rho ^{(N)}(x)$, with taking into account the wave equation, the expression (30) allows determination of an effective space-time distribution of charges that forms near field, i.e. allows the determination of dynamical form-factor of charge system:
$$\rho ^{(N)}(x)=\frac{1}{4\pi }\int d^{4}y\ j_{\mu }(y;g)[\partial _{\mu
}g(y)]\ \partial _{t}D^{(-)}(x-y)=\frac{1}{4\pi }\partial
_{t}D^{(-)}(x)\otimes _{x}J(x;g). \label{33}$$
Further analysis of (30) can be possible by substitution of the some expressions of FIS’ suggested in the Appendix.
Near field of point-like charge
===============================
Let’s consider the fixed charge $Q$ in the origo of coordinates. Its current density
$$j_{\mu }(x)=Q\ \delta _{\mu 0}\delta (r) \label{34}$$
leads with taking into account the FIS (61) to the generalized current function
$$J(x)=-\gamma Q\ \delta (r)sgn(t)e^{-\gamma |t|}. \label{35}$$
In the $(\omega ,\mathbf{r})$- representation
$$J(\omega ,\mathbf{r})=\frac{i\gamma Q\omega }{\pi (\omega ^{2}\ +\ \gamma
^{2})}\ \delta (\mathbf{r}) \label{36}$$
according to (30). It results in the following expressions for radial components of electric field strength in $(\omega ,\mathbf{r})$- and $%
(\omega ,\mathbf{k})$-representations:
$$E^{(N)}(\omega ,\mathbf{r};\gamma )=\frac{\theta (-\omega )\gamma Q}{2\pi
^{2}(\omega ^{2}\ +\ \gamma ^{2})}\partial _{r}\frac{sin(\omega r)}{r},
\label{37}$$
$$E^{(N)}(\omega ,\mathbf{k};\gamma )=\frac{\theta (-\omega )\gamma Q}{4\pi
^{3}(\omega ^{2}\ +\ \gamma ^{2})}\left\{ \frac{4\omega }{(k\ -\ i0)^{2}-\
\omega ^{2}}-\ln \left\vert \frac{k\ -\ \omega }{k\ +\ \omega }\right\vert
\right\} . \label{37'}$$
The inverse FT of (37) is expressed through the integral hyperbolic functions (cf. \[14\], Exp. (108-9), $\beta =\gamma (t-r)$; $\overline{\beta }%
=\gamma (t+r)$):
$$E^{(N)}(t,\mathbf{r};\gamma )=\frac{2Q}{(2\pi )^{3}}\partial _{r}\left\{
\frac{1}{r}[i\pi e^{-|\beta |}+2\left\{ \text{chi}(|\beta |)\text{sh}(\beta
)-\text{shi}(\beta )\text{ch}(\beta )\right\} ]-\frac{1}{r}[\beta
\rightarrow \overline{\beta }]\right\} . \label{38}$$
Let’s estimate the self energy of near field at the moment $t=0$:
$$W(0)=\frac{1}{8\pi }\int dr\ |E^{(N)}(t=0,\mathbf{r};\gamma )|^{2}.
\label{39}$$
Direct substitution of (38) into (39) leads to an excessively complicated expression. Therefore we shall consider more rough estimations for FT of (37) at $t=0$ for two frequencies regions taken separately:
$$E_{1}^{(N)}(t=0,\mathbf{r}|\gamma >>\omega )\approx -\frac{Q}{\pi ^{2}\gamma
r^{3}}; \label{40}$$
$$E_{2}^{(N)}(t=0,\mathbf{r}|\gamma <<\omega )\approx -\frac{Q\gamma }{2\pi
^{2}r}. \label{40'}$$
The expression (40) is related to low frequencies, i.e. to large distances, and therefore (39) can be integrated in the limits $(1/\gamma ,\infty )$. The expression (40’) corresponds to high frequencies and after substitution into (39) it can be integrated over $(0,1/\gamma )$. These integrations give very close expressions and their sum leads to the compound estimation:
$$W(0)\approx Q^{2}\gamma /6\pi ^{4}+Q^{2}\gamma /8\pi ^{4}\approx 3\cdot
10^{-3}Q^{2}\gamma \label{41}$$
or in the usual units and at $Q\rightarrow e$, $\alpha =e^{2}/\hbar c$, $%
[\gamma ]=\ $sec$^{-1}$$$W(0)/mc^{2}\approx 3\cdot 10^{-3}\alpha \lambda _{C}\ \gamma /c=3\cdot
10^{-3}r_{0}\gamma /c. \label{41'}$$
The estimation shows that on the distances of order of the Compton wavelength, $\gamma \rightarrow c/\lambda _{C}$, the near field energy is of order of some eV’s that corresponds to potentials of ionization. At $%
c/\gamma $ of the order of Bohr radius, $r_{B}=\alpha ^{-1}\lambda _{C}$, it gives decimal fractions of eV for near field energy in correspondence with interatomic bonds.
The estimation of charge distribution of near field represents curious. According to (231) and (37) in the $(\omega ,\mathbf{r})$-representation
$$\rho ^{(N)}(\omega ,\mathbf{r})=-\frac{Q\omega ^{2}}{4\pi }g(\omega
)D^{(-)}(\omega ,\mathbf{r})=\theta (-\omega )\frac{iQ}{(2\pi )^{3}}\frac{%
\gamma \omega ^{2}}{\omega ^{2}\ +\ \gamma ^{2}}\frac{\sin \omega r}{r}.
\label{42}$$
or$$\rho ^{(N)}(t,\ \mathbf{r})=\frac{1}{4\pi }\int d\tau \ g(\tau )\ \partial
_{t}^{2}D^{(-)}(\tau -t,r). \label{42'}$$
Its average value over $r$ or $t$ is equal, certainly, to zero, and the maximum is achieved at $\omega r=0$. On high frequencies, when $\omega
>>\gamma $,
$$\rho ^{(N)}(\omega ,\mathbf{r})\simeq Q\gamma D^{(-)}(\omega ,\mathbf{r}),
\label{43}$$
i.e. properties of near field come closer to far field features. In the low frequencies field, i.e. at $\omega <<\gamma $, an effective charge density in the near field $\rho ^{(N)}(\omega ,r)\rightarrow 0$.
Frustrated total internal reflection (FTIR)
===========================================
Let’s consider, via the expression (30), the phenomenon of FTIR of light wave $\mathbf{E}=\mathbf{E}_{0}e^{i\omega t}$ under an angle $\varphi $ ($%
\varphi >\varphi _{crit}$) from smooth dielectric surface ($z=0$) of medium with polarizability $\alpha .$
The index of refraction is formally expressed as
$$n(z)=n\ \theta (-z)+1\cdot \theta (z)=\frac{1}{2}[(n+1)-(n-1)sgn(z)].
\label{44}$$
In reality this idealized plane must be substituted by an effective layer of minimal depth, without an instantaneous jump from parameters of one medium to another ones, i.e. by a sufficiently smooth transitive layer between both media. With this aim it is necessary to choose FIS, smooth together with the first derivative, e.g. sgn$(z)\rightarrow g(z|\varphi )=\exp (-z^{2}/\Delta
z^{2})$ with $\Delta z^{-1}=4n(z)\omega \cos \varphi $.
The initial wave induces (or orients) dipoles $\mathbf{p}(t)=\alpha \mathbf{E%
}(t)$ on the surface plane, i.e. induces the “current” $j_{\mu }(y;g)\rightarrow (c/n)\delta _{\mu z}\ \rho
(t,\ \mathbf{r})$.
If to accept that a plane or layer interface of medium is strictly flat, the density of surface charges would be described as
$$\rho =\left\langle \mathbf{p}(t)\right\rangle \cdot \delta ^{\prime
}(z)\delta (x)\delta (y)=-\left\langle \mathbf{p}(t)\right\rangle \
z^{-1}\delta (z)\delta (x)\delta (y), \label{45}$$
i.e. as a double electric layer oscillating with a frequency of falling wave.
As this layer must be taken into account at estimations of other parameters of medium also, it is necessary to replace $\delta (z)$ in the last expression (45) on a $\delta $-like function in an agreement with the choice of FIS: e.g. $\delta (z)\rightarrow $ $\delta (z,\xi )=(\xi \surd \pi
)^{-1}\exp (-z^{2}/\xi ^{2})$.
Thus the expression (30) of near field strength at inter-surface layer will be of the order
$$\mathbf{E}^{(N)}(t,\mathbf{r}|\omega )=\alpha |\mathbf{E}_{0}|\int%
\nolimits_{-\infty }^{\infty }d\tau d\varsigma \ n^{-1}e^{-i\tau \omega
}\delta ^{\prime }(\varsigma ,\Delta z)[\partial _{\varsigma }g(\varsigma
|\Delta z)]\ \mathbf{\nabla }\partial _{\tau }D_{N}^{(-)}(\tau
-t;x,y,\varsigma -z), \label{46}$$
where the formal integration over time can be executed. It shows, that frequencies of near field “photons”, the evanescent “particles”, will coincide with frequencies $\omega $ of initial field and all their differences should be manifesting only in momenta.
As $\partial _{z}g(z|\varphi )=-2z\Delta z^{-2}g(z|\varphi )$ and $\Delta
z(\varsigma )$ has different values for $\pm $ arguments, the relation (46) can be rewritten as
$$\mathbf{E}^{(N)}(t,\mathbf{r}|\omega )=\frac{2i}{\sqrt{\pi }}\ \omega
e^{-i\omega t}\alpha |\mathbf{E}_{0}|\left\{ \left[ \frac{1}{\Delta z_{1}^{3}%
}\int\nolimits_{0}^{\infty }d\varsigma e^{(-2\varsigma ^{2}/\Delta
z_{1}^{2})}\mathbf{\nabla }D_{N}^{(-)}(\omega ;x,y,\varsigma -z)\right] +%
\frac{1}{n}\left[ \Delta z_{1}\rightarrow \Delta z_{2},\ z\rightarrow -z%
\right] \right\} , \label{47}$$
where it is taken into account that $\Delta z\prime =n\Delta z$ does not depend on $z$.
The distribution of near field, as shows (47), does not depend on depth of its tunneling relatively FTIR “plane”, in z-direction, and occurs instantaneously. For this reason the possible transformation of evanescent waves into extending waves occurs *simultaneously* in all forbidden depth (cf. \[15\]).
The expression (47) can be considered as the Fourier convolution over variable $z$ in the first term and over $(-z)$ in the second one. By separation of $z$-component of wave vector, $k=\{k_{\bot },q\}$ and with taking into account the Fourier transformation (FT) of “current” factor:
$$I(q,\Delta z)=\int\nolimits_{0}^{\infty }d\varsigma e^{(iq\varsigma
-2\varsigma ^{2}/\Delta z^{2})}=\Delta z\sqrt{\frac{\pi }{8}}e^{-q^{2}\Delta
z^{2}/8}[1-\ erf(-iq\Delta z/\surd 8)]. \label{48}$$
So we receive the final expression for FT of near field intensity as
$$E^{(N)}(t,\mathbf{k}|\omega )=2\pi ^{-1/2}\omega e^{-i\omega t}\alpha
|E_{0}|\Delta z^{-3}k\{I(-q,\Delta z)+n^{-4}I(q,\Delta
z)\}D_{N}{}^{(-)}(\omega ;k). \label{49}$$
At small values of parameter $q\Delta z$ the expression in braces in (49) becomes
$$\{\ldots \}\rightarrow \sqrt{\frac{\pi }{8}}\frac{1}{\Delta z^{2}}[\exp
(-q^{2}\Delta z^{2}/8)+\frac{1}{n^{3}}\exp (-n^{2}q^{2}\Delta z^{2}/8)],
\label{50}$$
i.e. it contains only $q$-components of wave vector of any magnitude. Therefore “evanescent photons” can possess, at the same frequency, a bigger or smaller momenta (compare \[16\]).
But for all that, due to the identity of frequencies, they can interfere with photons of inlet radiation. It means that such interference can locate objects, at supervision of interference picture closely enough to the refraction surface, with sizes about $|k_{z}-q|^{-1}$, i.e. smaller light wavelengths in vacuum. Just this effect is the physical basis of so-called near field optics \[4\] (we do not examine the evident complications associated with light polarization).
Let’s note that the use of another FIS, e.g. (67) instead of (65), leads to similar results with replacement $q^{2}\Delta z^{2}$ on $|q\Delta z|$ and insignificant change of numerical factors. The choice between different FIS’ can be made, at the given stage, by a comparison with experiments only.
Interaction of atoms in near field
==================================
** **
Energy of non-resonant interaction of two neutral atoms on distances smaller wavelength, but bigger their own sizes \[11\], is determined by the two-photon exchange (the fourth order of $S$-matrix) as
$$U(\mathbf{r})=\frac{i}{4\pi }\int\nolimits_{-\infty }^{\infty }d\omega \
\omega ^{4}\alpha _{1}(\omega )\alpha _{2}(\omega )[D_{ik}(\omega ,\mathbf{r}%
)]^{2}, \label{51}$$
where $\alpha _{i}(\omega )$ is the polarizability of cooperating atoms, scalar at the $S$-states.
The affinity of $D_{il}(\omega ,r)|_{NF}$ to the matrix element of dipole-dipole interaction is evident. Therefore for such distances and atomic frequencies that $\omega r<<1$, the calculations with inserting the decomposition (3) into (51) are precisely identical to the procedure \[11\] and lead to the Van-der-Waals energy of interaction proportional $R^{-6}$ and to the Casimir energy of interaction of atoms proportional to $R^{-7}$.
Thus, these interactions are describable by the propagator (6), i.e. they occur in the near field and, at least, are in part transferred superluminally.
However for resonant interaction between identical (motionless) atoms,
$$A_{1}^{\ast }+A_{2}\longleftrightarrow \ A_{1}+A_{2}^{\ast }, \label{52}$$
matrix element is nonzero still in the second order:
$$S^{(2)}=-\frac{1}{2}\int dt_{1}dt_{2}\ T\{V(t_{1})V(t_{2})\}, \label{53}$$
where $V=-\mathbf{E}(\mathbf{r}_{1})\mathbf{d}_{1}-\mathbf{E}(\mathbf{r}_{2})%
\mathbf{d}_{2}$. Therefore instead of (51) we have
$$U(\mathbf{r})=(i/2\pi )\int\nolimits_{-\infty }^{\infty }d\omega \ \omega
^{2}D_{ik}(\omega ,\mathbf{r})\ Re[\alpha _{ik}(\omega )], \label{54}$$
with the tensor of scattering of two-level, for simplicity, systems expressed through matrix elements of dipole momenta:
$$\alpha _{ik}(\omega )=\frac{(d_{i})_{01}(d_{k})_{10}}{\omega _{0}\ -\ \omega
\ -\ i\Gamma }+\frac{(d_{k})_{01}(d_{i})_{10}}{\omega _{0}\ +\ \omega \ -\
i\Gamma }. \label{55}$$
By the substitution of $D_{N}$ function into (54) it can be shown that the interaction (52) decreases as $R^{-3}$ (cf. \[17\]).
The full probability of process (52) in the near field is determined as
$$W\ \propto \int\nolimits_{-\infty }^{\infty }d\omega |D_{ik}(\omega ,\mathbf{%
r})\alpha _{ik}(\omega )|^{2}\rightarrow \int\nolimits_{-\infty }^{\infty
}d\omega |d_{1}|^{2}|d_{2}|^{2}|D_{ik}(\omega ,\mathbf{r})|_{NF}|^{2}\tau
(\omega )/\Gamma , \label{56}$$
where the expression of duration of scattering process is separating out:
$$\tau (\omega )=\Gamma /2[(\omega _{0}-\omega )^{2}+\Gamma ^{2}/4].
\label{57}$$
By carrying out integration in view of $\delta $-character of (57), using matrix elements of dipole operators $|d|^{2}=\hbar e^{2}f/2m\omega $, $f$ is the oscillator force, and substituting the expressions of singular functions (6-7), we receive, that the probability of process depends on distance between cooperating atoms as $R^{-6}$, i.e. it takes the form of the well-known half-empirical Förster law \[18\] (see, e.g., \[19\]):
$$W=\Gamma ^{-1}(R_{0}/R)^{6}, \label{58}$$
where $R_{0}$ is the so-called Förster radius.
With (6) it follows that the rate of process (52) in the time representation is represented by the square of near field singular function (6):
$$|D_{il}(t,\ \mathbf{r})|_{NF}|^{2}=\frac{1}{(4\pi r^{2})^{2}}\left\{ \theta
(t^{2}-r^{2})+(\frac{t}{r})^{2}\theta (r^{2}-t^{2})\right\} , \label{59}$$
that determines relative probabilities of excitation transfer with subluminal and superluminal speeds.
But the superluminal (instantaneous) interaction is possible, as was established in \[7\], at the tunneling only. Are there certain processes and certain frequencies ranges in condensed media similar to anomalous dispersion in optics?
In the articles \[20\] we had shown that at phase transitions of the first kind the liberated latent energy must be converted, at least partially, into the characteristic radiation with frequencies determined by released energy at establishing definite bonds. (This phenomenon is hard for observations, because surrounded substances quickly thermalize the emitted radiation, but its existence is confirmed by some experiments, e.g. \[21\] and references therein.)
This phenomenon can take place at the sight of discrepancy between energy of relating corresponding bonds and momenta of emitted virtual “photons”. Therefore their interaction with neighbors can have some similarity with anomalous dispersion regime, in particular with its instantaneous peculiarity. It allows to *assume* that the permanent interatomic bonds in condensed state, partially or completely, are instantaneous ones.
Conclusions
===========
Let us enumerate the results.
1\. The decomposition of canonical Green functions of QED leads to the appearance of propagators of far and near fields and an intermediate one describing transitions between them.
2\. Green function of near field corresponds to the Schwinger function, initially introduced for investigation of an electron self-field. The rule of transition from far field functions into near field ones and vice versa is considered.
3\. The Schwinger scheme of $A_{\mu }$ decomposition allows to discard at examination of near field phenomena a formal vacuum averaging of the classical Lorentz condition. At the same time it shows that the near field function is formed by longitudinal and scalar (temporal) components of the 4-vector $A_{\mu }$ or by two additional scalar fields, derivatives of which represent these components. Thus this scheme demonstrates the physical sense of “surplus” components and shows the inconsistency of their complete formal elimination by introduction of indefinite metrics.
4\. The performed analysis has shown, within the frame of QED, that the near zone of $A_{\mu }$ represents the nonlocal, but quickly decreasing field, so that the $\mathbf{E}$ and $\mathbf{B}$ fields remain local. Hence this analysis requires the introduction of the nonlocality in the small only.
It once more underlined the necessity and significance of notions of adiabatical switching on and off interaction for understanding details of QED interactions (cf., e.g. \[22\]).
5\. The general approach to FTIR phenomena is elaborated via *introduction*, along an analogue with the field theory, the function of interaction switching at wave transition into another medium. Such method can be applied to other phenomena of substance parameters changing. The used approach shows the superluminal features of FTIR.
6\. Instantaneous or, more widely, superluminal transfer of excitations can be or even must be detectable in other phenomena. In this connection it is shown that the common expressions for the Van-der-Waals potential and the Casimir forces can be naturally expressed through the Schwinger function. It means that these interactions can be, partly at least, instantaneous. The expression for excitation transfer on small distances (the Förster law) also has the same form.
Let’s recall, in this connection, the continued discussions of the temporal features of tunneling \[23\] that had induced a number of paradoxes (e.g., \[24, 25\]). Our consideration shows within the framework of QED, at least, that the tunnel transition must be executable within the scope of near field and, under some conditions, can be instantaneous \[7\].
All our consideration shows that the instantaneous transferring is not of very exotic, extraordinarily nature; its manifestations can occur in some phenomena, which may be considered as the “nonlocal in the small”, and therefore their temporal features should be investigated more carefully and widely. (It seems, for example, that such phenomena as the energy-time entanglements can be also connected with such nonlocality, e.g. \[27\].)
7\. The described features of near zone of point-like charge and, on the other hand, of optical transitive zones as the fields of longitudinal and scalar evanescent “photons” with possibilities of observability of their “superluminal” features, deprives the formal schemes of elimination of “superfluous” fields components, such as introduction the indefinite metrics, their general significance. All this demands anew returning to the principal problems of the QED gauges, opportunities of their transformations and their peculiarities.
8\. The revealed “nonlocality in the small” in the context of covariant field theory requires not only more scrupulous further research of its properties, but also the search of similar phenomena as in the QED, so, probably, in the theories of other fields.
**A**cknowledgments
===================
The author is grateful to Professors F. W. Hehl, G. Nimtz, I. I. Royzen and Dr. G. M. Rubistein for valuable comments and support.
Appendix: Functions of interaction switching
============================================
** **
In the Bogoliubov theory the temporal FIS’ $g(t,\ \mathbf{r})$ are not concretized, they must only satisfy the conditions:
a.$g(t,\mathbf{\ r})\in \lbrack 0,1]$, which can be slightly generalized as $|$$g(t,\ r)|\ \in \lbrack 0,1]$;
b.General covariance;
c.Limiting conditions: $g(x)\rightarrow 0$ at $x\rightarrow \pm
\infty $;
d.$g(-x_{\mu })=g(x_{\mu })$, that follows CPT invariance of $A_{\mu
} $.
The simplest FIS’ satisfying these conditions:
$$g_{1}(x)=e^{-\gamma |n_{\mu }x_{\mu }|};\quad g_{2}(x)=e^{-\gamma
^{2}(n_{\mu }x_{\mu })^{2}};\quad \ g_{3}(x)=e^{-\gamma ^{2}|x_{\mu }|^{2}}.
\label{60}$$
The existence of near field in classical theories allows its independence from $\hbar $. Therefore for classical problems it seems natural to express this parameter by the Thompson radius of electron: $%
\gamma =c/r_{0}\equiv mc^{3}/e^{2}$, this form leads to disappearance of interaction and near field at $e\rightarrow 0$. For bound electron its expression via the Compton wavelength of electron seems natural: $\gamma
^{\prime }=c/\lambda _{C}=mc^{2}/\hbar $.
For the part of our consideration the choice $n_{\mu }=\delta _{\mu 0}$ is sufficient:
$$g_{1}(x)=e^{-\gamma |t|};\quad g_{2}(x)=e^{-\gamma ^{2}t^{2}} \label{61}$$
and $g_{2}(x)=g_{3}(x)$ at $r=0$. Under the FT these FIS’ take the forms:
$$g_{1}(\omega )=\gamma /\pi (\omega ^{2}+\gamma ^{2});\quad g_{2}(\omega
)=(2\gamma \surd \pi )^{-1}\exp (-\omega ^{2}/4\gamma ^{2}), \label{62}$$
at $\gamma \rightarrow 0$ they aspire to $\delta (\omega )$.
To functions (62) different physical interpretations can be given. So, $%
g_{2}(\omega )$ corresponds to the normal law of probability with similar process of interaction switching, etc. The function $g_{1}(\omega )$ corresponds, excluding factor $\pi $, to the duration of elastic photon scattering on free electron. Such interpretation allows generalization of $%
g_{1}(\omega )$ for interaction with bound electron into two level systems:
$$g_{1}(\omega |\omega _{0})=\gamma /2\pi \{[(\omega -\omega _{0})^{2}+\gamma
^{2})]^{-1}+[(\omega +\omega _{0})^{2}+\gamma ^{2})]^{-1}\} \label{63}$$
or, in the $t$-representation,
$$g_{1}(t|\omega _{0})=e^{-\gamma |t|}\cos (\omega _{0}t). \label{64}$$
Let us introduce on the similar basis the space FIS’ determining the switching or alteration of interaction during particle (wave) transitions across determined (flat) space borders. So, at approach of light wave to refracting surface $z=0$ it can be *suggested* the FIS:
$$g_{1}(x)\rightarrow g_{1}(z)=e^{-\kappa |z|};\quad g_{2}(x)\rightarrow
g_{2}(z)=e^{-\kappa ^{2}z^{2}}, \label{65}$$
where $\kappa ^{-1}$ is the effective width of an intermediate surface layer depending on parameters of both substances and light flux.
At the total internal (or external) reflection of light of frequency $\omega
$ on intersection of media with indices of refraction $n_{1}$ and $n_{2}$ under angle $\varphi >\varphi _{crit}$, $z$-component of photon momentum $%
k_{z}=kn_{1}\cos \varphi $ changes sign and the alteration of momentum is equal
$$|\Delta k_{z}|\ =2kn_{1}\cos \varphi . \label{66}$$
In accordance with the uncertainty principle this alteration is executed in the layer $|\Delta z|\ =1/2|\Delta k_{z}|$, therefore the value $\kappa
=1/|\Delta z|$ and at $n=n_{1}/n_{2}$
$$g_{1}(t,r|\varphi )=e^{-4n|\omega ||z|\cos \varphi },\qquad \quad
g_{2}(t,r|\varphi )=e^{-(4n\omega z\cos \varphi )^{2}}. \label{67}$$
Such transient optical layer should exist at all processes of reflection and refraction, since at $\varphi <\varphi _{crit}$ and with refraction angle $%
\varphi ^{\prime }$ the change of photon momentum is
$$|\Delta k_{z}|=k|n_{1}\cos \varphi -n_{2}\cos \varphi \prime |,\ \label{68}$$
which again leads to (67). These processes, as it is known, can be of nonlocal character, described usually via surface polaritons (e.g., \[28\]).
References
----------
\*). E-mail: mark\_perelman@mail.ru
1\. E. Fermi. Rev.Mod.Phys., **4**, 87 (1932).
2\. G. Källen. *Quantum Electrodynamics*. London: 1972.
3\. Y. Aharonov, D. Bohm. Phys.Rev., **115**, 485 (1959).
4\. C. Girard and A. Dereux. Rep. Prog. Phys. **59**, 657 (1996); C. Girard, C. Joachim and S. Gauthier. Rep. Prog. Phys. **63**, 893 (2000).
5\. O. Keller. J.Opt.Soc.Am.B, **16**, 835 (1999); Phys.Rev.A, **62**, 022111 (2000).
6\. G. Nimtz and W. Heitman. Progr.Quant.Electr., **21**, 81 (1997); R. Y. Chiao and A. M. Steinberg. Phys.Scr.T, **76**, 61 (1998); E. Recami. Found.Phys., **31**, 1119 (2001); P. W. Milonni. J.Phys.B., **35**, R31 (2002); G. Nimtz. Progr.Quant.Electr., **27**, 417 (2003).
7\. M. E. Perel’man. Ann. Phys. (Leipzig), **14**, 733 (2005).
8\. M. E. Perel’man. Sov. Phys. Doklady, **19**, 26 (1974); Int. J. Theor.Phys. - in press; quant-ph/0510123.
9\. J. Schwinger. Phys.Rev., **75**, 651 (1949), **74**, 1439 (1948).
10\. N. N. Bogoliubov and D. V. Shirkov, *Introduction to the Theory of Quantized Fields*, 3rd edition (Wiley, New York, 1980).
11\. V. Berestetski, E. Lifshitz and L. Pitayevski, *Relativistic Quantum Theory* (Pergamon, Oxford, 1971).
12\. L. D. Landau and E. M. Lifshitz, *The Classical Field Theory* (Pergamon, Oxford, 1975).
13\. E. Wolf and H. T. Foley. Opt.Lett., **23**, 16 (1998).
14\. Ya. Brychkov and A. Prudnikov, *Integral Transformations of Generalized Functions* (Moscow: Nauka, 1977).
15\. K. Wynne and D. A. Jaroszynski. Opt.Lett., **24**, 25 (1999); D.R.Solli e.a. Phys.Rev.Lett, **91**, 143906 (2003).
16\. J. M. Vigoureux, L. D’Hooge and D. Van Labeke. Phys.Rev.A, **21**, 347 (1980).
17\. *Long Range Forces: Theory and Recent Experiments in Atomic Systems* (F. S. Levin, D. Misha, Ed’s). NY, Plenum, 1992.
18\. Th. Förster. Ann. Phys. (Leipzig), **2**, 55 (1948).
19\. S. Y. Buhmann e.a.: quant-ph/0403128.
20\. M. E. Perel’man. Phys.Lett. A, **32**, 411 (1971); Sov.Phys.-Doklady, **19,** 26 (1974); Astrophysics, **17**, 383 (1981); arXiv: cond-mat/0401544, Phil. Mag. (2006) - in press.
21\. L. M. Umarov and V. A. Tatarchenko. Crystallography, **29**, 1146 (1984).
22\. P. W. Milonni, R. J. Cook and J. R. Ackarhalt. Phys.Rev.A, **40**, 3764 (1989).
23\. L. A. MacColl. Phys.Rev., **40**, 621 (1932).
24\. E. O. Kane. In: *Tunneling Phenomena in Solids*. (E.Burstein, S.Lundqvist, Ed’s). NY: Plenum, 1969.
25\. F. E. Low and P.F. Mende, Ann. Phys. (NY) **210**, 380 (1991); M. Morgenstern et al. Phys.Rev.B, **63**, 201301 (2001).
26\. J. Broe and O. Keller. Opt.Comm. **194**, 83 (2001) and references therein.
27\. S. Faisel et al. Phys.Rev.Lett., **94**, 110501 (2005).
28\. A. Zangwill, *Physics at Surfaces* (Cambridge Univ. Press, London, 1988).
|
---
abstract: 'The onset of distortion in one-dimensional monatomic chains with partially filled valence bands is considered to be well-established by the Peierls theorem, which associates the distortion with the formation of a band gap and a subsequent gain in energy. Employing modern total energy methods on the test cases of lithium, sodium and carbon chains, we reveal that the distortion is not universal, but conditional upon the balance between distorting and stabilizing forces. Furthermore, in all systems studied, the electrostatic interactions between the electrons and ions act as the main driving force for distortion, rather than the electron band lowering at the Fermi level as is commonly believed. The main stabilizing force which drives the chains toward their symmetric arrangement is derived from the electronic kinetic energy. Both forces are affected by the external conditions, e.g. stress, and consequently the instability of one-dimensional nanowires is conditional upon them. This brings a new perspective to the field of one-dimensional metals, and may shed new light on the distortion of more complex structures.'
author:
- 'D. Kartoon'
- 'U. Argaman'
- 'G. Makov'
bibliography:
- 'Peierls\_refs.bib'
title: 'The driving force behind the distortion of one-dimensional monatomic chains – Peierls theorem revisited'
---
[^1]
[^2]
\[sec:level1\]Introduction
==========================
One-dimensional monatomic chains are of fundamental interest [@Peierls53; @Peierls91; @McAdon88; @Littlewood81], and a focus of applications in nanotechnology [@Zeng2008; @Casillas2014-C; @Casari2016-C; @Ayuela2002; @Cretu2013; @LaTorre2015]. In his seminal work [@Peierls53; @Peierls91], Peierls argued that one-dimensional evenly-spaced metallic chains can never be stable at zero Kelvin (neglecting the effect of zero-point motion). According to Peierls’ theorem, such systems will spontaneously undergo a transition into a more stable lower-symmetry insulating state. This transition causes the Fermi surface to coincide with the Brillouin zone boundaries, and it is driven by the opening of an energy gap at the zone boundaries. Peierls showed that the energy gain from such a distortion is $$\label{eq:Peierls}
\Delta E \propto -\tau^2 \cdot log (\tau)$$ where $\tau$ is the relative displacement of atoms from their symmetric (equally spaced) positions with a periodicity determined so that the Fermi surface and the edge of the Brillouin zone intersect. At sufficiently small distortions, the energy gain given in Eq. \[eq:Peierls\] is always greater than the repulsion between the atomic cores which is assumed to vary as $\tau^2$, thus making the one-dimensional chain inherently unstable. In particular, in the case of half-filled valence bands, the optimal distortion is dimerization of the chain, where every second atom is displaced by $\tau$, and according to Peierls the energy gain is the largest.
Experiments with one-dimensional carbon chains exhibit dimerization [@Casillas2014-C; @LaTorre2015], often attributed to Peierls distortion. Complex, three-dimensional crystal structures are also often considered to be distorted from higher symmetry lattices due to Peierls-like transition, also refered to as Jones theory [@Jones34; @Shick99; @Shang2007; @Gaspard2016]. However, several Peierls immune phases have been reported to appear in calculations of one-dimensional chains of different elements, e.g. [@Alemany2009; @Sen2006; @Khomyakov2006; @Ayuela2002; @McAdon88], but attempts to explore their origin are few [@Littlewood81; @McAdon88; @Johannes2008]. Littlewood and Heine [@Littlewood81] stressed the importance of the electron-electron interactions which were not taken into account in Peierls theorem, arguing that Eq. \[eq:Peierls\] is incorrect, but did not depart from its conceptual framework which considers the lowering of energy band at the Fermi level as the main cause of distortion. Johannes and Mazin [@Johannes2008] also analysed a canonical Peierls system of Na atoms and argued that any expected dimerization along the chain axis is energetically unfavourable and doubling of the primitive unit cell occurs only if a transformation into a two-dimensional pattern is allowed.
In this study we analyse the stability of quasi one-dimensional equally spaced monatomic chains from a total energy point-of-view, in accordance with Peierls. The chains are quasi one-dimensional in the sense that three-dimensional calculations are employed to describe one-dimensional lattices with one-dimensional distortions. We study the associated Born-Oppenheimer energy surface and its component contributions which, to our knowledge, were not studied previously in this context in the framework of density-functional theory (DFT) or similar methods. These modern, more accurate methods may contradict prevailing conceptions and give rise to surprising results.
\[sec:level1\]Methods
=====================
The total energy in Kohn-Sham DFT is given as $$\label{eq:Ecomps}
E[\rho]=T_{s}[\rho]+E_{xc}[\rho]+E_{Hartree}[\rho]+V_{nn}[\rho]+V_{ne}[\rho]$$ where $T_{s}$ is the noninteracting kinetic energy, $E_{xc}$ is the exchange-correlation energy, $E_{Hartree}$ and $V_{nn}$ are respectively the Hartree and Ewald potentials describing the coulombic repulsion between the electrons and between the ions, and $V_{ne}$ is the external potential due to the attraction between the electrons and the ions. Using the Kohn-Sham formulation, the energy can also be written as [@ParrYang]: $$\begin{aligned} \label{eq:KSEcomps}
E={} & \sum_{i}^{N}\varepsilon _{i}-\frac{1}{2}\int \frac{\rho (\mathbf{r})\rho ({\mathbf{r}}')}{\left | \mathbf{r}-{\mathbf{r}}' \right |}d\mathbf{r}d{\mathbf{r}}'+E_{Ewald} \\
& +E_{xc}[\rho ]-\int v_{xc}(\mathbf{r})\rho(\mathbf{r})d(\mathbf{r})
\end{aligned}$$ where the first term is the sum over the electronic band energies, the second is minus the Hartree energy to correct for overcounting, the third term is the Ewald energy, and the last two terms are the exchange-correlation adjustment.
In Peierls’ analysis, the electrostatic repulsion is described using an elastic approximation, and the energy gain is due to the lowering of the top of the valence band at the Brillouin zone boundaries evaluated in the nearly-free-electron approximation. Using DFT calculations we can evaluate both the complete electrostatic (classical) contribution (including the attraction between the ions and the electrons) and the overall quantum contribution, which consists mainly of the total kinetic energy of the electrons.
The electronic structure of the systems considered was calculated using a pseudopotentials plane-waves method and performed with the Quantum Espresso package [@QE]. The exchange-correlation functional was approximated by the PBE general gradient approximation (GGA) [@PBE]. Carbon and sodium pseudopotentials with 4 and 9 valence electrons respectively, were taken from the GBRV database [@GBRV] while lithium pseudopotential with 3 electrons was taken from the PSlibrary database [@PSlibrary]. The quasi one-dimensional chains along the z-axis were simulated in tetragonal supercells with vacuum on the x- and y-dimensions, two atoms per unit-cell and periodic boundary conditions. The k-points were aligned homogeneously in the z-direction of the reciprocal lattice vector and centred at the $\Gamma$ point. The calculated systems –- Li, Na and C –- were all set to their equilibrium atomic separation along the z-axis found in our calculations (2.97Å, 3.32Å and 3.83Å respectively) and subjected to one-dimensional distortion which doubled the unit cell along the main axis (dimerization).
\[sec:level1\]Results and discussion
====================================
The variation of the energy components with the distortion parameter $\tau$ for lithium and carbon monatomic chains is presented in Fig. \[fig:CLiTotE\]. The energy components were taken according to equations \[eq:Ecomps\] and \[eq:KSEcomps\] to be the kinetic energy, Hartree, Ewald, external potential and exchange-correlation (XC). In addition Fig. \[fig:CLiTotE\] shows the total energy, the sum of the coulomb energies (Classical) and the sum over the electronic band energies (E bands) $\sum_{i}^{N}\varepsilon _{i}$ which is defined as: $$\sum_{i}^{N}\varepsilon _{i}=\sum_{i}^{N}\left \langle \psi _{i}\left | -\tfrac{1}{2}\nabla^{2}+v_{eff}(\mathbf{r}) \right |\psi _{i} \right \rangle$$ where the effective potential is given by: $$v_{eff}(\mathbf{r})=v_{ext}(\mathbf{r})+\int \frac{\rho ({\mathbf{r}}')}{\left | \mathbf{r}-{\mathbf{r}}' \right |}d\mathbf{r}+v_{xc}(\mathbf{r})$$
As can be seen from Fig. \[fig:CLiTotE\]a-b, one-dimensional equally-spaced carbon chains are Peierls-unstable as expected and their total energy acquires a double-well shape, while lithium chains are stable, in contradiction to Peierls theorem. However, both systems exhibit the same qualitative behaviour which demonstrates the main problem with Peierls theorem: in contrast to previous notions, it is apparent that the classical contribution to the overall energy, comprised of the coulombic terms and drawn in Fig. \[fig:CLiTotE\]c-d, is the driving force of distortion rather than the quantum energy. In all systems studied, distorted and undistorted, the electrostatic attraction between the electrons and the ions favours the distorted structure and dominates the classical contribution, whereas the kinetic energy which dominates the quantum contribution favours the symmetric lattice.
![\[fig:CLiTotE\] Energy change of carbon and lithium chains due to a one-dimensional distortion. (a)-(b) Total energy and the sum of the electronic band energy (E bands) of carbon (a) and lithium (b) versus distortion parameter $\tau$. (c)-(d) Energy components: Hartree, Ewald, External potential, kinetic and exchange-correlation (XC) of carbon (c) and lithium (d) versus distortion parameter $\tau$. Classic energy denotes the sum of External potential, Ewald and Hartree energies. (e)-(f) Selected energy components divided by $\tau^{2}$ ](Ecomp_Li_C.eps){width="1.0\linewidth"}
The overall electronic energy ($E_{bands}$) as plotted in Fig. \[fig:CLiTotE\]a-b shows that it is minimized at the equally-spaced chain configuration rather than under distortion. This is one of the most striking results, since the lowering of the bands near the Fermi level is the main cause for distortion according to Peierls, as all other contributions to the energy from changes in the bands far from the Fermi level are neglected.
The lower panels of Fig. \[fig:CLiTotE\] show the behaviour of some of the energy components divided by the square of the distortion parameter versus the distortion parameter $\tau$. According to Peierls’ analysis, the electrostatic forces opposing the distortion are represented as an elastic energy which is proportional to $\tau^{2}$, and thus should appear as a horizontal line in these coordinates. However, different contributions to the electrostatic energy, in this case the Ewald and Hartree energies, demonstrate completely different behaviours: The Ewald energy, a sum over an infinite lattice of point-charges, demonstrates a nearly-perfect quadratic dependence on the distortion, whereas the Hartree energy deviates substantially from a $\tau^{2}$ behaviour, resembling qualitatively the $\log{}\tau$ behaviour of the bands energy. This inevitably makes the entire classical energy comprised of the coulombic forces deviate from $\tau^{2}$ behaviour, especially at small distortions, in contrast to Peierls’ analysis. A possible explanation for this behaviour is the realistic three-dimensional charge distribution and its distortion which was not originally taken into account.
To further study the charge distribution we calculate the electronic density, as demonstrated in Fig. \[fig:ChargeDen\]. The very different electronic structure of the two chains is apparent: in carbon the highest electron density occurs between neighbouring atoms, whereas in lithium the highest density is distributed in an almost-perfect sphere around each atom. For both carbon and lithium the distribution is not uniform, and the charge density does not spread equally along the chain but rather concentrates between pairs of neighbouring atoms. This charge concentration occurs due to the stronger attraction between the electrons and the ions in the region between adjacent atoms, which increases the energy gain from the interaction between the external potential and the electrons, similar to the function of a bonding orbital in the hydrogen molecule [@Pauling].
![\[fig:ChargeDen\] Spatial electronic density distribution for carbon (a) and for lithium (b) linear chains with $\tau=0.03$. Grey spheres represent positions of the ions, and connecting bars are drawn between pairs of nearest neighbours. Colours represent percentage of maximal density value between 0-100% (carbon) and 0-0.5% (lithium). The scale for lithium is very narrow (maximal value of 0.5%, all values above are depicted in red) as most of its charge is centred around the ion cores.](charge_density.eps){width="1.0\linewidth"}
The integrated local density of states (ILDOS) of the highest occupied band provides a good description of bonding in real space, especially for lithium where most of the charge density belongs to the lower bands and is concentrated spherically about the ion cores (Fig. \[fig:ILDOS\]). In carbon the highest occupied band, corresponding to the $\sigma$-bond of the $p_z$ orbital, is broken into almost discontinuous pairs, whereas in lithium the $\sigma$-bond of the s orbital remains a continuous chain.
![\[fig:ILDOS\] Spatial distribution of the integrated local density of states (ILDOS) of the highest occupied band in carbon (a) and lithium (b) linear chains with $\tau=0.03$. Grey spheres represent positions of the ions, and connecting bars are drawn between pairs of nearest neighbours. Colours represent percentage of maximal ILDOS value between 0-100% (carbon) and 0-40% (lithium).](ILDOS2D.eps){width="1\linewidth"}
In order to understand the behaviour of the band energy with distortion, we analyse in Fig. \[fig:Bands\] the electronic band structures of carbon (left) and lithium (right) chains both in their undistorted state and after a 3% distortion. As expected, the distortion opens an energy gap at the Fermi level at the edge of the Brillouin zone, although the gap is not symmetric below and above the Fermi level, as demonstrated in the inset of Fig. \[fig:Bands\]d. The opening of the gap causes a decrease in the electronic band energy and a metal-to-insulator transition. In carbon, this band-gap suffices to make the highest occupied band energy obtain a maximum value in the undistorted monatomic chain configuration (Fig. \[fig:Bands\]a). In contrast, in lithium the changes in the highest occupied band far from the Brillouin zone boundaries (inset of Fig. \[fig:Bands\]d) cancel the energy gain and result in a minimum on the equally spaced monatomic chain (Fig. \[fig:Bands\]b). In both materials, it is apparent that the distortion significantly affects the bands far below the Fermi level (see Fig. \[fig:Bands\]d), resulting in the increase of the overall band energy with distortion (Fig. \[fig:CLiTotE\]c and \[fig:CLiTotE\]d). It was previously observed [@Johannes2008] that in the closely related three-dimensional materials exhibiting a charge density wave, the distortion is not a result of Fermi surface nesting but rather the outcome of combined electronic and ionic interactions in agreement with the present results. These results deviate from the commonly accepted analysis of the band structure in the framework of the nearly-free-electron model used by Peierls, but could be explained within the same framework by using a stronger potential and taking additional higher order terms in the Fourier expansion of the potential.
![\[fig:Bands\] Electronic band energies of carbon and lithium chains. (a)-(b) The energy dependence on distortion of the different bands from the lowest energy band to the highest energy band for carbon (a) and lithium (b). (c)-(d) The band structure of both equilibrium and distorted carbon (c) and lithium (d). Blow-ups of the highest occupied band of lithium at the edge of the Brillouin zone and at the zone center are shown in the insets of (d).](bands.eps){width="1.0\linewidth"}
As has been experimentally observed, increasing pressure decreases the distortion (e.g. in carbon [@Cretu2013; @Artyukhov2014; @LaTorre2015]) and vice versa. Subjecting a stable chain of sodium, for example, to tension, destabilizes it, as was previously reported [@Khomyakov2006] and can be seen in Fig. \[fig:Na\_strech\]. As the total energy of sodium acquires a double-well shape with stretching, it is insightful to examine its classical (Hartree, Ewald and external) and quantum (kinetic and exchange-correlation) components in order to determine which is the driving force of the instability. It is apparent from Fig. \[fig:Na\_strech\] that both the classical and the quantum energy components increase their tendency toward distortion with increased lattice parameter. The combination of the classical driving force enhancement with tension and the decrease in resistance to distortion of the quantum energy components, results in the destabilisation of the stretched sodium chain. It is important to note that similarly to carbon, the energy gain does not result from the lowering of the band structure. This is demonstrated clearly in the case of sodium as the total band energy becomes more stable at the symmetric alignment with stretching (see inset in Fig. \[fig:Na\_strech\]).
![\[fig:Na\_strech\] Energy change per atom in a sodium one-dimensional chain with distortion $\tau$ for equilibrium lattice parameter a=$3.32\AA$ (solid lines) and stretched lattice parameter a=$3.81\AA$ (dotted lines) corresponding to a tensile stress. Quantum energy is the sum over the kinetic and exchange-correlation energies, classical energy is the sum of Ewald, Hartree and external energies. The sum over the electronic band energies is drawn in the inset.](Na_Ecomp_Vs_streched.eps){width="1.0\linewidth"}
\[sec:level1\]Conclusions
=========================
In conclusion, a detailed analysis of all the energy components involved in the dimerization of realistic one-dimensional chains shows that the energy gain is dominated by the coulomb energy, while the energy loss is mainly due to the kinetic energy. The opening of the energy gap on the boundary of the Brillouin zone at the Fermi level is just one contribution to the energy gain which cancels, in some cases, with other contributions from the band structure, including zone-centred states at the valence band and lower energy bands. The instability of one-dimensional chains is found to be dependent on the external stress and not a universal phenomenon as previously considered. This analysis can shed new light on the driving force for more complex distorted structures such as three dimensional structures.
[^1]: D. Kartoon and U. Argaman contributed equally to this work
[^2]: D. Kartoon and U. Argaman contributed equally to this work
|
---
abstract: 'Simion [@simion] conjectured the unimodality of a sequence counting lattice paths in a grid with a Ferrers diagram removed from the northwest corner. Recently, Hildebrand [@hildebrand] and then Wang [@wan:spc] proved the stronger result that this sequence is actually log concave. Both proofs were mainly algebraic in nature. We give two combinatorial proofs of this theorem.'
author:
- |
Miklós Bóna\
Department of Mathematics\
University of Florida\
Gainesville, FL 32611\
USA\
bona@math.ufl.edu\
and\
Bruce E. Sagan\
Department of Mathematics\
Michigan State University\
East Lansing, MI 48824-1027\
USA\
sagan@math.msu.edu
title: Two Injective Proofs of a Conjecture of Simion
---
Introduction
============
In this note we present two injective proofs of a strengthening of a conjecture of Simion [@simion]. To describe the result, let $\lambda=(\lambda_1, \lambda_2, \cdots ,\lambda_k)$ be the Ferrers diagram of a partition viewed as a set of squares in English notation. (See any of the texts [@bona; @sag:tsg; @sta:ec2] for definitions of terms that we do not define here.) The shape $\lambda$ will be fixed for the rest of this paper.
Consider a grid with the vertices labeled $(i,j)$ for $i,j\ge0$ as in Figure \[rgrid\].
Place $\lambda$ in the northwest corner of this array so that its squares coincide with those of the grid.
A [*northeastern lattice path*]{} is a lattice path on the grid in which each step goes one unit to the north or one unit to the east. Let $N(m,n)$ be the number of northeastern lattice paths from $(m,0)$ to $(0,n)$ that [*do not go inside $\lambda$*]{} (although they may touch its boundary), and let ${{\cal N}}(m,n)$ be the set of such paths. In particular, $N(m,n)=0$ if either the starting or ending point is inside $\lambda$.
Simion [@simion] conjectured that for all $m,n\ge0$ the sequence $$N(0,m+n), N(1,m+n-1), \ldots, N(m+n,0)$$ is unimodal. Lattice path techniques for proving unimodality were investigated by Sagan [@sagan], but the conjecture remained open at that point. Recently, Hildebrand [@hildebrand] proved the stronger result that this sequence is actually log concave by mostly algebraic means. Shortly thereafter, Wang [@wan:spc] simplified Hildebrand’s proof using results about Polya frequency sequences. In the present work, we will give two injective proofs of the stong version of Simion’s conjecture. The one in Sections \[seceasy\] and \[sechard\] employs ideas from Hildebrand’s proof while the one in Section \[direct\] is more direct. Our injections come from a method of Lindström [@lin:vri], later popularized by Gessel and Viennot [@gv:bdp; @gv:dpp], that can be used to prove total positivity results for matrices. For an exposition, see Sagan’s book [@sag:tsg pp. 158–163]. Bóna [@logc] has used related ideas to prove the log concavity of a sequence counting $t$-stack sortable permutations.
We end this section by reiterating the statement of the main theorem for easy reference. Notice that when $\lambda=\emptyset$ it specializes to the well-known result that the rows of Pascal’s triangle are log concave.
\[main\] Let $\lambda$ be the Ferrers diagram of a partition and let $N(m,n)$ be the number of northeastern lattice paths in the grid from $(m,0)$ to $(0,n)$ which do not intersect the interior of $\lambda$. Then for all $m,n\ge0$ the sequence $$N(0,m+n), N(1,m+n-1), \ldots, N(m+n,0)$$ is log concave.
A decomposition of the problem
==============================
This preliminary part of the first proof is from [@hildebrand]. We include it so that our exposition will be self contained. We need to prove that for all $m,n>0$ we have $$N(m-1,n+1) N(m+1,n-1) \leq N(m,n)^2.$$ To prove this, it suffices to show that $$N(m-1,n+1)N(m+1,n) \leq N(m,n) N(m,n+1),$$ because then, by symmetry, we also have $$N(m+1,n-1)N(m,n+1) \leq N(m,n) N(m+1,n).$$ Now multiplying the last two equations together and simplifying gives the first.
The second inequality can be proved by demonstrating another pair of equations, namely $$\label{easy}
N(m,n+1) N(m+1,n) \leq N(m,n) N(m+1,n+1),$$ and $$\label{hard}
N(m-1,n+1)N(m+1,n+1) \leq N(m,n+1)^2.$$ Multiplying these two equations together and cancelling gives the desired result.
The proof of (\[easy\]) {#seceasy}
=======================
In this section we prove that (\[easy\]) holds by constructing an injection $$\Psi: {{\cal N}}(m,n+1) \times {{\cal N}}(m+1,n) \rightarrow{{\cal N}}(m,n) \times {{\cal N}}(m+1,n+1).$$
Consider a path pair $(p,q)\in {{\cal N}}(m,n+1) \times {{\cal N}}(m+1,n)$. Then $p$ and $q$ must intersect. Let $C$ be their first (most southwestern) intersection point. Say that $C$ splits $p$ into parts $p_1$ and $p_2$, and splits $q$ into parts $q_1$ and $q_2$. Then the concatenation of $p_1$ and $q_2$ is a path in ${{\cal N}}(m,n)$, and the concatenation of $q_1$ and $p_2$ is a path in ${{\cal N}}(m+1,n+1)$. So define $\Psi(p,q)=(p_1q_2,q_1p_2)=(p',q')$. It is easy to see that the image of $\Psi$ is exactly all $(p',q')\in {{\cal N}}(m,n) \times {{\cal N}}(m+1,n+1)$ such that $p'$ and $q'$ intersect. It is also simple to verify that if $\Psi(p,q)=(p',q')$, then applying the same algorithm to $(p',q')$ recovers $(p,q)$. So $\Psi$ is injective. See Figure \[easyf\] for an example.
The proof of (\[hard\]) {#sechard}
=======================
In this section we construct an injection $$\Phi: {{\cal N}}(m-1,n)\times{{\cal N}}(m+1,n) \rightarrow {{\cal N}}(m,n)^2,$$ thus proving (\[hard\]).
Let $(p,q)\in {{\cal N}}(m-1,n) \times {{\cal N}}(m+1,n)$. If $P=(i,j)$ and $Q=(i,k)$ are vertices of $p$ and $q$, respectively, with the same first coordinate, define the [*vertical distance from $p$ to $q$ at $P$ and $Q$*]{} to be $k-j$. The vertical distance from $p$ to $q$ starts at 2 for their initial vertices and ends at 0 for their final ones. Since vertical distance can change by at most one with a step of a path, there must be some vertical distance equal to 1. Let $P$ and $Q$ be the first (most southwest) pair of points with vertical distance one.
It follows from our choice of vertices that $p$ must enter $P$ with an east step and $q$ must enter $Q$ with a north step. Let $p_1$ and $p_2$ be the portions of $p$ before and after $P$, respectively, and similarly for $q$. Now let $$\Phi(p,q)=(p_1'q_2,q_1'p_2)$$ where $p_1'$ is $p_1$ moved south one unit and $q_1'$ is $q_1$ moved north one unit. Since $P$ and $Q$ are the first pair of points at vertical distance one, $q_1'$ will not intersect ${\lambda}$ and the concatenations are valid paths in ${{\cal N}}(m,,n)$. In fact, the image of $\Phi$ is exactly all path pairs $(p',q')\in{{\cal N}}(m,n)^2$ that have a pair of points at vertical distance -1. Applying the same procedure to the first such pair inverts the map and so $\Phi$ is injective. See Figure \[newphi\] for an example.
This completes the first proof of Theorem \[main\].
A more direct proof {#direct}
===================
The reader may wonder if we can do away with splitting our problem into two parts, that is, equations (\[easy\]) and (\[hard\]). The answer is yes, and the necessary injection is just a modification $\overline{\Phi}$ of the map $\Phi$. This will give us a second, completely combinatorial, proof of our main theorem
Take a path pair $(p,q)\in {{\cal N}}(m-1,n+1)\times{{\cal N}}(m+1,n-1)$. Notice that $p$ and $q$ must intersect. So before the first intersection there must be a first pair of points $P, Q$ (on $p,q$ respectively) at vertical distance 1. Similarly, after the last intersection there must be a last pair of points $\overline{P},\overline{Q}$ at horizontal distance 1, where horizontal distance is defined analogously. Let $P$ and $\overline{P}$ divide $p$ into subpaths $p_1,p_2,p_3$ and use the same notation for $q$. Then define $$\overline{\Phi}(p,q)=(p_1'q_2,p_3'',q_1'p_2q_3'')$$ where $p_1'$ is $p_1$ moved south one unit, $p_3''$ is $p_3$ moved west one unit, and $q_1',q_3''$ are defined in the analogous way but moving in the opposite directions. It is a simple job to verify that $\overline{\Phi}$ is well-defined and injective just as we did with $\Phi$.
This completes the second proof of Theorem \[main\].
We have two final remarks. First of all, it is clear from the geometry of the situation that if ${\lambda}$ is self-conjugate then the sequence in Theorem \[main\] is also symmetric, but this does not hold in general. One might also wonder if this sequence has the stronger property that the associated polynomial generating function has only real zeros. This is not always true as can be seen by taking ${\lambda}=(1)$ and $m+n=4$. In this case the associated polynomial is $x(3x^2+5x+3)$ which has two complex roots. It might be interesting to determine for which shapes the real zero property holds.
[99]{} M. Bóna, ”A Walk Through Combinatorics”, World Scientific, 2002.
M. Bóna, Log-concavity for $t$-stack sortable permutations, [*J. Comb. Theory, Series A*]{}, submitted.
I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, [*Advances in Math.*]{} [**58**]{} (1985), 300–321.
I. Gessel and G. Viennot, Determinants, paths, and plane partitions, in preparation.
M. Hildebrand, Log Concavity of a Sequence in a Conjecture of Simion, [*J. Comb. Theory, Series A*]{}, [**97**]{} (2002) 108-116.
B. Lindström, On the vector representation of induced matroids, [*Bull. London Math. Soc. *]{} [**5**]{} (1973), 85–90.
B. Sagan, Unimodality and the Reflection Principle, [*Ars Combin.*]{}, [**48**]{} (1998), 65-72.
B. Sagan, “The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions,” 2nd edition, Springer-Verlag, New York, 2001.
R. Simion, Combinatorial Statistics and Noncrossing partitions, [*J. Comb. Theory, Series A*]{}, [**94**]{} (1994) 270-301.
R. P. Stanley, “Enumerative Combinatorics, Volume 2,” Cambridge University Press, Cambridge, 1999.
Y. Wang, A simple proof of a conjecture of Simion, preprint.
|
---
abstract: 'In ${{\rm PG}}(2,q^3)$, let $\pi$ be a subplane of order $q$ that is exterior to ${\ell_\infty}$. The exterior splash of $\pi$ is defined to be the set of $q^2+q+1$ points on ${\ell_\infty}$ that lie on a line of $\pi$. This article investigates properties of an exterior [order-${{q}}$-subplane]{} and its exterior splash. We show that the following objects are projectively equivalent: exterior splashes, covers of the circle geometry $CG(3,q)$, Sherk surfaces of size $q^2+q+1$, and scattered linear sets of rank 3. We compare our construction of exterior splashes with the projection construction of a linear set. We give a geometric construction of the two different families of sublines in an exterior splash, and compare them to the known families of sublines in a scattered linear set of rank 3.'
author:
- 'S.G. Barwick and Wen-Ai Jackson'
date: 20 April 2014
title: Exterior splashes and linear sets of rank 3
---
Corresponding Author: Dr Susan Barwick, University of Adelaide, Adelaide 5005, Australia. Phone: +61 8 8313 3983, Fax: +61 8 8313 3696, email: susan.barwick@adelaide.edu.au
Keywords: subplanes, sublines, linear sets, Sherk surfaces, circle geometries
AMS code: 51E20
Introduction
============
Let $\pi$ be a subplane of ${{\rm PG}}(2,q^3)$ of order $q$ that meets ${\ell_\infty}$ in 0 or 1 point. Each line of $\pi$, when extended to ${{\rm PG}}(2,q^3)$, meets ${\ell_\infty}$ in a point. The set of points of ${\ell_\infty}$ that lie on a line of $\pi$ is called the [*splash*]{} of $\pi$. If $\pi$ is tangent to ${\ell_\infty}$ at a point $T$, then the splash of $\pi$ consists of $T=\pi\cap{\ell_\infty}$, and $q^2$ further points. We call this splash a *tangent splash*, and these were investigated in [@BJ-tgt1; @BJ-tgt2]. If $\pi$ is exterior to ${\ell_\infty}$, then the splash ${{\mathbb S}}$ contains $q^2+q+1$ points, and we call it an [*exterior splash*]{}. Note that each line of $\pi$ contains a distinct point of ${{\mathbb S}}$, and conversely, each point of ${{\mathbb S}}$ lies on a unique line of $\pi$. More generally, we can define an exterior splash onto any exterior line. That is, let $\pi$ be a subplane of order $q$ of ${{\rm PG}}(2,q^3)$, and let $\ell$ be an exterior line of $\pi$, then the [*exterior splash of $\pi$ onto $\ell$*]{} is the set of $q^2+q+1$ points of $\ell$ that lie on lines of $\pi$.
In this article we will investigate exterior [order-${{q}}$-subplanes]{} and exterior splashes. In Section \[sec:group\] we consider the relationship of exterior splashes to other known objects. In particular, exterior splashes are projectively equivalent to scattered linear sets of rank 3; covers of circle geometries $CG(3,q)$; and Sherk surfaces of size $q^2+q+1$.
The rest of the article focuses on the relationship between an exterior [order-${{q}}$-subplane]{} and its exterior splash. In Section \[sec:orsl-splash\], we use [order-${{q}}$-sublines]{} contained in an exterior splash ${{\mathbb S}}$ in terms of an associated [order-${{q}}$-subplane]{} $\pi$. In an analogous manner to [@lavr10 Remark 20], we also discuss why projective bundles of conics of $\pi$ arise naturally as [order-${{q}}$-sublines]{} of ${{\mathbb S}}$. In Section \[sec:orsl-lin-ext\], we show that the two families of $q^2+q+1$ sublines in an exterior splash are equivalent in some sense, despite their apparent difference in definition.
A ${{\rm GF}}(q)$-linear set of rank 3 can be obtained by projecting an [order-${{q}}$-subplane]{} of ${{\rm PG}}(2,q^3)$ onto a line, see [@luna04]. In Section \[sec:proj-ext\], we investigate the relationship between this projection construction of a scattered linear set of rank 3, and our construction of an exterior splash. In Section \[sec:proj-tgt\], we revisit the tangent splash of a tangent [order-${{q}}$-subplane]{}, and look at the projection in this context.
Finally, in Section \[sec:common-splash\], we look at two exterior [order-${{q}}$-subplanes]{} that have a common exterior splash, and determine how they can intersect. Further, we show that there are exactly two [order-${{q}}$-subplanes]{} that have a common exterior splash and share a fixed [order-${{q}}$-subline]{}.
Notation and definitions
========================
In this section we introduce the notation we use, as well as defining linear sets, circle geometries and Sherk surfaces.
If $\pi$ is a subplane of ${{\rm PG}}(2,q^3)$ of order $q$, then we call $\pi$ an [*[order-${{q}}$-subplane]{}*]{}. If ${\ell_\infty}$ is an exterior line of $\pi$, then we say $\pi$ is an [*exterior*]{} [order-${{q}}$-subplane]{}. An [*[order-${{q}}$-subline]{}*]{} of ${{\rm PG}}(2,q^3)$ is a line of an [order-${{q}}$-subplane]{}, that is, it is isomorphic to ${{\rm PG}}(1,q)$. Points in ${{\rm PG}}(2,{{q}}^3)$ have homogeneous coordinates $(x,y,z)$ with $x,y,z\in{{\rm GF}}({{q}}^3)$. Let the line at infinity ${\ell_\infty}$ have equation $z=0$; so the affine points of ${{\rm PG}}(2,{{q}}^3)$ have coordinates $(x,y,1)$. We can construct ${{\rm GF}}(q^3)$ as a cubic extension of ${{\rm GF}}(q)$ using a primitive element $\tau$ with primitive polynomial $$\begin{aligned}
x^3-t_2x^2-t_1x-t_0,\label{t0t1t2}\end{aligned}$$ where $t_0,t_1,t_2\in{{\rm GF}}(q)$; so every element in ${{\rm GF}}({{q}}^3)$ can be uniquely written as $a_0+a_1\tau+a_2\tau^2$ with $a_0,a_1,a_2\in{{\rm GF}}({{q}})$.
We define linear sets of ${{\rm PG}}(1,q^3)$, more generally, linear sets of ${{\rm PG}}(n-1,q^t)$ are defined in [@lavr10]. The points of ${{\rm PG}}(1,q^3)$ can be considered as elements of the 2-dimensional vector space $V={{\rm GF}}(q^3)^2$ over ${{\rm GF}}(q^3)$. Let $U$ be a subset of $V$ that forms a 3-dimensional vector space over ${{\rm GF}}(q)$. Then the vectors of $U$ (considered as vectors over ${{\rm GF}}(q^3)$) form a ${{\rm GF}}(q)$-linear set of rank 3 of ${{\rm PG}}(1,q^3)$. A *scattered linear set of rank 3* is a ${{\rm GF}}(q)$-linear set of ${{\rm PG}}(1,q^3)$ of rank 3 and size $q^2+q+1$. By [@lavr10], all scattered linear sets of rank 3 are projectively equivalent. Scattered linear sets were introduced in [@blok00], and have recently been studied in [@dona14; @lavr10; @lavr13; @LMPT14].
In [@bruc73a; @bruc73b], Bruck gave a set of axioms for higher dimensional circle geometries. We look at the 3-dimensional case, namely $CG(3,q)$. The points of $CG(3,q)$ can be identified with the points of ${{\rm PG}}(1,q^3)$, and the *circles* of $CG(3,q)$ are identified with the [order-${{q}}$-sublines]{} of ${{\rm PG}}(1,q^3)$. The [*stability group*]{} of a circle is defined to be the subgroup of Aut$CG(3,q)={{\mbox{P}\Gamma {L}}}(2,q^3)$ that fixes the circle pointwise. For two distinct points of $CG(3,q)$, let $\phi(P,Q)$ denote the group generated by the stability groups of all circles containing $P$ and $Q$. A [*cover*]{} is defined to be an orbit under $\phi(P,Q)$ of any point $R$ distinct from $P,Q$. In [@bruc73b], it is shown that every cover of $CG(3,q)$ has $q^2+q+1$ points, and can be represented in one of the following two ways, using the identification of the points of $CG(3,q)$ with the field ${{\rm GF}}(q^3)\cup\{\infty\}$:
1. $\{x\in{{\rm GF}}(q^3){:}N(x-a)=f\}$, for some $a\in{{\rm GF}}(q^3)$, and $f\in{{\rm GF}}(q){\backslash}\{0\}$.
2. $\{x\in{{\rm GF}}(q^3)\cup\{\infty\}{:}N\left(\frac{x-a}{x-b}\right)=f\}$, for some $a,b\in{{\rm GF}}(q^3)$, and $f\in{{\rm GF}}(q){\backslash}\{0\}$.
where $N$ is the norm from ${{\rm GF}}(q^3)$ to ${{\rm GF}}(q)$, that is $N(x)=x^{q^2+q+1}$. The points $P,Q$ are called the [*carriers*]{} of the cover. The carriers for a cover of type I are $\{a,\infty\}$, and the carriers for a cover of type II are $\{a,b\}$. A cover of type II will contain $\infty$ if and only if $f=1$.
Sherk [@sher86], studied objects other than circles and covers in $CG(3,q)$. Representing points of $CG(3,q)$ as elements of ${{\rm GF}}(q^3)\cup\{\infty\}$, a [*Sherk surface*]{} is the set of points satisfying $$S(f,\alpha,\delta,g)=\{z\in{{\rm GF}}(q^3)\cup\{\infty\}{:}fN(z)+T(\alpha^{q^2}z^{q+1})+T(\delta z)+g=0\}$$ for $f,g\in{{\rm GF}}(q)$, $\alpha,\delta\in{{\rm GF}}(q^3)$, where $N,T$ are the norm and trace from ${{\rm GF}}(q^3)$ to ${{\rm GF}}(q)$. Sherk surfaces of size $q^2+q+1$ are precisely the Bruck covers of $CG(3,q)$.
Exterior splashes
=================
[sec:group]{}
An important Singer group {#sec:singer-gp}
-------------------------
We will need a particular Singer group acting on [order-${{q}}$-subplanes]{}. First we define the notion in ${{\rm PG}}(2,q^3)$ of conjugate points and lines with respect to an [order-${{q}}$-subplane]{}. Let $\pi$ be an [order-${{q}}$-subplane]{} of ${{\rm PG}}(2,q^3)$. There is a unique collineation group of order three that fixes every point of $\pi$, let $\zeta$ be a generator of this group. Then the points of ${{\rm PG}}(2,q^3){\backslash}\pi$ can be partitioned into sets of size three of form $\{P, \zeta(P), \zeta^2(P)\}$, called [*conjugate points with respect to $\pi$*]{}. Note that $\{P, \zeta(P), \zeta^2(P)\}$ are collinear if and only if they lie on the extension of an [order-${{q}}$-subline]{} of $\pi$. Similarly, the lines of ${{\rm PG}}(2,q^3){\backslash}\pi$ can be partitioned into sets of size three of form $\ell, \zeta(\ell), \zeta^2(\ell)$, called [*conjugate lines with respect to $\pi$.*]{} For example, let $\pi={{\rm PG}}(2,q)$, then $\zeta\colon (x,y,z)\mapsto (x^q,y^q,z^q)$ fixes every point of $\pi$ and acts semi-regularly on the remaining points of ${{\rm PG}}(2,q^3)$. If $P$ is a point of ${{\rm PG}}(2,q^3){\backslash}{{\rm PG}}(2,q)$, then the conjugate points with respect to the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,q)$ are $P,P^q,P^{q^2}$. If $\ell$ is a line of ${{\rm PG}}(2,q^3){\backslash}{{\rm PG}}(2,q)$, then the conjugate lines with respect to the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,q)$ are $\ell,\ell^q,\ell^{q^2}$.
We need the following result about the collineation groups acting on an exterior [order-${{q}}$-subplane]{}.
[propthm]{} Consider the collineation group $G={{\rm PGL}}(3,q^3)$ acting on ${{\rm PG}}(2,q^3)$. Let $I=G_{\pi,\ell}$ be the subgroup of $G$ fixing an [order-${{q}}$-subplane]{} $\pi$, and a line $\ell$ exterior to $\pi$. Then
1. $I$ is cyclic of order $q^2+q+1$, and acts regularly on the points and on the lines of $\pi$.
2. $I$ fixes exactly three lines: $\ell$, and its conjugates $m$, $n$ with respect to $\pi$. Further $I$ acts semi-regularly on the remaining line orbits.
3. $I$ fixes exactly three points: $E_1=\ell\cap m$, $E_2=\ell\cap n$, $E_3=m \cap n$ (which are conjugate with respect to $\pi$) and acts semi-regularly on the remaining point orbits.\
The points $E_1,E_2$ are called the [*carriers*]{} of $\pi$.
We first consider the homography $\phi$ of ${{\rm PG}}(2,q^3)$ with matrix $$T=\begin{pmatrix}0&1&0\\0&0&1\\t_0&t_1&t_2\end{pmatrix},$$ where $t_0,t_1,t_2$ are as in (\[t0t1t2\]) and $\phi(x,y,z)=T(x,y,z)^t$. The matrix $T$ has three eigenvalues $\tau$, $\tau^{{q}}$ and $\tau^{{{q}}^2}$ with corresponding eigenvectors $Q=(1,\tau,\tau^2)$, $Q^{{q}}$ and $Q^{{{{{q}}^2}}}$ respectively. Further, the eigenvalues of $T^i$ are $\tau^{i},\tau^{i{{q}}},\tau^{i{{q}}^2}$ with corresponding eigenvectors $Q,Q^{{q}},Q^{{{q}}^2}$ respectively. Now $\tau^i=\tau^{i{{q}}}$ if and only if $\tau^i\in{{\rm GF}}({{q}})$, if and only ${{q}}^2+{{q}}+1\bigm| i$, hence the eigenvalues are distinct. Thus, for $0<i<q^2+q+1$, $\phi^i$ fixes exactly three points $Q,Q^{{q}},Q^{{{q}}^2}$. Further, $\phi$ has order ${{q}}^2+{{q}}+1$, and $\phi$ acts semi-regularly on the points other than $Q,Q^{{q}},Q^{{{q}}^2}$. In particular, $\langle\phi\rangle$ is a Singer cycle of the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,q)$, and so acts regularly on the points and on the lines of $\pi$.
Now let $G={{\rm PGL}}(3,q^3)$, it is well known that $|G|={{q}}^9({{q}}^9-1)({{q}}^6-1)$, and that $G$ is transitive on the points of ${{\rm PG}}(2,{{q}}^3)$; on the lines of ${{\rm PG}}(2,{{q}}^3)$; and on the [order-${{q}}$-subplane]{}s of ${{\rm PG}}(2,{{q}}^3)$. Hence when considering the subgroup $K=G_\pi$ for some [order-${{q}}$-subplane]{} $\pi$, we can without loss of generality let $\pi={{\rm PG}}(2,q)$. Let $K=G_\pi={{\rm PGL}}(2,{{q}})$. Thus $K$ is transitive on the points and lines of $\pi$, and so this gives one point orbit of $K$. Further, $|K|=q^3(q^3-1)(q^2-1)$.
We now consider the point $P=(1,1,\tau)$ and calculate its orbit under $K$. Note that $P$ is not in $\pi$, but $P$ is on the line $[1,-1,0]$, which is a line of $\pi$, so $P$ lies on exactly one line of $\pi$. Consider the subgroup of $K$ fixing $P$, so consider a homography with matrix $$C=\begin{pmatrix}
a&b&c\\d&e&f\\g&h&i
\end{pmatrix},$$ with $a,\ldots,i\in{{\rm GF}}({{q}})$, such that $C(1,1,\tau)^t\equiv(1,1,\tau)$. Then $a+b+c\tau=d+e+f\tau$ (hence $c=f$ and $a+b=d+e$) and $(a+b+c\tau)\tau=g+h+i\tau$ giving $c=0$ (so $f=0$), $a+b=i$, and $g+h=0$. Hence $$C=\begin{pmatrix}
a&b&0\\d&a+b-d&0\\g&-g&a+b\end{pmatrix}=\begin{pmatrix}a&1-a&0\\d&1-d&0\\g&-g&1\end{pmatrix}$$ since we can take $a+b=1$ as det $C\neq0$. Now det $C\neq0$ if and only if $a\neq d$. Hence the size of $K_P$ is ${{q}}^2({{q}}-1)$. Thus by the orbit stabilizer theorem, $|P^K|=|K|/|K_P|=q(q^3-1)(q+1).$ This is the number of points of ${{\rm PG}}(2,q^3){\backslash}\pi$ that lie on exactly one line of $\pi$, hence these points lie in one orbit of $K$.
We now consider the points of ${{\rm PG}}(2,q^3){\backslash}\pi$ that lie on exactly zero lines of $\pi$. Consider the point $Q=(1,\tau,\tau^2)$, it does not lie on any line of ${{\rm PG}}(2,q)$. We calculate the size of the orbit of $Q$ under $K$. Note first that the subgroup of $K$ generated by the homography $\phi$ with matrix $T$ fixes $Q$. Now consider a homography of $K$ with matrix $$B=\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix},$$ $a,\ldots,i\in{{\rm GF}}(q)$ that fixes $Q$. We have $(a+b\tau+c\tau^2)\tau=d+e\tau+f\tau^2$ and $(d+e\tau+f\tau^2)\tau=g+h\tau+i\tau^2$. So we have $B=\big(\boldsymbol{x}, \boldsymbol{x}T, \boldsymbol{x}T^2\big)^t$ where $ \boldsymbol{x}=(a,b,c)$. Now $B$ has nonzero determinant if and only not all of $a,b,c$ are zero. Further, $\boldsymbol{x}=(a,b,c)$ and $\boldsymbol{x}=y(a,b,c)$ correspond to the same collineation if and only if $y\in{{\rm GF}}({{q}})$. Thus the subgroup $K_Q$ is of size ${{q}}^2+{{q}}+1$. Hence $K_Q$ is generated by the Singer cycle $\phi$ with matrix $T$. By the orbit stabilizer theorem, $|Q^K|=|K|/|K_Q|=q^3(q^2-1)(q-1)$. This is equal to the number of points of ${{\rm PG}}(2,q^3)$ that lie on zero lines of $\pi={{\rm PG}}(2,q)$. Hence these points all lie in one orbit of $K$. A similar argument shows that $K$ has three line orbits: the lines of $\pi$, the lines meeting $\pi$ in exactly one point, and the lines exterior to $\pi$.
As $K=G_\pi$ is transitive on the exterior lines of $\pi$, so we can without loss of generality consider the line $\ell=[1,\tau,\tau^2]$ which is exterior to $\pi={{\rm PG}}(2,q)$, and we calculate $I=G_{\pi,\ell}$ for this $\pi,\ell$. Consider the homography $\phi'$ which has matrix $T^{-t}$ (the transpose of the inverse of $T$). Note that $\phi'$ has order $q^2+q+1$ as $\phi$ does. Now $\phi'$ maps a line $[l,m,n]$ of ${{\rm PG}}(2,q^3)$ to the line $[l,m,n]T^t=\ell T^t$. Hence $\phi'$ fixes the line $\ell=[1,\tau,\tau^2]$. So using a similar argument to the above paragraph, we have $I=G_{\pi,\ell}$ is generated by the singer cycle $\phi'$. Hence $|I|=q^2+q+1$ and $I$ acts regularly on the points, and on the lines, of ${{\rm PG}}(2,{{q}})$, proving part 1.
As $T$ has entries in ${{\rm GF}}({{q}})$, we have $T=T^{{q}}$, hence $\phi'(\ell^q)=\ell^qT^t=(\ell T^t)^q=(\ell)^q=\ell^q$. That is $\phi'$ fixes $\ell^q$, similarly, $\phi'$ fixes $\ell^{q^2}$. Hence $\phi'$ will fix the three points $P=\ell\cap\ell^{q^2}$, $P^q=\ell\cap\ell^q$, $P^{q^2}=\ell^q\cap\ell^{q^2}$. A similar argument to that used when analysing the homography $\phi$ shows that $I$ acts semi-regularly on the remaining line (and point) orbits of ${{\rm PG}}(2,q^3)$, proving parts 2 and 3.
The Singer group $I={{\rm PGL}}(3,q^3)_{\pi,\ell}$ has a number of important consequences. We can use $I$ to define a special family of conics and dual conics in $\pi$ which play an important role in exterior splashes. A conic of $\pi$ whose extension to ${{\rm PG}}(2,q^3)$ contains the three conjugate points ${E}_1,{E}_2,{E}_3$ is called a $(\pi,\ell)$-[*special conic*]{} of $\pi$. Dually we can define a $(\pi,\ell)$-[*special-dual conic*]{} to be an dual conic of $\pi$ that contains the three conjugate lines $\ell,m,n$. Special conics and dual conics are irreducible. Further, the two incidence structures with points the points of $\pi$, lines the $(\pi,\ell)$-special conics (respectively dual conics) of $\pi$, and natural incidence are isomorphic to ${{\rm PG}}(2,q)$. In addition, a *projective bundle* of conics is defined in [@BBEF] to be a set of $q^2+q+1$ conics of ${{\rm PG}}(2,q)$ which pairwise meet in exactly one point. There are only three known classes of projective bundles, of which the $(\pi,\ell)$-special conics of $\pi$ form a [*circumscribed bundle*]{}, and the $(\pi,\ell)$-special-dual conics of $\pi$ form an [*inscribed bundle*]{}.
An example
----------
We will need an example of an [order-${{q}}$-subplane]{} to prove subsequent results. So the next result takes the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,q)$ of ${{\rm PG}}(2,q^3)$, finds a suitable exterior line $\ell$ to $\pi$, and calculates the exterior splash, the Singer group $I$, and the carriers for $\pi$.
[splash-eg]{} Let $\pi={{\rm PG}}(2,q)$ be an [order-${{q}}$-subplane]{} of ${{\rm PG}}(2,q^3)$.
1. The line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$ is exterior to $\pi$.
2. The exterior splash ${{\mathbb S}}$ of $\pi={{\rm PG}}(2,q)$ on ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$ is $${{\mathbb S}}=
\{{E}+\theta {E}^q{:}\theta^{q^2+q+1}=-1,\theta\in{{\rm GF}}(q^3)\}, \quad {\rm where\ } E=(1,\tau,\tau^2).$$
3. The Singer group $I={{\rm PGL}}(3,q^3)_{\pi,\ell}$ is generated by the homography $\phi$ with matrix $$T=\begin{pmatrix}0&1&0\\0&0&1\\t_0&t_1&t_2\end{pmatrix},$$ where $t_0,t_1,t_2$ are as in [(\[t0t1t2\])]{}, and $\phi(x,y,z)^t=T(x,y,z)^t$. The three fixed points of $\phi$ are the points $E=\ell^{q^2}\cap\ell=(1,\tau,\tau^2)$, $E^q,E^{q^2}$. The three fixed lines of $\phi$ are $\ell, \ell^q,\ell^{q^2}$. So $\pi$ has carriers ${E}=(1,\tau,\tau^2)$ and ${E}^q=(1,\tau^q,\tau^{2q})$.
We first show that the line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$ is exterior to $\pi={{\rm PG}}(2,q)$. The points of $\pi$ are of the form $(x,y,z)$, $x,y,z\in{{\rm GF}}(q)$, not all zero. This point lies on the line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$ if and only if $\tau\tau^qx-(\tau+\tau^q)y+z=0$ for some $x,y,z\in{{\rm GF}}(q)$, not all zero. As $1,\tau\tau^q,\tau+\tau^q$ of ${{\rm GF}}(q^3)$ are linearly independent over ${{\rm GF}}(q)$, $\tau\tau^qx-(\tau+\tau^q)y+z=0$ has no solutions with $x,y,z\in{{\rm GF}}(q)$, not all zero, thus $\pi$ is exterior to $\ell$. We can parameterize the line $\ell$ as $\ell=\{E+\theta E^q{:}\theta\in{{\rm GF}}(q^3)\cup\{\infty\}\}$, where $E=(1,\tau,\tau^2)$ (so $E$ has parameter $0$, and $E^q$ has parameter $\infty$). Consider a line $\ell_i$ of $\pi$, so $\ell_i=[l,m,n]$, $l,m,n\in{{\rm GF}}(q)$, not all zero. Then $\ell_i$ meets the line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$ in the point $X_i=(m+n(\tau+\tau^q),\ n\tau\tau^q-l,\ -l(\tau+\tau^q)-m\tau\tau^q).$ We can write this as $X_i={E}+\theta_i{E}^q$ where $\theta_i=-(l+m\tau+n\tau^{2})/(l+m\tau+n\tau^2)^q$. Note that $\theta_i^{q^2+q+1}=\theta_i^{q^2}\theta_i^q\theta_i=-1$. As there are $q^2+q+1$ solutions to the equation $\theta^{q^2+q+1}=-1$, the $q^2+q+1$ lines of $\pi$ meet $\ell$ in the $q^2+q+1$ points ${E}+\theta{E}^q$ with $\theta^{q^2+q+1}=-1$. That is, $\pi$ has exterior splash ${{\mathbb S}}=\{{E}+\theta {E}^q{:}\theta^{q^2+q+1}=-1,\theta\in{{\rm GF}}(q^3)\}$ on $\ell$, completing the proof of part 2. The proof of part 3 involves straightforward calculations.
Exterior splashes and linear sets
---------------------------------
The following result is important to our study of exterior [order-${{q}}$-subplanes]{} and exterior splashes.
[transsplashes]{} Consider the collineation group $G={{\rm PGL}}(3,{{q}}^3)$ acting on ${{\rm PG}}(2,q^3)$. The subgroup $G_\ell$ fixing a line $\ell$ is transitive on the [order-${{q}}$-subplanes]{} that are exterior to $\ell$, and is transitive on the exterior splashes on $\ell$.
Recall that the group $G={{\rm PGL}}(3,q^3)$ is transitive on the [order-${{q}}$-subplane]{}s of ${{\rm PG}}(2,{{q}}^3)$. From the proof of Theorem \[propthm\], the subgroup of $K$ fixing an [order-${{q}}$-subplane]{} $\pi$ is transitive on the exterior lines of $\pi$. Hence $G$ is transitive on pairs $(\pi,\ell)$ where $\pi$ is an [order-${{q}}$-subplane]{} and $\ell$ is a line exterior to $\pi$. Thus $G_\ell$ is transitive on the the [order-${{q}}$-subplanes]{} that are exterior to $\ell$, and hence $G_\ell$ is transitive on the exterior splashes of $\ell$.
The importance of this result is two fold. Firstly, it means that when proving results about exterior [order-${{q}}$-subplanes]{} and exterior splashes, we can without loss of generality prove results for a particular example [order-${{q}}$-subplane]{}. Secondly, this theorem shows that all exterior splashes are projectively equivalent. It is then straightforward to use Example \[splash-eg\], and the result that all scattered linear sets of rank 3 are projectively equivalent by [@lavr10], to prove that exterior splashes of ${{\rm PG}}(2,q^3)$ are projectively equivalent to scattered linear sets of rank 3.
[lin-set]{} Exterior splashes of ${{\rm PG}}(1,q^3)$ are projectively equivalent to scattered linear sets of rank 3.
We note that in concurrent, independent work, Lavrauw and Zanella [@lavr14] prove the more general result that exterior splashes of ${{\rm PG}}(1,q^n)$ are projectively equivalent to scattered linear sets of ${{\rm PG}}(1,q^n)$.
Circle geometries and Sherk surfaces
------------------------------------
[sec:def-carrier]{}
In this section we observe that exterior splashes are projectively equivalent to covers of the circle geometry $CG(3,q)$, and to Sherk surfaces of size $q^2+q+1$.
[spiscov]{} Let $\pi$ be an exterior [order-${{q}}$-subplane]{} with exterior splash ${{\mathbb S}}$ on ${\ell_\infty}={{\rm PG}}(1,q^3)$ and carriers $E_1,E_2$. Then ${{\mathbb S}}$ is projectively equivalent to a cover of the circle geometry $CG(3,q)$ with carriers $E_1,E_2$.
By Theorem \[transsplashes\], we can without loss of generality prove this for the exterior splash ${{\mathbb S}}$ of the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,{{q}})$ onto the exterior line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$. By Example \[splash-eg\], ${{\mathbb S}}$ has exterior splash ${{\mathbb S}}=\{{E}+\theta {E}^q{:}\theta^{q^2+q+1}=-1,\theta\in{{\rm GF}}(q^3)\}$ on $\ell$, and carriers $E_1={E}=(1,\tau,\tau^2)$ and $E_2={E}^q$ (corresponding to $\theta=0$ and $\theta=\infty$ respectively). Hence using the notation of Section 2, the exterior splash is a cover of type I with $a=0$, $f=-1$.
We note that this equivalence can also be deduced from the general theory of linear sets of pseudoregulus type in [@LMPT14]. In particular, an exterior splash ${{\mathbb S}}$ is equivalent to a scattered linear set of rank 3 of pseudoregulus type. The carriers of ${{\mathbb S}}$ are called transversal spaces of the pseudoregulus. There are two important corollaries for exterior splashes, we note that they both follow immediately from properties of circle geometries, and are also proved directly in [@LMPT14]. We state them here in the exterior splash context as they are important for understanding the carriers of an exterior splash.
[carriers-belong-splash]{}
1. Let $\pi_1,\pi_2$ be two exterior [order-${{q}}$-subplanes]{} with the same exterior splash ${{\mathbb S}}$ on ${\ell_\infty}$, then $\pi_1$ and $\pi_2$ have the same carriers. That is, an exterior splash has two unique carriers.
2. Given two carriers ${E}_1,{E}_2$ of ${\ell_\infty}$, there are exactly $q-1$ disjoint exterior splashes with carriers ${E}_1,{E}_2$.
[As the Sherk surfaces of size $q^2+q+1$ are precisely the Bruck covers of $CG(3,q)$, exterior splashes are also projectively equivalent to Sherk surfaces of size $q^2+q+1$. In fact, the Sherk surface corresponding to the exterior splash of the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,q)$ onto the exterior line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$ has parameters $S(1,0,0,1)$. This equivalence means we can adapt results about permutation groups of Sherk surfaces from [@sher86] to our setting, also see [@dona14]. In particular we will need the following collineation group. ]{}
[spl-gp-eg]{} Let ${{\mathbb S}}$ be the exterior splash of the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,q)$ onto the exterior line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$. Then ${{\mathbb S}}\equiv\{(\theta,1){:}\theta^{q^2+q+1}=-1,\theta\in{{\rm GF}}(q^3)\}$, where $(\theta,1)\equiv\theta E_1+E_2$. The group of collineations of $\ell$ fixing ${{\mathbb S}}$ has as generators two homographies $\Gamma$, $\Delta$ with matrices $$\begin{pmatrix}
\tau&0\\0&\tau^q
\end{pmatrix} \quad {\rm and}\quad
\begin{pmatrix}
0&1\\1&0
\end{pmatrix}$$ respectively; so $\Gamma$ fixes the carriers ${E}_1,{E}_2$, and $\Delta$ interchanges them.
Order-$q$-sublines of an exterior splash
========================================
[sec:orsl-splash]{}
We have so far discussed the relation between exterior splashes and other geometrical objects in their own right. For the remainder of the paper we will primarily consider the relationship between an exterior [order-${{q}}$-subplane]{} and its exterior splash. In this section we investigate the [order-${{q}}$-sublines]{} contained in an exterior splash. We first characterise the [order-${{q}}$-sublines]{} of an exterior splash [*with respect to*]{} an associated exterior [order-${{q}}$-subplane]{}. Then we compare this with the characterisation of [order-${{q}}$-sublines]{} of a scattered linear set discussed in [@lavr10].
Sublines in an exterior splash
------------------------------
[sec:orsl-extsplash]{}
We first note that the dual of [@lavr10 Corollary 19] can be generalised to the following result.
[secondlines]{} Let $\pi$ be an exterior [order-${{q}}$-subplane]{} of ${{\rm PG}}(2,q^3)$ with exterior splash ${{\mathbb S}}$ on ${\ell_\infty}$, and let $\Gamma$ be a dual $(q+1)$-arc of $\pi$. The lines of $\Gamma$ meet the exterior splash ${{\mathbb S}}$ in an [order-${{q}}$-subline]{} if and only if $\Gamma$ is $(\pi,{\ell_\infty})$-special-dual conic of $\pi$.
Using this lemma, it is straightforward to prove the following geometric interpretation of the [order-${{q}}$-sublines]{} of ${{\mathbb S}}$ in relation to an associated [order-${{q}}$-subplane]{}.
[defn:sline]{} Let $\pi$ be an exterior [order-${{q}}$-subplane]{} of ${{\rm PG}}(2,q^3)$, $q>2$, with exterior splash ${{\mathbb S}}$ on ${\ell_\infty}$. The $2(q^2+q+1)$ [order-${{q}}$-sublines]{} of ${{\mathbb S}}$ lie in two families of size $q^2+q+1$ as follows.
1. If $A$ is a point of $\pi$, then the pencil of $q+1$ lines of $\pi$ through $A$ meets ${\ell_\infty}$ in an [order-${{q}}$-subline]{} of ${{\mathbb S}}$, called a [*$\pi$-[pencil-subline]{}.*]{}
2. If $\Gamma$ is a $(\pi,{\ell_\infty})$-special-dual conic of $\pi$, then the lines of $\Gamma$ meet ${\ell_\infty}$ in an [order-${{q}}$-subline]{} of ${{\mathbb S}}$, called a [*$\pi$-[dual-conic-subline]{}*]{}.
[The $\pi$-[dual-conic-subline]{}s of ${{\mathbb S}}$ arise from an inscribed bundle of conics (see Section \[sec:singer-gp\]). We note that in [@lavr10 Remark 20], the authors show that irregular sublines of a scattered linear set of rank 3 arise from a circumscribed bundle of conics. ]{}
Sublines of a linear set and an exterior splash
-----------------------------------------------
[sec:orsl-lin-ext]{}
In this section we compare the construction of the two families of [order-${{q}}$-sublines]{} from [@lavr10] with that of Theorem \[defn:sline\]. In particular, we explain why the notions of regular and irregular [order-${{q}}$-sublines]{} of a linear set defined in [@lavr10] are not intrinsic properties of the linear set, but that the two classes can be interchanged.
We first prove that the characterisation of [order-${{q}}$-sublines]{} given in Theorem \[defn:sline\] is a property of the associated [order-${{q}}$-subplane]{}, not a property of the exterior splash.
[orsls-same]{} Consider an exterior splash ${{\mathbb S}}$ of an exterior [order-${{q}}$-subplane]{} $\pi$ onto ${\ell_\infty}$. Let ${\mathcal X}$ denote the $\pi$-[pencil-subline]{}s of ${{\mathbb S}}$, and let ${\mathcal Y}$ denote the $\pi$-[dual-conic-subline]{}s of ${{\mathbb S}}$. Then there exists an exterior [order-${{q}}$-subplane]{} $\pi'$ with the same exterior splash ${{\mathbb S}}$ on ${\ell_\infty}$, such that ${\mathcal X}$ contains the $\pi'$-[dual-conic-subline]{}s of ${{\mathbb S}}$ and ${\mathcal Y}$ contains the $\pi'$-[pencil-subline]{}s of ${{\mathbb S}}$.
By Theorem \[transsplashes\], all exterior splashes are projectively equivalent, so we can without loss of generality look at the exterior splash ${{\mathbb S}}$ of Example \[splash-eg\]. Now ${{\mathbb S}}$ contains the points $X_i=E+\theta_iE^q$, so points in ${{\mathbb S}}$ have coordinates equivalent to $(\theta_i,1)\equiv(
l+m\tau+n\tau^{2},-(l+m\tau+n\tau^2)^q)$ where $l,m,n\in{{\rm GF}}(q)$, not all zero. We can map this to the point $(l+m\tau+n\tau^{2},(l+m\tau+n\tau^2)^q,0)$, so, we can without loss of generality let ${{\mathbb S}}=\{(x,x^q,0){:}x \in {{\rm GF}}(q^3)\}$. (We note that this is equivalent to the canonical form for a linear set of pseudoregulus type given in [@LMPT14].) It is straightforward to calculate the [order-${{q}}$-sublines]{} of ${{\mathbb S}}$, they are ${\mathcal X}=\{x_P{:}P\in{{\rm PG}}(2,q)\}$, ${\mathcal Y}=\{y_P{:}P\in{{\rm PG}}(2,q)\}$, where $$\begin{aligned}
x_P&=&\{ \left(\,l+m\tau+n\tau^{2},(l+m\tau+n\tau^2)^q,0\right){:}l,m,n\in{{\rm GF}}(q),\ P\cdot[l,m,n]=0\}\\
y_P&=&\{\left((l+m\tau+n\tau^2)^q, l+m\tau+n\tau^{2},0\right){:}l,m,n\in{{\rm GF}}(q),\ P\cdot[l,m,n]=0\}.\end{aligned}$$ Let $\pi$ be an exterior [order-${{q}}$-subplane]{} with exterior splash ${{\mathbb S}}$. Consider the involutary homography $\Delta'$ of ${{\rm PG}}(2,q^3)$ with matrix $$\begin{pmatrix}
0&1&0\\
1&0&0\\
0&0&1
\end{pmatrix},$$ it fixes ${\ell_\infty}$. As the homography $\Delta$ of ${\ell_\infty}$ given in Lemma \[spl-gp-eg\] exchanges the two families ${\mathcal X}$, ${\mathcal Y}$, of [order-${{q}}$-subline]{}s of ${{\mathbb S}}$, $\Delta'$ exchanges the two sets of [order-${{q}}$-subline]{}s in ${{\mathbb S}}$, that is, $\Delta'$ maps ${\mathcal X}$ to ${\mathcal Y}$ and ${\mathcal Y}$ to ${\mathcal X}$. Thus $\Delta'$ does not fix $\pi$, as only elements of the Singer group $I={{\rm PGL}}(3,q^3)_{\pi,{\ell_\infty}}$ fix $\pi$ and ${\ell_\infty}$, and $I$ acts regularly on the elements of ${\mathcal X}$ by Theorem \[propthm\]. Thus $\Delta'$ fixes ${{\mathbb S}}$ and maps $\pi$ to another exterior [order-${{q}}$-subplane]{} $\pi'$. Hence $\Delta'$ maps the $\pi$-[pencil-subline]{}s to the $\pi'$-[pencil-subline]{}s. Thus if ${\mathcal X}$ contains the $\pi$-[pencil-subline]{}s, then ${\mathcal Y}$ contains the $\pi'$-[pencil-subline]{}s, as required.
Note that there is an analogous argument for scattered linear sets. Suppose that the exterior splash/scattered linear set ${{\mathbb S}}$ in the above proof is the projection of an [order-${{q}}$-subplane]{} $\alpha$ from a point $P$ onto ${\ell_\infty}$, then under the homography $\Delta'$, we obtain the same linear set ${{\mathbb S}}$ as the projection of an [order-${{q}}$-subplane]{} $\alpha'$ from a point $P'$, and the [order-${{q}}$-subline]{}s which are regular with respect to $\alpha$ are now irregular with respect to $\alpha'$. That is, the notion of regular and irregular sublines in a fixed scattered linear set ${{\mathbb S}}$ of rank 3 can be interchanged by considering a different associated [order-${{q}}$-subplane]{} that projects the linear set ${{\mathbb S}}$.
Projection of [order-${{q}}$-subplane]{}s
=========================================
[sec:project]{}
In this section we look further at the relationship between exterior splashes and linear sets. Lunardon and Polverino [@luna04] showed that a ${{\rm GF}}(q)$-linear set of rank 3 of ${{\rm PG}}(1,q^3)$ can be constructed by projecting an [order-${{q}}$-subplane]{} $\alpha$ of ${{\rm PG}}(2,q^3)$ from a point $P\notin\alpha$ onto ${\ell_\infty}\cong{{\rm PG}}(1,q^3)$. We compare this with our definition of the splash of an [order-${{q}}$-subplane]{} $\pi$ onto ${\ell_\infty}$, namely the intersection of the lines of $\pi$ with ${\ell_\infty}$. If $\pi$ is exterior to ${\ell_\infty}$, then the exterior splash of $\pi$ is equivalent to a ${{\rm GF}}(q)$-linear set of rank 3 and size $q^2+q+1$. If $\pi$ is tangent to ${\ell_\infty}$, then [@BJ-tgt1 Theorem 7.1] shows that the tangent splash of $\pi$ is equivalent to a ${{\rm GF}}(q)$-linear set of rank 3 and size $q^2+1$. This leads to the question: given an [order-${{q}}$-subplane]{} $\pi$, can the splash of $\pi$ on ${\ell_\infty}$ be the same set of points as the projection of $\pi$ onto ${\ell_\infty}$ from some point? We consider the case when $\pi$ is exterior to ${\ell_\infty}$ in Section \[sec:proj-ext\], and consider the case when $\pi$ is tangent to ${\ell_\infty}$ in Section \[sec:proj-tgt\].
Projection and exterior splashes
--------------------------------
[sec:proj-ext]{}
In this section we explain under what circumstances the exterior splash of an exterior [order-${{q}}$-subplane]{} $\pi$ is equal to the projection of $\pi$ onto ${\ell_\infty}$. We also discuss a conjecture regarding which exterior splashes of ${\ell_\infty}$ can be obtained by projecting a fixed exterior plane [order-${{q}}$-subplane]{} onto ${\ell_\infty}$.
We begin by showing that the Singer cycle of Theorem \[propthm\] acts regularly on the special conics, and special-dual conics of an [order-${{q}}$-subplane]{}.
[cgreg]{} Let $\pi$ be an [order-${{q}}$-subplane]{} of ${{\rm PG}}(2,q^3)$ exterior to ${\ell_\infty}$. The group $I={{\rm PGL}}(3,q^3)_{\pi,{\ell_\infty}}$ acts regularly on the set of $(\pi,{\ell_\infty})$-special conics of $\pi$, and acts regularly on the $(\pi,{\ell_\infty})$-special-dual conics of $\pi$.
By Theorem \[propthm\], $I$ fixes $\pi$ and hence acts on the projective plane ${\mathcal P}$ with points the points of $\pi$, and lines the set $\mathscr C$ of $(\pi,{\ell_\infty})$-special conics of $\pi$. Consider the orbit $\mathscr C^I$ of $\mathscr C$ under $I$. As $I$ is transitive on the points of $\pi$ there is a constant number $n$ of elements of $\mathscr C^I$ through each point of $\pi$. Count the pairs $(Q,{{\cal D}})$ where $Q$ is a point of $\pi$ on a $(\pi,{\ell_\infty})$-special conic ${{\cal D}}\in \mathscr C^I$. We have $$({{q}}^2+{{q}}+1)\times n=|\mathscr C^I|\times ({{q}}+1).$$ As ${{q}}^2+{{q}}+1$ and ${{q}}+1$ have no common factors, it follows that ${{q}}+1$ divides $n$. However, as ${\mathcal P}$ is a projective plane, $n\le {{q}}+1$. Hence $n={{q}}+1$ and so $|\mathscr C^I|={{q}}^2+{{q}}+1$. Similarly, $I$ acts regularly on the $(\pi,{\ell_\infty})$-special-dual conics of $\pi$.
[proj-spl]{} Let $\pi$ be an exterior [order-${{q}}$-subplane]{} with exterior splash ${{\mathbb S}}$ on ${\ell_\infty}$, carriers $E_1,E_2$, and third conjugate point $E_3$. Let $P$ be a point of ${{\rm PG}}(2,q^3){\backslash}\pi$, then the projection of $\pi$ from $P$ onto ${\ell_\infty}$ is equal to ${{\mathbb S}}$ if and only if $q$ is even, and $P=E_3$.
By Theorem \[transsplashes\], we can without loss of generality prove this for the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,q)$ with exterior splash ${{\mathbb S}}$ onto the exterior line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$. By Theorem \[propthm\], the group $I={{\rm PGL}}(3,q^3)_{\pi,\ell}$ fixing $\pi$ and $\ell$ is cyclic of order $q^2+q+1$; acts regularly on the points and lines of $\pi$; and by Theorem \[cgreg\] acts regularly on the $(\pi,\ell$)-special-dual conics of $\pi$. By Theorem \[defn:sline\], there are $2(q^2+q+1)$ [order-${{q}}$-subline]{}s in ${{\mathbb S}}$, divided into two families denoted ${\mathcal X}$ and ${\mathcal Y}$, one family contains the $\pi$-[pencil-subline]{}s of ${{\mathbb S}}$, and the other family contains the $\pi$-[dual-conic-subline]{}s. So $I$ acts regularly on the [order-${{q}}$-sublines]{} of ${\mathcal X}$, and acts regularly on the [order-${{q}}$-sublines]{} of ${\mathcal Y}$.
Let $P$ be a point of ${{\rm PG}}(2,q^3)$ not in $\pi$ or $\ell$, and let ${{\mathscr L}}_P$ be the projection of $\pi$ onto $\ell$ from $P$. Suppose that ${{\mathscr L}}_P={{\mathbb S}}$, that is, the projection of $\pi$ from $P$ is the same as the exterior splash of $\pi$ onto $\ell$. By [@lavr10], the $q^2+q+1$ [order-${{q}}$-subline]{}s of $\pi$ are projected onto [order-${{q}}$-subline]{}s of $\mathcal L_P={{\mathbb S}}$ that lie in the same family, ${\mathcal X}$ say. Further, the line joining $P$ to an [order-${{q}}$-subline]{} $d$ in the other family ${\mathcal Y}$ meets $\pi$ in a set of points which form a conic of $\pi$ whose extension to ${{\rm PG}}(2,q^3)$ contains $P$, and hence $P^q$ and $P^{q^2}$. That is, it is a $(\pi,PP^q)$-special conic, and so the $q^2+q+1$ $(\pi,PP^q)$-special conics of $\pi$ are projected to the $q^2+q+1$ [order-${{q}}$-subline]{}s in ${\mathcal Y}$.
As the group $I$ acts regularly on the [order-${{q}}$-sublines]{} of ${\mathcal Y}$, it induces a Singer cycle on the $q^2+q+1$ $(\pi,PP^q)$-special conics of $\pi$. All these conics have $P,P^q,P^{q^2}$ in common. If $P$ is not a fixed point of this Singer cycle, then the $q^2+q+1$ $(\pi,PP^q)$-special conics all have the images of $P$ in common, a contradiction as two distinct irreducible conics have at most four points in common. Thus $P$ is a fixed point of the Singer cycle, and $P\notin\ell$, so $P=\ell^q\cap\ell^{q^2}$. That is, if ${{\mathscr L}}_P={{\mathbb S}}$ then $P=\ell^q\cap\ell^{q^2}=(1,\tau^{q^2},\tau^{2q^2})$.
It remains to show that for the point $P=\ell^q\cap\ell^{q^2}$, ${{\mathscr L}}_P={{\mathbb S}}$ if and only if $q$ is even. By Example \[splash-eg\], we can parameterise the points of ${{\mathbb S}}$ as $E+\theta E^q$ for $\theta\in{{\rm GF}}(q^3)$, $\theta^{q^2+q+1}=-1$. Let $(x,y,z)$, $x,y,z\in{{\rm GF}}(q)$, be a point of $\pi$. The line joining this point to $P=(r,s,t)$ has coordinates $[sz-ty,\ tx-rz,\ ry-sx]$. It meets the line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$ in the point $E+\theta E^q$ with parameter $$\begin{aligned}
\theta&=&
\frac{x(s\tau^2-t\tau)+y(t-r\tau^2)+z(r\tau-s)}
{x(t\tau^q-s\tau^{2q})+y(r\tau^{2q}-t)+z(s-r\tau^q)}\ =\ \frac{f(x,y,z)}{g(x,y,z)}.\end{aligned}$$ If $P=(r,s,t)=(1,\tau^{q^2},\tau^{2q^2})$ we see that $f^{q^2}=g$, and so $\theta^{q^2+q+1}=1$. Now ${{\mathbb S}}$ is the set of points with parameter $\theta$, where $\theta^{q^2+q+1}=-1$, hence the projection ${{\mathscr L}}_P$ from $P=(1,\tau^{q^2},\tau^{2q^2})$ is the same as the exterior splash ${{\mathbb S}}$ of $\pi$ onto $\ell$ if and only if $q$ is even.
Suppose we fix an [order-${{q}}$-subplane]{} $\pi$ and an exterior line $\ell$. We consider the set $\mathscr S$ of all the projections of $\pi$ from a point $P\in{{\rm PG}}(2,q^3){\backslash}\{\pi,\ell\}$ onto $\ell$. Note that the projections in $\mathscr S$ are either tangent or exterior splashes. As there are less than $q^6$ possible points to project $\pi$ from, and more than $q^6$ exterior splashes, we will not obtain all the possible exterior splashes on $\ell$. We further examine this situation. As usual, let $I={{\rm PGL}}(3,q^3)_{\pi,\ell}$ with fixed points $E_1,E_2,E_3$ and fixed lines $\ell,m,n$. Let $J$ be the subgroup of ${{\rm PGL}}(3,q^3)$ that fixes $E_1,E_2,E_3$. Then it is straightforward to show that $I$ is a normal subgroup of $J$, and that further $J$ has seven orbits, namely $E_1$; $E_2$; $E_3$; $\ell{\backslash}\{E_1,E_2\}$; $m{\backslash}\{E_1,E_3\}$; $n{\backslash}\{E_2,E_3\}$; and the points ${{\rm PG}}(2,q^3){\backslash}\{\ell,m,n\}$. Moreover, the orbits of $I$ on $\ell$ are $E_1,E_2$ and $q-1$ exterior splashes, with similar orbits on $m$ and $n$. Finally the orbits of $I$ on ${{\rm PG}}(2,q^3){\backslash}\{\ell,m,n\}$ consist of [order-${{q}}$-subplanes]{}. We now look at the action of $I$ on the splashes in $\mathscr S$.
1. An exterior splash in $\mathscr S$ is either fixed under $I$, or has an orbit of size $q^2+q+1$ under $I$.
2. A tangent splash in $\mathscr S$ has an orbit of size $q^2+q+1$ under $I$.
For part 1, consider an exterior splash ${{\mathbb S}}\in\mathscr S$ with an orbit under $I$ of size less than $|I|=q^2+q+1$, so $I_{{\mathbb S}}$ is non-trivial. We show that in this case ${{\mathbb S}}$ is fixed by $I$. Let $g\in I_{{\mathbb S}}$, $g$ not the identity, so $g$ fixes $E_1$ and $E_2$. As $I$ is cyclic, it is abelian, so $\langle g\rangle$ is a normal subgroup of $I$. Thus $\langle g\rangle$ fixes either all the points in an orbit of $I$, or has no fixed points on an orbit of $I$. In the first case, $g$ fixes at least $(q^2+q+1)+2$ points of $\ell$, which is more points that an [order-${{q}}$-subline]{} of $\ell$ contains, hence $g$ fixes $\ell$ pointwise, and so $g$ is the identity on $\ell$. As $I$ is faithful on $\ell$, it follows that $g$ is the identity. Thus the second case occurs. From Lemma \[spl-gp-eg\] we can show that the full group (acting faithfully) on the splash is of size $2(q^2+q+1)$, consisting of $q^2+q+1$ involutions and the cyclic group $I$ of odd order, whose elements only fix the carriers of the splash. As $g\in I_{{{\mathbb S}}}\subset I$, it follows that $g$ has odd order and the only fixed points are the carriers of ${{\mathbb S}}$. This implies that ${{\mathbb S}}$ has carriers $E_1,E_2$, and is fixed by $I$. So an exterior splash in $\mathscr S$ has orbit of size 1 or $|I|=q^2+q+1$.
For part 2, as the exterior splash of $\pi$ on $\ell$ does not contain $E_1$ or $E_2$, by definition no line of $\pi$ contains $E_1$ or $E_2$, thus the tangent splash obtained by projecting $\pi$ from a point on a line of $\pi$ does not have centre $E_1$ or $E_2$. Thus the centre of the tangent splash is not a fixed point of $I$, hence its orbit under $I$ is of size $|I|$, and hence the $q^2+q+1$ tangent splash projections in $\mathscr S$ have different centres, and so are distinct.
Let ${{\mathbb S}}$ be an exterior splash in $\mathscr S$ with carriers $E_1,E_2$. Then if $q$ is even, ${{\mathbb S}}$ is the projection of $\pi$ from 1 point or $q^2+q+1$ points; and if $q$ is odd, ${{\mathbb S}}$ is the projection of $\pi$ from $q^2+q+1$ or $q^2+q+2$ points.
We can without loss of generality consider the [order-${{q}}$-subplane]{} $\pi={{\rm PG}}(2,q)$ and the exterior line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$. First note that the projection of $\pi$ from $E_3$ is an exterior splash with carriers $E_1,E_2$. As ${{\mathbb S}}$ occurs as a projection of $\pi$, either it is from $E_3$ (this would give the 1 point or the $q^2+q+2$ points case), or it is from a point $P$ belonging to an orbit $\theta$ under $I$ of size $q^2+q+1$. In this case, as ${{\mathbb S}}$ and $\pi$ are both fixed by $I$, it follows that every point of the orbit $\theta$ projects $\pi$ on to the same splash ${{\mathbb S}}$. Now suppose there is another point $W$ which also projects $\pi$ onto ${{\mathbb S}}$. Recall that the exterior splash ${{\mathbb S}}$ has two families of [order-${{q}}$-subline]{}s, ${\mathcal X}$ and ${\mathcal Y}$. An [order-${{q}}$-subline]{} $m$ of $\pi$ is projected by $P$ onto an [order-${{q}}$-subline]{} in one of these families, ${\mathcal X}$ say. Further, the images of $m$ under $I$ are projected onto [order-${{q}}$-sublines]{} in ${\mathcal X}$. Thus $W$ cannot project $m$ onto an [order-${{q}}$-subline]{} of ${\mathcal X}$, since the point of projection of an [order-${{q}}$-subline]{} is unique. Thus $W$ projects $m$ onto an [order-${{q}}$-subline]{} $y$ of the family ${\mathcal Y}$. Similar to the proof of Theorem \[proj-spl\], the lines joining $W$ to the points of $y$ meet $\pi$ in a set of points which form a conic of $\pi$ whose extension to ${{\rm PG}}(2,q^3)$ contains $W$, and hence $W^q$ and $W^{q^2}$. Then, similar to the proof of Theorem \[proj-spl\], it follows that $W$ is a fixed point of the Singer cycle $I$, and so $W=\ell^q\cap\ell^{q^2}$. By Theorem \[proj-spl\], if $q$ is odd, then there is at most one exterior splash with carriers $E_1,E_2$ with $q^2+q+2$ points of projection, and if $q$ is even, all the exterior splashes with carriers $E_1,E_2$ have at most $q^2+q+1$ points of projection.
We conject that the following fully describes the projection of an [order-${{q}}$-subplane]{} onto an exterior line.
Let $\pi$ be an [order-${{q}}$-subplane]{} in ${{\rm PG}}(2,q^3)$ with exterior line $\ell$. Let $\mathscr S$ be the set of all the projections of $\pi$ from a point $P\in{{\rm PG}}(2,q^3){\backslash}\{\pi,\ell\}$ onto $\ell$. Note that the projections in $\mathscr S$ are either tangent or exterior splashes. More specifically,
1. The points $P\in{{\rm PG}}(2,q^3){\backslash}\{\pi,\ell\}$ that lie on the extension of a line of $\pi$ project $\pi$ onto distinct tangent splashes. Hence there are $(q^2+q+1)(q^3-q-1)$ such tangent splashes in $\mathscr S$.
2. The exterior splashes in $\mathscr S$ can be divided into three distinct groups. Let ${{\mathbb S}}_0$ be the exterior splash of $\pi$ onto $\ell$, and let ${{\mathbb S}}_1,\ldots,{{\mathbb S}}_{q-2}$ be the distinct exterior splashes on $\ell$ that have the same carriers ${E}_1,{E}_2$ as ${{\mathbb S}}_0$.
1. 1. If $q$ is even, then ${{\mathbb S}}_0$ is a projection of $\pi$ from $E_3$, and ${{\mathbb S}}_i$, $i=1,\ldots,q-2$ is a projection from exactly $q^2+q+1$ points which lie in one orbit of $I$.
2. If $q$ is odd, then ${{\mathbb S}}_i$, $i=1,\ldots,q-2$ is the projection of $\pi$ from exactly $q^2+q+1$ points which lie in one orbit of $I$, with the exception of one of these which is also a projection of $\pi$ from $E_3$.
2. The remaining exterior splashes in ${\mathcal S}$ are either:
1. projections of $\pi$ from exactly two points which are not in the same orbit under $I$; or
2. projections of $\pi$ from exactly one point.
Projection and tangent splashes
-------------------------------
[sec:proj-tgt]{}
Let $\pi$ be an [order-${{q}}$-subplane]{} that is tangent to ${\ell_\infty}$ at the point $T$, that is, $\pi$ is a tangent [order-${{q}}$-subplane]{}. The lines of $\pi$ meet ${\ell_\infty}$ in a set ${\mathscr S_T}$ of $q^2+1$ points (including $T$) called a [*tangent splash*]{} with [*centre*]{} $T$. For a detailed study of tangent splashes, see [@BJ-tgt1], in particular, by [@BJ-tgt1 Theorem 7.1], tangent splashes are projectively equivalent to ${{\rm GF}}(q)$-linear sets of rank 3 and size $q^2+1$. We can construct such a linear set by projecting an [order-${{q}}$-subplane]{}. We show that the linear set obtained by projecting a tangent [order-${{q}}$-subplane]{} $\pi$ onto ${\ell_\infty}$ can never equal the tangent splash of $\pi$ onto ${\ell_\infty}$.
[proj-count]{} Let $\ell,m$ be lines of ${{\rm PG}}(2,q^3)$, and $b$ an [order-${{q}}$-subline]{} of $m$ disjoint from $\ell$. Then each [order-${{q}}$-subline]{} of $\ell$, disjoint from $m$, is the projeciton of $b$ from exactly one point $P$ not on $\ell$ or $m$.
We first show that the subgroup $K$ of ${{\rm PGL}}(2,q^3)$ (acting on ${{\rm PG}}(1,q^3)$) fixing a point $P$ of ${{\rm PG}}(1,q^3)$ is regular on the [order-${{q}}$-subline]{}s not containing $P$. Note that ${{\rm PGL}}(2,q^3)$ acts faithfully on an [order-${{q}}$-subline]{} $b$ of ${{\rm PG}}(1,q^3)$, as any such homography fixing $3$ points is the identity. Thus $|{{\rm PGL}}(2,q^3)_b|=|PGL(1,q)|=q(q^2-1)$. As ${{\rm PGL}}(2,q^3)$ is sharply 3-transitive on the points of ${{\rm PG}}(1,q^3)$, it is transitive on the [order-${{q}}$-subline]{}s of ${{\rm PG}}(1,q^3)$. Consider the subline $b={{\rm PG}}(1,q)$ of ${{\rm PG}}(1,q^3)$. From above, the subgroup $H$ of ${{\rm PGL}}(2,q^3)$ fixing $b$ is of order $q(q^2-1)$. If we consider a homography acting on $P=(\tau,1)^t$, a straightforward calculation shows that $H_P$ is just the identity, hence by the orbit stabilizer theorem, $H$ is transitive on the points of ${{\rm PG}}(1,q^3)$ not on $b$. Now suppose $P$ is a point not on $b$. If there is an element of $K={{\rm PGL}}(2,q^3)_P$ which fixes $b$, then this element belongs to $H$ and fixes the point $P$ outside $b$, and so is the identity. Thus $K$ acts semiregularly on the $q^3(q^3-1)$ [order-${{q}}$-subline]{}s not through $P$, but as $|K|=q^3(q^3-1)$, this action is transitive, and hence regular.
Consider the group $L$ of axial homographies with axis $m$ and centre on $\ell$. There are $q^3-1$ non-identity elations in $L$, and $q^3(q^3-2)$ non-identity homologies in $L$, so $|L|=q^3(q^3-1)$. As an element of $L$ fixes $\ell$ and $P=\ell\cap m$, it induces a homography $\sigma$ on $\ell$ which fixes $P$. As the subgroup $K$ above is regular on the [order-${{q}}$-subline]{}s not containing $P$, $\sigma$ is either the identity, or acts semi-regularly on the [order-${{q}}$-subline]{}s of $\ell$ not through $P$. If $\sigma$ acts as the identity on $\ell$, then as it has axis $m$, it follows that it is the identity on ${{\rm PG}}(2,q^3)$. Hence $L$ acts semi-regularly and hence regularly on the [order-${{q}}$-subline]{}s of $\ell$ not through $P$.
Now consider a point $X$ projecting $b$ onto an [order-${{q}}$-subline]{} $c$ on $\ell$. Under the action of $L$, $b$ is fixed pointwise (as $b$ is contained in the axis of the elements of $L$) and $c$ is mapped to the $q^3(q^3-1)$ [order-${{q}}$-subline]{}s on $\ell$ not through $P$. Further, as $L$ acts semi-regularly on the $q^3(q^3-1)$ points of ${{\rm PG}}(2,q^3)$ not on $\ell\cup m$, it follows that $L$ is regular on the points not on $\ell\cup m$. Thus distinct points of ${{\rm PG}}(2,q^3)$ not on $\ell\cup m$ project $b$ onto distinct [order-${{q}}$-subline]{}s of $\ell$.
[proj-tgt]{} Let $\pi$ be a tangent [order-${{q}}$-subplane]{} with tangent splash ${\mathscr S_T}$ on ${\ell_\infty}$. Then ${\mathscr S_T}$ is not the projection of $\pi$ onto ${\ell_\infty}$ from a point $P$.
We let ${{\mathscr L}}_{X,P}$ denote the projection of the set of points $X$ of ${{\rm PG}}(2,q^3)$ onto ${\ell_\infty}$ from the point $P$. Note that if we project the tangent [order-${{q}}$-subplane]{} $\pi$ onto ${\ell_\infty}$ from a point $P\in{\ell_\infty}$ or $P\in\pi$, then ${{{\mathscr L}}_{\pi,P}}$ does not have enough points to be the tangent splash ${\mathscr S_T}$ of $\pi$. Similarly, if $P$ is a point not on a line of $\pi$, then ${{{\mathscr L}}_{\pi,P}}$ has too many points to be a tangent splash. Suppose we project from a point $P\notin\pi$ that is on a line $\ell$ of $\pi$ not through the centre $T$ of ${\mathscr S_T}$, and suppose ${{{\mathscr L}}_{\pi,P}}={\mathscr S_T}$. Let $b$ be an [order-${{q}}$-subline]{} of $\pi$ not through $T$, and not on $\ell$, then ${{\mathscr L}}_{b,P}$ is an [order-${{q}}$-subline]{} of ${\mathscr S_T}$ not through the centre $T$, contradicting [@BJ-tgt1 Corollary 7.3]. So this case cannot occur.
So, suppose we project $\pi$ from a point $P\notin\pi$ that lies on a line of $\pi$ through the centre $T$, and suppose ${{{\mathscr L}}_{\pi,P}}={\mathscr S_T}$. Let $J$ be the subgroup of ${{\rm PGL}}(3,q^3)_\pi\cong{{\rm PGL}}(3,q)$ that fixes $T$ and ${\ell_\infty}$. Consider $J_P$, the subgroup of $J$ fixing the projection point $P$. From the proof of [@BJ-tgt1 Theorem 4.2], $J$ contains only central collineations with centre $T$ and axes belonging to $\pi$. So $P$ is fixed by a non-identity element $\sigma$ of $J$ only if it lies on the axis of $\sigma$. So $J_P$ is the set of $q$ elations with centre $T$ and axis $TP$. Hence $|P^J|=|J|/|J_P|=q(q-1)$.
Let $b$ be an [order-${{q}}$-subline]{} of $\pi$ not through $T$. Consider the projection $c={{\mathscr L}}_{b,P}$. If $Q$ is another point in $P^J$, then by Lemma \[proj-count\], it follows that ${{\mathscr L}}_{b,Q}$ is not $c$. Thus the projections of $b$ under the $q(q-1)$ different points in $P^J$ is the orbit of $c$ under $J$, and is of size $q(q-1)$.
However, each point $A$ of $\pi$ determines a unique [order-${{q}}$-subline]{} of ${\mathscr S_T}$ via the intersection of the lines of $\pi$ through $A$ with ${\ell_\infty}$, and all the [order-${{q}}$-subline]{}s in the tangent splash ${\mathscr S_T}$ are determined this way (by [@BJ-tgt1 Corollary 7.4]). As the points of $\pi$ under $J$ fall into $q+1$ orbits, each consisting of $q$ points lying on a line through $T$, it follows that $|c^J|=q$, contradicting the previous paragraph.
Order-$q$-subplanes with a common exterior splash
=================================================
[sec:common-splash]{}
In this section we investigate the intersection of two exterior [order-${{q}}$-subplanes]{} that have a common exterior splash. We begin with a counting result.
[propsplash]{} Given an exterior splash ${{\mathbb S}}$ on ${\ell_\infty}$, there are $2{{q}}^6({{q}}^3-1)$ [order-${{q}}$-subplanes]{} with exterior splash ${{\mathbb S}}$.
By Theorem \[spiscov\], the number of exterior splashes is equal to the number of covers of a circle geometry $CG(3,q)$, which is $\frac12{{q}}^3({{q}}^3+1)({{q}}-1)$. Counting quadrangles gives that there are $q^9(q^3-1)(q^3+1)(q-1)$ [order-${{q}}$-subplanes]{} exterior to a given line ${\ell_\infty}$. Further, by Theorem \[transsplashes\], the group of homographies of ${{\rm PG}}(2,q^3)$ fixing ${\ell_\infty}$ is transitive on the [order-${{q}}$-subplane]{}s exterior to ${\ell_\infty}$. Hence the number of [order-${{q}}$-subplanes]{} with a fixed exterior splash is the total number of [order-${{q}}$-subplane]{}s exterior to $\ell$ divided by the number of exterior splashes on $\ell$, that is, $2q^6(q^3-1)$.
[ext2]{} Let ${{\mathbb S}}$ be an exterior splash of ${\ell_\infty}$ and let $\ell$ be an [order-${{q}}$-subline]{} exterior to ${\ell_\infty}$, whose extension to ${{\rm PG}}(2,q^3)$ contains a point of ${{\mathbb S}}$. Then there are exactly two [order-${{q}}$-subplanes]{} that contain $\ell$ and have exterior splash ${{\mathbb S}}$.
Let $\pi$ be an [order-${{q}}$-subplane]{} with exterior splash ${{\mathbb S}}$, and let $\ell$ be a line of $\pi$. Consider the subgroup $I=G_{\pi,{\ell_\infty}}$ of $G={{\rm PGL}}(3,q^3)$ fixing $\pi$ and ${\ell_\infty}$. By Theorem \[propthm\] the group $I$ fixes ${{\mathbb S}}$ and is transitive on the points of ${{\mathbb S}}$. By [@BJ-tgt2 Theorem 2.5], the subgroup of $G$ fixing $\ell$ point-wise is transitive on the exterior [order-${{q}}$-subline]{}s on a line $m\ne\ell$. Thus, given an exterior splash ${{\mathbb S}}$, the group fixing ${{\mathbb S}}$ is transitive on the exterior [order-${{q}}$-subline]{}s lying on a line through a point of ${{\mathbb S}}$. Hence the number $n$ of exterior [order-${{q}}$-subplane]{}s containing an [order-${{q}}$-subline]{} $\ell$ lying on a line through a point of a given exterior splash ${{\mathbb S}}$ is a constant. Thus if we count the pairs $(\pi,\ell)$ where $\pi$ is an exterior [order-${{q}}$-subplane]{} with a given exterior splash ${{\mathbb S}}$, containing any of the ${{q}}^3({{q}}^3-1)$ [order-${{q}}$-subline]{}s $\ell$ lying on any of the ${{q}}^3$ lines through a point of a given exterior splash ${{\mathbb S}}$, using Theorem \[propsplash\], we get $$2{{q}}^6({{q}}^3-1)\times({{q}}^2+{{q}}+1)=({{q}}^2+{{q}}+1)\times{{q}}^3\times {{q}}^3({{q}}^3-1)\times n$$ and so $n=2$ as required.
Recall that three collinear points lie in a unique [order-${{q}}$-subline]{}, and a quadrangle lies in a unique [order-${{q}}$-subplane]{}. Hence two [order-${{q}}$-subplanes]{} in ${{\rm PG}}(2,q^3)$ can meet in either $0,1,2$ or $3$ points, or in $q+1$ points lying on an [order-${{q}}$-subline]{}, or in $q+2$ points containing an [order-${{q}}$-subline]{}. In Lemma \[ext2\], we showed that there are exactly two exterior [order-${{q}}$-subplanes]{} that share an [order-${{q}}$-subline]{} and have the same exterior splash. We now show that these two [order-${{q}}$-subplanes]{} do not have any further common points.
[orsp-common-splash]{} Let $\pi_1$, $\pi_2$ be two exterior [order-${{q}}$-subplanes]{} with a common exterior splash ${{\mathbb S}}$ on ${\ell_\infty}$. Then $\pi_1,\pi_2$ meet in at most $q+1$ points, that is, $\pi_1,\pi_2$ meet in $0,1,2$ or $3$ points, or in an [order-${{q}}$-subline]{}.
By Theorem \[transsplashes\] and Example \[splash-eg\], we can without loss of generality let $\pi_1={{\rm PG}}(2,q)$ with exterior splash ${{\mathbb S}}_1=\{{E}_1+\theta {E}_2{:}\theta^{q^2+q+1}=-1,\theta\in{{\rm GF}}(q^3)\}$ on the exterior line ${\ell= [-\tau\tau^q, \tau^q+\tau,-1]}$ where ${E}_1=(1,\tau,\tau^2)$, ${E}_2={E}_1^q=(1,\tau^q,\tau^{2q})$ are the carriers of ${{\mathbb S}}_1$. Let $\pi_2$ be an [order-${{q}}$-subplane]{} exterior to $\ell$ with exterior splash ${{\mathbb S}}_2$ on $\ell$. We show that if ${{\mathbb S}}_1={{\mathbb S}}_2$ and $\pi_1,\pi_2$ contain $q+2$ common points, then $\pi_1=\pi_2$. By Theorem \[propthm\], as the group $I$ fixing $\pi_1$ and $\ell$ (and hence ${{\mathbb S}}_1$) is transitive on the lines of $\pi_1$, without loss of generality we can suppose $\pi_2$ contains the [order-${{q}}$-subline]{} $m=\{M_b=(0,1,b){:}b\in{{\rm GF}}(q)\cup\{\infty\}\}$ of $\pi_1$. Suppose that $\pi_2$ and $\pi_1$ contain a further common point $P$ not on $m$, so $P=(\mu,\nu,\omega)$ with $\mu,\nu,\omega\in{{\rm GF}}(q^3)$, with $\mu\ne 0$. As $P\in\pi_1$, we have $\nu/\mu,\omega/\mu\in{{\rm GF}}(q)$.
Let $Q=(0,0,1)=M_\infty\in m$, so the line $PQ$ contains an [order-${{q}}$-subline]{} of $\pi_2$. Without loss of generality, by suitable choice of $\mu,\nu,\omega\in{{\rm GF}}(q^3)$, let that [order-${{q}}$-subline]{} be determined by the three points $P,\ Q$ and $T=P+Q=(\mu,\nu,\omega+1)$. That is, $\pi_2$ is determined by the quadrangle $(0,1,1)$, $(0,1,0)$, $P=(\mu,\nu,\omega)$ and $T=(\mu,\nu,\omega+1)$. Further, $\pi_1$ and $\pi_2$ both contain the [order-${{q}}$-subline]{} $m$ and the point $P$.
We will show that, if ${{\mathbb S}}_2={{\mathbb S}}_1$, then $\mu\in{{\rm GF}}(q)$, and hence $\nu,\omega\in{{\rm GF}}(q)$ and so $T\in\pi_1$. The [order-${{q}}$-subline]{} $n$ defined by $P,Q,T=P+Q$ has points $N_c
=P+cQ=(\mu,\nu,\omega+c)$ for $c\in{{\rm GF}}(q)$. As the points $M_b,N_c$ lie in $\pi_2$ for $b,c\in{{\rm GF}}(q)$, the line $M_bN_c$ meets $\ell$ in a point $X_{bc}$ of ${{\mathbb S}}_2$. Now $M_bN_c$ has coordinates $[c-b\nu+\omega,\ b \mu ,\ -\mu]$, and it meets $\ell$ in the point $X_{bc}={E}_1+\theta_{bc}{E}_2$ with $$\theta_{bc}
=-\frac{c+b(\mu\tau-\nu)-(\mu\tau^2-\omega)}{c+b(\mu\tau^q-\nu)-(\mu\tau^{2q}-\omega)}.$$ The point $X_{bc}$ is in the exterior splash ${{\mathbb S}}_1$ if and only if $\theta_{bc}^{q^2+q+1}=-1$. We can regard this equation as a multivariate polynomial of degree 3 in $b$ and $c$. Note that if a multivariate polynomial of degree $k<q$ over ${{\rm GF}}(q^3)$ is identically zero over ${{\rm GF}}(q)$, then the polynomial is the zero polynomial. If $q=2$ or $3$, a computer search using Magma [@magma] verifies the result holds, so we assume $q>3$. Expanding, and calculating the coefficients of $bc^2$ and $c^2$, and equating them to zero gives $$\begin{aligned}
0&=&\kappa\tau+\kappa^q\tau^q+\kappa^{q^2}\tau^{q^2}\\
0&=&\kappa\tau^2+\kappa^q\tau^{2q}+\kappa^{q^2}\tau^{2q^2},\end{aligned}$$ respectively, where $\kappa=\mu-\mu^{q^2}$, and so $\kappa+\kappa^q+\kappa^{q^2}=0$. That is, the point $M_bN_c\cap\ell$ lies in the exterior splash ${{\mathbb S}}_1$ for all $b,c\in{{\rm GF}}(q)$ if $$\begin{pmatrix}
1&1&1\\
\tau&\tau^q&\tau^{q^2}\\
\tau^2&\tau^{2q}&\tau^{2q^2}\end{pmatrix}
\begin{pmatrix}\kappa\\ \kappa^q\\ \kappa^{q^2}\end{pmatrix}
=\begin{pmatrix}0\\0\\0\end{pmatrix}.$$ This $3\times 3$ matrix is invertible, so the only solution is $\kappa=\kappa^q=\kappa^{q^2}=0$, that is $\mu=\mu^q$, and so $\mu\in{{\rm GF}}(q)$. Hence $\pi_2$ has exterior splash ${{\mathbb S}}_1$ if and only if $\mu\in{{\rm GF}}(q)$. As $\nu/\mu,\omega/\mu\in{{\rm GF}}(q)$, it follows that $T\in\pi_2\cap\pi_1$. Thus $\pi_2$ and $\pi_1$ share the quadrangle $P,T,(0,1,0),(0,1,1)$ and so $\pi_2=\pi_1$ as required.
Suppose $\pi_1,\pi_2$ are two exterior [order-${{q}}$-subplanes]{} with a common exterior splash ${{\mathbb S}}$ that meet in $q+1$ points. We conject that the two families of [order-${{q}}$-sublines]{} of ${{\mathbb S}}$ with respect to $\pi_1,\pi_2$ are *swapped*. That is, $\pi_1,\pi_2$ correspond to the [order-${{q}}$-subplanes]{} $\pi$, $\pi'$ with [order-${{q}}$-sublines]{} behaving as in Theorem \[orsls-same\].
Conclusion
==========
[sec:con]{}
In this article we looked at the exterior splash as a set of points of ${{\rm PG}}(1,q^3)$, and showed its equivalence to ${{\rm GF}}(q)$-linear sets of rank 3 and size $q^2+q+1$, Sherk surfaces of size $q^2+q+1$, and covers of the circle geometry $CG(3,q)$. We also investigated properties of an exterior [order-${{q}}$-subplane]{} and its corresponding exterior splash, in particular relating [order-${{q}}$-sublines]{} of an exterior splash to the corresponding [order-${{q}}$-subplane]{}. Further, we looked at how our construction of exterior splashes related to the projection construction of linear sets.
In future work, we look in the Bruck-Bose representation of ${{\rm PG}}(2,q^3)$ in ${{\rm PG}}(6,q)$, and study exterior splashes in this context, further utilising properties of scattered linear sets of rank 3, as well as properties of hyper-reguli in ${{\rm PG}}(5,q)$.
[999]{}
R.D. Baker, J.M.N. Brown, G.L. Ebert and J.C. Fisher. Projective bundles. [*Bull. Belg. Math. Soc*]{}, [**3**]{} (1994) 329–336.
S.G. Barwick and W.-A. Jackson. A characterisation of tangent subplanes of PG$(2,q^3)$. [*Des. Codes Cryptogr.*]{}, [**71**]{} (2014) 541–545.
S.G. Barwick and W.-A. Jackson. An investigation of the tangent splash of a subplane of PG$(2,q^3)$. [*Des. Codes Cryptogr.*]{}, [**76**]{} (2015) 451–468.
S.G. Barwick and W.-A. Jackson. The tangent splash in PG$(6,q)$. Discrete Math., [**338**]{} (2015) 1178–1190.
A. Blokhuis and M. Lavrauw. Scattered spaces with respect to a spread in ${{\rm PG}}(n,q)$. [*Geom. Dedicata.*]{} [**81**]{} (2000) 231–243.
W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, [*J. Symbolic Comput.*]{}, [****]{} 24 (1997), 235Ð265.
R.H. Bruck. Construction problems of finite projective planes. [*Conference on Combinatorial Mathematics and its Applications*]{}, University of North Carolina Press, (1969) 426–514.
R.H. Bruck. Circle geometry in higher dimensions. [*A Survey of Combinatorial Theory*]{}, eds. J. N. Srivastava et al, Amsterdam (1973) 69–77.
R.H. Bruck. Circle geometry in higher dimensions. II. [*Geom. Dedicata*]{}, [**2**]{} (1973) 133–188.
C. Culbert and G.L. Ebert. Circle Geometry and three-dimensional subregular translation planes. [*Innov. Incidence Geom.*]{}, [**1**]{} (2005) 3–18.
. Scattered linear sets generated by collineations between pencils of lines, [*J. Algebraic. Comb.*]{}, [**40**]{} (2014) 1121–1134.
M. Lavrauw and G. Van de Voorde. On linear sets on a projective line. [*Des. Codes Cryptogr.*]{}, [**56**]{} (2010) 89–104.
M. Lavrauw and G. Van de Voorde. Scattered linear sets and pseudoreguli. [*Electronic J. Combin.*]{}, [**20**]{} (2013).
M. Lavrauw and C. Zanella. Subgeometries and linear sets on a projective line. [*Finite Fields Appl.*]{}, to appear.
. Maximum scattered linear sets of pseudoregulus type and the Segre variety ${\cal S}_{n,n}$. [*J. Algebraic. Comb.*]{}, [**39**]{} (2014) 807–831.
G. Lunardon and O. Polverino. Translation ovoids of orthogonal polar spaces. [*Forum Math.,*]{} [**16**]{} (2004) 663–669.
F.A. Sherk. The Geometry of ${{\rm GF}}(q^3)$. [*Canad J. Math.*]{}, [**38**]{} (1986) 672–696.
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---
abstract: 'Alice and Bob want to know if two strings of length $n$ are almost equal. That is, do they differ on *at most* $a$ bits? Let $0\le a\le n-1$. We show that any deterministic protocol, as well as any error-free quantum protocol ($C^*$ version), for this problem requires at least $n-2$ bits of communication. We show the same bounds for the problem of determining if two strings differ in *exactly* $a$ bits. We also prove a lower bound of $n/2-1$ for error-free $Q^*$ quantum protocols. Our results are obtained by lower-bounding the ranks of the appropriate matrices.'
author:
- |
[Andris Ambainis]{} [^1]\
[Univ. of Waterloo]{}
- |
[William Gasarch]{} [^2]\
[Univ. of MD at College Park]{}
- |
[Aravind Srinivasan]{} [^3]\
[Univ. of MD at College Park]{}
- |
[Andrey Utis]{} [^4]\
[Univ. of MD at College Park]{}
title: 'Lower bounds on the Deterministic and Quantum Communication Complexity of ${HAM_n^{(a)}}$'
---
=cmr10 =cmr9 =cmr8 =cmr7 ß=cmss10 =cmcsc10
\[section\]
\[theoremfoo\][Lemma]{}
\[theoremfoo\][Conjecture]{} \[theoremfoo\][Line of Research]{}
\[theoremfoo\][Convention]{}
\[theoremfoo\][Porism]{}
\[theoremfoo\][Corollary]{}
\[theoremfoo\][Claim]{}
\[theoremfoo\][Open Problem]{}
\[theoremfoo\][Potential Analogue]{}
\[theoremfoo\][Note]{}
\[theoremfoo\][Nota Bene]{}
\[theoremfoo\][Notation]{}
\[theoremfoo\][Empirical Note]{}
\[theoremfoo\][Example]{}
\[theoremfoo\][Definition]{}
\[theoremfoo\][Proposition]{}
\[theoremfoo\][Fact]{}
15=0.75em 16=0.75em
Introduction
============
Given $x,y\in {\{0,1\}^{{n}}}$ one way to measure how much they differ is the Hamming distance.
If $x,y\in {\{0,1\}^{{n}}}$ then ${{\rm HAM}}(x,y)$ is the number of bits on which $x$ and $y$ differ.
If Alice has $x$ and Bob has $y$ then how many bits do they need to communicate such that they both know ${{\rm HAM}}(x,y)$? The trivial algorithm is to have Alice send $x$ (which takes $n$ bits) and have Bob send ${{\rm HAM}}(x,y)$ (which takes ${\left\lceil {\lg (n+1)}\right\rceil}$ bits) back to Alice. This takes $n+{\left\lceil {\lg (n+1)}\right\rceil}$ bits. Pang and El Gamal [@pang] showed that this is essentially optimal. In particular they showed that ${{\rm HAM}}$ requires at least $n +\lg(n+1-\sqrt n)$ bits to be communicated. (See [@abdel; @commdoc; @metzner; @orlit] for more on the communication complexity of ${{\rm HAM}}$. See [@securemulti] for how Alice and Bob can approximate ${{\rm HAM}}$ without giving away too much information.)
What if Alice and Bob just want to know if ${{\rm HAM}}(x,y)\le a$?
Let $n\in{{\sf N}}$. Let $a$ be such that $0\le a\le n-1$. ${HAM_n^{(a)}}:{\{0,1\}^{{n}}} \times {\{0,1\}^{{n}}} {\rightarrow}{\{0,1\}}$ is the function
$${HAM_n^{(a)}}(x,y) = \cases { 1 & if ${{\rm HAM}}(x,y)\le a$\cr
0 & otherwise.\cr
}$$
The problem ${HAM_n^{(a)}}$ has been studied by Yao [@qfingerprinting] and Gavinsky et al [@qpublic]. Yao showed that there is an $O(a^2)$ public coin simultaneous protocol for ${HAM_n^{(a)}}$ which yields (by Newman [@Newman], see also [@commcomp]) an $O(a^2+\log n)$ private coin protocol and also an $O(2^{a^2}\log n)$ quantum simulataneous message protocol with bounded error [@qfingerprinting]. Gavinsky et al. give an $O(a\log n)$ public coin simultaneous protocol, which yields an $O(a\log n)$ private coin protocol. For $a \gg \log n$ this is better than Yao’s protocol.
All of the protocols mentioned have a small probability of error. How much communication is needed for this problem if we demand no error? There is, of course, the trivial $(n+1)$-bit protocol. Is there a better one?
In this paper we show the following; in the list of results below, the “$c$” (in the “$c \sqrt{n}$” terms) is some positive absolute constant.
1. \[det:n-2\] For any $0 \leq a \leq n-1$, ${HAM_n^{(a)}}$ requires at least $n-2$ bits in the deterministic model.
2. \[det:n\] For $a\le c \sqrt{n}$, ${HAM_n^{(a)}}$ requires at least $n$ bits in the deterministic model.
3. \[quant:n-2\] For any $0 \leq a \leq n-1$, ${HAM_n^{(a)}}$ requires at least $n-2$ bits in the quantum model with Alice and Bob share an infinite number of EPR pairs, using a classical channel, and always obtain the correct answer.
4. For $a\le c \sqrt{n}$, ${HAM_n^{(a)}}$ requires at least $n$ bits in the quantum model in item \[quant:n-2\].
5. \[quant:n/2minus1\] For any $0 \leq a \leq n-1$, ${HAM_n^{(a)}}$ requires at least $\frac{n}{2}-1$ bits in the quantum model with Alice and Bob share an infinite number of EPR pairs, using a quantum channel, and always obtain the correct answer.
6. \[quant:n/2\] For $a\le c \sqrt{n}$, ${HAM_n^{(a)}}$ requires at least $n/2$ bits in the quantum model in item \[quant:n/2minus1\].
Note that if $a=n$ then $(\forall x,y)[{HAM_n^{(a)}}(x,y)=1$, hence we do not include that case.
What if Alice and Bob need to determine if ${{\rm HAM}}(x,y)=a$ or not?
Let $n\in{{\sf N}}$. Let $a$ be such that $0\le a\le n$. ${HAM_n^{(=a)}}:{\{0,1\}^{{n}}} \times {\{0,1\}^{{n}}} {\rightarrow}{\{0,1\}}$ is the function
$${HAM_n^{(=a)}}(x,y) = \cases { 1 & if ${{\rm HAM}}(x,y)\le a$\cr
0 & otherwise.\cr
}$$
We show the exact same results for ${HAM_n^{(=a)}}$ as we do for ${HAM_n^{(a)}}$. There is one minor difference: for ${HAM_n^{(a)}}$ the $a=n$ case had complexity 0 since all pairs of strings differ on at most $n$ bits; however, for ${HAM_n^{(=a)}}$ the $a=n$ case has complexity $n+1$ as it is equivalent to equality.
All our results use the known “log rank” lower bounds on classical and quantum communication complexity: Lemmas \[le:rank\] and \[le:qrank\]. Our approach is to lower-bound the ranks of the appropriate matrices, and then to invoke these known lower bounds.
Definitions, Notations, and Useful Lemmas
=========================================
We give brief definitions of both classical and quantum communication complexity. See [@commcomp] for more details on classical, and [@qsurvey] for more details on quantum.
Let $f$ be any function from ${\{0,1\}^{{n}}} \times {\{0,1\}^{{n}}}$ to ${\{0,1\}}$.
1. A *protocol* for computing $f(x,y)$, where Alice has $x$ and Bob has $y$, is defined in the usual way (formally using decision trees). At the end of the protocol both Alice and Bob know $f(x,y)$.
2. $D(f)$ is the number of bits transmitted in the optimal deterministic protocol for $f$.
3. $Q^*(f)$ is the number of bits transmitted in the optimal quantum protocol where we allow Alice and Bob to share an infinite number of EPR pairs and communicate over a quantum channel.
4. $C^*(f)$ is the number of bits transmitted in the optimal quantum protocol where we allow Alice and Bob to share an infinite number of EPR pairs and communicate over a classical channel.
5. $M_f$ is the $2^n \times 2^n$ matrix where the rows and columns are indexed by ${\{0,1\}^{{n}}}$ and the $(x,y)$-entry is $f(x,y)$.
Let $\lg$ denote the logarithm to the base two. Also, as usual, if $x < y$, then ${x \choose y}$ is taken to be zero.
The following theorem is due to Mehlhorn and Schmidt [@ranklower]; see also [@commcomp].
\[le:rank\] If $f:{\{0,1\}^{{n}}} \times {\{0,1\}^{{n}}} {\rightarrow}{\{0,1\}}$ then $D(f)\ge \lg({\rm rank}(M_f))$.
Buhrman and de Wolf [@qlogrank] proved a similar theorem for quantum communication complexity.
\[le:qrank\] If $f:{\{0,1\}^{{n}}} \times {\{0,1\}^{{n}}} {\rightarrow}{\{0,1\}}$ then the following hold.
1. $Q^*(f)\ge \frac{1}{2}\lg({\rm rank}(M_f))$.
2. $C^*(f)\ge \lg({\rm rank}(M_f))$.
The Complexity ${HAM_n^{(a)}}$ for $a\le O(\sqrt n)$ {#se:hamasq}
====================================================
We start by presenting results for general $a$, and then specialize to the case where $a \leq c \sqrt{n}$.
Let $M_a$ be $M_{{HAM_n^{(a)}}}$, the $2^n\times 2^n$ matrix representing ${HAM_n^{(a)}}$.
$M_a$ has $2^n$ orthogonal eigenvectors.
This follows from $M_a$ being symmetric.
We know that $M_a$ has $2^n$ eigenvalues; however, some of them may be 0. We prove that $M_a$ has few 0-eigenvalues. This leads to a lower bound on $D({HAM_n^{(a)}})$ by Lemma \[le:rank\].
\[de:vz\] Let $z\in {\{0,1\}^{{n}}}$.
1. $v_z \in R^{2^n}$ is defined by, for all $x\in {\{0,1\}^{{n}}}$, $v_z(x) = (-1)^{\sum_i x_i z_i }$. The entries $v_z(x)$ of $v_z$ are ordered in the natural way: in the same order as the order of the index $x$ in the rows (and columns) of $M_a$.
2. We show that $v_z$ is an eigenvector of $M_a$. Once that is done we let ${eig}(z)$ be the eigenvalue of $M_a$ associated with $v_z$.
\[le:main\]
1. The vectors $\{v_z: ~z\in {\{0,1\}^{{n}}}\}$ are orthogonal.
2. For all $z\in {\{0,1\}^{{n}}}$, $v_z$ is an eigenvector of $M_a$.
3. If $z$ has exactly $m$ 1’s in it, then $$eig(z)=\sum_{j=0}^a {\ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ }{m\choose k} {n-m \choose j-k} (-1)^k.$$
The first assertion (orthogonality) follows by simple counting. We now prove the final two assertions together. Let $z\in {\{0,1\}^{{n}}}$ have exactly $m$ ones in it.
Fix a row in $M_a$ that is indexed by $x\in {\{0,1\}^{{n}}}$. Denote this row by $R_x$. We need the following notation: $$\begin{array}{rl}
L_a = & \{ y \mid {{\rm HAM}}(x,y) \le a \}\cr
E_j = & \{ y \mid {{\rm HAM}}(x,y) = j \}\cr
\end{array}$$ We will show that $R_x \cdot v_z$ is a constant multiple (independent of $x$) times $v_z(x)$. Now, $$R_x\cdot v_z = \sum_{y\in {\{0,1\}^{{n}}}} {HAM_n^{(a)}}(x,y) v_z(y) = \sum_{y\in L_a} v_z(y) = \sum_{y\in L_a} (-1)^{\sum_i y_iz_i }.$$ We would like to have this equal $b \times v_z(x)$ for some constant $b$. We set it equal to $b\times v_z(x)$ and deduce what $b$ works. So, suppose $$b \times v_z(x) = \sum_{y\in L_a} (-1)^{\sum_i y_iz_i }.$$
We have $$\begin{aligned}
b & = & \frac{1}{v_z(x)} \sum_{y\in L_a} (-1)^{\sum_i y_iz_i } \nonumber \\
& = & v_z(x) \sum_{y\in L_a} (-1)^{\sum_i y_iz_i} \nonumber \\
& = & (-1)^{\sum_i x_iz_i} \sum_{y\in L_a} (-1)^{\sum_i y_iz_i } \hbox{\ \ \ (by the definition of $v_z(x)$)} \nonumber \\
& = & \sum_{y\in L_a} (-1)^{\sum_i (x_i+y_i)z_i} \nonumber \\
& = & \sum_{y\in L_a} (-1)^{\sum_i |x_i-y_i|z_i } \hbox{\ \ \ (since $x_i+y_i \equiv |x_i-y_i| \pmod 2$) } \nonumber \\
& = & \sum_{j=0}^a \sum_{y\in E_j} (-1)^{\sum_i |x_i-y_i|z_i } \hbox{\ \ \ (since $L_a = \bigcup_{j=0}^a E_j$)}. \label{eqn:b}\end{aligned}$$
We partition $E_j$. If $y\in E_j$ then $x$ and $y$ differ in exactly $j$ places. Some of those places $i$ are such that $z_i=1$. Let $k$ be such that the number of places where $x_i\ne y_i$ and $z_i=1$.
[Upper Bound on $k$:]{} Since there are exactly $m$ places where $z_i=1$ we have $k\le m$. Since there are exactly $j$ places where $x_i\ne y_i$ we have $k\le j$. Hence $k\le\min\{j,m\}$.
[Lower Bound on $k$:]{} Since there are exactly $n-m$ places where $z_i=0$, we have $j-k\le n-m$. Hence $k\ge \max\{0,j+m-n\}$.
In summary, the only relevant $k$ are $\max\{0,j+m-n\} \le k \le \min\{j,m\}$. Fix $j$. For
$\max\{0,j+m-n\}\le k\le \min\{j,m\}$, let $D_{j,k}$ be defined as follows: $$D_{j,k} = \{ y \mid ((y \in E_j) \wedge
(\hbox{on exactly $k$ of the coordinates
where $x_i\ne y_i$, we have $z_i = 1$})) \}.$$
Note that
$$E_j = \bigcup_{k=0}^{\min\{j,m\}} D_{j,k}$$ and $|D_{j,k}|= {m\choose k} {n-m \choose j-k}$. So, by (\[eqn:b\]), $$b = \sum_{j=0}^a \sum_{y\in E_j} (-1)^{\sum_i |x_i-y_i|z_i }
= \sum_{j=0}^a {\ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ }\sum_{y\in D_{j,k}} (-1)^{\sum_i |x_i-y_i|z_i }.$$
By the definition of $D_{j,k}$ we know that for exactly $k$ of the values of $i$ we have both $|x_i-y_i|=1$ and $z_i=1$. On all other values one of the two quantities is 0. Hence we have the following: $$\begin{aligned}
b & = & \sum_{j=0}^a {\ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ }\sum_{y\in D_{j,k}} (-1)^k \\
& = & \sum_{j=0}^a {\ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ }|D_{j,k}| (-1)^k \\
& = & \sum_{j=0}^a {\ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ }{m\choose k} {n-m \choose j-k} (-1)^k.\end{aligned}$$
Notice that $b$ is independent of $x$ and is of the form required.
Let $$F(a,n,m)= \sum_{j=0}^a {\ \ \sum_{k=\max\{ 0,j+m-n \} }^{\min\{j,m\}}\ \ }{m\choose k} {n-m \choose j-k} (-1)^k.$$
The following lemma will be used in this section to obtain a lower bound when $a=O(\sqrt n)$, and in Section \[se:gen\] to obtain a lower bound for general $a$.
\[le:uslea\]
1. $D({HAM_n^{(a)}}) \ge \lg \sum_{m: F(a,n,m)\ne 0 } { n \choose m }.$
2. $Q^*({HAM_n^{(a)}}) \ge \frac{1}{2}\lg \sum_{m: F(a,n,m)\ne 0 } { n \choose m }.$
3. $C^*({HAM_n^{(a)}}) \ge \lg \sum_{m: F(a,n,m)\ne 0 } { n \choose m }.$
By Lemma \[le:main\], the eigenvector $v_z$ has a nonzero eigenvalue if $v_z$ has $m$ 1’s and $F(a,n,m)\ne0$. The rank of $M_a$ is the number of nonzero eigenvalues that correspond to linearly independent eigenvectors. This is $\sum_{m: F(a,n,m)\ne 0 } { n \choose m }.$ The theorem follows from Lemmas \[le:rank\] and \[le:qrank\].
\[le:lea\] The number of values of $m$ for which $F(a,n,m)=0$ is $\le a$.
View the double summation $F(a,n,m)$ as a polynomial in $m$. The $j$th summand has degree $k+(j-k)=j$. Since $j\le a$ the entire sum can be written as a polynomial in $m$ of degree $a$. This has at most $a$ roots.
\[th:main\] There is a constant $c > 0$ such that if $a\le c\sqrt n$ then the following hold.
1. $D({HAM_n^{(a)}}) \ge n$.
2. $Q^*({HAM_n^{(a)}}) \ge n/2$.
3. $C^*({HAM_n^{(a)}})\ge n$.
By Lemma \[le:uslea\] $D(f),Q^*(f) \ge \lg (\sum_{m: F(a,n,m)\ne 0 } { n \choose m })$ and $C^*(f) \ge \frac{1}{2}\lg (\sum_{m: F(a,n,m)\ne 0 } { n \choose m })$.
Note that $$2^n = \sum_{m: F(a,n,m)\ne 0 } { n \choose m } +\sum_{m: F(a,n,m)=0 } { n\choose m}.$$ By Lemma \[le:lea\] $|\{ m : F(a,n,m)=0 \}|\le a$. Hence, $$\sum_{m: F(a,n,m)=0 } { n\choose m} \le |\{ m : F(a,n,m)=0 \}|\cdot\max_{0\le m\le n} {n\choose m}
\le a{ n \choose {n/2} } \le \frac{a2^n}{\sqrt n}.$$ So, if $a\le \frac{1}{4}\sqrt n$, then $$\sum_{m: F(a,n,m)\ne 0 } { n \choose m } \ge 2^n - \frac{a2^n}{\sqrt n}
\geq 2^n - 2^{n-2}.$$ Hence, $$\lg \left(\sum_{m: F(a,n,m)\ne 0 } { n \choose m }\right) \ge
\lg(2^n - 2^{n-2}); ~~
\mathrm{i.e.}, ~
\left\lceil \lg \left(\sum_{m: F(a,n,m)\ne 0 } { n \choose m }\right)
\right\rceil \ge n.$$
The Complexity of ${HAM_n^{(=a)}}$ for $a\le O(\sqrt{n})$ {#se:hameasq}
=========================================================
We again start by deducing results for general $a$, and then specialize to the case where $a \leq c \sqrt{n}$.
Let $M_{=a}$ be $M_{{HAM_n^{(=a)}}}$, the $2^n\times 2^n$ matrix representing ${HAM_n^{(=a)}}$.
The vectors $v_z$ are the same ones defined in Definition \[de:vz\]. We show that $v_z$ is an eigenvector of $M$. Once that is done we let ${eig}(z)$ be the eigenvalue of $M$ associated to $z$.
The lemmas needed, and the final theorem, are very similar (in fact easier) to those in the prior section. Hence we just state the needed lemmas and final theorem.
\[le:maina\]
1. For all $z\in {\{0,1\}^{{n}}}$ $v_z$ is an eigenvector of $M_{=a}$.
2. If $z$ has exactly $m$ 1’s in it then $$eig(z)={\ \ \sum_{k=\max\{ 0,a+m-n \} }^{\min\{a,m\}}\ \ }{m\choose k} {n-m \choose a-k} (-1)^k.$$
\[defn:f\] $$f(a,n,m)= {\ \ \sum_{k=\max\{ 0,a+m-n \} }^{\min\{a,m\}}\ \ }{m\choose k} {n-m \choose a-k} (-1)^k.$$
Note, from our convention that “if $x < y$, then ${x \choose y}$ is taken to be zero”, that we can also write $$f(a,n,m)= \sum_{k=0}^{a} {m\choose k} {n-m \choose a-k} (-1)^k.$$
The following lemma will be used in this section to obtain a lower bound when $a=O(\sqrt n)$, and in Section \[se:gen\] to obtain a lower bound for general $a$.
\[le:useqa\]
1. $D({HAM_n^{(=a)}}) \ge \lg \sum_{m: f(a,n,m)\ne 0 } { n \choose m }.$
2. $Q^*({HAM_n^{(=a)}}) \ge \lg \sum_{m: f(a,n,m)\ne 0 } { n \choose m }.$
3. $C^*({HAM_n^{(=a)}}) \ge \frac{1}{2} \cdot \lg \sum_{m: f(a,n,m)\ne 0 } { n \choose m }.$
\[le:ea\] The number of values of $m$ for which $f(a,n,m)=0$ is $\le a$.
\[th:maine\] There is a constant $c > 0$ such that if $a\le c\sqrt n$ then the following hold.
1. $D({HAM_n^{(=a)}}) \ge n$.
2. $Q^*({HAM_n^{(=a)}}) \ge n/2$.
3. $C^*({HAM_n^{(=a)}})\ge n$.
The Complexity of ${HAM_n^{(a)}}$ and ${HAM_n^{(=a)}}$ for General $a$ {#se:gen}
======================================================================
We now consider the case of general $a$. As above, we will show that $F(a,m,n)$ and $f(a,m,n)$ are nonzero for many values of $m$. This will imply that the matrices $M_a$ and $M_{=a}$ have high rank, hence ${HAM_n^{(a)}}$ and ${HAM_n^{(=a)}}$ have high communication complexity. We will use general generating-function methods to derive facts about these sums. A good source on generating functions is [@wilfgen].
One of our main results will be Lemma \[le:singlesum\], which states that if $0 \leq a \leq m < n$, then “$f(a,m,n)=0$” implies “$f(a,m+1,n) \neq 0$”. The idea behind our proof of Lemma \[le:singlesum\] will be the following: we will show a relationship between the sum $f(a,m,n)$ and a certain new sum $h(a,m,n)$. Then we will derive generating functions for $f$ and $h$, and translate this relationship into a relation between their generating functions. Finally, we will show that this relation cannot hold under the assumption that $f(a,m,n)=f(a,m+1,n)=0$, thus reaching a contradiction. Some auxiliary results needed for this are now developed in Section \[sec:aux\].
Auxiliary Notation and Results {#sec:aux}
------------------------------
$[x^b]g(x)$ is the coefficient of $x^b$ in the power series expansion of $g(x)$ around $x_0=0$.
$t^{(i)}(x)$ is the $i$’th derivative of $t(x)$.
We will make use of the following lemma, which follows by an easy induction on $i$:
\[lem:deriv\] Let $t(x)$ be an infinitely differentiable function. Let $T_1(x)=(x-1)t(x)$, and
$T_2(x)=(x+1)t(x)$. Then for any $i\geq 1$:\
$T_1^{(i)}(x)=(x-1)t^{(i)} + i\cdot t^{(i-1)}(x)$\
$T_2^{(i)}(x)=(x+1)t^{(i)} + i\cdot t^{(i-1)}(x)$
For the rest of Section \[sec:aux\], the integers $a, m, n$ are arbitrary subject to the constraint $0 \leq a \leq m \leq n$, unless specified otherwise.
1. $h(a,m,n)=\sum_{i=0}^{a} {m \choose i}{n-m \choose a-i}\frac{(-1)^i}{m-i+1}$.
2. $g(x)=\frac{x^{m+1} - (x-1)^{m+1}}{m+1}\cdot (x+1)^{n-m}$.
We will show an interesting connection between $h$ and $f$.
\[cl:one\] Suppose $f(a,m,n)=0$. Then $f(a,m+1,n)=0$ iff $h(a,m,n)=0$.
$$\begin{array}{rl}
f(a,m+1,n)=&\sum_{i=0}^{a} {m+1 \choose i}{n-m-1 \choose a-i}(-1)^i\cr
=&\frac{m+1}{n-m}\sum_{i=0}^{a} {m \choose i}{n-m \choose a-i}(-1)^i
\cdot \frac{n-m-a+i}{m-i+1}\cr
=&\frac{m+1}{n-m}((n+1-a)\sum_{i=0}^{a} {m \choose i}{n-m \choose a-i}\frac{(-1)^i}{m-i+1})-\sum_{i=0}^{a} {m \choose i}{n-m \choose a-i}(-1)^i)\cr
=&\frac{m+1}{n-m}((n+1-a)h(a,m,n)-f(a,m,n))\cr
\end{array}$$ Thus, if $f(a,m,n)=0$, then $f(a,m+1,n)=0$ iff $h(a,m,n)=0$.
We next show a connection between $g(x)$ and $h$.
\[cl:two\] $h(a,m,n)=(-1)^m\cdot [x^a]g(x)$.
$$\begin{array}{rl}
g(x)=&\frac{x^{m+1} - (x-1)^{m+1}}{m+1}\cdot (x+1)^{n-m}\cr
=&\frac{x^{m+1} - \sum_{i=0}^{m+1}{m+1 \choose i}x^i(-1)^{m+1-i}}{m+1}\cdot (x+1)^{n-m}\cr
=&(-1)^m\sum_{i=0}^{m}{m \choose i}x^i\frac{(-1)^i}{m+1-i}\cdot (x+1)^{n-m}\cr
=&(-1)^m\sum_{i=0}^{m}{m \choose i}x^i\frac{(-1)^i}{m+1-i}\cdot\sum_{j=0}^{n-m}{n-m \choose j}x^j\cr
\end{array}$$ Therefore, $h(a,m,n)=(-1)^m\cdot [x^a]g(x)$.
Next, define an auxiliary function $\phi(u,v,w)$ as the $w$’th derivative of the function $(x+1)^u(x-1)^v$ evaluated at $x=0$. We now relate $\phi$ and $h$.
\[cl:three\] $h(a,m,n)=0$ iff $\phi(n-m, m+1, a) = 0$.
By Claim \[cl:two\]
$$\begin{array}{rl}
h(a,m,n)=&(-1)^m\cdot [x^a]g(x)\cr
=&\frac{(-1)^m}{m+1}([x^a](x^{m+1}\cdot (x+1)^{n-m})-[x^a]((x-1)^{m+1}\cdot (x+1)^{n-m})).\cr
\end{array}$$
But $[x^a](x^{m+1}\cdot (x+1)^{n-m})=0$, since $a<m+1$. So
$$\begin{array}{rl}
h(a,m,n)=&\frac{(-1)^{m+1}}{m+1}[x^a]((x-1)^{m+1}\cdot (x+1)^{n-m})\cr
=&\frac{(-1)^{m+1}}{m+1}\cdot\frac{\phi(n-m, m+1, a)}{a!}.\cr
\end{array}$$
Thus, $h(a,m,n)=0$ iff $\phi(n-m, m+1, a) = 0$.
Now we can relate the zeroes of $f$ with those of $\phi$:
\[cl:four\] $f(a,m,n)=0$ iff $\phi(n-m, m, a)=0$.
$$\begin{array}{rl}
(x-1)^m(x+1)^{n-m}=&\sum_{i=0}^m {m \choose i}x^i(-1)^{m-i} \cdot \sum_{j=0}^{n-m}{n-m \choose j}x^j\cr
=&(-1)^m\sum_{i=0}^m {m \choose i}x^i(-1)^i \cdot \sum_{j=0}^{n-m}{n-m \choose j}x^j\cr
=&(-1)^m\sum_{b=0}^{n}\sum_{k=0}^b {m \choose k}{n-m \choose b-k}(-1)^k x^b\cr
=&(-1)^m\sum_{b=0}^{n} f(b,m,n)\cdot x^b.\cr
\end{array}$$
So $f(a,m,n)=\frac{(-1)^m}{a!}\cdot\phi(n-m, m, a)$, thus $f(a,m,n)=0$ iff $\phi(n-m, m, a)=0$.
\[cl:five\] Suppose $m < n$ and $\phi(n-m, m, a)=0$. Then
$$\phi(n-m-1, m+1, a)=0 \hbox{ iff } \phi(n-m, m+1, a)=0.$$
This claim follows from Claims \[cl:one\], \[cl:three\], and \[cl:four\].
We are now able to prove a recursive relation between values of $\phi$:
\[cl:six\] If $k>0$, $a > 0$, and $\phi(k, m, a)=\phi(k,m,a-1)=0$, then
$\phi(k-1, m, a)=\phi(k-1,m,a-1)=0$.
Suppose $\phi(k, m, a)=\phi(k,m,a-1)=0$. By Lemma \[lem:deriv\], $$\label{eqn:phi-k-m+1}
\phi(k,m+1,a)=-\phi(k,m,a)+a\cdot\phi(k,m,a-1)=0.$$
By Claim \[cl:five\], since $\phi(k, m, a)=0$, we know that $$\phi(k-1,m+1,a)=0 \hbox{ iff }\phi(k,m+1,a)=0.$$ Now, (\[eqn:phi-k-m+1\]) yields $\phi(k-1,m+1,a)=0$. Applying Lemma \[lem:deriv\] again, we obtain:
$$\begin{array}{rl}
0 = & \phi(k-1,m+1,a)=-\phi(k-1,m,a)+ a \cdot\phi(k-1,m,a-1); \cr
0 =& \phi(k,m,a)=\phi(k-1,m,a)+ a \cdot \phi(k-1,m,a-1)\cr
\end{array}$$ Solving the equations, we get $$\phi(k-1,m,a) = \phi(k-1,m,a-1)=0.$$ Thus the claim is proved.
The main results {#sec:main-results}
----------------
We are now ready to prove our main lemma.
\[le:singlesum\] Let $0 \leq a \leq m < n$, and suppose $f(a,m,n)=0$. Then $f(a,m+1,n) \neq 0$.
The lemma holds trivially for $a = 0$, since both $f(a,m,n)$ and $f(a,m+1,n)$ are nonzero if $a = 0$. So suppose $a \geq 1$. Suppose $f(a,m,n)=f(a,m+1,n)=0$. Then by Claims \[cl:four\] and \[cl:five\], we know that $$\phi(n-m, m, a)=\phi(n-m-1, m+1, a)=\phi(n-m, m+1, a)=0.$$ By Lemma \[lem:deriv\], $$\phi(n-m, m+1, a)= -\phi(n-m,m,a)+a\cdot\phi(n-m,m,a-1),$$ i.e., $\phi(n-m,m,a-1) = 0$. Hence $\phi(n-m,m,a-1)= \phi(n-m,m,a)=0$. Now, an iterative application of Claim \[cl:six\] eventually yields $\phi(0,m,a)=\phi(0,m,a-1)=0$. By definition, $\phi(0,m,a)$ is the $a$’th derivative of $$(x-1)^m=\sum_{i=0}^{m} {m \choose i}x^i(-1)^{m-i}$$ evaluated at $x=0$. But $m \geq a$, so this is clearly not zero. Thus we have reached a contradiction, and Lemma \[le:singlesum\] is proved.
\[thm:eqcomplexity\] For large enough $n$ and all $0 \leq a \leq n$ the following hold.
1. $D({HAM_n^{(=a)}}) \geq n-2$.
2. $Q^*({HAM_n^{(=a)}})\geq \frac{n}{2}-1$.
3. $C^*({HAM_n^{(=a)}})\ge n-2$.
By Lemma \[le:useqa\], $$D(f), C^*(f) \ge \lg(\sum_{m: f(a,m,n)\ne 0} { n \choose m })$$ and $$Q^*(f) \ge \frac{1}{2}\lg(\sum_{m: f(a,m,n)\ne 0} { n \choose m }).$$
First suppose $a \leq n/2$. We have $$\label{eqn:small-a-large-m}
\sum_{m: f(a,m,n)\ne 0} {n \choose m} \geq
\sum_{m \geq n/2: f(a,m,n)\ne 0} {n \choose m}.$$ Let us lower-bound the r.h.s. of (\[eqn:small-a-large-m\]). First of all, since the r.h.s. of (\[eqn:small-a-large-m\]) works in the regime where $m \geq n/2 \geq a$, Lemma \[le:singlesum\] shows that no two consecutive values of $m$ in this range satisfy the condition “$f(a,m,n) = 0$”. Also, for $m \geq n/2$, ${n \choose m}$ is a non-increasing function of $m$. Thus, if we imagine an adversary whose task is to keep the r.h.s. of (\[eqn:small-a-large-m\]) as small as possible, the adversary’s best strategy, in our regime where $m \geq n/2$, is to make $f(a,m,n) = 0$ exactly when $m \in S$, where $$\label{eqn:S}
S \doteq \{\lceil n/2 \rceil, \lceil n/2 \rceil + 2,
\lceil n/2 \rceil + 4, \ldots \}.$$
Now, $$\label{eqn:m-large}
2^{n-1} \leq \sum_{m \geq n/2} {n \choose m} \leq
2^{n-1} + O(2^n / \sqrt{n}).$$ (We need the second inequality to handle the case where $n$ is even.) Also, recall that an $(1 - o(1))$ fraction of the sum $\sum_{m \geq n/2} {n \choose m}$ is obtained from the range $n/2 \leq m \leq n/2 + \sqrt{n \log n}$, for instance. (Here and in what follows, “$o(1)$” denotes a function of $n$ that goes to zero as $n$ increases.) In this range, the values of ${n \choose m}$ for any two consecutive values of $m$ are within $(1 + o(1))$ of each other. In conjunction with (\[eqn:m-large\]), this shows that $$\sum_{m \geq n/2: f(a,m,n)\ne 0} {n \choose m}
\geq \sum_{m \geq n/2: m \not\in S} {n \choose m} \geq
(1/2 - o(1)) 2^{n-1}.$$ Thus, $$\left\lceil \lg\left(\sum_{m \geq n/2: f(a,m,n)\ne 0} {n \choose m}\right)
\right\rceil \geq n-2,$$ completing the proof for the case where $a \leq n/2$.
Now we apply symmetry to the case $a> n/2$: note that Alice can reduce the problem with parameter $a$ to the problem with parameter $n - a$, simply by complementing each bit of her input $x$. Thus, the same communication complexity results hold for the case $a > n/2$.
\[le:zero\] Let $0 \leq a < m < n$, and suppose $F(a,m,n)=0$. Then $F(a, m+1, n) \neq 0$.
We have $f(j,m,n)=(-1)^m[x^j]((x-1)^m(x+1)^{n-m})$. By definition, $$\begin{array}{rl}
F(a,m,n)=&\sum_{j=0}^a f(j,m,n)\cr
=&(-1)^m \sum_{j=0}^a [x^j]((x-1)^m(x+1)^{n-m})\cr
=&(-1)^m [x^a]((x-1)^m(x+1)^{n-m}\cdot\sum_{j=0}^{\infty}x^j)\cr
=&(-1)^m [x^a]((x-1)^m(x+1)^{n-m}\cdot\frac{1}{1-x})\cr
=&(-1)^{m-1} [x^a]((x-1)^{m-1}(x+1)^{n-m})=f(a,m-1,n-1).\cr
\end{array}$$ So $F(a,m,n)=F(a,m+1,n)=0$ iff $f(a,m-1,n-1)=f(a,m,n-1)=0$. But the latter is impossible by Lemma \[le:singlesum\], thus the lemma is proved.
For large enough $n$ and all $0 \leq a \leq n-1$, the following hold.
1. $D({HAM_n^{(a)}}) \geq n-2$.
2. $Q^*({HAM_n^{(a)}}) \geq \frac{n}{2}-1$.
3. $C^*({HAM_n^{(a)}}) \geq n-2$.
The proof is identical to that of Theorem \[thm:eqcomplexity\] except for one point. In that proof we obtained the $a> n/2$ case easily from the $a\le n/2$ case. Here it is also easy but needs a different proof. Let $a> n/2$ and, for all $x\in {\{0,1\}^{{n}}}$, let ${\overline{x}}$ be obtained from $x$ by flipping every single bit. Note that
${HAM_n^{(a)}}(x,y)=1$ iff ${{\rm HAM}}(x,y)\le a$ iff ${{\rm HAM}}({\overline{x}},y)\ge n-a$ iff NOT(${{\rm HAM}}({\overline{x}},y)\le (n-a)-1$ iff ${{\rm HAM}}_{n-a-1}({\overline{x}},y)=1$.
Since $n-a-1 \le n/2$ we have that a lower bound for the $a\le n/2$ case implies a lower bound for the $a>n/2$ case.
Open Problems
=============
We make the following conjectures.
1. For all $n$, for all $a$, $0\le a\le n-1$, $D({HAM_n^{(a)}}), C^*({HAM_n^{(a)}}), Q^*({HAM_n^{(a)}}) \ge n+1$.
2. For all $n$, for all $a$, $0\le a\le n$, $D({HAM_n^{(=a)}}), C^*({HAM_n^{(=a)}}), Q^*({HAM_n^{(=a)}}) \ge n+1$.
[10]{}
K. Abdel-Ghaffar and A. E. Ababdi. An optimal strategy for comparing file copies. *IEEE Transactions on Parallel and Distributed Systems*, 5:87–93, 1994.
H. Buhrman and R. de Wolf. Communication complexity lower bounds by polynomials. In *Proc. of the 16th IEEE Conf on Complexity Theory*. IEEE Computer Society Press, 2001.
G. Cormode, M. Paterson, S. Sahinalp, and U. Vishkin. Communication complexity of document exchange. In *Proc. of the 11th ACM Sym. on Discrete Algorithms*, 2000.
R. de Wolf. Quantum communication and complexity. *Theoretical Comput. Sci.*, 12:337–353, 2002.
J. Feigenbaum, Y. Ishai, T. Malkin, K. Nissim, M. Strauss, and R. Wright. Secure multiparty computation of approximations. In *Proc. of the 28th ICALP (LNCS 2076)*, volume 2076 of [ *Lecture Notes in Computer Science*]{}, pages 927–938, Berlin, 2001. Springer-Verlag.
D. Gavinsky, J. Kempe, and R. de Wolf. Quantum communication cannot simulate a public coin, 2004. arxiv.org/abs/quant-ph/0411051.
E. Kushilevitz and N. Nisan. *Communication Complexity*. Cambridge University Press, 1997.
K. Mehlhorn and E. Schmidt. Las [V]{}egas is better than determinism for [VLSI]{} and distributed systems. In *Proc. of the 14th ACM Sym. on Theory of Computing*, pages 330–337, 1982.
J. Metzner. Efficient replicated remote file comparison. *IEEE Transactions on Computers*, 40:651–659, 1991.
I. Newman. Private vs. common random bits in communication complexity. *Inf. Process. Lett.*, 39:67–71, 1991.
A. Orlitsky. Interactive communication: balanced distributions, correlated files, and average-case complexity. In *Proc. of the 32st IEEE Sym. on Found. of Comp. Sci.*, pages 228–238, 1991.
K. Pang and A. E. Gamal. Communication complexity of computing the [H]{}amming distance. *SIAM Journal of Computing*, 15, 1986.
H. Wilf. *Generatingfunctionology*. Academic Press, 1994.
A. Yao. On the power of quantum fingerprinting. In *Proc. of the 35th ACM Sym. on Theory of Computing*, 2003.
[^1]: University of Waterloo, Dept. of Combinatorics and Optimization and Instuitute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L 3G1 `ambainis@uwaterloo.ca`, Partially supported by IQC University Professorship and CIAR.
[^2]: University of Maryland, Dept. of Computer Science and Institute for Advanced Computer Studies, College Park, MD 20742. `gasarch@cs.umd.edu`, Partially supported by NSF grant CCR-01-05413.
[^3]: University of Maryland, Dept. of Computer Science and Institute for Advanced Computer Studies, College Park, MD 20742. `srin@cs.umd.edu`, Partially supported by NSF grant CCR-020-8005.
[^4]: University of Maryland, Dept. of Computer Science, College Park, MD 20742. `utis@cs.umd.edu`
|
---
abstract: 'We report the magnetic and calorimetric measurements in [*single crystal*]{} samples of the square lattice $J_{1}-J_{2}$ quantum antiferromagnet [BaCdVO(PO$_4$)$_2$]{}. An investigation of the scaling of magnetization reveals a “dimensionality reduction” indicative of a strong degree of geometric frustration. Below a characteristic temperature of $T^{\ast}\simeq150$ mK we observe the emergence of an additional strongly fluctuating quantum phase close to full magnetic saturation. It is separated from the magnetically ordered state by first- and second-order phase transitions, depending on the orientation of the applied magnetic field. We suggest that this phase may indeed be related to the theoretically predicted spin-nematic state.'
author:
- 'K. Yu. Povarov'
- 'V. K. Bhartiya'
- 'Z. Yan'
- 'A. Zheludev'
bibliography:
- 'd:/The\_Library.bib'
title: Thermodynamics of a frustrated quantum magnet on a square lattice
---
Introduction
============
The quest for the so-called spin-nematic state in magnetic insulators started more than three decades ago, but continues to this day [@AndreevGrishchuk_JETP_1984_SpinNematics; @Chubukov_PRB_1991_Chains; @ShannonMomoi_PRL_2006_J1J2squarecircle; @ZhitomirskyTsunetsugu_EPL_2010_nematic; @ButtgenNawa_PRB_2014_LiCuVO4yetanothernematic; @Orlova_PRL_2017_moreNematicLiCuVO4]. This exotic magnetic order spontaneously breaks rotational symmetry, while keeping time-reversal symmetry intact. It can be understood as a quantum condensate of bound magnon pairs [@Chubukov_PRB_1991_Chains; @ShannonMomoi_PRL_2006_J1J2squarecircle; @ZhitomirskyTsunetsugu_EPL_2010_nematic]. The key characteristics of any potential host system are competing ferro- (FM) and antiferromagnetic (AF) interactions and extreme quantum fluctuations. The baseline model is the $S=1/2$ square lattice Heisenberg Hamiltonian with FM nearest-neighbor exchange $J_1$ and AF next-nearest-neighbor coupling $J_2$ [@Shannon_EPJB_2004_GenericJ1J2theory; @ShannonMomoi_PRL_2006_J1J2squarecircle; @ShindouMomoi_PRB_2009_J1J2nematicSlavebosons; @Shindou_PRB_2011_J1J2nematicProjective; @SchmidtThalmeier_PhysRep_2017_2Dfrustratedreview] sketched in Fig. \[FIG:Magnetic\](a) alongside its phase diagram. The classical critical point at $J_{2}/J_{1}=-1/2$ separates FM and columnar-AF states, but becomes destabilized by quantum fluctuations and is replaced by a novel region in its vicinity. The resulting state can be understood as a magnon bound pair condensate occurring at zero field — the spin-nematic [@ShannonMomoi_PRL_2006_J1J2squarecircle; @Shindou_PRB_2011_J1J2nematicProjective]. Even outside the narrow $J_{2}/J_{1}$ parameter range in which spin nematic is supposed to exist at zero field, this state can be further stabilized in a magnetized system. Magnon pair condensation and hence spin nematicity can be induced by an external magnetic field close to the saturation point. This result turns out to hold well away from optimal parameter set $J_{2}/J_{1}=-1/2$ and even in the presence of additional couplings in the Hamiltonian [@ShannonMomoi_PRL_2006_J1J2squarecircle; @Ueda_JPSJ_2015_NematicInField].
![(a) The frustrated $S=1/2$ Heisenberg square lattice model and typical “circular” representation of its ground state as a function of the ferromagnetic $J_{1}$ to antiferromagnetic $J_{2}$ exchange ratio. The positions of [BaCdVO(PO$_4$)$_2$]{} as well as of few other similarly structured materials are shown (after Ref. [@NathTsirlin_PRB_2008_BaCdVO(PO4)2]). (b) Magnetic susceptibilities along the different directions of [BaCdVO(PO$_4$)$_2$]{} crystal, scaled with their $g-$factors. (c) Isothermal magnetization along different directions at $T=0.55$ K. The inset shows the region around $\mu_{0}H_{\text{SF}}\simeq0.5$ T in more detail. The dashed lines show the estimate for $M_{\text{sat}}=g\mu_{B}/2$ value. All the magnetic data is corrected for the diamagnetic background. []{data-label="FIG:Magnetic"}](Magnetic.pdf "fig:"){width="50.00000%"}\
Despite the vast body of theoretical work, experimentally the spin nematic state on a frustrated square lattice remains elusive. One obvious problem is that the tensorial order parameter is invisible to most conventional magnetism probes. What is an even bigger obstacle, is that potential model compounds are few and hard to synthesize. The most promising known candidate is [BaCdVO(PO$_4$)$_2$]{} [@NathTsirlin_PRB_2008_BaCdVO(PO4)2; @Tsirlin_PRB_2009_FSQLmagnetization]. The applicability of the $J_{1}-J_{2}$ model to this compound has been validated by density functional theory calculations [@TsirlinRosner_PRB_2009_FSQLvanadatesSummary]. The material features strong geometric frustration ($J_{2}/J_{1}\simeq-0.9$) and easily accessible energy scales (saturation field about $4$ T, AF order below $T_{N}=1.05$ K). The high-temperature thermodynamics is consistently described by $J_{1}=-3.6$ K and $J_{2}=3.2$ K [@NathTsirlin_PRB_2008_BaCdVO(PO4)2]. For lack of other candidates, [BaCdVO(PO$_4$)$_2$]{} has been a subject of intense theoretical studies, including specific predictions for inelastic neutron scattering [@Smerald_PRB_2015_INSnematic] and nuclear magnetic resonance [@SmeraldShannon_PRB_2016_nematicNMR]. Disappointingly, a lack of single crystals has severely impeded experimental studies. To date, no empirical evidence of a spin-nematic phase or any related unconventional magnetism has been reported in this material.
In the present paper we describe the unusual magnetic and thermodynamic properties of *single-crystal* samples of [BaCdVO(PO$_4$)$_2$]{}. We map out the anisotropic magnetic phase diagram and study the “dimensionality reduction” and peculiar scaling of magnetization near the field-induced quantum phase transition. We accomplish this by employing magnetization, specific heat, and the magnetocaloric effect studies. In what may be the first sign of spin nematicity, we report evidence of an additional low-temperature field-induced strongly fluctuating quantum regime just below saturation. In an axially symmetric geometry the new state emerges in a first-order transition, and is preceded by substantial precursor transverse fluctuations in the magnetically ordered state.
Experiment details
==================
Material
--------
High-quality single crystals of [BaCdVO(PO$_4$)$_2$]{} were grown using the self-flux Bridgman method from the melt of presynthesized BaCdP$_2$O$_7$ and vanadium dioxide at $1000^{\circ}$ C. The details of the method will be published elsewhere. The crystal structure \[orthorhombic $P_{bca}$ ($D^{15}_{2h}$, No $61$), $a=8.84$, $b=8.92$, $c=19.37$ Å\] was validated using single-crystal x-ray diffraction on a Bruker APEX-II instrument, and found to be totally consistent with that reported previously [@Meyer_ZNatur_1997_CrystVanadates].
![A single 16-mg crystal sample of [BaCdVO(PO$_4$)$_2$]{} used in magnetization measurements. A snapshot from a Bruker APEX II single-crystal x-ray diffractometer.[]{data-label="FIG:xtalpic"}](SuppFigs_xtal.pdf "fig:"){width="30.00000%"}\
The single crystals of [BaCdVO(PO$_4$)$_2$]{} have the appearance of green transparent square plates. The plate corresponds to the crystallographic $ab$ plane, with the directions of $a$ and $b$ axes being usually along the diagonals. Correspondingly, the $c$ axis is normal to the plane. One of the crystals from the present study is shown in Fig. \[FIG:xtalpic\].
We would like to note that the highly symmetric shape of the samples ensures that no crystal morphology-related effects can be expected to make a difference between $\mathbf{H}\parallel\mathbf{a}$ and $\mathbf{H}\parallel\mathbf{b}$ configurations.
Techniques
----------
The magnetization measurements were performed with the $7$ T SQUID magnetometer \[Quantum Design Magnetic Property Measurement System (MPMS)\] in the temperature range $1.8-300$ K. Further extension to the temperatures of about $0.5$ K was achieved with the help of $^3$He cryostat inset iQuantum iHelium3. The magnetic susceptibility $\chi=M/H$ was measured at a small field $0.1$ T. The crystal shown in Fig. \[FIG:xtalpic\] was used in all the magnetic measurements.
Specific heat measurements were carried out on a standard Quantum Design relaxation calorimetry option and the $^3$He-$^4$He dilution refrigerator inset for Quantum Design Physical Properties Measurement System (PPMS). Two measurement geometries $\mathbf{H}\parallel\mathbf{a}$ and $\mathbf{H}\parallel\mathbf{b}$ were realized by mounting a $2.3$-mg flat single-crystal sample on a small silver foil holder with Apiezon N grease. The measurement procedure consists of giving a gentle heat pulse to the sample platform. Then the temperature rise is observed during the pulse, and subsequent temperature fall is observed as the heater is turned off. The resulting $T(t)$ curve typically has a characteristic “shark fin” shape, and specific heat can be calculated from the curvature.
The magnetocaloric effect measurements were performed in the same setup by directly reading the resistivity of the sample thermometer as the function of slowly varying magnetic field. It was done either with an Agilent E4980A LCR meter, or with a Stanford Research SR830 lock-in amplifier.
Results and discussion
======================
Magnetization studies
---------------------
### Susceptibility above $T_N$
Above $T_N$ the susceptibilities \[Fig. \[FIG:Magnetic\](b)\] show qualitatively identical behavior: typical Curie–Weiss tail at high temperatures, followed by a rounded maximum at $T\simeq2.5$ K and then gradual decrease down to $T_{N}=1.05$ K marked by a kink.
The Curie–Weiss part of the susceptibility curve at high temperatures can be used for accurate determination of the $g$ factors and the diamagnetic background:
$$\label{EQ:CWfit}
\chi_{\alpha}(T)=\chi_{\alpha}^{0}+\frac{(g_{\alpha}/2)^{2}C}{T+\Theta},$$
where $C=0.375$ K emu/mol is the Curie constant for the $S=1/2$ case with $g=2.00$. The analysis given by Eq. (\[EQ:CWfit\]) was performed in a temperature window between $30$ and $200$ K. The Curie–Weiss temperature is found to be $\Theta=-0.90(2)$ K; we have enforced the equal value for all three directions. The values of $g$ factors and diamagnetic background susceptibilities $\chi_{\alpha}^{0}$ are summarized in Table \[TAB:CWfit\]. The obtained $g-$factor values are rather isotropic and consistent with powder EPR estimates [@Foerster_2011_PhDthesis]. In Fig. \[FIG:Magnetic\](b) one can see that background subtracted susceptibilities normalized by $g^2$ show a perfect overlap in absence of magnetic order.
Direction $g_\alpha$ $\chi^{0}_{\alpha}$ (emu/mol)
-------------- -- ------------ -- -------------------------------
$\mathbf{a}$ 1.95(1) $-1.79(2)\times10^{-4}$
$\mathbf{b}$ 1.97(1) $-2.45(2)\times10^{-4}$
$\mathbf{c}$ 1.92(1) $-2.32(2)\times10^{-4}$
: Results of the Curie–Weiss analysis of the high-temperature susceptibility.[]{data-label="TAB:CWfit"}
### Easy-axis anisotropy
Below the Néel temperature $T_{N}=1.05$ K the susceptibilities shown in Fig. \[FIG:Magnetic\](b) start to show rather different behavior. At low temperatures $\chi_{b}(T)$ and $\chi_{c}(T)$ remain more or less constant, while $\chi_{a}(T)$ shows a rapid decrease upon cooling. This suggests a collinear magnetic structure with spins along the $\mathbf{a}$ axis. This interpretation is backed by isothermal magnetization $M(H)$ scans at $T=0.55$ K \[Fig. \[FIG:Magnetic\](c)\]. For the $\mathbf{H}\parallel
\mathbf{a}$ case (and only for that geometry) there is a pronounced magnetization jump around $\mu_{0}H_{\text{SF}}\simeq0.5$ T. This behavior is characteristic of a spin-flop transition driven by the weak Ising-like anisotropy, $\mathbf{a}$ being the magnetic easy axis. Thus, using a simple Heisenberg model to describe the system can only be done with caution. Below we shall refer to experiments with $\mathbf{H}\parallel\mathbf{a}$ as the *axial* geometry, and to those with a field in perpendicular directions as *transverse*.
### Convex shape and “dimensionality reduction”
![Scaling of magnetization, observed near the saturation field ($\mathbf{H}\parallel\mathbf{a}$ case). (a) Raw data, taken in the interval $0.5-4$ K. (b) Same data, scaled according to Eq. (\[EQ:MHscaling\]). The observed exponents are $1/\varphi=1.6(3)$ and $m=0.8(1)$. The inset shows the empirical $\chi^2$ goodness of overlap with highlighted boundary at which the optimal value increases by 50%.[]{data-label="FIG:Mscale"}](Mscale.pdf "fig:"){width="50.00000%"}\
The most striking feature of the measured $M(H)$ curves is their extreme convex shape close to the saturation. As known from the numerical studies of the $J_{1}-J_{2}$ model [@ThalmeierZhitomirsky_PRB_2008_J1J2magnetization; @Tsirlin_PRB_2009_FSQLmagnetization], it serves as a reliable indicator of the significant magnetic frustration, indirectly confirming the nearly critical positioning of [BaCdVO(PO$_4$)$_2$]{} on the Fig. \[FIG:Magnetic\](b) phase diagram.
We note that the measured convex magnetization curve is reminiscent of the cusp singularity occurring at saturation in the AF spin chains [@JeongRonnow_PRB_2015_CupzNscaling; @Breunig_SciAdv_2017_CupzNscaling]. This feature — square root cusp at the saturation magnetization $M_{\text{sat}}-M(H)\propto\sqrt{H_{c}-H}$ — is endemic to one dimension, yet it appears in our essentially two-dimensional (2D) material. This is another signature of the frustration, known as the “dimensionality reduction”. For example, a similar effect is responsible for low-temperature crossover to effectively 2D behavior in a nominally 3D material BaCuSi$_2$O$_6$ (“Han purple”) [@SebastianHarrison_Nature_2006_HanPurpleDimRed]. In the case of the frustrated square lattice a qualitative prediction is given by Jackeli and Zhitomirsky [@JackeliZhitomirsky_PRL_2004_FrustratedBEC]: close to the points of perfect frustration $|J_{2}/J_{1}|=1/2$ 1D-like behavior with a square root magnetization cusp is indeed present in a 2D material at saturation. A simple explanation is, close to $H_c$ the low-energy part of the spin-wave spectrum defining the low-$T$ behavior features a *continuous circle of degenerate minima* as the result of strong frustration. This effectively reduced the problem to a one-dimensional one, rendering the low-energy spectrum as being pseudo-1D [@JackeliZhitomirsky_PRL_2004_FrustratedBEC].
Thus, verifying the zero temperature $M_{\text{sat}}-M(H)\propto\sqrt{H_{c}-H}$ prediction would be a strong signature of nearly critical $J_{2}/J_{1}$ coupling ratio in [BaCdVO(PO$_4$)$_2$]{}. However, in a realistic experiment we are dealing with the finite temperatures that make the cusp rounded and hide away the associated power law. Below we will show that by considering the quantum critical behavior of longitudinal magnetization close to $H_c$ it is nonetheless possible to relate the “hidden” zero temperature cusp to the available finite temperature data.
### Quantum critical scaling: Theory
The basic assumption that we need to make is that the hyperscaling holds at the quantum critical point. This would be the case if the “dimensionality reduction” scenario takes place indeed. Then in the vicinity of the transition the free energy can be expressed as
$$\label{EQ:FscalingSM}
F(T,H)=\lambda^{b}\mathcal{F}\left[\lambda^{z}T,\lambda^{1/\nu}(H-H_{c})\right]+F_{0}(T,H).$$
Here $\mathcal{F}(x,y)$ is some *a priori* unknown function of two variables and $\lambda$ is an arbitrary positive number. This term reflects the singular part of the free energy. We do not even need to make any assumption about the particular value of the exponent $b$ (which is usually set to be $d+z$ — the effective dimensionality of the quantum phase transition). The nonsingular part of free energy $F_{0}(T,H)$ is important at $H\gg H_{c}$ and can be approximated as $-M_{\text{sat}}(H-H_{c})$. Then, one can express the magnetization reduction as:
$$\begin{aligned}
\label{EQ:dFdHscalingSM}
-M_{\text{sat}}+M(T,H)=-\left(\frac{\partial (F-F_{0})}{\partial
H}\right)_{T}\\\nonumber
=-\lambda^{b+1/\nu}\mathcal{M}_{0}\left[\lambda^{z}T,\lambda^{1/\nu}(H_{c}-H)\right].\end{aligned}$$
Once again, $\mathcal{M}_{0}(x,y)$ is the unknown function of two variables. Now, at finite temperatures by setting $\lambda=T^{-1/z}$ one arrives at the following general scaling relation:
$$\label{EQ:MHscaling}
1-M(H,T)/M_{\text{sat}}=T^{m}\mathcal{M}\left(\frac{g\mu_{B}\mu_{0}(H-H_{c})}{T^{1/\varphi}}\right).$$
Here $\varphi=\nu z$ is the *crossover exponent* describing the interplay between the thermal and quantum fluctuations in the transition vicinity. The second exponent $m$ also has a simple physical meaning. It describes the temperature dependence of magnetization reduction at $H=H_{c}$.
The exponents $\varphi$ and $m$ are also crucial for characterizing the $T=0$ behavior. To see this, one needs to set $\lambda=(H_{c}-H)^{-\nu}$ in Eq. (\[EQ:dFdHscalingSM\]). Then the zero-temperature magnetization cusp is described as
$$\label{EQ:MHcusp}
1-M(H)/M_{\text{sat}}\propto(H_{c}-H)^{m\varphi}.$$
So this is the $m\varphi$ product that defines the low-temperature “cusp singularity”, and this is the quantity that needs to be found experimentally in order to verify the “dimensionality reduction” prediction by Jackeli and Zhitomirskii [@JackeliZhitomirsky_PRL_2004_FrustratedBEC].
Finally, we note that the above discussion yields a generalized version of the “zero scale universality” behavior at $z=2$ critical point in one dimension [@Sachdev_PRB_1994_ZeroScaleU]. The difference is, unlike in the former case neither the numeric values of the corresponding exponents (e.g., $m=1/2$, $\varphi=1$) nor the functional form of $\mathcal{M}(x)$ are predefined.
### Quantum critical scaling: Experiment
The manifestation of the experimentally accessible magnetization quantum critical behavior is contained in Eq. (\[EQ:MHscaling\]). To verify this relation and determine exponents $\varphi$ and $m$ we studied the $H-T$ scaling of magnetization near saturation in the axial geometry of [BaCdVO(PO$_4$)$_2$]{}. $M(H,T)$ data measured vs applied field at different temperatures are shown in Fig. \[FIG:Mscale\](a). Equation (\[EQ:MHscaling\]) suggests that all measurements are expected to collapse onto a single curve if rescaled with appropriate exponents. In order to determine the latter, for the data in Fig. \[FIG:Mscale\](a) we defined an empirical goodness of overlap criterion [@PovarovSchmidiger_PRB_2015_DimpyScaling; @*HaelgHuvonen_PRB_2015_NTENPscaling]. Overall, this criterion is similar to a standard $\chi^2$ with the only difference that the “theoretical” curve with respect to which the deviation of datapoints is calculated is not predefined, but empirically created on the fly at each iteration of the fit. A more detailed description of the algorithm is contained in Appendix \[APP:scaling\].
Using $\mu_{0}H_{c}=3.95(2)$ T obtained in calorimetric measurements as described below, we plot $\chi^2$ as a function of $m$ and $1/\varphi$ in the inset in Fig. \[FIG:Mscale\](b). The best overlap is found for $m=0.8(1)$ and $1/\varphi=1.6(3)$, and results in a spectacular data collapse shown in Fig. \[FIG:Mscale\](b) (main panel). The measured exponents are quite distinct from those in the pure one-dimensional case, where $m=1/2$ and $\varphi=1$ [@Sachdev_PRB_1994_ZeroScaleU; @JeongRonnow_PRB_2015_CupzNscaling]. Nonetheless, the observed exponent describing the magnetization cusp in the $T= 0$ limit \[as given by Eq. (\[EQ:MHcusp\])\] is the same, namely, $m\varphi=0.5\pm0.15$, and agrees well with this prediction made for the perfectly frustrated square lattice [@JackeliZhitomirsky_PRL_2004_FrustratedBEC].
![Low temperature specific heat in [BaCdVO(PO$_4$)$_2$]{} for axial and transverse geometries of the magnetic field. (a-e) $C_{p}(H,T)/T$ as the function of $T$ for different magnetic fields. Dashed lines show the power laws that can be identified in the data. (f-j) $C_{p}(H,T)$ at fixed temperature as the function of $H$. Arrows indicate excess specific heat appearing at low temperatures above the field-induced phase transition. In all the plots the dotted lines show the Zeeman effect based estimate of nuclear contribution, Eqs. (\[EQ:NucGen\],\[EQ:NucContribs\]). Please note that the panels on the left have logarithmic scale, while the panels on the right have linear scale. In Appendix \[APP:addendas\] one can also see alternative ways of plotting this data.[]{data-label="FIG:Cp_panels"}](Cp_panels_SUPP.pdf "fig:"){width="50.00000%"}\
Calorimetric studies
--------------------
### Specific heat measurements
Further unusual behavior of [BaCdVO(PO$_4$)$_2$]{} was revealed by the specific heat measurements. In both the $\mathbf{H}\parallel\mathbf{a},\mathbf{b}$ orientations zero-field-cooling data shows a pronounced lambda anomaly at $T_{N}=1.05$ K followed by a power-law decrease in $C_{p}(T)/T$ as shown in Fig. \[FIG:Cp\_panels\](a). This behavior is fully consistent with the previously reported powder data [@NathTsirlin_PRB_2008_BaCdVO(PO4)2].
Tracking the phase transition to reconstruct the $H-T$ phase diagram is often easier in constant-$H$ scans, shown in Figs. \[FIG:Cp\_panels\](f)-(j). However, these data reveal a striking difference between axial and transverse geometries. The first key result of our calorimetry studies is that in the axial case, the field-induced transition becomes *discontinuous* at low temperatures. Above $T^{\ast}\simeq0.15$ K both geometries yield a sharp $C_{p}(H)$ peak, marking a second-order transition \[Figs. \[FIG:Cp\_panels\](f)-\[FIG:Cp\_panels\](h)\] at a critical field $H_c$. In the vicinity of $H_c$ and at all fields above it the data for the two geometries are virtually indistinguishable. In contrast, below $T^{\ast}$ the character of the anomaly in the axial geometry changes. As shown in Figs. \[FIG:Cp\_panels\](i),\[FIG:Cp\_panels\](j) it rapidly evolves from a peak to a steplike feature, similar to the step found at the spin flop (a textbook example of discontinuous transition in a magnet).
The second and perhaps the most important finding of our calorimetry experiments is that there is an additional anomalous contribution to specific heat at the lowest temperatures [*above*]{} $H_c$ in both geometries. It can be seen in both constant-$T$ \[Fig. \[FIG:Cp\_panels\](j)\] and constant-$H$ scans \[Fig. \[FIG:Cp\_panels\](d)\]. At 100 mK it persists as a plateau all the way up to $\mu_{0}H_{c}^{\ast}\simeq5.2$ T, but vanishes at higher fields \[Fig. \[FIG:Cp\_panels\](e)\].
A very straightforward illustration of vanishing high-field specific heat is shown in Fig. \[FIG:relaxations\] for the axial geometry case. It demonstrates the relaxation curves obtained with the fixed measurement time of $300$ s and magnitude of heating pulse $P=15.8$ pW applied for $150$ s. The difference between the measurements at fields of $4.5$ T$\gtrsim\mu_{0}H_{c}$ and $8$ T$\gg\mu_{0}H_{c}$ at $T=100$ mK is apparent. At $4.5$ T the relaxation curve indeed has the characteristic “shark fin” shape, which means substantial specific heat present (as the rather long measurement period is comparable to the characteristic relaxation time). In contrast, at $8$ T after turning the heater on or off, in a few seconds the system ends up in the stationary regime. This is the clear signature of very short relaxation time and almost absent specific heat in the sample.
.[]{data-label="FIG:relaxations"}](SuppFigs_raw.pdf "fig:"){width="50.00000%"}\
### Magnetocaloric effect measurements
The discontinuous character of the low-temperature field-induced transition in the axial case is also confirmed by the measurements of the magnetocaloric effect. Utilizing the same experimental setup as for the relaxation calorimetry, we monitor the sample temperature during slow magnetic field sweeps, while keeping the heat bath temperature constant. In this so-called equilibrium regime [@AczelKohama_PRL_2009_Sr3Cr2O8BEC] the excess thermal power created due to the sample’s entropy change is balanced by the temperature gradient between the sample and the bath across the weak heat link. The evolution of the resulting sample’s $T(H)$ curves for up and down magnetic field sweeps is shown in Fig. \[FIG:MCE\]. The first-order spin-flop transition manifests itself as a highly asymmetric peaklike feature at all the temperatures. This is a direct consequence of the entropy discontinuity. In contrast, at elevated temperatures the magnetocaloric anomaly at $H_{c}$ is very symmetric, as it should be for a continuous transition [@GarstRosch_PRB_2005_MagnetocaloricTheory; @SchmidtThalmeier_PRB_2007_J1J2squareMCeffect]. However, below around $T^{\ast}$ this anomaly rapidly becomes rather asymmetric as well, confirming the change of the transition type.
![Magnetocaloric effect in [BaCdVO(PO$_4$)$_2$]{} at low temperatures in the axial geometry. The $T(H)$ dependencies taken at different temperatures with the field sweeping rate of $\pm5\times10^{-4}$ T/s.[]{data-label="FIG:MCE"}](MCE_LT.pdf "fig:"){width="50.00000%"}\
### Possible nuclear specific heat contributions
Although observations of divergent low-temperature specific heat due to nuclear magnetism with extremely low-energy scale are common, below we will show that the simple picture is qualitatively inconsistent with the present data.
The simplest model of nuclear specific heat assumes the energy levels in the spinful nuclei being split due to Zeeman effect. Then the nuclear contribution is approximately given as:
$$\label{EQ:NucGen}
C_{p}^{\text{Nuc}}(T,H)=\mathfrak{A}\left(\frac{\mu_{0}H}{T}\right)^2.$$
The material-dependent amplitude coefficient is calculated as follows:
$$\label{EQ:NucContribs}
\mathfrak{A}=\sum\limits_{i}\mathfrak{A}_{i}=\sum\limits_{i}N_{A}n_{i}\alpha_{i}\frac{I_{i}(I_{i}+1)(\gamma_{i}\hbar)^{2}}{3k_{B}}.$$
The summation goes through all the spinful types of nuclei present in the material; $n_i$ is the stoichiometric coefficient in the chemical formula and $\alpha_i$ is the abundance of the particular isotope. The data on the isotopes abundance, nuclear spins $I_i$, and corresponding nuclear gyromagnetic ratios $\gamma_i$ are found in Ref. [@Mason_1987_MultNMR], for example. The isotope data and the corresponding contribution to the nuclear specific heat prefactor relevant to [BaCdVO(PO$_4$)$_2$]{} are summarized in Table \[TAB:Isotopes\]. The overall $C_{p}^{\text{Nuc}}(T,H)$ prefactor is estimated as $\mathfrak{A}=1.5671\times10^{-5}$ J K/mol T$^{-2}$, with about 80% of it stemming from the magnetic ion $^{51}$V having nuclear spin $I=7/2$. This means that for the consistent description of $C_{p}^{\text{Nuc}}$ the quadrupolar splitting and hyperfine interactions on the $^{51}$V site also need to be taken into account. These parameters are unknown at the moment, and therefore Eqs. (\[EQ:NucGen\],\[EQ:NucContribs\]) should be seen only as the crude estimate of possible effect magnitude. As one can see from Fig. \[FIG:Cp\_panels\], the low-temperature specific heat in [BaCdVO(PO$_4$)$_2$]{} is completely at odds with this simple estimation.
Nonetheless, since nuclear spin $I=7/2$ is also carried by the magnetic $S=1/2$ $^{51}$V$^{4+}$ ions, some complex behavior induced by hyperfine coupling close to the quantum critical point can not be fully ruled out. There are some experimental [@Ronnow_Science_2005_nuclearQCP] and theoretical [@TsvelikZaliznyak_PRB_2016_NuclearNecklace] studies of hyperfine coupled settings, but not for the strongly frustrated 2D case.
Isotope $\alpha_i$ $n_i$ $\gamma_i$ (rad/s T$^{-1}$) $I_i$ $\mathfrak{A}_i$ (J K/mol T$^{-2}$)
------------ -- ------------ -- ------- -- ----------------------------- -- ------- -- -------------------------------------
$^{135}$Ba 0.06590 1 $2.6755\times10^{7}$ 3/2 $2.8604\times10^{-8}$
$^{137}$Ba 0.11320 1 $2.9930\times10^{7}$ 3/2 $6.1488\times10^{-8}$
$^{111}$Cd 0.12750 1 $-5.7046\times10^{7}$ 1/2 $5.0318\times10^{-8}$
$^{113}$Cd 0.12260 1 $-5.9609\times10^{7}$ 1/2 $5.2829\times10^{-8}$
$^{50}$V 0.00240 1 $2.6721\times10^{7}$ 6 $1.1638\times10^{-8}$
$^{51}$V 0.99760 1 $7.0492\times10^{7}$ 7/2 $1.2625\times10^{-5}$
$^{31}$P 1.00000 2 $1.0839\times10^{8}$ 1/2 $2.8497\times10^{-6}$
$^{17}$O 0.00037 9 $-3.6280\times10^{7}$ 5/2 $6.2013\times10^{-9}$
: The relevant isotope data (from Ref. [@Mason_1987_MultNMR]) and corresponding calculated contribution \[Eq. (\[EQ:NucContribs\])\] to the nuclear specific heat due to Zeeman splitting.[]{data-label="TAB:Isotopes"}
### Magnetic phase diagram of [BaCdVO(PO$_4$)$_2$]{}
Leaving aside the certainly exotic scenario of interplay between the nuclear and electronic spins, we face the conclusion that the found excess specific heat is of purely electron spin origin. Apart from either electronic or nuclear spins no other degrees of freedom may give a field-dependent contribution to the specific heat of an insulating material at these low temperatures. We conclude that in [BaCdVO(PO$_4$)$_2$]{}at the lowest temperatures $H_c$ does [*not*]{} correspond to the full saturation. Indeed, the latter would open a Zeeman gap in the spectrum and suppress any magnetic specific heat. Instead, $H_c$ indicates the appearance of a [*new quantum regime with substantial low-energy fluctuations*]{}.
The magnetic $H-T$ phase diagram of [BaCdVO(PO$_4$)$_2$]{}in Fig. \[FIG:AB\_Cpmap\] summarizes the findings. We distinguish conventional paramagnetic (PM), field polarized (FP), and AF states (and its post-spin-flop version SF). At intermediate temperatures a quantum critical (QC) regime is observed above $H_{c}$. The new low-temperature field-induced states are labeled as LT. While they are separated from the ordered states by obvious phase transitions, their finite-$T$ boundaries cannot be clearly identified in our calorimetry data. Thus, we simply identify the crossover line below which the anomalous behavior becomes pronounced. The appearance of the LT regime, already intriguing on its own, becomes especially interesting if considered in the context of predictions made for the spin-nematics.
![Magnetic phase diagram for $\mathbf{H}\parallel\mathbf{a,b}$. The background shows the false color map of $C_{p}(T,H)/T$, thin and thick black solid lines represent the phase transitions (of second or first order correspondingly), and gray dashed lines mark crossovers. Points are the ordered phase boundary data obtained from $C_{p}$ anomalies. The phases are as follows: PM, paramagnetic; FP, fully polarized; AF, antiferromagnetic; SF, antiferromagnetic after the spin flop; QC, quantum critical regime; LT, unconventional low temperature regime. Crossover lines marking the QC regime follow $T\propto|H-H_{c}|^{\varphi}$ with the same crossover exponent $\varphi$ found from scaling Eq. (\[EQ:MHscaling\]). []{data-label="FIG:AB_Cpmap"}](AB_Cpmap.pdf "fig:"){width="50.00000%"}\
### Observed anomalies in context of spin-nematics
One can find interesting possible connections of the observed anomalies to the expected behavior of the two-dimensional $S=1/2$ spin-nematic materials. First, we would like to note that the location of the LT regimes is *in principle* consistent with the expectations for the spin-nematic state in the frustrated square lattice model. The anomalous specific heat found in [BaCdVO(PO$_4$)$_2$]{} samples is endemic to the very low temperatures compared to the typical interactions of the order of a few kelvin in the material. Nonetheless, this is indeed the temperature range in which the anomalous behavior due to magnons pairing up is expected to take place from the theory point of view. Exact diagonalization studies of the frustrated model with $J_{2}/J_{1}=-0.4$ suggest significant nematic-type contributions to the specific heat to occur at temperatures order of magnitude lower than typical $J$’s in the system [@ShannonMomoi_PRL_2006_J1J2squarecircle]. In the case of [BaCdVO(PO$_4$)$_2$]{} the relevant energy scale can be suppressed even further below $0.1J_{1}\simeq0.3$ K, as the system deviates from the idealized $J_{2}/J_{1}=-0.4$ zero-field case. This looks very consistent with our present observations shown in Fig. \[FIG:Cp\_panels\](d). At the same time the spin-nematic precursor behavior may in principle be present at any field below the true saturation point — as long as there are fluctuating transverse spin components. There are indications that spin fluctuations associated with the new low-temperature high-field phases are present already in the ordered states. There too we find anomalous contributions to specific heat below the crossover temperature $T^{\ast}\simeq0.15$ K \[Figs. \[FIG:Cp\_panels\](a)-\[FIG:Cp\_panels\](c)\]. They roughly follow $C_{p}(T)\propto T^{-1.5}$ and are particularly strong in the spin-flop state. This suggests their transverse character. The field $H_{c}^{\ast}$ at which the spin fluctuations vanish is consistent with the effective magnetic energy scale of the material $J_{\text{eff}}=\sqrt{J_{1}^{2}+J_{2}^2}$ [@Shannon_EPJB_2004_GenericJ1J2theory].
A second interesting observation is related to the boundary between conventional antiferromagnetic and LT phases. As shown above, in the axial geometry the new high-field state is entered from the spin-flop AF phase through a [*discontinuous transition*]{}. Incidentally, this is exactly the type of behavior expected for the spin-nematic phase predicted to emerge just below full saturation [@SmeraldShannon_PRB_2016_nematicNMR]. The AF and spin-nematic phases have competing order parameters, and therefore the transition between them has to be first order. In the transverse geometry, the spin-nematic phase should not exist in a field due to a lack of axial symmetry [@Zhang_PRB_2017_NematicAnisotropy]. However, this is not supposed to impede the associated fluctuations completely. While strong spin fluctuations persist irrespective of field orientation in [BaCdVO(PO$_4$)$_2$]{}, they may result in nematic order only in the axial geometry. This may explain why the transition at $H_c$ remains continuous in the transverse case and becomes discontinuous in the axial — the only case expected to support the nematic long-range ordering.
To summarize, it is very tempting to consider the observed anomalous regime LT as the precursor of the true spin-nematic long-range order. This would simultaneously explain the small energy scale associated with the new state as well as the field-direction-dependent transition type. However, at this stage we still cannot fully rule out the possibility of interference between the electronic and the nuclear spins going beyond the simple model described by Eq. (\[EQ:NucContribs\]).
Conclusions
===========
The high hopes for finding the unconventional magnetism in the frustrated $S=1/2$ square lattice magnet [BaCdVO(PO$_4$)$_2$]{} appear to be well justified. In addition to the experimentally quantified “dimensionality reduction” effect serving as the indicator of strong frustration we have also found anomalously strong contributions to the specific heat in the vicinity of saturation field at lowest temperatures. Although the possibility of their origin from the interplay of electronic and nuclear magnetism is not yet fully ruled out, these anomalies show qualitative consistency in the order of phase transition and energy scale with the predicted spin-nematic behavior. Future efforts aimed at understanding the origins of the novel regime will have to specifically focus on the lowest possible temperatures.
This work was supported by Swiss National Science Foundation, Division II. We would like to thank Prof. Oleg Starykh (University of Utah) for enlightening discussions, Stanislaw Galeski and Dominic Blosser (ETH Zürich) for help with the low-temperature magnetocaloric measurements, and Dr. Severian Gvasaliya (ETH Zürich) for technical assistance.
Definition of scaling $\chi^{2}$ criterion {#APP:scaling}
==========================================
![The scaling analysis of the magnetization data (see Fig. \[FIG:Mscale\] of the main text). Left: optimal scaling exponents $1/\varphi=1.55$, $m=0.76$. Right: non-optimal scaling exponents $1/\varphi=1$, $m=1$. Green curve represents the empirical data interpolation with respect to which the $\chi^2$ costs are calculated.[]{data-label="FIG:X2"}](SuppFigs_X2.pdf "fig:"){width="50.00000%"}\
The hypothesis that is being tested for the magnetization data present in Fig. \[FIG:Mscale\] of the main text is that it follows the universal behavior in the vicinity of $H_{c}$:
$$\label{EQ:MHscalingSM}
1-M(T,H)/M_{\text{sat}}=T^{m}\mathcal{M}\left(\frac{g\mu_{0}\mu_{B}(H_{c}-H)}{T^{1/\varphi}}\right).$$
This means that for correctly chosen exponents $m$ and $\varphi$ the set of datapoints $X=\frac{g\mu_{0}\mu_{B}(H-H_{c})}{T^{1/\varphi}}$ and $Y=[1-M(H)/M_{\text{sat}}]/T^{m}$ should lie close to some hypothetical curve. In principle, if this hypothetical curve $Y_{0}(X)$ is known, the problem of calculating the abstract “goodness of overlap” can be reduced to the very standard problem of calculating of the “goodness of fit” of the data $Y(X)$ by theory $Y_{0}(X)$.
The key idea in the present approach, where no *a priori* scaling curve is postulated, is to construct $Y_{0}(X)$ “on the fly” based on the current $Y(X)$ data. This is achieved by interpolating the scattered $Y(X)$ with cubic splines. It guarantees the smoothness of the resulting curve and at the same time gives a bit more flexibility than polynomial interpolation used, e.g., in Ref. [@JeongRonnow_PRB_2015_CupzNscaling] in a similar situation. The examples of such an empirical interpolation curve for cases with good and poor choices of scaling exponents are shown in Fig. \[FIG:X2\].
A remark needs to be made regarding the normalization of cost function in the case described above. The $\chi^{2}$ value is usually normalized with the number of degrees of freedom, which is typically the number of datapoints. However, in the present situation individual degrees of freedom are rather represented by the individual $M(H)$ scans at fixed temperatures. For any separately taken scan the interpolation procedure would by definition provide an ideal overlap with the “empirical curve”, and it is the optimization in the presence of multiple such datasets that constitutes the essence of the procedure. Then the cost function, being the equivalent of a standard normalized error-bar-weighted $\chi^{2}$ is calculated as follows:
$$\label{EQ:X2}
\chi^{2}=\frac{1}{N_{\text{Datasets}}-1}\sqrt{\sum\limits_{X_{i}}\left(\frac{Y(X_{i})-Y_{0}(X_{i})}{\Delta Y(X_{i})}\right)^{2}}.$$
Addenda in the specific heat measurements {#APP:addendas}
=========================================
{width="80.00000%"}\
Finally, we would like to present a proof that the observed anomalous specific heat is sample related. One simple consideration is that the addenda contribution is somewhat different in both $\mathbf{H}\parallel\mathbf{a}$ and $\mathbf{H}\parallel\mathbf{b}$ cases (as different amounts of grease and a different piece of silver foil holder was used), while the observed extra specific heat is well matched. But even more valuable is the direct comparison, given in Fig. \[FIG:Maddenda\] for the $\mathbf{H}\parallel\mathbf{a}$ setup. One can see that the background specific heat contribution is very small. Apart from a tiny Shottky anomaly close to $H=0$ it is dominated by $C_{p}\propto
T$ linear specific heat of the silver foil.
|
---
abstract: 'We present optical light curves of 19 radio quiet (RQ) broad absorption line (BAL) QSOs and study their rapid variability characteristics. Systematic CCD observations, aided by a careful data analysis procedure, have allowed us to clearly detect any such microvariability exceeding 0.01–0.02 mag. Our observations cover a total of 13 nights ($\sim$72 hours) with each quasar monitored for about 4 hours on a given night. Our sample size is a factor of three larger than the number of radio-quiet BALQSOs previously searched for microvariability. We introduce a scaled $F-$test statistic for evaluating the presence of optical microvariability and demonstrate why it is generally preferable to the statistics usually employed for this purpose. Considering only unambiguous detections of microvariability we find that $\sim$11 per cent of radio-quiet BALQSOs (two out of 19 sources) show microvariability for an individual observation length of about 4 hr. This new duty cycle of 11% is similar to the usual low microvariability fraction of normal RQQSOs with observation lengths similar to those of ours. This result provides support for models where radio-quiet BALQSO do not appear to be a special case of the RQQSOs in terms of their microvariability properties.'
author:
- |
Ravi Joshi$^{1}$[^1], Hum Chand$^{1}$, Alok C. Gupta$^{1}$ and Paul J. Wiita$^{2}$\
$^{1}$Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital, 263129, India\
$^{2}$Department of Physics, The College of New Jersey, PO Box 7718, Ewing, NJ 08628, USA
date: 'Accepted 2010 November 25. Received 2010 November 02; in original form 2010 September 07'
title: Optical microvariability properties of BALQSOs
---
\[firstpage\]
galaxies: active – galaxies: photometry – galaxies: jet – quasars: general
Introduction
============
Significant variability in brightness over a few minutes to several hours (less than a day) is commonly known as microvariability, intra-night optical variability (INOV) or intra-day variability. Optical microvariability is a well known property of radio-loud (RL) active galactic nuclei (AGN), particularly of its blazar subclass (e.g., Gupta et al. 2008 and references therein). Over past two decades there have been rather extensive searches for this phenomenon in blazars, other types of RLQSOs, and the far more numerous radio quiet quasars (RQQSOs) (e.g., Miller, Carini & Goodrich 1989; Carini et al. 1992, 2007; Gopal-Krishna et al. 1993b, 2000, 2003; de Diego et al. 1998; Romero, Cellone & Combi 1999; Sagar et al. 2004; Stalin et al. 2004; Montagni et al. 2006; Goyal et al. 2010). In the case of blazars these studies have provided useful constraints on the relativistic jet based models that are used to explain the origin of the large variations that help define the category (e.g., Marscher, Gear & Travis 1992; Rani et al. 2010). Since RQQSOs lack jets of significant power and extent, the microvariability seen in them may arise from processes on the accretion disc itself, and thus could possibly be used to probe the properties of the discs (e.g., Gopal-Krishna, Sagar & Wiita 1993a). However, so far there has been a lack of systematic effort to exploit microvariability properties to understand the nature of the substantial quasar sub-class with broad absorption lines (BALs), the BALQSOs.
These BALQSOs are AGN characterized by the presence of strong absorption troughs in their optical spectra. They constitute about 10–15 per cent of optically selected quasars (e.g., Reichard et al. 2003; Hewett & Foltz 2003). The BALs are attributed to material flowing outwards from the nucleus with velocities of 5000 to 50000 km s$^{-1}$ (Green et al. 2001). BALQSOs are classified mainly into two subclasses based on the material predominantly producing the BAL troughs. High ionization BAL quasars (HiBALs) have broad absorption from C IV, Si IV, N V and O IV lines. About 10 per cent of BALQSOs also show, along with HiBAL features, broad absorptions from lower ionization lines such as Mg II or Al III; these are called low-ionization BAL quasars (LoBALs). Any complete model of quasars and AGNs needs to explain self-consistently a wide range of their observational properties which also include: the presence of other emission/absorption lines, the fraction of quasars showing broad absorption lines and the fraction among them showing continuum variability such as microvariability.
Carini et al. (2007) have compiled a sample of 117 radio-quiet objects that have been searched for their microvariability. Of these, 47 are classified as Seyfert galaxies, 64 as QSOs, and 6 as BALQSOs. In their entire sample 21.4 per cent of the objects were found to exhibit microvariability, but among objects classified as Seyfert galaxies, QSOs and BALQSOs, microvariability was seen in 17 per cent, 23 per cent and 50 per cent, respectively (Carini et al. 2007). In addition, Rabbette et al. (1998) have noted that two radio-quiet BALQSOs displayed short term X-ray variability. The observed high fraction of microvariations in BALQSOs suggests that it might be worthwhile to expend more of the observing time devoted to microvariability on the BALQSO class if one wants to understand physical processes in or near the accretion disc. Clearly, the present sample size of BALQSOs is very small compared to those of the non-BALQSO classes, and no useful conclusions about their nature can be drawn from them. Therefore it is important to increase the sample of BALQSOs, so as to be able to arrive at firmer conclusions about the fraction showing microvariability. We note that if BALQSOs do really show a substantially higher duty cycle for microvariability than do non-BAL RQQSOs, this would shed light on the question of whether or not radio-quiet BALQSOs are special cases of the RQQSOs, especially in terms of their microvariability properties. For instance, if even weak jets dominate the rapid variability (e.g., Gopal-Krishna et al. 2003) then a higher duty cycle for microvariability will give indirect support for the hypothesis that BALQSOs are viewed at angles nearly perpendicular to their accretion discs (e.g., Ghosh & Punsly 2007). This is because jet fluctuations originating in relativistic jets pointing close to our line-of-sight are amplified in magnitude and compressed in timescale (e.g., Gopal-Krishna et al. 2003). Whereas, if BALQSOs show only the usual low microvariability duty cycles of normal RQQSOs (around 20 per cent) and the fluctuations still arise in weak jets, that would provide indirect support for alternative models, such as those where the BAL outflows come out closer to the disc plane (e.g., Elvis 2000). In conjunction with X-ray and optical spectral properties, such variation information is very useful in constraining various physical models for the origin of microvariability (e.g., Czerny et al. 2008) and the nature of BALQSOs itself (e.g., Weymann et al. 1991; Elvis et al. 2000). To address these questions, we have recently started a pilot program to make an extensive search for optical microvariability of BALQSOs.
This paper is organized as follows. In Section 2 we describe the main aspects of our sample selection criteria, while Section 3 briefly describes our observations and the data reductions. In Sections 4 and 5 we present our analysis and results, respectively. Section 6 gives a discussion and our conclusions.
[@llccccccrccl@]{} & & &[g$_{i}$]{} &[M$_{i}$]{} &[$z_{em}$]{} &[R[^2]]{} &[Type[^3]]{} &[Ref [^4]]{} &[Date of Obs]{}\
WFM91 0226$-$1024 &02$^h$ 28$^m$ 39.20$^s$ &$-$10$^h$ 11$^m$ 10.0$^s$ &15.16$^{\star}$ &$-$30.7$^{\star}$ &2.256& 0.38 &HiBAL & 3 & 22.12.2009\
J073739.96$+$384413.2 &07$^h$ 37$^m$ 39.96$^s$ &$+$38$^h$ 44$^m$ 13.2$^s$ &16.99 &$-$28.04 &1.399& ND &LoBAL & 1 & 22.12.2009\
J084044.41$+$363327.8 &08$^h$ 40$^m$ 44.41$^s$ &$+$36$^h$ 33$^m$ 27.8$^s$ &16.59 &$-$28.36 &1.225& 0.66 &LoBAL & 1 & 06.01.2010\
J084538.66$+$342043.6 &08$^h$ 45$^m$ 38.66$^s$ &$+$34$^h$ 20$^m$ 43.6$^s$ &16.96 &$-$29.03 &2.149& ND &HiBAL & 1 & 07.01.2010\
J090924.01$+$000211.0 &09$^h$ 09$^m$ 24.01$^s$ &$+$00$^h$ 02$^m$ 11.0$^s$ &16.68 &$-$29.12 &1.864& ND &HiBAL & 4 & 25.01.2010\
J094443.13$+$062507.4 &09$^h$ 44$^m$ 43.13$^s$ &$+$06$^h$ 25$^m$ 07.4$^s$ &16.25 &$-$27.40 &0.695& 0.14 &LoBAL & 1 & 16.02.2010\
J094941.10$+$295519.2 &09$^h$ 49$^m$ 41.10$^s$ &$+$29$^h$ 55$^m$ 19.0$^s$ &16.04 &$-$28.56 &1.665& ND &HiBAL & 2 & 07.01.2010\
J100711.81$+$053208.9 &10$^h$ 07$^m$ 11.81$^s$ &$+$05$^h$ 32$^m$ 08.9$^s$ &16.21 &$-$29.71 &2.143& ND &HiBAL & 1 & 25.03.2010\
J111816.95$+$074558.1 &11$^h$ 18$^m$ 16.95$^s$ &$+$07$^h$ 45$^m$ 58.1$^s$ &16.27 &$-$29.34 &1.735& ND &MiBAL & 4 & 25.01.2010\
J112320.73$+$013747.4 &11$^h$ 23$^m$ 20.70$^s$ &$+$01$^h$ 37$^m$ 47.0$^s$ &15.84 &$-$29.32 &2.130& ND &LoBAL & 2 & 17.01.2010\
J120051.52$+$350831.6 &12$^h$ 00$^m$ 51.52$^s$ &$+$35$^h$ 08$^m$ 31.6$^s$ &16.79 &$-$28.77 &1.717& 0.23 &HiBAL & 1 & 12.05.2010\
J120924.07$+$103612.0 &12$^h$ 09$^m$ 24.07$^s$ &$+$10$^h$ 36$^m$ 12.0$^s$ &16.53 &$-$25.69 &0.394& 0.33 &LoBAL & 1 & 08.05.2010\
J123820.19$+$175039.1 &12$^h$ 38$^m$ 20.19$^s$ &$+$17$^h$ 50$^m$ 39.1$^s$ &16.86 &$-$25.99 &0.449& 1.03 &LoBAL & 2 & 09.05.2010\
J125659.92$+$042734.3 &12$^h$ 56$^m$ 59.90$^s$ &$+$04$^h$ 27$^m$ 34.0$^s$ &15.80 &$-$28.19 &1.025& ND &LoBAL & 2 & 16.02.2010\
J151113.84$+$490557.4$\dagger$&15$^h$ 11$^m$ 13.84$^s$ &$+$49$^h$ 05$^m$ 57.4$^s$ &16.49 &$-$28.37 &1.359& 0.91 &LoBAL & 1 & 23.04.2010\
J152350.42$+$391405.2$\dagger$&15$^h$ 23$^m$ 50.42$^s$ &$+$39$^h$ 14$^m$ 05.2$^s$ &16.68 &$-$26.77 &0.661& 1.01 &LoBAL & 1 & 24.04.2010\
J152553.89$+$513649.1 &15$^h$ 25$^m$ 53.89$^s$ &$+$51$^h$ 36$^m$ 49.1$^s$ &16.85 &$-$30.03 &2.882& ND &HiBAL & 1 & 12.05.2010\
J154359.44$+$535903.2 &15$^h$ 43$^m$ 59.44$^s$ &$+$53$^h$ 59$^m$ 03.2$^s$ &17.03 &$-$29.28 &2.370& ND &HiBAL & 1 & 09.05.2010\
J160207.68$+$380743.0 &16$^h$ 02$^m$ 07.70$^s$ &$+$38$^h$ 07$^m$ 43.1$^s$ &16.10 &$-$28.62 &1.594& ND &LoBAL & 1 & 14.06.2010\
\
Source selection criteria
=========================
Our sample is chosen from the BALQSO catalogues compiled by Trump et al. (2006), Scaringi et al. (2009) and Gibson et al. (2009) which are based on Sloan Digital Sky Survey (SDSS) Data Releases 3 and 5 (DR3: Schneider et al. 2005; DR5: Adelman-McCarthy et al. 2007; Schneider et al. 2007). In addition, we also included one brighter BALQSO from the compilation by Weymann et al. (1991). Most of the sources were selected in such a way that both optical and X-ray spectral data are available for them in archives. All were at declinations that allowed for the observations to be made at relatively low air masses. We also required our candidate sources to have g$_{i} \leq 17$. This constraint means that even with a 1-m class telescope we could obtain a good enough signal to noise ratio to detect fluctuations of $<0.02$ mag with a reasonably good time resolution of $< 10$ minutes. We also limit the BALQSOs to have absolute magnitudes M$_{i} < -24.5$, so that the flux contribution from the host galaxy can be assumed to be negligible (Miller et al. 1990).
Our final sample consists of a total of 19 BALQSOs, as listed in Table \[tab:sample\]. Among these BALQSOs 8 are classified as HiBALs, 10 are LoBALs and 1 is a mini BAL. The whole sample covers a redshift range of $ 0.39< z_{em} < 2.9$.
Observations and Data Reductions
================================
Photometric observations
------------------------
Our observations of each of the BALQSOs were carried out continuously for $\sim$ 4h in the R passband, mainly using the 1.04-m Sampurnanand telescope located at the Aryabhatta Research Institute of observational sciencES (ARIES), Nainital, India. It has Ritchey-Chretien (RC) optics with a f$/$13 beam and is equipped with a cryogenically cooled CCD detector with a 2048 pixel $\times$ 2048 pixel chip mounted at the Cassegrain focus (Sagar 1999). The readout noise of the CCD chip is 5.3 e$^{-}$/pixel and it has a gain of 10 e$^{-}$$/$ Analog to Digital Unit (ADU). Each pixel of the CCD chip has a dimension of 24 $\mu$m$^{2}$, corresponding to 0.37 arcsec$^{2}$ on the sky, and so covers a total field of $\sim$ 13$^{\prime}$ $\times$ 13$^{\prime}$. To improve the signal to noise ratio, observations were carried out in a 2 pixel $\times$ 2 pixel binning mode. The typical seeing during our observing runs at ARIES was $\sim$ 3$^{\prime\prime}$. In addition, two sources were observed with 2.01-m Himalayan Chandra Telescope (HCT) located at the Indian Astronomical Observatory (IAO), Hanle, India. It is also of the RC design with a f$/$9 beam at the Cassegrain focus[^5]. The detector was a cryogenically cooled 2048 $\times$ 4096 chip, of which the central 2048 $\times$ 2048 pixels were used. The pixel size is 15 $\mu$m$^{2}$ so that the image scale of 0.29 arcsec$/$pixel covers an area of about 10$^{\prime}$ $\times$ 10${^\prime}$ on the sky. The readout noise of this CCD is 4.87 e$^{-}$/pixel and the gain is 1.22 e$^{-}$$/$ADU. The CCD was used in an unbinned mode. The typical seeing during our observations at IAO was $\sim$ 1.5$^{\prime\prime}$.
We chose an R filter for this observational program because it is at the maximum response of the CCD system; thus the time resolution achievable for each object is maximized. As most of our sources have $g_{i}\sim 16-17$, the best time resolution we could achieve was of the order of 3 minutes, and we almost always managed data points spaced less than 8 minutes apart, so very rapid fluctuations could be picked up. We also took care to select sources and fields of view so as to ensure availability of at least two, but usually more, comparison stars on the CCD frame that were within around 1 mag of the QSO’s brightness. This allowed us to identify and discount any comparison star which itself varied during a given night and hence ensured reliable differential photometry of the QSO. Observations were made on a total of 13 nights for this program during December 2009 – June 2010, as specified in Table \[tab:sample\].
[@ccc ccc ccc cc@]{} & & & &\
& & & & & & & & & &\
(1)&(2) &(3) &(4) &(5) &(6) &(7)&(8)&(9) &(10) &(11)\
WFM91 0226$-$1024$^*$ &02$^h$28$^m$31.17$^s$& $-$10$^d$17$^m$15.4$^s$& 02$^h$28$^m$40.65$^s$& $-$10$^d$15$^m$50.3$^s$&16.30&—–&—–& —– &—— & —–\
J073739.96$+$384413.2 &07$^h$37$^m$22.23$^s$& $+$38$^d$48$^m$55.5$^s$& 07$^h$37$^m$24.63$^s$& $+$38$^d$41$^m$29.5$^s$&17.00&15.68&16.81& $-$0.01& $-$0.74& $-$0.73\
J084044.41$+$363327.8 &08$^h$40$^m$21.03$^s$& $+$36$^d$37$^m$26.6$^s$& 08$^h$40$^m$32.32$^s$& $+$36$^d$33$^m$34.5$^s$&16.59&16.32&16.17& $+$0.02& $-$0.12& $-$0.14\
\
J084538.66$+$342043.6 &08$^h$45$^m$42.98$^s$& $+$34$^d$17$^m$46.3$^s$& 08$^h$45$^m$45.14$^s$& $+$34$^d$17$^m$07.3$^s$&16.95&16.05&16.66& $-$0.53& $-$1.20& $-$0.67\
J090924.01$+$000211.0 &09$^h$09$^m$17.27$^s$& $+$00$^d$04$^m$01.3$^s$& 09$^h$09$^m$13.86$^s$& $-$00$^d$01$^m$24.6$^s$&16.67&15.66&16.52& $-$0.82& $-$0.39& $+$0.43\
J094443.13$+$062507.4 &09$^h$44$^m$40.39$^s$& $+$06$^d$28$^m$14.9$^s$& 09$^h$44$^m$34.10$^s$& $+$06$^d$30$^m$33.0$^s$&16.24&15.62&16.76& $-$0.32& $-$1.19& $-$0.87\
\
J094941.10$+$295519.2 &09$^h$49$^m$23.32$^s$& $+$29$^d$54$^m$13.5$^s$& 09$^h$49$^m$42.06$^s$& $+$30$^d$01$^m$07.1$^s$&16.06&15.80&15.44& $-$0.79& $-$0.93& $-$0.14\
J100711.81$+$053208.9 &10$^h$07$^m$17.82$^s$& $+$05$^d$37$^m$05.1$^s$& 10$^h$07$^m$03.42$^s$& $+$05$^d$34$^m$07.0$^s$&16.27&15.37&14.55& $-$0.18& $-$0.47& $-$0.29\
J111816.95$+$074558.1 &11$^h$18$^m$13.91$^s$& $+$07$^d$46$^m$28.7$^s$& 11$^h$18$^m$05.79$^s$& $+$07$^d$51$^m$21.5$^s$&16.15&15.94&15.59& $-$1.17& $-$0.49& $+$0.68\
\
J112320.73$+$013747.4 &11$^h$23$^m$21.59$^s$& $+$01$^d$45$^m$10.2$^s$& 11$^h$23$^m$30.68$^s$& $+$01$^d$39$^m$55.6$^s$&15.84&15.98&15.67& $-$0.23& $-$0.57& $-$0.34\
J120051.52$+$350831.6 &12$^h$01$^m$13.18$^s$& $+$35$^d$09$^m$03.9$^s$& 12$^h$01$^m$21.41$^s$& $+$35$^d$04$^m$18.7$^s$&16.79&15.26&15.12& $-$0.40& $-$0.26& $+$0.14\
J120924.07$+$103612.0 &12$^h$09$^m$07.90$^s$& $+$10$^d$34$^m$14.0$^s$& 12$^h$09$^m$16.06$^s$& $+$10$^d$38$^m$38.0$^s$&16.50&15.82&15.07& $-$0.66& $-$1.08& $-$0.42\
\
J123820.19$+$175039.1 &12$^h$38$^m$47.21$^s$& $+$17$^d$56$^m$01.7$^s$& 12$^h$38$^m$19.22$^s$& $+$17$^d$46$^m$07.9$^s$&16.42&15.62&15.15& $-$1.16& $-$0.27& $+$0.89\
J125659.92$+$042734.3 &12$^h$56$^m$47.73$^s$& $+$04$^d$25$^m$25.1$^s$& 12$^h$56$^m$59.64$^s$& $+$04$^d$31$^m$49.3$^s$&16.04&15.13&14.85& $-$0.34& $-$0.32& $-$0.02\
J151113.84$+$490557.4 &15$^h$11$^m$06.35$^s$& $+$49$^d$08$^m$07.8$^s$& 15$^h$11$^m$34.13$^s$& $+$49$^d$07$^m$17.0$^s$&16.49&16.07&16.12& $-$0.74& $-$0.19& $+$0.55\
\
J152350.42$+$391405.2 &15$^h$23$^m$59.73$^s$& $+$39$^d$17$^m$05.9$^s$& 15$^h$23$^m$51.32$^s$& $+$39$^d$11$^m$48.4$^s$&16.65&16.35&15.64& $-$1.01& $-$0.11& $+$0.90\
J152553.89$+$513649.1 &15$^h$25$^m$57.63$^s$& $+$51$^d$34$^m$51.9$^s$& 15$^h$26$^m$10.32$^s$& $+$51$^d$36$^m$11.1$^s$&16.84&16.78&15.75& $-$1.03& $-$0.28& $+$0.75\
J154359.44$+$535903.2 &15$^h$44$^m$29.09$^s$& $+$53$^d$58$^m$16.5$^s$& 15$^h$44$^m$19.22$^s$& $+$53$^d$58$^m$03.2$^s$&17.05&16.48&16.06& $-$0.40& $-$0.48& $-$0.08\
J160207.68$+$380743.0 &16$^h$01$^m$34.59$^s$& $+$38$^d$09$^m$31.4$^s$& 16$^h$01$^m$33.19$^s$& $+$38$^d$05$^m$14.2$^s$&16.83&15.66&16.16& $-$0.47& $-$0.34& $+$0.13\
Data Reduction
--------------
The raw photometric data was first pre-processed using standard routines in the Image Reduction and Analysis Facility [^6] (IRAF) software. We generated a master bias frame for the observing night by taking the median of all bias frames taken on that night. This master bias frame was subtracted from all the twilight sky flat image frames as well as from the source image frames taken on that night. The routine step of dark frame subtraction was not performed because the CCDs used in our observations were cryogenically cooled to $-$120$^\circ$ C; at that temperature the amount of thermal charge deposition is negligible for our brief exposure times. Then the master flat was generated by median combining of several flat frames (usually more than 5 taken on the twilight sky) in that passband. Next, the normalized master flat was generated. Each source image frame was flat-fielded by dividing by the normalized master flat in the respective band to remove pixel-to-pixel inhomogeneities. Finally, cosmic ray removal was done from all source image frames using the task *cosmicrays* in IRAF.
Photometry
----------
The instrumental magnitudes of the comparison stars and the target source are obtained from the data by using Dominion Astronomical Observatory Photometry (DAOPHOT II) software to perform the concentric circular aperture photometric technique (Stetson 1987, 1992). Aperture photometry was carried out with four aperture radii, to wit, $\sim$ 1$\times$FWHM, 2$\times$FWHM, 3$\times$FWHM and 4$\times$FWHM. Utmost caution has been taken to deal with the seeing, and we have taken the mean full width at half maximum $(FWHM)$ of 5 fairly bright stars on each CCD frame in order to choose the apertures for the photometry of that individual frame. The data reduced with different aperture radii were found to be in good agreement. However, it was noticed that the best S/N was almost always obtained with aperture radii of 2$\times$FWHM, so we adopted that aperture for our final analysis.
Analysis
========
Selection of comparison stars {#subs:stars}
-----------------------------
The comparison stars are chosen on the basis of their proximity in both location and magnitude to the quasar. Preference was given to those stars having magnitudes similar to that of the monitored quasar, so that the errors in the Differential Light Curves (DLCs) will not be dominated by any faint object (see Sect. \[subs:stat\]). The locations of the two best comparison stars for each BALQSO are given in columns 2–5 of Table \[tab:color\].
In addition, our atmosphere acts like a colour filter of variable transparency, so the photometry of two stars (or quasar-star pair) of different colours will be affected by different amounts because of the changing air-mass during the monitoring (e.g., Eq. 2, in Stalin et al. 2004). Therefore, for ascertaining the variability properties from DLC, the colours of the two objects in the DLC really should be similar. We list V$-$R colour differences for all pairs of objects we observe in columns 9–11 of Table \[tab:color\]. For most of the quasar-star pairs the V$-$R colour differences are smaller than unity, except for seven pairs (out of a total of 38) where the differences are the range of 1.0–1.20. Similarly, for the star-star pairs, the colour differences for all pairs are smaller than unity. Stalin et al. (2004) report a detailed investigation quantifying the effect of colour differences, and they show that the effect of colour differences of this amount on DLCs will be negligible for a specific band (see also Carini et al. 1992).
We also used the star-star DLCs to identify any spikes in them (unusually sharp rise or fall of the DLC over a single time bin), assuming that the true star-star DLC is overall non-variable or at worst reflects small stellar oscillations. Such spikes may arise from improper removal of cosmic rays, cirrus clouds or some unknown instrumental cause. Such outliers can sometimes significantly alter the nominal statistics on short-term variations, especially when DLCs do not have enough data points. We typically have $\sim$30 individual temporal data points in our sample; see Table \[tab:res\], column 2. We removed such outliers if they were more than 3$\sigma$ from the mean, by applying a mean clip algorithm on the comparison star-star DLCs. In cases where we find any such outliers we have censored those time bin data points from our analysis of quasar-star DLCs as well. Only DLCs once freed from any such outliers have been used for carrying out our statistical analysis of microvariability. However, we should stress that such outliers in our comparison star-star DLCs were usually not present and never exceeded two data points.
Statistics to quantify microvariations {#subs:stat}
--------------------------------------
### C-test statistics {#subsubs:ctest}
To quantify microvariation of a DLC, by far the most commonly used statistic is the so-called $C$-statistic (e.g., Jang & Miller 1995; Romero et al. 1999). This technique uses a variability parameter $C$, which is an average of $C1$ and $C2$ with
$$C1 = \frac{\sqrt{Var(q-s1)}}{\sqrt{Var(s1-s2)}} \hspace{0.2 cm} {\rm
and} \hspace{0.2 cm} C2 = \frac{\sqrt{Var(q - s2)}}{\sqrt{Var(s1 - s2)}}.
\label{eq:cvalue}$$
Here $Var (q-s1)$, $Var (q-s2)$ and $Var (s1-s2)$ are the variance of observational scatters of the differential instrumental magnitudes of the quasar$-$star1, quasar$-$s2 and star1$-$star2, respectively. The normally adopted criterion to claim that variability is present is $C
\geq 2.576$, which corresponds to a nominal confidence level of $\ge
0.99$.
### F-test statistics {#subsubs:ftest}
Despite the very common use of these $C$-statistics, de Diego (2010) has pointed out it has severe problems. Because it considers the ratio of two standard deviations rather than of variances it does not describe a normally distributed variable and it is not properly centered with the mean expected value being zero; hence it is not a good statistic and de Deigo (2010) concludes that the nominal critical value for the presence of variability (i.e., 2.576) is usually too conservative.
Another statistical method that can be used to quantify the presence of microvariability is the $F$-test, which has been recently been shown to be a more powerful and reliable tool for detecting microvariability (de Diego 2010). The $F$ value is computed as $$\label{eq.ftest}
F_1=\frac{Var(q-s1)}{Var(s1-s2)} \nonumber \\
F_2=\frac{Var(q-s2)}{ Var(s1-s2)},$$ where $Var(q-s1)$, $Var(q-s2)$ and $Var(s1-s2)$ are the variances of the quasar-star1, quasar-star2 and star1-star2 DLCs, respectively. These $F$ values are then compared individually with the critical $F$ value, $F^{(\alpha)}_{\nu_{QS}\nu_{SS}}$, where $\alpha$ is the significance level set for the test, and $\nu_{QS}$ and $\nu_{SS}$ are the degrees of freedom of the quasar-star and star-star DLCs, respectively. The smaller the $\alpha$ value, the more improbable that the result is produced by chance. Thus values of $\alpha=$ 0.0001, 0.001 or 0.01 (the last assumed in our analysis) roughly correspond to $5\sigma$, 3$\sigma$ or a 2.6$\sigma$ detections, respectively. If $F$ is larger than the critical value, the null hypothesis (i.e., no variability) is discarded. Here we also note that having two F-values, $F_1$ and $F_2$, allows us two choices in deciding the presence of variability: (i) to take the average of $F_1$ and $F_2$, and compare it with critical $F$ value; (ii) to compare $F_1$ and $F_2$ separately with the critical $F$ value. We prefer the latter option, as it serves as further validation for the F-test; for if one DLC indicates variability and one doesn’t, these mixed signals bring into question the reality of the putative variability.
### Scaled F-test statistics {#subsubs:fscaled}
Although the $F$-test is certainly better than the $C$-test, it should be noted that for the $F$-test to give a truly reliable result the error due to random noise in the quasar–star and star-star DLCs should be of a similar order, apart from any additional scatter in the quasar-star DLC due to possible QSO variability. For instance, if both comparison stars are either brighter (fainter) than the monitored quasar, then a false alarm detection (non-detection) is possible due to the very small (large) photon noise variance of the star-star DLC compared to the quasar-star DLCs. This in practice can happen, as sometimes it is difficult to fulfill the desiderata of having non-variable comparison stars within the quasar CCD image frame that are very similar in magnitude to the QSO.
In our sample we have tried to choose non-variable comparison stars in proximity to the magnitude of the quasar (see Sect. \[subs:stars\]), but it was not possible to fulfill this requirement for all quasars. So sometimes in performing the $F$-test we may have to compare the variance of star-star DLCs involving stars substantially brighter than the quasar, where scatter due to photon noise is very small, with the noisier quasar-star DLC. In such cases, the standard statistics of the $F$-test do seem to give too much weight to even very nominal fluctuations in a quasar-star DLC. A sensible way to deal with this real problem is to scale the star-star variance by a factor, $\kappa$, which is proportional to the ratio of the noise in the quasar-star and star-star DLCs. One logical choice along these lines is to consider the ratio of the average squared error in the quasar-star and star-star DLCs i.e., $$\kappa=\left[\displaystyle{\frac{\sum_\mathbf{i=0}^{N}\sigma^2_{i,err}(q-s)/N}{\sum_{i=0}^{N}\sigma^2_{i,err}(s1-s2)/N}}\right] \equiv \frac{\langle\sigma^2(q-s)\rangle}{\langle\sigma^2(s1-s2)\rangle},
\label{eq:kappa}$$ where $\sigma^2_{i,err}(q-s)$ and $\sigma^2_{i,err}(s1-s2)$ are, respectively, the errors on individual points of the quasar-star and star-star DLCs, as returned by the DAOPHOT/IRAF routine. Then the scaled F-value, $F^{s}$, can be computed as, $$\label{eq.fstest}
F_{1}^{s}=\frac{Var(q-s1)}{\kappa Var(s1-s2)}, \nonumber \\
F_{2}^{s}=\frac{Var(q-s2)}{\kappa Var(s1-s2)}.$$
Here scaling the variance of the star-star DLC by $\kappa$ basically amounts to normalizing the variance of the DLC by the mean of the squared errors of its individual points (i.e., by $\langle\sigma^2\rangle$ in Eq. \[eq:kappa\]). This is sensible, as we know that for no intrinsic variability present in a light curve, the variance gives an estimate of the square of the mean errors (i.e., $\langle\sigma^2\rangle$) of the light curve. These $\langle\sigma^2\rangle$s of the light curves depend on the brightnesses of the observed objects, so to remove any effects of brightness on the variances (used in the standard F-test as Var(q$-$s1)/Var(s1$-$s2)) of light curves it should be better to use the variances that have been normalised by their $\langle\sigma^2\rangle$ values.
The value of scale factor, $\kappa$, used in the [*scaled*]{} F-test (Eq. \[eq.fstest\]) will be near unity if the quasar and stars are of similar magnitude, and as a result it will give a similar F-value as is given by the standard F-test (see Eq. \[eq.ftest\]). On the other hand, if both comparison stars are either brighter (fainter) than the monitored quasar, then $\kappa$ will be larger (smaller) than unity. As a result, $\kappa$ will reasonably scale the variance of comparison star-star DLCs for any magnitude difference between stars and quasar, and hence avoid the problem with the standard $F$-test which does seem to give too much weight to even very nominal fluctuations in a quasar$-$star DLC, when it is compared to brighter star$-$star DLCs.
Other alternatives to the standard F-test are the use of one-way analysis of variance (ANOVA) or a $\chi^2$ test (e.g., de Diego 2010). For an appropriate use of ANOVA the number of data points in the DLC needs to be large enough so as to have many points in each subgroup used for the analysis; however, this is not possible for our observations as we typically have only around 30 data points in our light curves. For the appropriate use of a $\chi^2$ test, the errors of individual data points need to have Gaussian distributions and those errors should be accurately estimated. It has been claimed in the literature that errors returned by photometric reduction routines in IRAF and DAOPHOT usually are underestimated, often by factors of 1.3–1.75 (Gopal-Krishna et al. 2003; Sagar et al. 2004; Bachev et al. 2005), which makes the use of a $\chi^2$ test less desirable for such real photometric light curves. However, as our scale factor depends on the ratio of average squared errors, this possible caveat does not affect our scaled $F$-test analysis.
In conclusion, we propose that by applying such scaling to the variance of the star-star DLC we can perform a [*scaled*]{} $F$-test, where our scale factor is designed so that it: (i) takes care of the difference in magnitude between the QSO and star in quasar-star and star-star DLCs; (ii) retains the requirement that both the variance being compared in a $F$-test should have a $\chi^2$ distribution, which is not the case in $C$-statistics; and (iii) cancels out the problem of uncertain error underestimation by DAOPHOT/IRAF routines reported by many other authors, in that our scaling factors depend on ratios of averaged squared errors. Therefore we report our final results based on this scaled $F$-test. However it is also worthwhile to compute $C$-values and standard $F$-values to facilitate the comparison of results for variability based on them with those based on our newly proposed [*scaled*]{} $F$-test.
[@ccc ccc ccc cc cc@]{} &[N]{} &[T]{} &[C-test]{} & & &[$\sqrt\kappa$[[^7]]{}]{} &[$\sqrt { \langle \sigma^2_{i,err} \rangle}$]{}\
& &[(hr)]{}&C-value &[$F_{1}$]{},[$F_{2}$]{}&[$F_{1}^{s}$]{},[$F_{2}^{s}$]{}&[$F_{c}(0.95)$]{}&[$F_{c}(0.99)$]{} &[C-test]{}&F-test &F$_s$-test &$\frac{}{}$ &(Q-S)\
(1)&(2) &(3) &(4) &(5) &(6) &(7)&(8)&(9) &(10) &(11) &(12)&(13)\
WFM91 0226$-$1024 & 31 & 4.04 &2.20, 2.47 & 4.84, 6.10& 1.80, 2.25& 1.84 & 2.39 & Pv,Pv& V,V&Nv,Pv& 1.64 & 0.02\
J073739.96$+$384413.2 & 40 & 5.54 &1.45, 1.83 & 2.11, 3.35& 1.73, 1.78& 1.70 & 2.14 & Nv,Nv& Pv,V&Pv,Pv& 1.24 & 0.01\
J084044.41$+$363327.8 & 33 & 3.86 &1.24, 0.84 & 1.54, 0.71& 1.02, 0.51& 1.80 & 2.32 & Nv,Nv&Nv,Nv&Nv,Nv& 1.21 & 0.01\
\
J084538.66$+$342043.6 & 32 & 3.90 &1.89, 1.99 & 3.57, 3.96& 1.07, 1.08& 1.82 & 2.35 & Nv,Pv& V,V&Nv,Nv& 1.87 & 0.01\
J090924.01$+$000211.0 & 33 & 3.50 &1.74, 1.56 & 3.04, 2.44& 2.35, 1.25& 1.80 & 2.32 & Nv,Nv& V,V& V,Nv& 1.27 & 0.02\
J094443.13$+$062507.4 & 25 & 2.79 &1.37, 1.65 & 1.87, 2.71& 2.80, 2.05& 1.98 & 2.66 & Nv,Nv& Nv,V& V,Pv& 0.98 & 0.01\
\
J094941.10$+$295519.2 & 36 & 3.50 &2.17, 1.82 & 4.71, 3.31& 2.56, 1.94& 1.76 & 2.23 & Pv,Nv& V,V& V,Pv& 1.33 & 0.01\
J100711.81$+$053208.9 & 33 & 3.88 &1.49, 1.80 & 2.23, 3.25& 0.46, 0.75& 1.80 & 2.32 & Nv,Nv& Pv,V&Nv,Nv& 2.14 & 0.04\
J111816.95$+$074558.1 & 36 & 3.74 &1.56, 1.73 & 2.44, 3.00& 1.77, 2.58& 1.76 & 2.23 & Nv,Nv& V,V& Pv,V& 1.13 & 0.01\
\
J112320.73$+$013747.4 & 42 & 3.48 &1.17, 1.21 & 1.38, 1.47& 1.12, 1.47& 1.68 & 2.09 & Nv,Nv&Nv,Nv&Nv,Nv& 1.05 & 0.01\
J120051.52$+$350831.6 & 30 & 3.86 &3.25, 3.24 &10.57,10.50& 1.64, 1.64& 1.86 & 2.42 & V,V& V,V&Nv,Nv& 2.53 & 0.02\
J120924.07$+$103612.0 & 39 & 4.10 &3.37, 3.06 &11.36, 9.34& 2.85, 2.41& 1.72 & 2.16 & V,V& V,V& V,V& 1.98 & 0.02\
\
J123820.19$+$175039.1 & 30 & 3.77 &1.97, 1.98 & 3.87, 3.92& 0.81, 0.74& 1.86 & 2.42 & Pv,Pv& V,V&Nv,Nv& 2.24 & 0.01\
J125659.92$+$042734.3 & 33 & 3.84 &2.32, 2.20 & 5.40, 4.84& 2.99, 2.68& 1.80 & 2.32 & Pv,Pv& V,V& V,V& 1.34 & 0.01\
J151113.84$+$490557.4 & 66 & 3.70 &1.36, 1.37 & 1.84, 1.88& 1.37, 1.79& 1.51 & 1.79 & Nv,Nv& V,V& Nv,V& 1.09 & 0.01\
\
J152350.42$+$391405.2 & 43 & 2.75 &1.94, 1.83 & 3.75, 3.35& 1.97, 1.96& 1.67 & 2.08 & Nv,Nv& V,V&Pv,Pv& 1.34 & 0.01\
J152553.89$+$513649.1 & 24 & 3.00 &1.16, 1.12 & 1.35, 1.25& 0.73, 1.05& 2.01 & 2.72 & Nv,Nv&Nv,Nv&Nv,Nv& 1.22 & 0.01\
J154359.44$+$535903.2 & 25 & 2.90 &1.94, 2.01 & 3.75, 4.05& 1.33, 1.64& 1.98 & 2.66 & Nv,Pv& V,V&Nv,Nv& 1.62 & 0.02\
J160207.68$+$380743.0 & 32 & 4.07 &2.12, 2.34 & 4.50, 5.48& 2.01, 2.08& 1.82 & 2.35 & Pv,Pv& V,V&Pv,Pv& 1.56 & 0.01\
Results {#sec:res}
=======
Differential light curves (DLCs) {#subsec:dlc}
--------------------------------
The R-band differential light curves (DLCs) of our sample are shown in Figures \[fig:s01to04\]-\[fig:s11to19\]. For each quasar the upper panel gives the star-star DLCs of the two best comparison stars and the two lower panels give the quasar-star DLCs. We first give brief notes on each individual source and then present our results based on all three statistical tests, the $C$-test, the [*standard*]{} $F$-test and the [*scaled $F$-test*]{}. As discussed in the previous section our final results will be based on scaled $F$-test but the results based on the other two tests will facilitate their intercomparisons, allowing us to discuss their relative merits.
Brief notes on individual sources
---------------------------------
### \[WFM91\] 0226$-$1024
\[WFM91\] 0226$-$1024 is a high ionization BAL (HiBAL) QSO, having balnicity index (BI) =7344 km s$^{-1}$, and detachment index (DI) = 4.72 km s$^{-1}$ (Weymann et al. 1991). This BALQSO was reported as a normal QSO and earlier photometric monitoring to search for its microvariability over $\sim$3.4 hr did not yield any positive microvariability detection (Bachev et al. 2005). Over our observational run of $\sim$ 4 hr a peak in the DLC is apparent by visual inspection. However, due to the rather high error bars in the DLCs, the $C$-test and the $F$-test have indicated this source is a possible variable and variable, respectively. But our scaled $F$-test shows it is not variable; nonetheless as the scaled $F$-test with one standard star has shown it as possible variable, this BALQSO is a prime candidate for additional monitoring. For the remainder of the sources we will not discuss the details of the differences between the different statistical tests, reserving a general discussion for the next subsection.
### J073739.96$+$384413.2
J073739.96$+$384413.2 is a Low ionization (LoBAL) QSO. We have monitored this source over a span of more than $\sim$ 5 hr. Statistical analysis of its DLC shows it is a probably variable source.
### J084044.41$+$363327.8
Becker et al. (1997) reported the discovery of this unusual LoBAL QSO. A spectropolarimetry study by Brotherton et al. (1997) reveals that it is a highly polarized BALQSO, with the continuum polarization rising steeply toward shorter wavelengths, while keeping a constant position angle in the continuum. This source was observed for $\sim$3.8 hours, but no evidence of microvariations were detected in its DLC.
### J084538.66$+$342043.6
The source, also known as CSO230, has a black hole with mass $16.4\times10^9M_{\odot}$ , estimated using its H$\beta$ broad line width (e.g., Yuan et al. 2003). It is HiBAL QSO having balnicity index of $2564\pm 1.17$ km s$^{-1}$, and absorption index (AI) of $4091\pm1.28$ km s$^{-1}$ (Trump et al. 2006). This source has been extensively studied spectroscopically. Barlow et al. (1992) has studied its spectral variability during four epochs over a 17-month time span. They found three distinct levels in the broad absorption lines of Si IV 1397Å and C IV 1549Å . A broad-band monitoring effort during this period showed that the continuum level remained constant to within 10 percent. The source remained non-variable during our observational run of $\sim$4 hr.
### J090924.01$+$000211.0
This is a HiBAL, with balnicity index of 71$\pm0.90$ km s$^{-1}$(Trump et al. 2006). This is binary quasar system (Hennawi et al. 2006). We observed this source for $\sim3.5$ hr. Statistical analysis of its DLC does not give a good indication of rapid variability according to the scaled F-test.
### J094443.13$+$062507.4
This is LoBAL quasar with balnicity index of 820$\pm0.53$ km s$^{-1}$(Trump et al. 2006). This source was found to be probably variable during the course of our $\sim$ 2.7 hr observation, which makes it a potential source for further microvariability study.
### J094941.10$+$295519.2
This source is a prime candidate for microvariability and an intensive search over a long time span (from 1993–1996) was performed by Gopal-Krishna et al. (2000) in their programme to search for intranight optical variability in RQQSOs. They found evidence of an $\sim$0.05 mag probable variation and marginal evidence of $\sim$0.03 mag variation over observations lasting 2.5 hr and 4.5 hr, respectively. In addition, Jang et al. (2005) monitored this source for two nights for durations of 3.9 hr and 2.0 hr respectively, but they did not find any sign of variability. Our scaled $F$-test analysis indicates that it possibly exhibited microvariability during our observation lasting $\sim$ 3.5 hr.
### J100711.81$+$053208.9
This is a HiBAL QSO with balnicity index of 2901$\pm0.88$ km s$^{-1}$ (Trump et al. 2006). For this source the quasar-star DLC is a noisier than usual. We did not find any signature of microvariability in its DLC during our observation of $\sim$ 3.8 hr. To reach a firmer conclusion as to its rapid variability this source merits additional observations.
### J111816.95$+$074558.1
PG 11514$+$081, also known as the “triple quasar”, was the second gravitational lens found (Kristian et al. 1993), to have three components with identical spectra (at a redshift of 1.722). Hubble Space Telescope observations resolved the system PG 11514$+$081 into four point sources and a red extended lens galaxy (Kristian et al. 1993). The source was observed for $\sim$ 3.7 hr and found to be a probably variable source, which makes it an excellent candidate for future microvariability investigations.
### J112320.73$+$013747.4
Meylan and Djorgovski (1989) reported that this quasar is probably lensed by a galaxy at z$\sim0.6$. The UV line profile structure found with the International Ultraviolet Explorer in this gravitational lens candidate indicates pronounced BAL structure in the high-ionization resonance lines of O VI 1033Å and N V 1240Å. Michalitsianos et al. (1997) performed a comparison of far-UV spectra, with data separated by nearly 10 months, that indicated that changes occurred in both absorption and ionization levels associated with BAL structure in the QSO. We found this source to be non-variable during our $\sim$ 3.4 hr observation.
### J120051.52$+$350831.6
This source is a HiBAL with a balnicity index of 4600$\pm2.48$ km s$^{-1}$ (Lamy et al. 2004). This source did not show any significant microvariation over an observational run of $\sim$3.8 hr and is non-variable according to the scaled $F$-test. Although the $C$-statistic showed it as a strong contender to have presented microvariability, that result appears to have been induced because of its relatively bright comparison stars, as discussed above.
### J120924.07$+$103612.0
Significant variations were noticed in the DLC over our observational run of $\sim$ 4 hr. Note that a coherent variability trend can be seen in both the quasar–star DLCs. Statistical analyses using the $C$-test, $F-$test and scaled $F-$test all strongly indicate the presence of microvariability.
### J123820.19$+$175039.1
This LoBAL is in the Large Bright Quasar Survey, and was also detected in the Chandra BAL quasar survey (Green et al. 2001). We did not find any signature of microvariation in its DLC over an observational period of $\sim$ 3.77 hr.
### J125659.92$+$042734.3
This source has been extensively studied for optical microvariability. Barbieri et al. (1984) did not find any signature of variability in their observations. In their search for intranight optical variability in RQQSOs. Gopal-Krishna et al. (2000) observed this source twice for 5 hr each time and on one of those nights, during which they had unfortunately sparse sampling, saw a hint of microvariation. We have investigated this source for $\sim$ 3.8 hr, and the statistical analysis of its DLCs showed clear evidence of microvariability.
### J151113.84$+$490557.4
This LoBAL quasar has a balnicity index of 802$\pm1.33$ km s$^{-1}$ (Trump et al. 2006). We observed this source for $\sim$ 3.5 hr but found no overall evidence of microvariability, although one star-QSO DLC was nominally variable.
### J152350.42$+$391405.2
This is a LoBAL QSO having a balnicity index of 7147$\pm1.66$ km s$^{-1}$ (Trump et al. 2006). This bright quasar was found in the third Hamburg Quasar Survey (Hagen et al. 1999). This source appeared to be variable in a 20 cm radio study (Becker et al. 2000). We found it to be probably variable over the course of an observing run of $2.7$ hr.
### J152553.89$+$513649.1
This source, also known as CSO 755, is a strongly polarized ($\sim3.9$ per cent) BALQSO (Glenn et al. 1994). A strongly polarized continuum and unpolarized emission lines indicate that its polarization arises by scattering very near the central source (Glenn et al. 1994). XMM-Newton spectroscopy of this luminous quasar gives a photon index of $\Gamma =1.83^{+0.07}_{-0.06}$ and a flat (X-ray bright) intrinsic optical-X-ray spectral slope of $\alpha _{ox}=-1.51$ (Shemmer et al. 2005). The source shows evidence for intrinsic absorption, having a column density of N(H) $\sim 1.2 \times 10^{22}$ cm$^{-2}$. This is among the lowest X-ray columns measured for a BALQSO (Shemmer et al. 2005). We detected no signature of microvariability over a short run of $\sim$ 2.9 hr.
### J154359.44$+$535903.2 {#notes:J154359}
J154359.3$+$535903 is also known as SBS 1542$+$541 as this source was discovered in the Second Byurakan Survey (Stepanyan et al. 1991). It has many interesting properties: its BAL has a very high degree of ionization (Telfer et al. 1998), an associated absorption system and damped Ly$\alpha$ (DLA) absorption system, and a strong X-ray absorption (Green et al. 2001). This bright high-redshift HiBAL QSO ‘has very highly ionized BALs (including O VI, Ne VIII, and Si XII; Telfer et al. 1998) and appears to have an X-ray brightness typical for a non-BAL of its optical luminosity. Bechtold et al. (2002) has found intervening metal absorption systems at $z$ = 1.41, 0.1558, and 0.72 along its line of sight. We found this source to be non-variable during our observation of $\sim 4$ hr.
### J160207.68$+$380743.0
This source was continuously observed for $\sim3.7$ hr. We found this as a probably variable source, which makes this source another potentially good candidate for microvariability studies in the future.
Variability results based on different statistical test {#subsec:resvar}
-------------------------------------------------------
The results of our analysis are summarized in Table \[tab:res\]; we applied both the $C$-statistic and the scaled $F$-test, as discussed above (e.g., see Sect \[subs:stat\]). In the first three columns we list the object name, number of data points ($N_{points}$) used in the DLC and the duration of our observation. The fourth column lists the pair of $C$-values based on star1 and star2 (Eq. \[eq:cvalue\]) while the fifth and sixth columns list the pair of $F$-values in the standard and scaled $F$-test. Columns 7 and 8, respectively, give $F_{c}$ for 0.95 and 0.99 confidence levels. Columns 9, 10 and 11 respectively, list the pairs of variability statuses using star1 and star2 based on $C$-statistics, the standard $F$-test and the scaled $F$-test. The status, based on both star1 and star2 are listed separately rather than using their average value so as to impose as additional validation: is the variability status based on individual stars are consistent with one another or not? In these pairs of variable status indicators using a quasar-star DLC, the quasar is marked as variable (‘V’) for a $C$-value $\ge 2.576$ or $F$-value $\ge
F_{c}(0.99)$, which corresponds to a confidence level $\ge 0.99$. The quasar is marked as ‘probably variable’ (Pv) if the $C$ value of quasar-star DLC is in the range 1.950 to 2.576 or if the $F$-value is between $F_{c}(0.95)$ and $F_{c}(0.99)$. Those sources for which the $C$-values are less than 1.95, or the $F$-value are less than $F_{c}(0.95)$ are marked Non-variable (‘Nv’). Column 12 lists the square root of scaling factor, $\sqrt\kappa$, where it is computed by $\kappa=\langle\sigma^2(q-s)\rangle/\langle\sigma^2(s1-s2)\rangle$ (as in Eq. \[eq:kappa\]), and has been used to scale the variance of the star-star DLCs while computing the $F$-value in the scaled-$F$-test. The last column gives our photometric accuracy, [$\sqrt { \langle \sigma^2_{i,err} \rangle}$ ]{} in the quasar-star DLCs, which typically are between 0.01$-$0.02mag.
As can be seen from Columns 9 – 11 of Table \[tab:res\] the variability status indicators based on quasar-star1 and quasar-star2 are often not consistent with one another. The importance of our choice to mark the variable status separately based on individual star vs quasar DLCs can be illustrated by taking the example of J094941.10+295519.2. Based on the $C$-test its DLC with respect to star1 shows it as a probable variable but with star2 as non-variable. The standard $F$-test terms it as variable based on both star1 and star2 DLCs. However, this QSO’s status using the scaled $F$-test is variable based on the first star and probably variable using the second star. The average of the scaled $F$-value for this source comes out to be 2.25, which is just above the critical $F$-value of 2.23 for 0.99 confidence, and hence it would be classified as a variable source if we used that average criterion. However, an examination the DLC of this source in top right panel of Fig. \[fig:s05to10\] by eye indicates that there is no variation that can defined coherently by more than 2 points. Therefore, to exclude such questionable variability and to be on the conservative side for unambiguous microvariable detection, only those sources should be termed as variable for which both quasar-star1 and quasar-star2 DLCs mark the source as variable (i.e ‘V,V’ in Table \[tab:res\]). Probably variable sources are taken as those for which either both the status are of probable variable (i.e., ‘Pv,Pv’ in Table \[tab:res\]) or one quasar-star DLC marks it as a probable variable and the other as a variable (i.e., ‘Pv,V’ or ‘Pv,V’ in Table \[tab:res\]). Sources termed as non-variable (‘Nv’) are those for which at least one of the status based on quasar-star1 and quasar-star2 DLC marked them as non-variable (i.e., at least one ‘Nv’ status in Table \[tab:res\]).
Column (9) of Table \[tab:res\] indicates that the $C$-statistics shows two sources as variable and four as probably variable. The scaled $F$-test shows two sources as variable and six sources as probably variable. As we have discussed above (in section \[subsubs:fscaled\]), that scaled $F$-test is better for our work (and probably also better in many observations made by others) than the standard $F$-test due to differences between the magnitudes of the quasars and their comparison stars. This is also evident from column (10) of Table \[tab:res\] which shows that the standard $F$-test would give 13 sources as variable and two as a probably variable, indicating that this test certainly suffers from the problem related to small variances of the brighter star-star DLCs, at least for our sample of BALQSOs.
Although the $C$-test and the [*scaled*]{} $F$-test both give two sources as variable, it is not difficult to appreciate the scaled $F$-test merits over the $C$-test by taking specific examples. For instance, J120051.52$+$350831.6 is a BALQSOs with a $C$-value of 3.25 from quasar-star1 DLC and 3.24 from quasar-star2 DLC, which seems to make it a clear case of INOV detection, particularly since the $C$-statistic is usually conservative. However, by looking at the DLCs for this source in the top left panel of Fig. \[fig:s11to19\], it is clear even by eye that: (i) there is likely to have been a random fluctuation (not a coherent one) for the last 9 points of the DLCs; and (ii) its comparison stars are about 1.5 mag brighter than this quasar, which makes the variance of the star-star DLC very small (due to small photon noise). As a result the $C$-value will be artificially very high, leading to false detections. This flaw also crops up in the standard $F$-test, but is eliminated in the scaled $F$-test which termed this BALQSO as a non-variable source (not even probably variable). Another source, J120924.07$+$103612.0, has a $C$-value of 3.37 from the quasar-star1 DLC and 3.06 from the quasar-star2 DLC but these are probably so high because of the $\sim$1.2 mag brighter comparison stars; however, this BALQSO also shows a coherent variability trend (even by eye), and is also termed as variable by the scaled $F$-test. These empirical examples, and the fact that the scaled $F$-test detects two cases of unambiguous variability in comparison to the $C$-test which makes only one unambiguous detections (after eliminating the false positive case mentioned above) clearly shows that the scaled $F$-test, beside being more sensitive than the $C$-test to small amplitude variability, is also sufficiently robust to eliminate nearly any false alarm detections.
Therefore, finally, we rely on the result given by the scaled $F$-test, by which we find two unambiguous detections of microvariability in our sample of 19 BALQSOs up to an accuracy of 0.01-0.02mag (see columns 9,10, 11 and 13 of Table \[tab:res\]). As a result, our sample shows that about 10-11 per cent of BALQSOs (i.e., 2 out of 19 sources) certainly showed microvariability (at a confidence level of 0.99).
Discussion and Conclusions
==========================
As noted in the introduction, there have been rather extensive examinations of the frequency of optical microvariability for RQQSOs as well as blazars and other RLQSOs. The typical duty cycle (DC) for blazars is 60–65 per cent (e.g., Gupta et al. 2005), while for normal quasars it has been found to be around 20–25 per cent (e.g., Carini et al. 2007). For both these classes the number of sources in each total sample was quite large, so these values should be reasonably reliable, and support the hypothesis that most of these rapid variations arise, or at least are amplified, in the relativistic jets (e.g., Jang & Miller 1995; Gopal-Krishna et al. 2003). The interesting class of radio-quiet BALQSOs was reported to have a 50 per cent DC but this sample had only 6 members (Carini et al. 2007). Therefore, one of the reasons for the difference in DC results could be poor statistics in the previous study and better statistics now with a sample about a factor of three larger. Apart from sample size, some of the difference might be due to differences in the typical length of the observation. As long known, lengthier observations of blazars are more likely to reveal variability (e.g., Carini 1990), which was also later shown to be the case for RQQSOs (Gupta et al. 2005, Carini et al. 2007). Carini et al. (2007) found that RQQSOs that were monitored for about 6-7 hr showed the highest fraction of microvariability, typically around 24 per cent. In the 4 hour observation range, which is where most of our observations fall, less than 10 percent of sources were found to have microvariability. Therefore the fact that we found about 10-11% DC for radio-quiet BALQSOs is in agreement with the results from the literature for other radio-quiet non-BALQSOs indicates that the BAL nature does not have an effect on the presence of microvariability.
In addition, as we discussed in Sections \[subs:stat\] and \[subsec:resvar\], this DC fraction also depends on what statistical test has been used to decide on the significance of microvariation (see de Diego 2010 for details). Most of these previous studies have used the $C$-test, which has been shown recently to be an unreliable and usually too conservative method to detect microvariation, when compared to proper statistics such as the $F$-test (de Diego 2010). So, to allow comparison with earlier results, we also computed duty cycles (DCs) of BALQSOs in our sample using the $C$-test, which shows only one out of 19 source as variable (excluding one false detection, as discussed in Sec. \[subsec:resvar\]), resulting in a DC of about 5 per cent. As a result the DCs of all classes of AGN may increase if their DLCs are analyzed with the scaled $F$-test rather than with the usually more conservative $C$-test, which, for the 4 hour observation range, were reported at less than 10 percent for RQQSOs (Carini et al. 2007). Therefore, after taking into account the observation length and the dependence on statistical test used, it seems that the DC of radio-quiet BALQSOs is likely to be of a similar value to the DC of non-BAL RQQSOs, but without redoing all past analyzes with the scaled $F$-test we cannot be certain of this assertion. Apart from this comparison using the $C$-test, all our final results and conclusions are based on the more reliable scaled $F$-test (see Sect. \[subsec:resvar\]), which gives our new result of an approximately 11 per cent DC for radio-quiet BALQSOs.
The phenomenon of microvariability was first noticed for blazars, and for them microvariability almost certainly arises from a relativistic jet. However, given the lower DCs for RQ AGN it is still unclear if the nature of intranight variability is the same in these objects, or if it arises from processes in the accretion disc itself (e.g., Mangalam & Wiita 1993; Chakrabarti & Wiita 1993), and thus could possibly be used to probe the accretion disc (e.g., Gopal-Krishna et al. 2000 and references therein). Recent modeling suggests that even for RQQSOs, jet based models should be the most efficient way to produce microvariability (e.g., Czerny et al. 2008). Such jet-based models should predict a difference in equivalent widths of emission lines of variable and non-variable sources, as they should be smaller in the former, due to its dilution by jet components. However, recently this hypothesis was shown to be unlikely based on an analysis of spectra of a set of RQQSOs that had already been searched for microvariability (Chand et al. 2010). Some other possibilities include disc based models where variation can be due to: variable hard radiation from near the disc center that is reprocessed into the UV-optical region (e.g., Ulrich et al. 1997); instabilities in the accretion flow itself producing multiple hot-spots (e.g., Mangalam & Wiita 1993); disco-seismological modes within the disc (e.g. Nowak & Wagoner 1991, 1992). In the first case above the variable hard radiation instead may also come from a hot corona above the accretion disc (Merloni & Fabian 2001).
The shortest variability time scale in this scenario can be associated with the light-crossing time, which will be larger for higher central black hole masses (e.g see Bachev et al. 2005). In a scenario with instabilities in an accretion flow, if one assumes that the inner part of the flow operates through an optically thin advective mode, then the border between it and outer thin disc may be good candidate for the region where these instabilities may occur (e.g., Gracia et al. 2003; Krishan, Ramadurai & Wiita 2003). The time scale may also be associated with the much longer accretion time scale. Observational estimation of such variability time scales could possibly be used to distinguish between above various disc based scenarios. However, for luminous QSOs such as those in our sample that should have black hole masses in excess of $10^8$ M$_{\odot}$, even the fastest disc based time scale can be a few hours. As our monitoring of each source rarely exceeded 4 hours and was sometimes unevenly sampled, it is not possible for us to obtain reliable estimates of the variability time scales for the minority of variable sources. Therefore we are not able to distinguish between the various disc based scenarios mentioned above, nor can we cleanly distinguish between jet based and disc based models.
Our larger sample (a factor of three improvement) of radio-quiet BALQSOs, aided by more robust detection criteria, have allowed us to conclude that the fraction of radio-quiet BALQSO showing microvariability are about 11% for an observation length of about 4 hr. This new DC of 11% is similar to the usual low microvariability fraction of normal RQQSOs with observation length similar to our observation, though we note those DCs were obtained using the $C$-statistic and not our scaled $F$-test. This similarity in microvariability frequency provides some support for models where radio-quiet BALQSO do not appear to be a special case of the RQQSOs.
Further extension this type of study to radio-loud BALQSOs will be important in obtaining good values for the microvariability percentage for all types of BALQSOs. Such additional observations will also help in understanding that whether we are viewing BALQSOs closer to the disc plane or to the perpendicular to the disc. In the latter case higher DCs are expected assuming that the cause of microvariation is related to the relativistic jet. In addition, to investigate whether and how X-ray and optical microvariability are correlated in BALQSOs, it will be useful to carry out future simultaneous multi-color optical and X-ray monitoring observations (e.g., Ram[í]{}rez et al. 2010). Finally, to ascertain whether or not such microvariation depends on spectral properties, additional optical and X-ray spectral analyses of the BALQSOs already searched for microvariability could place important constraints on the possible origin of such microvariations.
Acknowledgments {#acknowledgments .unnumbered}
===============
The help rendered by the observer at the 1.04-m ARIES telescope, Nainital, and technical staff at the 2.01-m HCT CREST is gratefully acknowledged. We would like to thank the anonymous referee for constructive comments on the manuscript.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
Adelman-McCarthy J. K., et al., 2007, ApJS, 172, 634 Bachev R., Strigachev A., Semkov E., 2005, MNRAS, 358, 774 Barbieri C., Romano G., 1984, AcA, 34, 117 Barlow T. A., Junkkarinen V. T., Burbidge E. M., Weymann R. J., Morris S. L., Korista K. T., 1992, ApJ, 397, 81 Bechtold J., Dobrzycki A., Wilden B., Morita M., Scott J., Dobrzycka D., Tran K.-V., Aldcroft T. L., 2002, ApJS, 140, 143 Becker R. H., Gregg M. D., Hook I. M., McMahon R. G., White R. L., Helfand D. J., 1997, ApJ, 479, L93 Becker R. H., White R. L., Gregg M. D., Brotherton M. S., Laurent-Muehleisen S. A., Arav N., 2000, ApJ, 538, 72 Brotherton M. S., Tran H. D., van Breugel W., Dey A., Antonucci R., 1997, ApJ, 487, L113 Carini M. T., 1990, PhD Thesis, Georgia State Univ. Carini M. T., Miller H. R., Noble J. C., Goodrich B. D., 1992, AJ, 104, 15 Carini M. T., Noble J. C., Taylor R., Culler R., 2007, AJ, 133, Chand H., Wiita P. J., Gupta A. C., 2010, MNRAS, 402, 1059 Chakrabarti S. K., Wiita P. J., 1993, ApJ, 411, 602 Czerny B., Siemiginowska A., Janiuk A., Gupta A. C., 2008, MNRAS, 386, 1557 de Diego J. A., 2010, AJ, 139, 1269 de Diego J. A., Dultzin-Hacyan D., Ramirez A., Benitez E., 1998, ApJ, 501, 69 Elvis M., 2000, ApJ, 545, 63 Ghosh K. K., Punsly B., 2007, ApJ, 661, L139 Gibson R. R., et al., 2009, ApJ, 692, 758 Glenn J., Schmidt G. D., Foltz C. B., 1994, AAS, 26, 875 Gopal-Krishna, Sagar R., Wiita P. J., 1993a, MNRAS, 262, 963 Gopal-Krishna, Wiita P. J., Altieri B., 1993b, A&A, 271, 89 Gopal-Krishna, Gupta A. C., Sagar R., Wiita P. J., Chaubey U. S., Stalin C. S., 2000, MNRAS, 314, 815 Gopal-Krishna, Stalin C. S., Sagar R., Wiita P. J., 2003, ApJ, 586, L25 Goyal A., Gopal-Krishna, Joshi S., Sagar R., Wiita P. J., Anupama G. C., Sahu D. K., 2010, MNRAS, 401, 2622 Gracia J., Peitz J., Keller C., Camenzind M., 2003, MNRAS, 344, 468 Green P. J., Aldcroft T. L., Mathur S., Wilkes B. J., Elvis M., 2001, ApJ, 558, 109 Gupta A. C., Joshi U. C., 2005, A&A, 440, 855 Gupta A. C., Fan J. H., Bai J. M., Wagner S. J., 2008, AJ, 135, 1384 Hagen H.-J., Engels D., Reimers D., 1999, A&AS, 134, 483 Hennawi J. F., et al., 2006, AJ, 131, 1 Hewett P. C., Foltz C. B., 2003, AJ, 125, 1784 Jang M., 2005, Ap&SS, 295, 397 Jang M., Miller H. R., 1995, ApJ, 452, 582 Krishan V., Ramadurai S., Wiita P. J., 2003, A&A, 398, 819 Kristian J., et al., 1993, AJ, 106, 1330 Lamy H., Hutsem[é]{}kers D., 2004, A&A, 427, 107 Mangalam A. V., Wiita P. J., 1993, ApJ, 406, 420 Marscher A. P., Gear W. K., Travis J. P., 1992, in Valtaoja E., Valtonen M., eds, Variability of Blazars. Cambridge Univ. Press, Cambridge, p. 85 Merloni A., Fabian A. C., 2001, MNRAS, 321, 549 Meylan G., Djorgovski S., 1989, ApJ, 338, L1 Michalitsianos A. G., Falco E. E., Munoz J. A., Kazanas D., 1997, ApJ, 487, L117 Miller H. R., Carini M. T., Goodrich B. D., 1989, Nature, 337, 627 Montagni F., et al., 2006, A&A, 451, 435 Nowak M. A., Wagoner R. V., 1991, ApJ, 378, 656 Nowak M. A., Wagoner R. V., 1992, ApJ, 393, 697 Rabbette M., McBreen B., Smith N., Steel S., 1998, A&AS, 129, 445 Ram[í]{}rez A., Dultzin D., de Diego J. A., 2010, ApJ, 714, 605 Rani B., Gupta A. C., Joshi U. C., Ganesh S., Wiita P. J., 2010, ApJ, 719, L153 Reichard T. A., et al., 2003, AJ, 126, 2594 Romero G. E., Cellone S. A., Combi J. A., 1999, A&AS, 135, 477 Sagar R. 1999, Curr. Sci, 77 , 643 Sagar R., Stalin C. S., Gopal-Krishna, Wiita P. J., 2004, MNRAS, 348, 176 Scaringi S., Cottis C. E., Knigge C., Goad M. R., 2009, MNRAS, 399, 2231 Schneider D. P., et al., 2005, AJ, 130, 367 Schneider D. P., et al., 2007, AJ, 134, 102 Schneider D. P., et al., 2010, yCat, 7260, 0 Shemmer O., Brandt W. N., Gallagher S. C., Vignali C., Boller T., Chartas G., Comastri A., 2005, AJ, 130, 2522 Stalin C. S., Gopal Krishna, Sagar R., Wiita P. J., 2004, JApA, 25, 1 Stepanyan D. A., Lipovetskii V. A., Chavushyan V. O., Erastova L. K., Shapovalova A. I., 1991, Ap, 34, 163 Stetson P. B., 1992, ASPC, 25, 297 Telfer R. C., Kriss G. A., Zheng W., Davidsen A. F., Green R. F., 1998,ApJ, 509, 132 Trump J. R., et al., 2006, ApJS, 165, 1 Ulrich M. H., 1997, LNP, 487, 329 V[é]{}ron-Cetty M.-P., V[é]{}ron, P. 2006, A&A, 455, 773 Weymann R. J., Morris S. L., Foltz C. B., Hewett P. C., 1991, ApJ, 373, 23 Yuan M. J., Wills B. J., 2003, ApJ, 593, L11
\[lastpage\]
[^1]: E-mail: ravi@aries.res.in (RJ); hum@aries.res.in (HC); acgupta30@gmail.com (ACG); wiitap@tcnj.edu (PJW)
[^2]: Ratio of the radio \[5 GHz\] flux to the optical \[2500Å\] flux taken from SDSS DR7 (Schneider et al. 2010); ND means no radio detection.
[^3]: BAL type: for HiBAL, LoBAL see text; MiBAL = Mini Broad Absorption Line Quasar.
[^4]: References: (1) Gibson et al. (2009); (2) Scaringi et al. (2009); (3) Weymann et al. (1991); (4) Trump et al. (2006)
[^5]: http://www.iiap.res.in/$\sim$iao
[^6]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^7]: Here $\kappa=\langle\sigma^2(q-s)\rangle/\langle\sigma^2(s1-s2)\rangle$ (as in Eq. \[eq:kappa\]), is used to scale the variance of star-star DLCs for the scaled F-test.
|
---
abstract: 'The Border Gateway Protocol (BGP) is globally used by Autonomous Systems (ASes) to establish route paths for IP prefixes in the Internet. Due to the lack of authentication in BGP, an AS can *hijack* IP prefixes owned by other ASes (i.e., announce illegitimate route paths), impacting thus the Internet routing system and economy. To this end, a number of hijacking detection systems have been proposed. However, existing systems are usually third party services that -inherently- introduce a significant delay between the hijacking detection (by the service) and its mitigation (by the network administrators). To overcome this shortcoming, in this paper, we propose ARTEMIS, a tool that enables an AS to *timely* detect hijacks on its own prefixes, and *automatically* proceed to mitigation actions. To evaluate the performance of ARTEMIS, we conduct real hijacking experiments. To our best knowledge, it is the first time that a hijacking detection/mitigation system is evaluated through extensive experiments in the real Internet. Our results (a) show that ARTEMIS can detect (mitigate) a hijack within a few seconds (minutes) after it has been launched, and (b) demonstrate the efficiency of the different control-plane sources used by ARTEMIS, towards monitoring routing changes.'
author:
- |
Pavlos Sermpezis, Gavriil Chaviaras, Petros Gigis, and Xenofontas Dimitropoulos\
\
title: 'Monitor, Detect, Mitigate: Combating BGP Prefix Hijacking in Real-Time with ARTEMIS'
---
Introduction {#sec:intro}
============
The inter-domain routing in the Internet takes place over the -globally adopted- Border Gateway Protocol (BGP) [@BGPv4]. Autonomous Systems (ASes) use BGP to advertise routing paths for IP prefixes to their neighboring ASes. Since BGP is a distributed protocol and authentication of advertised routes is not always feasible, it is possible for an AS to advertise illegitimate route paths for IP prefixes. These paths can propagate and “infect" many ASes, or even the entire Internet, impacting thus severely the Internet routing system and/or economy [@hijack-YouTube; @hijack-ChinaTelecom; @hijack-BitCoins; @Ramachandran-BGP-spammers-CCR-2006; @Vervier-mind-blocks-NDSS-2015].
This phenomenon, called *BGP prefix hijacking*, is frequently observed [@Vervier-mind-blocks-NDSS-2015], and usually caused by router misconfigurations [@hijack-YouTube; @hijack-ChinaTelecom] or malicious attacks [@hijack-BitCoins; @Ramachandran-BGP-spammers-CCR-2006; @Vervier-mind-blocks-NDSS-2015]. Some examples of real BGP hijacking cases include: (a) a Pakistan’s ISP in 2008, due to a misconfiguration, hijacked the YouTube’s prefixes and disrupted its services for more than $2$ hours [@hijack-YouTube]; (b) China Telecom mistakenly announced $\sim37000$ IP prefixes (corresponding to $15\%$ of the whole BGP table) in 2010, causing routing problems in the Internet [@hijack-ChinaTelecom]; and (c) hackers performed several hijacking attacks, through a Canadian ISP, to redirect traffic and steal thousands dollars worth of bitcoins in 2014 [@hijack-BitCoins].
To prevent prefix hijackings, several proactive mechanisms for enhancing the BGP security have been proposed [@Kent-secure-BGP-JSAC-2000; @Subramanian-listen-whisper-NSDI-2004; @bgpsec-specification-2015; @rpki-rfc; @Karlin-PGBGP-ICNP-2006]. These mechanisms need to be globally deployed to be effective. However, despite the standardization efforts [@bgpsec-specification-2015; @rpki-rfc], their deployment is slow due to political, technical, and economic challenges, leaving thus the Internet vulnerable to BGP hijacks.
Therefore, currently, reactive mechanisms are used for defending against prefix hijackings: after a hijacking is detected, network administrators are notified (e.g, through mailing lists [@Shi-Argus-IMC-2012], or dedicated services [@Lad-Phas-Usenix-2006]), in order to proceed to manual actions towards its mitigation (e.g., reconfigure routers, or contact other ASes to filter announcements). A number of systems have been proposed for detecting prefix hijacking, based on control plane (i.e., BGP data) and/or data plane (i.e,. pings/traceroutes) information [@Chi-cyclops-CCR-2008; @Lad-Phas-Usenix-2006; @bgpmon; @Zhang-Ispy-CCR-2008; @Zheng-lightweight-hijacks-2007; @Shi-Argus-IMC-2012; @Hu-accurate-hijacks-SP-2007]. Most of them, are designed to operate as third-party services (external to an AS) that monitor the Internet, and upon the detection of a suspicious incident, notify the involved ASes. Although this approach has been shown to be able to detect suspicious routing events in many cases, two main issues still remain unsolved: (i) the detection might not be accurate, since the suspicious routing events might not correspond to hijacks, but be caused by, e.g., traffic engineering; and (ii) the mitigation is not automated, increasing thus significantly the time needed to resolve a hijack.
In this paper, we propose a reactive mechanism/system, which we call ARTEMIS (*Automatic and Real-Time dEtection and MItigation System*)[^1], that aims to be operated by an AS itself, rather than a third-party, to timely detect and mitigate hijackings against its *own* prefixes in an automatic way. ARTEMIS (i) exploits the most recent advances in control-plane monitoring to detect in near real-time prefix hijackings, and (ii) immediately proceeds to their automatic mitigation (Section \[sec:artemis\]).
We then conduct several real hijacking experiments in the Internet using the PEERING testbed and analyze the effect of various network parameters (like, type of hijacking, hijacker / defender-AS location and connectivity) on the performance of ARTEMIS. We show that it is possible to detect and mitigate prefix hijacking within [*few seconds*]{} from the moment the offending announcement is first made. This is a major improvement compared to present approaches, which require slow procedures, like manual verification and coordination. The timely mitigation of ARTEMIS, prevents a hijacking from spreading to just, e.g., 20%-50% of the ASes that would be affected otherwise (Section \[sec:evaluation\]).
Finally, we discuss related work in hijacking detection systems and measurement studies, and compare it to our study, in order to highlight the new capabilities that are offered with ARTEMIS (Section \[sec:related\]). We conclude our paper by discussing the potential for future applications and extensions of ARTEMIS (Section \[sec:conclusion\]).
ARTEMIS {#sec:artemis}
=======
In this section, we first present the different sources that are used by ARTEMIS for control-plane *monitoring* (Section \[sec:sources\]), and then describe the *detection* (Section \[sec:detection\]) and *mitigation* (Section \[sec:mitigation\]) services.
Control-Plane Data Sources {#sec:sources}
--------------------------
For the monitoring service, ARTEMIS combines multiple control-plane sources to (a) accelerate the detection of a hijacking (i.e., minimum time of all sources), and (b) have a more complete view of the Internet (i.e., from the vantage points of all the sources). ARTEMIS receives control-plane information from publicly available sources, namely, the BGPmon tool [@bgpmon], the live-streaming service of RIPE-RIS [@ripe-ris-real-time], and the Periscope platform [@periscope].
ARTEMIS supports the BGPstream tool [@bgpstream] as well. However, during our experiments, the [BGPstream]{} service was unavailable, and, thus, we do not use it in this paper.
In the following, we present the main features of these control-plane sources.
**BGPmon [@bgpmon]** is a tool that monitors BGP routing information in real-time. It is connected to, and collects BGP updates and routing tables (RIBs) from BGP routers of: (a) the RouteViews sites and (b) a few dozen of peers around the world; at the time we conducted our study, BGPmon had $43$ vantage points, in total. BGPmon provides the live BGP data, as an XML stream.
**RIPE RIS streaming service [@ripe-ris-real-time].** The RIPE’s Routing Information System (RIS) is connected to route collectors (RCs) in several locations around the world, and collects BGP updates and RIBs. In the standard RIPE RIS [@ripe-ris], the data can be accessed via the raw files (in MRT format) or RIPEstat. The delay for BGP updates is $\sim 5min$ and $\sim 8h$ for RIBs. Recently, RIPE RIS offers a streaming service [@ripe-ris-real-time] that provides live information from $4$ RCs via websockets. The live streaming service of RIPE RIS, which we use in ARTEMIS, has currently $3$ RCs in Europe and $1$ RC in Africa; all of them are located in large IXPs.
**Periscope [@periscope]** is a platform that provides a common interface for issuing measurements from Looking Glass (LG) servers. Through Periscope, a user can send a command to a number of chosen LGs to ask for control-plane (*show ip bgp*) or data-plane (*traceroute/ping*) information. The status and the output of the measurements can be retrieved at any time (even before its completion). Periscope currently provides access to $1691$ LG servers.
**BGPstream [@bgpstream]** is an open-source framework for live (and historical) BGP data analysis. It enables users to quickly inspect raw BGP data from the command-line, or through a Python and C/C++ API. BGPstream provides live access to RouteViews and RIPE RIS data archives. While the delay of acquiring the data from these two services is considerable ($5$min and $15$min, respectively, for BGP updates), BGPstream recently introduced a service for live access to a stream of BGP data from BMP-enabled RouteViews collectors (with only $\sim 1$min delay). In total, BGPstream receives data from $76$ route collectors, from all its providers.
Prefix Hijacking Detection with ARTEMIS {#sec:detection}
---------------------------------------
The detection service of ARTEMIS aims to detect hijacks in (i) *real-time* and (ii) *without false positives*, while monitoring the (iii) *entire Internet* in a (iv) *light-weight* fashion.
The detection service continuously receives from the $3$ control-plane sources (see Section \[sec:sources\]) information about the BGP route paths for the monitored prefixes, as they are seen at the different vantage points (e.g., route collectors, LG servers). This routing information is compared with a local file that defines the legitimate origin-ASNs for each IP prefix that is owned by the operator of ARTEMIS; any violation denotes a hijacking. Since operator has full knowledge on the legitimate origin-ASNs for its prefixes, the detection service returns *no false positives*.
With the combination of $3$ sources, the detection can take place when an illegitimate route path is received by any of the sources. This is always faster than using only one source, and can decrease the time needed for detection. Using multiple sources gives also the possibility to benefit from the *large number of vantage points* they have around the globe. This is important, because a hijacking might affect only a part of the Internet, due to BGP policies and shortest-path routing [@Shi-Argus-IMC-2012; @Lad2-Understanding-resiliency-hijacks-DSN-2007].
Finally, ARTEMIS aims to impose limited load on the used third-party services, so that potentially 100s ASes (that run ARTEMIS) could use them in parallel. ARTEMIS needs to receive only the data (i.e., the part of the BGP tables, or specific BGP updates) that correspond to the local prefixes. As a result, *the imposed load is low*, since (a) BGPmon and RIPE RIS (as well as, BGPstream) are services/tools designed and optimized to provide streams of live data to many users simultaneously, (b) and Periscope has already a limit in the rate of requests to avoid overloading of LG servers. Similarly, *the consumption of network resources is very low*, allowing thus a single AS to monitor many prefixes.
Automatic Prefix Hijacking Mitigation {#sec:mitigation}
-------------------------------------
The goals of a mitigation mechanism are to be (i) *fast* and (ii) *efficient*, and (iii) have *low impact* on the Internet routing system.
Currently mitigation relies on manual actions, e.g., after a network administrator is notified for a prefix hijacking, she proceeds to reconfiguration of the BGP routers, or contacts other administrators to filter the hijacker’s announcements. As it becomes evident, this manual intervention introduces a significant delay (e.g., in the YouTube hijacking incident in 2008 [@hijack-YouTube], a couple of hours were needed for the mitigation of the problem). Hence, our primary focus is to accelerate the mitigation. To this end, we implement an automatic mitigation mechanism, which starts the mitigation *immediately* after the detection, i.e., without manual intervention.
Specifically, when ARTEMIS detects a hijacking in a prefix, let *10.0.0.0/23*, it proceeds to its de-aggregation: it sends a command to the BGP routers of the AS to announce the two more-specific prefixes, i.e., *10.0.0.0/24* and *10.0.1.0/24*. The sub-prefixes will disseminate in the Internet and re-establish legitimate route paths, since more specific prefixes are preferred by BGP. Prefix de-aggregation, as described above, is efficient for */23* or less specific (i.e., */22, /21, ...*) prefixes. However, when it comes to hijacking of */24* prefixes, the de-aggregation might not be always efficient, since prefixes more specific than */24* are filtered by most routers [@Bush-Internet-optometry-IMC-2009]. Although this is a shortcoming of the de-aggregation mechanism, it is not possible to overcome it in an automatic way (manual actions are needed); to our best knowledge, only solutions that require the cooperation of more than one ASes could be applied [@Zhang-practical-defenses-CoNext-2007; @Kotronis-Routing-Centralization-ComNets-2015].
The de-aggregation mechanism of ARTEMIS, increments the number of entries in the BGP routing table by $1$ per hijacked prefix. However, since the number of concurrent hijackings is not expected to be large, and the duration of a hijacking is limited, the imposed overhead is low.
Finally, since ARTEMIS monitors continuously the control-plane of the Internet, from many vantage points, it becomes possible to monitor in real-time the process of the mitigation. This enables a network administrator to see how efficient the mitigation is, and if needed to proceed to further (e.g., manual) actions or to rely exclusively on the de-aggregation mechanism.
Evaluation with a real AS {#sec:evaluation}
=========================
In this section, we conduct experiments in the Internet, to investigate (a) the overall performance of ARTEMIS, and (b) the efficiency of the different sources presented in Section \[sec:sources\] for monitoring the control-plane of the Internet. In Section \[sec:experimental-setup\] we provide the details for the setup of our experiments, and present the results in Section \[sec:experiments-results\].
Experimental Setup {#sec:experimental-setup}
------------------
In our experiments, we conduct *real* hijackings in the Internet. We use the PEERING testbed [@Schlinker-PEERING-HotNets-2014; @peering-website], which provides the possibility to announce IP prefixes from real ASNs to the Internet; both the IP prefixes and the ASNs are owned by PEERING, hence, our experiments have no impact on the connectivity of other ASes.
Specifically, we create a virtual AS in PEERING, and connect it to one or more real networks. This AS (which we call “legitimate” AS) announces an IP prefix and uses ARTEMIS to continuously monitor this prefix. We also create another virtual AS (the “hijacker” AS) in PEERING, connect it to a real network in a different location, and hijack the prefix of the legitimate AS.
### The PEERING testbed
PEERING is a testbed that enables researchers to interact with the Internet’s routing system. It connects with several real networks at universities and Internet exchange points around the world. The users of PEERING can announce IP prefixes using multiple ASNs owned by PEERING as the origin-AS.
In our experiments, we use the connections of PEERING to three real networks/sites (Table \[table:peering-sites\])[^2]. We are given authorization to announce the prefix *184.164.228.0/23* (as well as, its sub-prefixes), and use the AS numbers *61574* (for the legitimate AS) and *61575* (for the hijacker AS).
[Organization]{} [Location]{} [ASN]{} [\#providers]{} [\#customers]{} [\#peers]{}
------------- ------------------------------------------ -------------------------- ---------- ----------------- ----------------- -------------
[**AMS**]{} [Amsterdam Internet Exchange (AMS-IX)]{} [Amsterdam, NL]{} [1200]{} [-]{} [-]{} [509]{}
[**ISI**]{} [Los Nettos Regional Network]{} [Los Angeles (CA), US]{} [226]{} [4]{} [19]{} [19]{}
[**GAT**]{} [Georgia Institute of Technology]{} [Atlanta (GA), US]{} [2637]{} [4]{} [1]{} [6]{}
### Types of prefix hijacking attack
We test ARTEMIS in two different types of hijacking attacks: (a) exact prefix hijacks, and (b) sub-prefix hijacks.
**Exact prefix hijacking** is a common attack type where the hijacker announces the same prefix that is announced by the legitimate AS. Since shortest route paths are typically preferred, only *a part of the Internet* that is closer to the hijacker (in number of AS-hops) switches to route paths towards the hijacker. Exact prefix hijacks typically infect a few tens or hundreds of ASes [@Shi-Argus-IMC-2012], from small stub networks to large tier-1 ISPs [@Lad2-Understanding-resiliency-hijacks-DSN-2007]. In our experiments, the legitimate AS announces the prefix *184.164.228.0/23*; then the hijacker announces the same prefix. To mitigate the attack, the legitimate AS, announces the sub-prefixes *184.164.228.0/24* and *184.164.229.0/24*.
**Sub-prefix hijacking** contributes around 10% of all stable hijackings in the Internet [@Shi-Argus-IMC-2012]. The hijacker announces a more specific prefix, which is covered by the prefix of the legitimate AS. Since in BGP more specific prefixes are preferred, *the entire Internet* switches to routing towards the hijacker for the announced sub-prefix. We configure ARTEMIS to monitor the *184.164.228.0/22* prefix[^3]. The hijacker announces the prefix *184.164.228.0/23*. The attack is mitigated by de-aggregating the hijacked prefix, i.e., the legitimate AS announces the two */24* prefixes.
### Experiments
The experiment process comprises the following steps:
**(1)** The legitimate AS (*AS61574*) announces the IP prefix, and we wait $20$min for BGP convergence.
**(2)** The hijacker AS (*AS61575*) announces the IP prefix (or, sub-prefix).
**(3)** ARTEMIS detects the hijacking.
**(4)** ARTEMIS starts the mitigation, i.e., the legitimate AS announces the de-aggregated sub-prefixes.
**(5)** We monitor the mitigation process for $30$min, and end the experiment by withdrawing all announcements.
We conduct experiments for a number of different scenarios, varying the (a) *location/site* of the legitimate and hijacker ASes, and (b) *number of upstream providers* of the legitimate AS. We repeat each scenario/experiment $10$ times. The experiments took place in May-June 2016.
While normally ARTEMIS proceeds immediately after a hijacking detection to its mitigation, in some experiments we add a $30$min delay between steps 3 and 4, i.e., we defer the mitigation. This allows us to investigate the efficiency of the different control-plane sources, i.e., how much time each of them needs to detect the hijacking.
### Configuration of the control-plane sources
*BGPmon* provides to ARTEMIS a stream of all the updates it receives from its peers. Hence, configuration is not needed; filtering and detection are internal services of ARTEMIS.
*RIPE RIS* needs only the information about the monitored prefix, and returns to ARTEMIS only the BGP messages that correspond to announcements for this prefix.
In *Periscope*, due to the limit on the rate of measurements per user, only a subset of the total 1691 LG servers can be used. To conform to the rate-limit, we use $18$ LG servers, which we select based on their performance (response time, availability) and location. The selected set consists of $11$ LGs in Europe, $2$ in Asia, $4$ in North America, and $1$ in Australia.
Results {#sec:experiments-results}
-------
### Performance of control-plane sources
The performance of ARTEMIS depends on the control-plane sources it uses. Therefore, to obtain an initial understanding about the capabilities and limitations of ARTEMIS, we present in Fig. \[fig:detection-mititgation\] experimental results that demonstrate the efficiency and characteristics of the different control-plane sources in hijacking detection (Fig. \[fig:boxplots-detection-delay-per-tool\]) and mitigation monitoring (Fig. \[fig:mitigation-AS-vs-time\]).
Fig. \[fig:boxplots-detection-delay-per-tool\] shows how much time is needed by BGPmon, RIPE RIS, and Periscope to observe an illegitimate route, after it has been announced from the hijacker AS. We present the distribution of the times (among different experiments) for both attack types: prefix and sub-prefix hijacking.
A first observation is that the streaming services (BGPmon and RIPE RIS) observe the hijack in $\leq1$min in most cases, and are significantly faster than Periscope ($1$-$2$min), which monitors the control-plane by periodically issuing measurements from LG servers. This is due to the response delay of the LGs, as well as, a limit in the minimum time interval between consequent measurements imposed by Periscope.
The detection delay in the sub-prefix attack case (SP) is -on average- lower than in prefix hijacking (P). This is because a sub-prefix hijacking appears in the whole Internet, whereas prefix hijacking affects only a fraction of it. This partial infection of the Internet can be faster observed by BGPmon that has more vantage points than RIPE RIS, as is indicated by the lower mean value and variance of BGPmon in the prefix hijacking case.
In Fig. \[fig:mitigation-AS-vs-time\], we show the mitigation progress as it has been observed by the ASes with a vantage point, i.e., an RC feed or an LG server, in all sources. The average number of ASes that have been infected by the hijacker and switched back to the legitimate routes, are $29$ and $15$ in the SP and P case, respectively. Despite the differences, both attacks can be quickly mitigated; $45\%$ (SP) and $50\%$ (P) of the ASes re-establish legitimate routes in $10$sec after the mitigation was launched, while almost complete mitigation is achieved in less than $1$min. Furthermore, Figs. \[fig:boxplots-detection-delay-per-tool\] and \[fig:mitigation-AS-vs-time\] hint to an interesting trade-off: more vantage points (and, thus, ASes) can be monitored by Periscope, however, this comes with an increase in the detection delay compared to BGPmon and RIPE RIS.
### Effect of network connectivity
We now proceed to test the efficiency of ARTEMIS under various scenarios of network connectivity. Fig. \[fig:detection-location\] illustrates the effect of the (i) hijacker site and (ii) number of upstream providers of the legitimate AS[^4]. In the prefix hijacking case (Fig. \[fig:boxplots-delay-vs-location-prefix\]), when the hijacking is triggered by a well connected site, as in the case of AMS that it peers with $88$ real networks, the detection of the hijacking can be done in around $10$sec. When the connectivity of the hijacker AS is low, as in the GAT case that there are less than a dozen of directly connected networks, the detection delay is always higher than $15$sec and can need up to $1$min (the average detection delay is around $30$sec). These findings are intuitive and consistent with the conclusions of the simulation study in [@Lad2-Understanding-resiliency-hijacks-DSN-2007]; adding to this, they quantify for the first time the effects of the hijacker’s connectivity with real experiments.
In Fig. \[fig:boxplots-delay-vs-location-prefix\], we can also observe that when the connectivity of the legitimate AS increases, i.e., $2$ upstream providers, the detection delay (slightly) increases as well. This is due to the fact that with $2$ upstream providers, more ASes are closer to the legitimate AS (in terms of AS-hops) than the hijacker, and thus the effect of prefix hijacking is lower (and, consequently, its detection becomes more difficult).
In contrast to the prefix hijacking case, when the hijacker announces a sub-prefix (Fig. \[fig:mboxplots-delay-vs-location-subprefix\]), the connectivity of the involved networks does not play a crucial role. The effect of the hijacking is large and the detection is always completed within $10$sec, and on average it needs only $3$sec!
In [@Shi-Argus-IMC-2012] it is shown that the “detection delay” of Argus (a state-of-the-art hijacking detection system) is less than $10$sec for $>60\%$ hijacks. However, this delay, let $T_{dd}$, refers to the time needed to infer that an observation of an illegitimate route corresponds to a hijacking attack; i.e., if Argus uses the same control-plane sources as ARTEMIS, the total detection delay of Argus is $T_{Argus} = T_{ARTEMIS}+T_{dd}\geq T_{ARTEMIS}$.
### Gains of automatic mitigation
After presenting the hijacking detection efficiency, we study the gains of the automatic mitigation of ARTEMIS. Specifically, Fig. \[fig:percentage-mititgation\] shows the percentage of infected ASes in relation to the time since the hijacking has been launched. Each curve corresponds to a different “mitigation start time” $T_{start}$, i.e., the time between the hijacker’s announcement and the de-aggregation. The two bottom lines[^5] correspond to the near real-time automatic mitigation with ARTEMIS (we selected representative scenarios; cf. Fig. \[fig:detection-location\]). The two top lines are assumed to correspond to a timely (but not real-time) mitigation, e.g., with manual actions. As it can be seen, ARTEMIS can significantly decrease the impact of a hijacking. For instance, in scenarios where the detection delay of ARTEMIS is $10$sec, the fraction of infected ASes is $20\%$ and $50\%$ in the prefix and sub-prefix hijacking, respectively, while even a timely mitigation starting $1$min after the hijacking is not able to prevent the infection of all ASes. Moreover, with ARTEMIS the attack is completely mitigated in $T_{total}\leq2$min, whereas a mitigation that started after all ASes are infected (i.e., top two lines) needs around $1.5$min *after the detection*, i.e., $T_{total} = T_{start}+1.5$min.
This fast and effective mitigation that ARTEMIS can achieve, is particularly important for short hijacking attacks, whose frequency increases [@Shi-Argus-IMC-2012], and which can still cause serious problems [@hijack-BitCoins].
Related Work {#sec:related}
============
Detection of Prefix Hijacking
-----------------------------
Detection mechanisms can be classified based on the type of information they use for detecting prefix hijackings as: (i) control-plane, (ii) data-plane, and (iii) hybrid approaches.
Control-plane approaches [@Chi-cyclops-CCR-2008; @Lad-Phas-Usenix-2006; @bgpmon] collect information, like BGP updates or tables, from route servers and/or looking glass servers (LGs), from which they detect incidents that can be caused by prefix hijackings. When, for example, a change in the origin-AS of a prefix, or a suspicious change in a route path, is observed, an alarm is raised. Data plane approaches [@Zhang-Ispy-CCR-2008; @Zheng-lightweight-hijacks-2007] use ping/traceroute measurements to detect a prefix hijacking. They continuously monitor the data plane connectivity of a prefix and raise an alarm for hijacking, when significant changes in the reachability of the prefix [@Zhang-Ispy-CCR-2008] or in the paths leading to it [@Zheng-lightweight-hijacks-2007], are observed. A main shortcoming of data-plane mechanisms, is that a significant (minimum) number of active measurements is required to safely characterise an event as hijacking. Hence, these systems cannot be implemented in a light-weight fashion; and if deployed by every AS, they could introduce a large overhead [@Shi-Argus-IMC-2012]. Finally, hybrid approaches [@Hu-accurate-hijacks-SP-2007; @Shi-Argus-IMC-2012] combine control and data plane information to detect, with higher accuracy [@Shi-Argus-IMC-2012], multiple types of prefix hijacking [@Hu-accurate-hijacks-SP-2007; @Shi-Argus-IMC-2012].
Argus [@Shi-Argus-IMC-2012] is the most recent among the aforementioned detection systems, and has few false positives/negatives, and near real-time detection. However, Argus is based only on BGPmon for control-plane information [@bgpmon], whereas ARTEMIS receives data from multiple sources (BGPmon, RIPE-RIS, BGPstream, Periscope), which leads to a faster detection in more than $60\%$ of the cases (as we observed in our experiments).
The main difference between ARTEMIS and previous detection mechanisms is that most of the previous approaches are *notification systems*. They are designed to be operated by a third party and to monitor [*all*]{} the prefix the Internet. Upon detection of a suspicious event, they notify the involved ASes about a possible hijacking. This process has two shortcomings: (a) it yields many false positives, since suspicious events can usually be due to a number of legitimate reasons, like traffic engineering, anycast, congestion of the data-plane, etc.; and (b) it introduces significant delay, between the detection and mitigation of an event, as the network administrators of the involved AS need to be notified and then have to manually verify the incident. On the contrary, ARTEMIS is designed to detect hijacks against owned prefixes (for which the origin-AS information is known) and thus, overcomes the accuracy limitations, and eliminates the notification/verification delay.
Among previous works, only [@Zhang-Ispy-CCR-2008] is designed for detection of hijacks against owned prefixes. However, it is a pure data-plane mechanism, which as discussed above, introduces significant overhead (especially, if deployed by many ASes). In contrast, ARTEMIS, which is a pure control-plane mechanism, can be deployed simultaneously in many ASes without significant overhead for the control-plane resources/tools (see discussion in Section \[sec:detection\]). At the time the first of these mechanisms were proposed, the capability of the available BGP feeds for providing real-time information was limited. However, currently there exist several state-of-the-art *publicly available* control-plane sources/tools [@bgpmon; @ripe-ris-real-time; @bgpstream; @Giotsas-Periscope-PAM-2016] that enable pure control-plane mechanisms, like ARTEMIS, to detect a prefix hijacking event in near real-time (a few seconds [@bgpmon; @ripe-ris-real-time], or minutes [@Giotsas-Periscope-PAM-2016], as we show in Section \[sec:evaluation\]). To our best knowledge ARTEMIS is the first approach to exploit the streaming interfaces of RIPE RIS [@ripe-ris-real-time] and Periscope [@Giotsas-Periscope-PAM-2016] for prefix hijacking detection.
Measurement Studies
-------------------
Previous studies have taken measurements either over real hijacking incidents that happened in the Internet [@Shi-Argus-IMC-2012; @Lad2-Understanding-resiliency-hijacks-DSN-2007; @Khare-concurrent-hijacks-IMC-2012] or through simulations [@Zhang-Ispy-CCR-2008; @Lad2-Understanding-resiliency-hijacks-DSN-2007; @Zheng-lightweight-hijacks-2007; @Zhang-practical-defenses-CoNext-2007]. While the former correspond to the behavior of the Internet and capture the real effects of hijacking, they are limited to the investigation of a few known incidents, which do not span all possible cases. The latter are able to perform an investigation over a wider range of scenarios and study the effect of different parameters, but do not capture accurately real-world effects, since the topology, routing, and policies of the Internet cannot be perfectly replicated in simulations. Moreover, due to the absence of the ground-truth, i.e., if a detected routing change is indeed a hijacking or not, previous studies refer to third party sources, e.g., the Route Origin Authorizations (ROA) or Internet Routing Registries (IRR), for the validation of their results. However, such information is usually incomplete and/or inaccurate [@Shi-Argus-IMC-2012], and, this might have an impact on the findings. Closer to our study is [@Zhang-Ispy-CCR-2008] that tested its performance in self-triggered hijacks (for self-owned prefixes) in the Internet. Nevertheless, only few experiments (15 hijacks) were conducted, whereas in this paper we conduct a large number of experiments (spread over 4 weeks) with varying network parameters (location and connectivity of the hijacking/legitimate AS) and types of hijacks.
Conclusion {#sec:conclusion}
==========
We have presented ARTEMIS, a system for near real-time detection and automatic mitigation of BGP hijacking attacks. The evaluation with extensive real hijacking experiments, showed that ARTEMIS can detect hijacks in a few seconds, and completely mitigate them in less than $2$min.
In this initial implementation of ARTEMIS, we detect *origin-AS* inconsistencies in route paths, and combat them using the *prefix de-aggregation* method. Although not a panacea, prefix de-aggregation can be also effective for adjacency / policy [@Shi-Argus-IMC-2012] or last-hop anomalies [@Lad-Phas-Usenix-2006], or even path interception attacks [@caida-hijacks-project]. To extend ARTEMIS towards this direction, it suffices to modify only the detection algorithm; the monitoring and mitigation services can remain intact.
Finally, since the detection service of ARTEMIS is built on top (and, independently) of the monitoring service, the employed monitoring methodology and results (e.g., Fig. \[fig:detection-mititgation\]) are generic and can be useful in a number of application related to control-plane monitoring, e.g., to provide visibility to an AS of the impact of the routing changes it triggers (anycasting, traffic engineering, etc).
[10]{}
S. Hares, Y. Rekhter, and T. Li, “A border gateway protocol 4 (bgp-4).” <https://tools.ietf.org/html/rfc4271>, 2006.
.
.
<https://www.wired.com/2014/08/isp-bitcoin-theft/>.
A. Ramachandran and N. Feamster, “Understanding the network-level behavior of spammers,” [*ACM SIGCOMM Computer Communication Review*]{}, vol. 36, no. 4, pp. 291–302, 2006.
P.-A. Vervier, O. Thonnard, and M. Dacier, “Mind your blocks: On the stealthiness of malicious bgp hijacks.,” in [*Proc. NDSS*]{}, 2015.
S. Kent, C. Lynn, and K. Seo, “Secure border gateway protocol (s-bgp),” [ *IEEE Journal on Selected Areas in Communications*]{}, vol. 18, no. 4, pp. 582–592, 2000.
L. Subramanian, V. Roth, I. Stoica, S. Shenker, and R. Katz, “Listen and whisper: Security mechanisms for bgp,” in [*Proc. NSDI*]{}, 2004.
M. Lepinski, “Bgpsec protocol specification,” 2015.
M. Lepinski, R. Barnes, and S. Kent, “An infrastructure to support secure internet routing,” 2012.
J. Karlin, S. Forrest, and J. Rexford, “Pretty good bgp: Improving bgp by cautiously adopting routes,” in [*Proc. IEEE ICNP*]{}, 2006.
X. Shi, Y. Xiang, Z. Wang, X. Yin, and J. Wu, “Detecting prefix hijackings in the [I]{}nternet with [A]{}rgus,” in [*Proc. ACM IMC*]{}, 2012.
M. Lad, D. Massey, D. Pei, Y. Wu, B. Zhang, and L. Zhang, “Phas: A prefix hijack alert system.,” in [*Usenix Security*]{}, 2006.
Y.-J. Chi, R. Oliveira, and L. Zhang, “Cyclops: the as-level connectivity observatory,” [*ACM SIGCOMM Computer Communication Review*]{}, vol. 38, no. 5, pp. 5–16, 2008.
“[BGP]{}mon.” <http://www.bgpmon.io>.
Z. Zhang, Y. Zhang, Y. C. Hu, Z. M. Mao, and R. Bush, “i[SPY]{}: detecting ip prefix hijacking on my own,” in [*ACM SIGCOMM CCR*]{}, vol. 38, pp. 327–338, 2008.
C. Zheng, L. Ji, D. Pei, J. Wang, and P. Francis, “A light-weight distributed scheme for detecting ip prefix hijacks in real-time,” in [*ACM SIGCOMM Computer Communication Review*]{}, vol. 37, pp. 277–288, 2007.
X. Hu and Z. M. Mao, “Accurate real-time identification of ip prefix hijacking,” in [*IEEE Symposium on Security and Privacy*]{}, pp. 3–17, 2007.
G. Chaviaras, P. Gigis, P. Sermpezis, and X. Dimitropoulos, “[ARTEMIS]{}: Real-time detection and automatic mitigation for bgp prefix hijacking,” in [*Proc. ACM SIGCOMM Demo*]{}, 2016.
“[RIPE RIS]{} - [S]{}treaming [S]{}ervice.” .
<http://www.caida.org/tools/utilities/looking-glass-api/>.
“[BGP]{}stream.” <https://bgpstream.caida.org/>.
“[RIPE RIS]{}.” <http://ris.ripe.net/>.
M. Lad, R. Oliveira, B. Zhang, and L. Zhang, “Understanding resiliency of internet topology against prefix hijack attacks,” in [*Proc. IEEE/IFIP Dependable Systems and Networks*]{}, 2007.
R. Bush, O. Maennel, M. Roughan, and S. Uhlig, “Internet optometry: assessing the broken glasses in internet reachability,” in [*Proc. ACM IMC*]{}, 2009.
Z. Zhang, Y. Zhang, Y. C. Hu, and Z. M. Mao, “Practical defenses against bgp prefix hijacking,” in [*Proc. ACM CoNEXT*]{}, 2007.
V. Kotronis, A. G[ä]{}mperli, and X. Dimitropoulos, “Routing centralization across domains via sdn: A model and emulation framework for [BGP]{} evolution,” [*Computer Networks*]{}, pp. –, 2015.
B. Schlinker, K. Z. I., Cunha, N. Feamster, and E. Katz-Bassett, “Peering: An as for us,” 2014.
“The [PEERING]{} testbed.” <https://peering.usc.edu>.
“[AS]{}-rank, [CAIDA]{}.” <http://as-rank.caida.org>.
V. Giotsas, A. Dhamdhere, and K. Claffy, “Periscope: Unifying looking glass querying,” in [*Proc. PAM*]{}, Springer, 2016.
V. Khare, Q. Ju, and B. Zhang, “Concurrent prefix hijacks: Occurrence and impacts,” in [*Proc. ACM IMC*]{}, 2012.
, “Detecting and characterizing internet traffic interception based on bgp hijacking.” <https://www.caida.org/funding/hijacks/hijacks_proposal.xml>, 2014.
[^1]: A demo of ARTEMIS is to appear in ACM Sigcomm 2016 [@ARTEMIS-Demo-Sigcomm-2016].
[^2]: PEERING peers with $88$ organizations in AMS-IX [@peering-website]. Statistics for the number providers, customers, and peers for each AS are from [@as-rank-website].
[^3]: Since we have access only to the */23* prefix, we do not announce the */22* prefix; we only assume it is announced. However, this does not affect the outcome of the experiments.
[^4]: Our results do not significantly variate with the location of the legitimate AS or the number of upstream providers of the hijacker.
[^5]: With blue and red; or, $10$sec and $30$sec in Fig. \[fig:salvaged-AS-prefix\], and $5sec$ and $10sec$ in Fig. \[fig:salvaged-AS-subprefix\].
|
---
abstract: 'Recent advances in artificial intelligence have been strongly driven by the use of game environments for training and evaluating agents. Games are often accessible and versatile, with well-defined state-transitions and goals allowing for intensive training and experimentation. However, agents trained in a particular environment are usually tested on the same or slightly varied distributions, and solutions do not necessarily imply any understanding. If we want AI systems that can model and understand their environment, we need environments that explicitly test for this. Inspired by the extensive literature on animal cognition, we present an environment that keeps all the positive elements of standard gaming environments, but is explicitly designed for the testing of animal-like artificial cognition. All source-code is publicly available (see appendix).'
author:
- |
Benjamin Beyret,^1,4^ Jos[é]{} Hern[á]{}ndez-Orallo,^2,4^ Lucy Cheke,^3,4^ Marta Halina,^3,4^\
[ **Murray Shanahan,^1,4^ Matthew Crosby^1,4^** ]{}\
^1^Imperial College London, UK ^2^Universitat Polit[è]{}cnica de Val[è]{}ncia, Spain\
^3^University of Cambridge, UK ^4^Leverhulme Centre for the Future of Intelligence, UK\
bb1010@ic.ac.uk, jorallo@upv.es, {lgc23, mh801}@cam.ac.uk, {m.shanahan, m.crosby}@imperial.ac.uk
bibliography:
- 'references.bib'
title: |
The Animal-AI Environment:\
Training and Testing Animal-Like Artificial Cognition
---
Introduction
============
From chess [@silver_2017] to Go [@silver_2016], from older Atari games [@bellemare_2012] to Starcraft 2 [@vinyals_2017], we have seen a wide variety of challenging environments where AI now outperforms humans. Successes such as these in deep reinforcement learning (DRL) have been driven by the introduction of game environments and physics simulators [@todorov_2012], and have even resulted in the transfer of trained agents to the real world for robotic manipulations [@openai_2018]. These impressive results are only a first step towards agents that can robustly interact with their environments. In these settings, most tasks used for training are identical, or extremely similar, to the ones used for testing. Whilst the agent has not always been exposed to the test tasks during training, it has probably been exposed to tasks drawn from the same distribution. This leads to agents that reach superhuman performance on specific problems, but are unable to generalise [@packer2018assessing].
AI systems can solve and outperform humans in many complex tasks, but, whilst progress is ongoing, a [*single*]{} agent cannot compare to a human’s ability to adapt and generalise [@espeholt2018impala]. Humans are no longer the best Go players, but they are still the best at functioning across a wide number of environments and adapting to unexpected situations. Even non-human animals exhibit the ability to solve a wider array of tasks than AI systems are currently capable of [@herrmann2007humans].
In animal cognition research, ethologists observe organisms in their natural environments whilst comparative psychologists design carefully-controlled studies in order to probe animals’ cognitive and behavioural capacities [@thorndike1911animal]. Over the last century the testing process has been refined to eliminate confounding factors, noise and other non-cognitive elements that are interfere with identifying the targeted skill [@shaw2017cognitive]. A wide range of standardised tests have been developed, along with some well-known experimental paradigms (radial mazes, landmark arenas, etc.). With these tools, comparative psychologists can adapt and customise tasks to fit a wide range of species, finding the best way to measure a skill or clarify some particular question about behaviour.
Following the above-mentioned limitations of current AI testbeds and the theoretical and methodological tools available in comparative psychology, we introduce the Animal-AI environment, which is used in the Animal-AI Olympics competition, to test trained agents on tasks inspired by animal cognition research. The environment includes all the beneficial parts of gaming environments: it has a simple reward function, action space and complex yet deterministic state transition function based on a simulated physics engine. It is also set up to test multiple cognitive abilities in a single agent. The environment consists of a small arena with seven different classes of objects that can be placed inside. These include reward objects (corresponding to food) and building blocks which can be combined to create complex cognitive tests.
The paper is structured as follows. We first present current benchmarks and related environments and identify a need for cognitive behavioural testing. We then introduce experimental paradigms from comparative cognition and show how they can be adapted for AI. Following this, we present the Animal-AI environment, illustrating how it can be used as a training and testing platform. Finally, we introduce the Animal-AI testbed, which is used to test a wide range of cognitive skills in the Animal-AI Olympics competition.
Related Benchmarks in AI
========================
Progress in DRL in recent years has been fuelled by the use of games and game-inspired simulated environments [@castelvecchi2016tech; @hernandez2017new]. In these settings, for an agent or policy to be successful it must integrate perception of the scene (usually in 2D or 3D) with the right choice of action sequences that allow the agent to achieve the goal conditions. Some important benchmarks are simply collections of existing games, such as the very popular *Arcade Learning Environment* (ALE) [@bellemare2015arcade; @machado2018revisiting], with dozens of (2D) Atari 2600 games. In a similar vein, OpenAI Gym [@brockman2016openai] provides a common interface to a collection of RL tasks including both 2D and 3D games. As machine learning methods conquer more games, there seem to be endless new (and old) games ready to be put forward as the next challenge [@jaderberg2018human]. Moving towards generalisation, we see games designed to include variations or set stages, where some skill transfer between training and testing is necessary. [*CoinRun*]{} [@cobbe2018quantifying] is a 2D arcade-style game aimed at generalisation and transfer. It is designed to be simple enough so that individual levels are easy to solve when trained on, but the task is to create an agent that can solve unseen variations. Another example, [*Obstacle Tower*]{} [@juliani2019obstacle] is a new 3D game based on Montezuma’s revenge, one of the harder (for AI) Atari games, whose stages are generated in a procedural way. The procedural generation ensures that the agent can be tested on unseen instances.
While the above collections and game-like environments give the possibility of adding new tasks or task variations, other platforms are designed to be customisable to allow for a wide variety of challenges. For instance, using the video game definition language (VGDL), new real-time 2D games can be created relatively easily. This has led to several *General Video Game AI* (GVGAI) competitions, with new games for each edition [@perez20162014]. [*ViZDoom*]{} [@kempka2016vizdoom] is a research platform with customisable scenarios based on the 1993 first-person shooting video game Doom that has been used to make advancements in model-based DRL [@ha2018world]. Microsfoft’s [*Malmo*]{} [@johnson2016malmo], which is based on the block-based world of Minecraft, also makes it possible to create new tasks, ranging from navigation and survival to collaboration and problem solving. Finally, [*DeepMind Lab*]{} [@beattie2016deepmind] is an extensible 3D platform with simulated real-world physics built upon id Software’s Quake III Arena. DeepMind Lab is primarily set up for large navigational problems or tests based on visual cues. In contrast, the Animal-AI environment is useful for setting up small navigation problems, and also supports interactive experiments that test an agent’s ability to reason about the physical environment based on its (simulated) physics, like those used in comparative psychology.
Without being comprehensive (as new platforms appear very regularly), this AI-evaluation landscape shows some common characteristics. (1) Most traditional platforms are devised to set new challenges assuming some incremental improvement over current AI technology. (2) To be considered a real challenge, oversimplified tasks that look like toy problems are avoided (even if AI cannot currently solve them). (3) When many tasks are integrated into a benchmark, it is not always the same trained agent (but the same algorithm) that is evaluated on them, with retraining for each task. (4) Only a few more recent platforms are aimed towards generalisation (of the same agent) where the training and the test sets differ –but not always and retraining is not allowed. (5) Most challenges are task-oriented instead of ability-oriented [@hernandez2016aire], without explicitly identifying the skill to be measured. We challenge (1)-(5), introducing an environment that presents a new paradigm for AI. We use tests that are simple in the context of comparative psychology, but are challenging for AI and include many ability-oriented tasks aimed at testing generalisation of a single agent.
Another ability-oriented approach is *bsuite*, which presents a series of reinforcement learning tasks designed to be easily scalable and to provide a measure for a number of core capabilities [@osband2019]. These tests are deliberately simple to allow for a more accurate measure of the ability being tested. In contrast, our approach, inspired by biological intelligence, focuses on measuring cognitive skills scaffolded by an agent’s ability to perceive, navigate, and interact with its environment. Our tests are also deliberately simple to a human observer, but because they are built up from perception, they are much more complicated to solve.
Animal Cognition Benchmarks
===========================
The study of animal behaviour includes areas such as ethology, behavioural ecology, evolutionary psychology, comparative psychology and more recently, comparative cognition [@wasserman2006comparative; @shettleworth2009cognition]. In the latter two, animals are usually evaluated in carefully designed and controlled conditions, using standardised procedures. Radial arm mazes, for instance, are frequently employed to study spatial memory in rats (Figure \[fig:radial-a\]). An experiment is usually constrained by three design principles: (1) Solving the task requires some basic functions that are taken for granted or have been previously demonstrated in the animal, such as being able to move, recognise shapes and colours, desire rewards, etc. (2) Beyond these basic functions, the cognitive skill or capacity under investigation (e.g., spatial memory) is required to perform successfully on the task. (3) All other confounding factors are eliminated or controlled for, such as cues that would allow subjects to solve the task (e.g., odors in a maze) without relying on the target skill. Other experimental confounds, such as personality traits or motivation, are also avoided or taken into account in the statistical analysis of the results [@shaw2017cognitive]. Note that these three principles make experimental tasks used in comparative cognition very different from the AI testbeds reviewed in the previous section.
[0.22]{} ![Two different apparatus commonly used in comparative cognition. Food is placed in the arms of the maze for (\[fig:radial-a\]) and inside a tube for (\[fig:cylinder\]). Whether or not the animal collects food, and more importanlty how it does it, demonstrates various skills in animals.[]{data-label="fig:animalcogexperiments"}](figures/radialmaze.pdf "fig:"){width="\textwidth"}
[0.22]{} ![Two different apparatus commonly used in comparative cognition. Food is placed in the arms of the maze for (\[fig:radial-a\]) and inside a tube for (\[fig:cylinder\]). Whether or not the animal collects food, and more importanlty how it does it, demonstrates various skills in animals.[]{data-label="fig:animalcogexperiments"}](figures/detour.pdf "fig:"){width="\textwidth"}
An experiment is then designed to analyse or evaluate a particular cognitive function. For instance, the task might be to find food in an eight-arm maze (fig. \[fig:radial-a\]), and the way in which participants succeed or fail in the task provides insight into their cognitive abilities. If an animal retraces its steps down previously explored arms, for example, then this suggests it is not exhibiting spatial memory compared to an animal that can solve the maze systematically. The three principles above are combined with a strict protocol for evaluation. Trials can also vary in difficulty (e.g., number of food items in the maze, set of objects and their arrangement, time for each trial or between them) and presentation (e.g., training episodes). Recently, so-called “mini test batteries" [@shaw2017cognitive] are becoming popular in comparative cognition. They include a small number of different tasks that target a particular function (e.g., spatial memory), standardised across species.
AI Evaluation Inspired by Animal Cognition
==========================================
The route for measuring cognitive functions (or functional domains) through mini test batteries and combining them to analyse more complex patterns or profiles of behaviour (such as general intelligence) is a powerful methodology that can be extended (with the appropriate adaptations) for the evaluation of AI systems. Consequently, we introduce the Animal-AI environment as a new paradigm for training and testing AI algorithms. We provide a platform which allows researchers to design experimentation protocols in a way similar to animal cognition research. As discussed above, when testing animals for cognitive skills it is customary that the animal has to move around and interact with a set of objects in order to retrieve one or more pieces of food.
In order to make experiments comparable between the Animal-AI environment and the animal cognition literature, we mimic the way an experimenter would build a testing environment for an animal. In practice this means that the arena must follow the following principles: (1) the agent is enclosed in a fixed size arena with relatively simple objects so that we exclude as many confounding factors as possible, (2) physics must be realistic and visually reproduce how objects behave in the real world (e.g., gravity, collisions, friction, etc.). In addition, from an engineering standpoint we would expect that (3) reproducing experiments from animal cognition in our environment should be simple and fast and our environment must integrate easily with standard ML paradigms. We designed the environment following these principles. The agent itself is a simple sphere that can only “walk" forward and backward and turn left and right. It cannot jump nor fly, but can climb on objects that present a slope such as ramps and cylinders. In order to simplify the problem from an AI point of view, the agent only has monocular vision through pixel inputs, and to partially mimic an animal’s interoceptive abilities, it also has a sense of its own speed in its local reference frame. These simplifying assumptions are motivated in part by the principle of avoiding confounding factors originating from complex bodies or large sets of actions, as in other platforms.
Unlike with animals, we cannot assume that AI systems are prebuilt with an understanding of intuitive physics or the ability to manipulate objects. Even if they can navigate basic versions of the environment, we cannot assume that artificial systems can avoid obstacles or push objects, which is a fundamental component of many animal cognition tests. We therefore define a playground where these affordances are common and provide this as a training arena so that agents can bring with them capabilities similar to animals.
Environment Design and Specification
====================================
The environment is provided as an executable that can be interfaced with using the provided Python API, or manually in “play mode” where the user can control an agent. We built on top of Unity ML-Agents [@mlagents] in order to provide an extra layer of abstraction as well as additional features. The environment is composed of one or more arenas, each containing a unique agent acting independently of all other agents and which can be independently configured. In figure \[fig:arena\_all\] we present one arena and the objects the user can place in it, while the next section explain how this can be done. The supplementary material contains full technical details of the arena and objects, such as their size, weight, colour, reward value, etc.
The Arena and Objects
---------------------
[0.45]{} ![Arena displaying all possible objects: agent (blue sphere at the bottom), immovable objects (left), rewards (middle) and movable objects (right).[]{data-label="fig:arena_all"}](figures/arenas/ArenaAllObjects.png "fig:"){width="\textwidth"}
The arena itself is made of a tiled ground and four walls represented by wooden fences, as well as an agent (blue sphere), and can hold various objects from seven categories. As shown in figure \[fig:arena\_all\], the objects are:
- [**Immovable objects**]{}: these objects cannot be pushed by the agent. They separate and possibly occlude parts of the environment, allowing for the creation of different configurations that the agent has to explore to find food. They are shown on the left in figure \[fig:arena\_all\] and make three categories (from bottom to top):
- ramps the agent can climb,
- walls, both opaque and transparent, that the agent must move around, and
- cylinders, both opaque and transparent, that the agent can move through.
All objects can be resized and rotated freely. They can be used individually, but can also be thought of as building blocks from which complex constructions can be built, such as those in Figure \[fig:animalcogexperiments\].
- [**Reward objects**]{}: these represent pieces of food or positive rewards. They ensure that the agent has comparable motivations to in animal cognition, where food is the goal in many experiments. We also include negative rewards, testing an agent’s ability to avoid aversive stimuli. The reward objects are shown in the middle of Figure \[fig:arena\_all\] and can be divided into two categories (from bottom to top, excluding the blue agent):
- spheres that have positive (green or gold) or negative (red) rewards (green and red terminate an episode, gold does not unless it is the last positive reward item in the arena),
- orange zones on the ground, *hot zones*, that give negative rewards to the agent standing on it, but do not terminate an episode, and red zones, *death zones*, also give negative reward but terminate an episode and are useful for setting up no-go areas similar to those in elevated maze experiments for example [@pellow1985validation].
It is important that the food items are spherical. The simulated physics allows us to make use of this in the experiments.
- [**Movable objects**]{}: Some experiments in comparative psychology require the animal to interact with objects either via grabbing, pushing or pulling, using them as [*tools*]{}. For simplicity, the Animal-AI agent does not have any means of holding onto objects, but it can push them by moving into them. In Figure \[fig:arena\_all\] these objects are on the right and make up the last two categories of objects (from bottom to top):
- Cardboard boxes that can be be pushed around, one heavier than the other,
- A set of U-shaped and L-shaped objects, all of identical weight that have different affordances to the boxes.
On top of the physical configuration of objects in an arena, we also allow the experimenter to *switch the lights on and off* as is done in some landmark navigation experiments. This can also be used to simulate experiments where items need to be intentionally obscured from view for a short period of time. With animals, this is normally done by adding a screen between the animal and the testing apparatus. In our environment the lights-off condition is implemented by replacing the visual observations sent to the agent by a black image.
Reinforcement Learning Setup
----------------------------
From a reinforcement learning perspective, for a single agent, we define observations, actions and rewards as follows (see supplementary material for more details):
- **Observations**: The agent is equipped with a single camera which shows it a pixel grid with resolution $k \times k \times 3$ where $4 \leq k \leq 512$. The agent also perceives its speed along the axes of its local referential (forward, right, up).
- **Action space**: The agent can take actions $(m,t) \in \{0,1,2\}\times\{0,1,2\}$ where $m$ applies a force forward ($1$), backward ($2$) or no force ($0$), and $r$ is an instantaneous rotation to the right ($1$), left ($2$) or no rotation ($0$).
- **Rewards**: are obtained when a food item is collected or when the agent touches a *hot* or *death zone*. The agent also receives a small negative reward at each step, equal to $\frac{-1}{T}$ where $T$ is the number of timesteps in the episode (or no reward if $T=0$ which is the setting for an infinite length episode).
We offer an API to interact with the environment either as a Gym [@brockman2016openai] or an ML-Agents environment [@mlagents].
Task Configurations
-------------------
A crucial part of our environment is the way a user can configure tasks for both training and evaluating agents. The main way of doing so is via the use of human readable configuration files, which can precisely define the position of objects in an arena, and also allows for randomisation of most parameters (position, rotation, size, colour). The user can change the configuration of the objects in the arena between each training or testing episode via the API. This allows for agents to be trained on specifically designed task sequences, with new arenas introduced based on any user-defined trigger.
Note that with the environment, and with the Animal-AI Olympics competition that uses it, we do not place any restrictions on configurations used for training. This is a new paradigm in AI evaluation that reflects the fact that we currently do not know the most efficient setups for learning real-world skills. Part of the problem is to understand the types of environments that lead to learning of transferable skills or to find ways to learn in noisy environments without explicit direction from a reward function. For example, in our competition setup, all tests must remain secret until the end, so that no participants can train on test configurations directly.
A common configuration that has been used for training in the competition is to spawn objects randomly, increasing the number and types of objects as the agent learns. A key element for progress in Animal-AI will be to design more sophisticated training environments that allow the agent to experience a wide range of the affordances of the environment in an efficient manner and without explicit reward feedback. It is possible to build files of increasing difficulties to learn from, as in curriculum learning [@bengio_2009]. In the following section we show first how this might work for maze navigation alone, and then proceed to outline our complete test battery of 300 tests inspired by comparative cognition.
Example: Maze Navigation
========================
Mazes are a simple and yet widely used problem for testing spatial memory and navigation skills of both animals [@olton1979mazes] and AI agents [@johnson2016malmo; @beattie2016deepmind]. In the current RL paradigm, it is standard practice to train an agent in an environment that presents many configurations of a maze and then test on unseen variations drawn from the same probability distribution. This setup allows agents to solve particular mazes, but does not necessarily lead to agents that learn transferable skills that will be applicable to other types of maze.
In contrast, in animal cognition it is usually the case that we are interested in behaviour on first presentation of a particular type of maze. Training on similar mazes would be considered an experimental confound that detracts from the ability to test for navigation capabilities. To train an agent that can acquire true navigation skills we would use a set of training configurations that do not contain the types of mazes we want to test the agent on. This can easily be configured using the Animal-AI environment.
[0.22]{} ![Samples of various types of mazes we want to evaluate an agent on.[]{data-label="fig:mazes"}](figures/arenas/maze2x2.png "fig:"){width="\textwidth"}
[0.22]{} ![Samples of various types of mazes we want to evaluate an agent on.[]{data-label="fig:mazes"}](figures/arenas/maze3x3.png "fig:"){width="\textwidth"}
\
[0.22]{} ![Samples of various types of mazes we want to evaluate an agent on.[]{data-label="fig:mazes"}](figures/arenas/maze_all_random.png "fig:"){width="\textwidth"}
[0.22]{} ![Samples of various types of mazes we want to evaluate an agent on.[]{data-label="fig:mazes"}](figures/arenas/maze_circular.png "fig:"){width="\textwidth"}
Figure \[fig:mazes\](a-d) shows samples of different types of mazes that we would like to evaluate a trained agent on in order to test its navigation skills. The experimenter needs to design a twofold training procedure made of a trainable agent (similar to the classic RL paradigm) and also a set of arena configurations to train the agent on. For a given algorithm, several training sets will lead to agents acquiring various skills.
[0.22]{} ![Parts of the curriculum used to train a PPO agent to navigate around walls.[]{data-label="fig:curriculum"}](figures/arenas/maze_curriculum1.png "fig:"){width="\textwidth"}
[0.22]{} ![Parts of the curriculum used to train a PPO agent to navigate around walls.[]{data-label="fig:curriculum"}](figures/arenas/maze_curriculum2.png "fig:"){width="\textwidth"}
\
[0.22]{} ![Parts of the curriculum used to train a PPO agent to navigate around walls.[]{data-label="fig:curriculum"}](figures/arenas/maze_curriculum3.png "fig:"){width="\textwidth"}
[0.22]{} ![Parts of the curriculum used to train a PPO agent to navigate around walls.[]{data-label="fig:curriculum"}](figures/arenas/maze_curriculum4.png "fig:"){width="\textwidth"}
For this example, we compare two agents trained using PPO (same hyper-parameters for both, see appendix) [@schulman_2017]. One agent is trained to solve a randomised $2\times2$ maze (shown in Figure \[fig:mazes:2x2\]), while the other is trained on a curriculum of arenas with increasing numbers of walls randomly placed along the $X$ and $Z$ axes, as shown in Figure \[fig:curriculum\]. We switch the curriculum from one level of difficulty to the next when the agent reaches an $85\%$ success rate over 600 episodes. The first agent has seen maze-like configurations, with regularity and openings in the walls, but only of a single type. The second agent has not seen anything maze-like, but has experienced environments with a range of complexities.
[0.45]{} ![Cumulative reward per episode for PPO agents trained on a curriculum (red) and a $2\times2$ maze (blue) on the training set– mean and standard deviation over 4 random seeds. The maximum reward per episode is $2$ (obtaining no reward yields $-1$).[]{data-label="fig:training"}](figures/graph.png "fig:"){width="\textwidth"}
[|c||C|C||C|C||C|C||C|C| ]{} & & & &\
& A.R. & S.R & A.R. & S.R & A.R. & S.R & A.R. & S.R\
Curriculum & 0.70 & 64% & 0.92 & 70% & 0.4 & 53% & 0.03 & 39%\
$2\times2$ maze & 1.82 & 99% & 1.18 & 78% & 0.51 & 55% & 0.12 & 41%\
We are interested in evaluating the two trained agents on the set of experiments defined above (Figure \[fig:mazes\]). In Figure \[fig:training\] we show that both agents reached equilibrium, maximising the rewards they can obtain on their two training sets respectively. Note that the red curve shows the agent learning over a curriculum, therefore the difficulty of the training tasks increase with the number of timesteps (hence the initial spike in rewards obtained). We then select the best agents: the best one overall for the agent trained on a $2\times2$ maze, and the best performer on the hardest task of the curriculum. In Table \[results\] we show both the average rewards and the success rates of each of the two best agents, evaluated on the four types of mazes defined in Figure \[fig:mazes\]. As expected, the agent trained on the $2\times2$ maze outperforms the other agent on the same problem and reaches 99% accuracy. This represents a standard reinforcement learning problem and solution.
The agent trained on $2\times2$ mazes has a significant drop in performance as we move through tests a-d, away from similarity to its training set. Even though it has seen many variations of $2\times2$ mazes, it loses considerable performance when moving to 3x3 mazes. The curriculum agent, whilst not achieving the same level of performance in the $2\times2$ maze, has much less of a drop in performance on the harder problems, achieving near parity with the standard agent on the circular maze.
These results give an idea as to how the environment can be used to train and test for animal-like cognitive skills and also how they can transfer across different problem instances. However, introducing variability into training sets adds new considerations for AI research, and also means more extensive testbeds in order to draw valid conclusions.
Artificial Cognition
====================
Drawing conclusions about the cognitive abilities of animals requires caution, even from the most well-designed experiments. Moving to AI makes this even harder. We have seen that the Animal-AI environment introduces a new paradigm in the field of AI research by dissociating the training and testing environments, making the design of a training curriculum part of the entire training process. In the previous toy example we could, for example, train an agent using PPO on a training curriculum $\mathcal{A}$, and then another agent using Rainbow [@hessel_2017] on a training curriculum $\mathcal{B}$, yielding success rates $s_1$ and $s_2$ on the same test set. However, even if $s_1 > s_2$, we could not conclude that PPO is better than Rainbow as we cannot isolate whether the algorithm used or the training curriculum is the source of the better results. Therefore, instead of comparing training algorithms, one now needs to compare the entire package of algorithm and curriculum combined.
The methods of comparative psychology provide us with tools for controlling confounds like the above. In order to interpret the results of a test trial in comparative cognition, one must consider a participant’s evolutionary and ontogenetic history, previous training, and familiarity with the experimental setup. As noted above, when a participant performs successfully on a cognitive task, researchers carefully rule out plausible alternative explanations - ones in which the participant might have learned to solve the task using capacities other than the targeted skill. Ruling out such alternatives requires proper experimental design and statistical analysis [@bausman2018not].
One partial solution to dealing with this number of confounds is to define test batteries [@herrmann2007humans; @shaw2017cognitive] that increase the number of skills an individual agent is tested on. Compared to animals, experiments with AI systems are less costly and time intensive, so each skill can also require solving a wide variety of tests. An agent scoring well on only a few tests would likely have overfitted, or found a lucky solution. With a full test battery, an agent’s successes can be compared across tests to build a profile of its behaviour and capabilities.
The First Animal-AI Test Battery
================================
The Animal-AI test battery has been released as part of a competition alongside the Animal-AI Environment. The competition format allows us to assess performance on completely hidden tasks. If we had first released the full details of the test battery then we would have lost the opportunity to know how well AI systems could do with zero prior knowledge of the tests. Once the competition has ended we will release the full details of the tests. At this point, it will be even more important for anyone working with the test battery to disclose their training setup so that it can be determined how much they are training to solve the specific tests, and how much to learn robust food retrieval behaviour.
We designed 300 experiment configurations split over categories testing for 10 different cognitive skills. We then challenged the AI community to design both training environments and learning algorithms and submit agents that they believe display these cognitive skills. We briefly introduce the ten categories here:
**1. Basic food retrieval:** Most animals are motivated by food and this is exploited in animal cognition tests. This category tests the agent’s ability to reliably retrieve food in the presence of only food items. It is necessary to make sure agents have the same motivation as animals for subsequent tests.
**2. Preferences:** This category tests an agent’s ability to choose the most rewarding course of action. Almost all animals will display preferences for more food, or easier to obtain food, although exact details differ between species. Tests are designed to be unambiguous as to the correct course of action based on the rewards in our environment.
**3. Obstacles:** This category contains objects that might impede the agent’s navigation. To succeed in this category the agent will have to explore its environment, a key component of animal behaviour.
**4. Avoidance:** This category identifies an agent’s ability to detect and avoid negative stimuli. This is a critical capacity shared by biological organisms, and is also an important prerequisite for subsequent tests.
**5. Spatial Reasoning:** This category tests an agent’s ability to understand the spatial affordances of its environment. It tests knowledge of some of the simple physics by which the environment operates, and also the ability to remember previously visited locations.
**6. Robustness:** This category includes variations of the environment that look superficially different, but for which affordances and solutions to problems remain the same.
**7. Internal Models:** In these tests, the lights may turn off and the agent must remember the layout of the environment to navigate in the dark. Most tests are simple in nature, with light either alternating or only going off after some time.
**8. Object Permanence:** This category checks whether the agent understands that objects persist even when they are out of sight, as they do in the real world and in our environment.
**9. Numerosity:** This category tests the agent’s ability to make more complex decisions to ensure it gets the highest possible reward. These include situations where decisions must be made between different sets of rewards.
**10. Causal Reasoning:** This category tests the agent’s ability to plan ahead and contains situations where the consequences of actions need to be considered. All the tests in this category have been passed by some non-human animals, and these include some of the most striking examples of intelligence from across the animal kingdom.
By organising these problems into categories, we can get a profile for each agent, which is more informative than a simple overall test score. Creating agents that can solve such a wide range of problems is a challenging task. We expect this to require sustained effort from the AI community to solve, and that doing so will lead to many innovations in reinforcement learning.
Current State-of-the-art
------------------------
Human performance on the tests is close to 100%, and most of the tests have been successfully completed by some animals. Some of the easier tests are solvable by nematodes, whilst the most complex tests in the competition are only solved by animals such as corvids and chimps. Each category contains a variety of test difficulties. For example, the first tests in the basic food category spawn a single food in front of the agent whereas the final tests contain multiple moving food objects.
[0.45]{} ![Radar plot showing the performance of the top four entries to the Animal-AI Olympics at the half-way point.[]{data-label="fig:AAIO-results"}](figures/Radar2.png "fig:"){width="\textwidth"}
Current state-of-the-art in AI scores around 40%. Figure \[fig:AAIO-results\] shows the results of the top four competitors of the competition at the moment. Methods used were all variations of reinforcement learning. We can see that certain categories, corresponding to those closer to traditional problems in reinforcement learning, are easier to solve than others. These include navigation-based problems and problems involving choosing between types of reward. On the other hand, those that require planning or that necessarily involve keeping accurate models of the world, such as for object permanence, are much harder to solve and will be an important challenge for ongoing research.
Conclusions
===========
The Animal-AI environment is a new AI experimentation and evaluation platform that implements ideas from animal cognition in order to better train AI agents that possess cognitive skills. It is designed to contain only the necessary ingredients for performing cognitive testing built up from perception and navigation. We start with simple yet crucial tasks that many animals are able to solve. These include reward maximisation, intuitive physics, object permanence, and simple causal reasoning. By mimicking tasks and protocols from animal cognition we can better evaluate the progress of domain-general AI.
The Animal-AI platform is currently in its first iteration, and will continue to be developed as AI progresses and solves the original testbed. There are many tasks that some animals can solve that were considered too complex (for AI to solve) for the first iteration. Once it becomes possible to train agents to solve the first tasks, we will introduce more complex tests. In future versions we also plan to include more complicated substrates and objects such as liquids, soft bodies and ropes which are common in animal experiments. A more complex physics engine would allow for embodied interactions between the agent and its environment, creating many more opportunities for animal-like tests. We will also add the possibility for multi-agent training which could allow for social interactions and social learning and also include tests that require online learning to solve correctly.
Having said this, we expect the current testbed to remain an open challenge and provide a pathway for AI research for years to come. We have seen incredible results in game-like environments in recent years, and we hope this momentum can translate to solving the kind of problems that animals solve on a daily basis when navigating their environment or foraging for food. The Animal-AI environment is set up to make this transition as easy as possible, and to provide a series of increasingly difficult challenges towards animal-like artificial cognition. Finally, this platform helps bring the communities of AI and comparative cognition closer together, with the potential to disentangle capabilities and behaviours that are associated in animals, but can be disassociated in AI.
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abstract: 'We show that the frequency spectrum of two dimensional photonic crystals is strongly influenced by Fano resonances which can be excited already in the linear array of dielectric cylinders. To support this claim, we calculate the transmission of electromagnetic wave through linear array of dielectric cylinders and show that frequencies of observed Fano resonances coincides with position of narrow frequency bands found in the spectra of corresponding two-dimensional photonic crystals. Split of frequency band or overlap of two bands, observed in the band structure of photonic structures are also associated with Fano resonances.'
author:
- Peter Markoš
title: Fano resonances and band structure of two dimensional photonic structures
---
Introduction
============
Frequency spectrum of photonic crystals consists of large number of continuous bands. This is a direct consequence of spatial periodicity of the permittivity $\varepsilon$ which defines the structure. In the limit of infinitesimally small variations of the permittivity, all frequency bands could be constructed from the dispersion relation of electromagnetic wave in homogeneous medium by reduction of the momenta $\vec{q}$ to the first Brillouin zone [@sakoda; @costas; @iok]. With increasing permittivity contrast, frequency gaps open at the edges of Brillouin zone [@joan-pc]. We will call the resulting frequency bands periodic (${\fp}$) bands.
For higher permittivity contrast, the frequency spectrum is more complicated. It contains, besides the ${\fp}$ bands, also other, usually very narrow (almost dispersionless) frequency bands. As an example, we show in Fig. \[uvod\](a) the frequency spectrum of two dimensional square array of dielectric cylinders embedded in vacuum. Only five of 17 displayed frequency bands are $\fp$ bands.
![(Color online) (a) Frequency spectrum of the square array of dielectric cylinders with radius $R=0.4a$ ($a$ is the spatial periodicity in $x$ and $y$ directions) and permittivity $\varepsilon=12$ calculated by the plane wave expansion method [@sakoda]. In the frequency region $a/\lambda <0.9$ we found five ${\cal P}$ bands (black full lines), five even (dashed red lines) and seven odd (dot blue lines) $\ff$ bands. Inset shows schematically the structure in the $xz$ plane, red arrow indicates the propagation direction of the electromagnetic wave. (b) Transmission coefficient of the plane wave incident on the linear array of cylinders (The geometry of the problem is shown in (c)). The incident angle is $\theta=0$ (red line) and $\theta = \pi/100$ (blue line). Note the one-to-one correspondence between resonant frequencies of excited Fano resonances and ${\cal F}$ bands observed in the frequency spectrum of the infinite 2D structure. (c) Transmission of incident electromagnetic wave through linear array of dielectric cylinders. (d) Geometry of the photonic structure composed from $N$ rows of dielectric cylinders. ($N=24$ throughout this Paper.) Both structures displayed in (c) and (d) are infinite in the $x$ direction. []{data-label="uvod"}](pm-fig1-ab.pdf "fig:"){width="0.64\linewidth"} ![(Color online) (a) Frequency spectrum of the square array of dielectric cylinders with radius $R=0.4a$ ($a$ is the spatial periodicity in $x$ and $y$ directions) and permittivity $\varepsilon=12$ calculated by the plane wave expansion method [@sakoda]. In the frequency region $a/\lambda <0.9$ we found five ${\cal P}$ bands (black full lines), five even (dashed red lines) and seven odd (dot blue lines) $\ff$ bands. Inset shows schematically the structure in the $xz$ plane, red arrow indicates the propagation direction of the electromagnetic wave. (b) Transmission coefficient of the plane wave incident on the linear array of cylinders (The geometry of the problem is shown in (c)). The incident angle is $\theta=0$ (red line) and $\theta = \pi/100$ (blue line). Note the one-to-one correspondence between resonant frequencies of excited Fano resonances and ${\cal F}$ bands observed in the frequency spectrum of the infinite 2D structure. (c) Transmission of incident electromagnetic wave through linear array of dielectric cylinders. (d) Geometry of the photonic structure composed from $N$ rows of dielectric cylinders. ($N=24$ throughout this Paper.) Both structures displayed in (c) and (d) are infinite in the $x$ direction. []{data-label="uvod"}](pm-fig1-cd.pdf "fig:"){width="0.3\linewidth"}
The aim of this Paper is to explain physical origin these additional frequency bands. We prove that they originate from Fano resonances [@fano; @miro] which can be excited in linear chain of dielectric cylinders by incident electromagnetic wave. We find numerically the spectrum of these resonances and demonstrate how each resonance develops into narrow Fano ($\ff$) band in corresponding two-dimensional photonic crystal.
The band structure shown in Fig. \[uvod\](a) was calculated for the square array of thick dielectric cylinders. For other structures, for instance photonic crystals composed from thinner cylinders, mutual coupling of $\ff$ and original $\fp$ band occurs which results in irregularities of the band spectrum. Two of them, namely the split of the $\fp$ band, and an overlap of two bands, will be discussed later.
Fano resonances in photonic structures [@sfan] were studied mostly in process of interaction of two dimensional photonic slabs with incident electromagnetic wave [@fan] and were used for experimental identification of spectra of leaky modes in photonic structures [@astr]. For the case of individual dielectric cylinder, Fano resonances were observed as a result of the coupling of Mie resonant states [@mie; @hulst] with an incident electromagnetic wave in spherical [@trib] and cylindrical dielectric object [@rybin]. Fano resonances play an important role in design of metamaterials [@rr; @luk] and influence considerably the transport properties of disordered systems [@fano-nature]. Here, we are interested in Fano resonances which could be excited by plane electromagnetic wave incident to the linear periodic array of dielectric cylinders (Fig. \[uvod\](c)) as a result of interference of leaky guided modes of periodic array of cylinders [@joan-pc] with incident electromagnetic wave. These resonances can be characterized by the resonant frequency and lifetime which is inversely proportional to the width of the resonant peak. Surprisingly, they are narrower than resonances excited in individual dielectric cylinders. If the electromagnetic wave propagates across $N$ chains of cylinders (Fig. \[uvod\](d)) then resonances excited in individual chains couple together and create narrow transmission band obtained in the frequency spectra of photonic crystals.
Structure and method
====================
We concentrate on periodic photonic structures composed from dielectric cylinders parallel to the $z$ axis. Cylinders possess a frequency independent permittivity $\varepsilon=12$ and permeability $\mu = 1$. The embedding medium is vacuum. Radius of cylinders is $R$ and the distance between two nearest-neighbor cylinders is $a$.
The first structure is the linear chain of dielectric cylinders lying in the $y=0$ axis (Fig. \[uvod\](c)). Incident electromagnetic wave of wavelength $\lambda$ excites in this structure Fano resonances which manifests themselves as sharp irregularities in the frequency dependence of the transmission coefficient visible in Fig. \[uvod\](b). Transmission coefficient is calculated also for finite photonic slab constructed from $N=24$ rows of the same cylinders located in planes $y = (n-1)a$ ($n=1,2,\dots,24$, see Fig. \[uvod\](d)). Observed transmission spectra will be compared with the band structure of an infinite square array of cylinders calculated by plane wave expansion method [@sakoda].
Transmission coefficient is calculated by the transfer matrix method [@pendry; @pre]. For more detailed analysis of the spatial distribution of electric field, we use another algorithm based on the expansion of electromagnetic field into cylinder functions [@hulst; @stratton; @on; @oua]. If an incident electromagnetic wave is polarized with electric field $E_z$ parallel to cylinders (this polarization is considered throughout this Paper) then the intensity of electric field scattered at a cylinder centered in $(xy)=(0,0)$ can be expressed as $$\label{eq:inc}
\begin{array}{rclcl}
E_z^{\rm in}(r,\phi) &\!\!\!=\!\!\!& {\cali}_0(r)\alpha_{0}^+
&\!\!\!\!+\!\!\!& 2\displaystyle{\sum_{k>0} }\alpha_{k}^+{\cali}_k(r) \cos(k\phi)\\
&&& + &
2i\displaystyle{\sum_{k>0}} \alpha_{k}^-{\cali}_k(r) \sin(k\phi)\\
E_z^{\rm out}(r,\phi) &\!\!\!=\!\!\!& {\calh}_0(r)\beta_{0}^+
&\!\!\!\!+\!\!\!& 2\displaystyle{\sum_{k>0} }\beta_{k}^+{\calh}_k(r) \cos(k\phi)\\
&&& + &
2i\displaystyle{\sum_{k>0}} \beta_{k}^-{\calh}_k(r) \sin(k\phi)
\end{array}$$ for $r<R$ and $r>R$, respectively. Similar expressions for radial and tangential components of magentic field could be derived from Maxwell equations [@stratton]. In Eq. (\[eq:inc\]), ${\cali}_k(r) = J_k(2\pi n r/\lambda)$, ${\calh}_k(r) = H_k(2\pi r/\lambda)/H'_k(2\pi R/\lambda)$, [@pozn] $J_k$, $H_k$ are Bessel and Hankel function, $\lambda=2\pi c/\omega$ determines the wavelength of electromagnetic field in vacuum and $n=\sqrt{\varepsilon\mu}$ is the index of refraction.
For another cylinder, centered at $x=n_xa$ and $y=n_ya$ the field $E_z$ is again expressed by Eq. \[eq:inc\] but with new set of coefficients $\alpha(n_x,n_y)$, $\beta(n_x,n_y)$ and cylindrical coordinates $r$ and $\phi$ associated with the center of the cylinder.
In numerical simulations, we calculate coefficients $\alpha$ and $\beta$ from the requirement of continuity of tangential components of the intensity of electric and magnetic field at the boundary of cylinders. Note that spatial periodicity of the structure along the $x$ direction considerably reduces the number of unknown coefficients since coefficients $\alpha(n_x,n_y)$ and $\beta(n_x,n_y)$ fulfill the Bloch theorem $$\label{eq:bloch}
\alpha(n_x,n_y) = \alpha(0,n_y)e^{iqan_x},~~~~
\beta(n_x,n_y) = \beta(0,n_y)e^{iqan_x}$$ where $q$ is the transverse component of the wave vector. This enables us to reduce the number of unknown coefficients $\beta^\pm_k(n_y)$ to $N\times (2N_B+1)$, where $N_B$ is the highest order of Bessel function used in numerical calculations ($N_B=12$ in most cases). To find the transmission coefficient $T$, Poynting vector is calculated on the opposite side of the structure. Details of the method are given elsewhere [@pm-15].
. Only even resonances are excited since the electromagnetic wave propagates perpendicularly to the cylinder chain [@pc-asym; @sak]. (b) The frequency dependence of parameters $\beta_k^\pm$. (c) Transmission coefficient for $N=24$ rows of cylinders. Fano resonances observed in linear chain develop themselves to narrow frequency bands. (d) Detail of the the frequency dependence of the transmission coefficient shown in (a) and (c) in the vicinity of the Fano resonance of coefficient $\beta_3^-$ ($a/\lambda= 0.562$). []{data-label="r40"}](pm-fig2.pdf){width="0.9\linewidth"}
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${\fp}$ bands: 1st 2nd 3rd 4th 5th
![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-01a.png "fig:"){width="0.11\linewidth"} ![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-028a.png "fig:"){width="0.11\linewidth"} ![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-046a.png "fig:"){width="0.11\linewidth"} ![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-063a.png "fig:"){width="0.11\linewidth"} ![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-08a.png "fig:"){width="0.11\linewidth"}
${\ff}$ bands: $2^+$ $3^-$ $4^+$ $0,2^+$ $5^-$
![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-041a.png "fig:"){width="0.11\linewidth"} ![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-056a.png "fig:"){width="0.11\linewidth"} ![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-071a.png "fig:"){width="0.11\linewidth"} ![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-0765a.png "fig:"){width="0.11\linewidth"} ![(Color online) Comparison of the field distribution $E_z$ in dielectric cylinder for frequencies in the center of the five lowest $\fp$ bands (top) and five lowest ${\cal F}$ bands (bottom). Electromagnetic wave propagates through photonic slab of the thickness of 24 rows of cylinders. For each frequency, only field inside the $10$th cylinder along the $y$ direction is shown. []{data-label="r40-fano"}](r04-0853a.png "fig:"){width="0.11\linewidth"}
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Results
=======
![(Color online) (a) Linear chain of thin dielectric cylinders ($R=0.1a$) and (b) Transmission coefficient $T$ and the coefficient $\beta_0$ as a function of the frequency $a/\lambda$. Owing to Fano resonance the transmission decreases to zero at $a/\lambda\approx 0.67$. (c) The band structure of two-dimensional square array. Fano resonance causes the broad gap between the second and the third bands. (d) The transmission coefficient of the electromagnetic wave propagating through an array of 24 rows of cylinders. (e) The intensity of electric field $E_z$ inside six cylinders along the $y$ direction at band edges X$_{2,3}$ and $\Gamma_{2,3}$. []{data-label="r10"}](pm-fig4-a.pdf "fig:"){width="0.232\linewidth"} ![(Color online) (a) Linear chain of thin dielectric cylinders ($R=0.1a$) and (b) Transmission coefficient $T$ and the coefficient $\beta_0$ as a function of the frequency $a/\lambda$. Owing to Fano resonance the transmission decreases to zero at $a/\lambda\approx 0.67$. (c) The band structure of two-dimensional square array. Fano resonance causes the broad gap between the second and the third bands. (d) The transmission coefficient of the electromagnetic wave propagating through an array of 24 rows of cylinders. (e) The intensity of electric field $E_z$ inside six cylinders along the $y$ direction at band edges X$_{2,3}$ and $\Gamma_{2,3}$. []{data-label="r10"}](pm-fig4-bcd.pdf "fig:"){width="0.72\linewidth"}
[(e)]{}
![(Color online) (a) Linear chain of thin dielectric cylinders ($R=0.1a$) and (b) Transmission coefficient $T$ and the coefficient $\beta_0$ as a function of the frequency $a/\lambda$. Owing to Fano resonance the transmission decreases to zero at $a/\lambda\approx 0.67$. (c) The band structure of two-dimensional square array. Fano resonance causes the broad gap between the second and the third bands. (d) The transmission coefficient of the electromagnetic wave propagating through an array of 24 rows of cylinders. (e) The intensity of electric field $E_z$ inside six cylinders along the $y$ direction at band edges X$_{2,3}$ and $\Gamma_{2,3}$. []{data-label="r10"}](61020.png "fig:"){width="0.9\linewidth" height="0.04\linewidth"} ![(Color online) (a) Linear chain of thin dielectric cylinders ($R=0.1a$) and (b) Transmission coefficient $T$ and the coefficient $\beta_0$ as a function of the frequency $a/\lambda$. Owing to Fano resonance the transmission decreases to zero at $a/\lambda\approx 0.67$. (c) The band structure of two-dimensional square array. Fano resonance causes the broad gap between the second and the third bands. (d) The transmission coefficient of the electromagnetic wave propagating through an array of 24 rows of cylinders. (e) The intensity of electric field $E_z$ inside six cylinders along the $y$ direction at band edges X$_{2,3}$ and $\Gamma_{2,3}$. []{data-label="r10"}](50001.png "fig:"){width="0.9\linewidth" height="0.04\linewidth"} ![(Color online) (a) Linear chain of thin dielectric cylinders ($R=0.1a$) and (b) Transmission coefficient $T$ and the coefficient $\beta_0$ as a function of the frequency $a/\lambda$. Owing to Fano resonance the transmission decreases to zero at $a/\lambda\approx 0.67$. (c) The band structure of two-dimensional square array. Fano resonance causes the broad gap between the second and the third bands. (d) The transmission coefficient of the electromagnetic wave propagating through an array of 24 rows of cylinders. (e) The intensity of electric field $E_z$ inside six cylinders along the $y$ direction at band edges X$_{2,3}$ and $\Gamma_{2,3}$. []{data-label="r10"}](40011.png "fig:"){width="0.9\linewidth" height="0.04\linewidth"} ![(Color online) (a) Linear chain of thin dielectric cylinders ($R=0.1a$) and (b) Transmission coefficient $T$ and the coefficient $\beta_0$ as a function of the frequency $a/\lambda$. Owing to Fano resonance the transmission decreases to zero at $a/\lambda\approx 0.67$. (c) The band structure of two-dimensional square array. Fano resonance causes the broad gap between the second and the third bands. (d) The transmission coefficient of the electromagnetic wave propagating through an array of 24 rows of cylinders. (e) The intensity of electric field $E_z$ inside six cylinders along the $y$ direction at band edges X$_{2,3}$ and $\Gamma_{2,3}$. []{data-label="r10"}](30006.png "fig:"){width="0.9\linewidth" height="0.04\linewidth"} $\longrightarrow~~y$
, to odd Fano bands. Note an overlap of the 3rd and 4th bands, shown in detail in the right panel. This overlap results from the coupling of two bands displayed by green dot-dashed and red dashed line. (b) Frequency dependence of the transmission coefficient for the linear chain of cylinders. Three Fano resonances could be easily identified. (c) Top panel shows the transmission coefficient for 24 rows of cylinders as a function of frequency $a/\lambda$. Two lower panels present detail of the frequency dependence of the transmission coefficient in the vicinity of the Fano resonance. [@xx] []{data-label="eps12-T"}](pm-fig5-a.pdf "fig:"){width="0.4\linewidth"} , to odd Fano bands. Note an overlap of the 3rd and 4th bands, shown in detail in the right panel. This overlap results from the coupling of two bands displayed by green dot-dashed and red dashed line. (b) Frequency dependence of the transmission coefficient for the linear chain of cylinders. Three Fano resonances could be easily identified. (c) Top panel shows the transmission coefficient for 24 rows of cylinders as a function of frequency $a/\lambda$. Two lower panels present detail of the frequency dependence of the transmission coefficient in the vicinity of the Fano resonance. [@xx] []{data-label="eps12-T"}](pm-fig5-bc.pdf "fig:"){width="0.50\linewidth"}
Isolated Fano bands
-------------------
Consider first an array of thick dielectric cylinders with radius $R=0.4a$. Figures \[uvod\](b) and \[r40\](a) shows the transmission coefficient of plane electromagnetic wave propagating through linear chain of cylinders. A series of very narrow resonances could be identified. Similar maxima and minims has been found numerically in the reflection coefficient [@on]. Following [@fan] we interpret these resonances as Fano resonances which results from the interference of incident plane wave with leaky guided modes excited in the periodic cylinder row [@joan-pc]. Indeed, these Fano resonances are accompanied by sharp maxima of coefficients $\beta$ defined in Eq. \[eq:inc\] (Figure \[r40\](b)). More detailed analysis of resonances is given in Sect. \[sect:fano\].
Comparison of Figures \[uvod\](a) and (b) confirms that Fano resonances develop to narrow Fano bands in the spectra of 2D structures. This is also shown in Figs. \[r40\](c,d) which present the frequency dependence of the transmission coefficient of plane wave propagating through the slab composed from $N=24$ rows of cylinders.
Different character of $\fp$ and $\ff$ bands is clearly visible from the spatial symmetry of the electric field $E_z$ shown in Fig. \[r40-fano\]. The top panel displays the field $E_z$ for frequencies chosen in the center of five lowest ${\fp}$ band shown in Fig. \[r40\](c). As expected, the field symmetry changes when the frequency increases from one frequency band to the next one [@sakoda; @joan-pc]. The bottom panel shows the field $E_z$ for the five lowest resonant frequencies identified in Fig. \[r40\](b). As shown in Fig. \[r40\](d), these frequencies correspond to the center of $\ff$ bands. The symmetry of the field is unambiguously determined by the order of excited Fano resonance.
Overlap of two bands
--------------------
For thinner cylinders, the $\fp$ band and the $\ff$ band can overlap. Then, the resulting frequency spectrum depends on the $q$-dependence of the frequency in two bands and on the strength of their mutual coupling. Consider a simple model of two bands $2V_1\cos q$ and $2V_2\cos q$, (centered at the same frequency for simplicity) coupled together with coupling constant $2A$. Resulting spectrum has a form $$\omega_{1,2}(q) = (V_1+V_2)\cos q \pm \sqrt{|A|^2+(V_1-V_2)^2\cos^2q}$$ Two bands $\omega_1(q)$ and $\omega_2(q)$ are separated by gap if $4V_1V_2<|A|^2$. This happens either when $|A|$ is large or when one of two bands is narrow.
*Band splitting*. We found the above mentioned band splitting in the frequency spectrum of the square array of thin ($R=0.1a$) dielectric cylinders (Fig. \[r10\]). The transmission coefficient through a linear chain of cylinders (Fig. \[r10\](b)) decreases to zero for the frequency $a/\lambda\approx 0.67$. This decrease is accompanied by an increase of coefficient $\beta_0$ (Eq. \[eq:inc\]). We interpret this decrease of the transmission as a result of excitation of broad Fano resonance.
Figure \[r10\](c) shows the band structure of the square array of thin cylinders. Note that both the second and the third bands have a minimum at the X point. These two bands result from the coupling of the ${\cal P}$ band with broad ${\cal F}$ band displayed by dot dashed (X$_2\Gamma_3$) and dashed ($\Gamma_2$X$_3$) lines, respectively.
This statement is supported also by the analysis of the spatial symmetry of electric field within two bands shown in Fig. \[r10\](e). Note that the symmetry of electric field changes along the line X$_2\Gamma_2$. Also, the field in points X$_2$ and $\Gamma_3$ have the same symmetry; of course, the same holds for pair X$_3$ and $\Gamma_2$. While the field at X$_2$ and $\Gamma_3$ possesses the symmetry of the ${\cal P}$ band, field close to inner band edges X$_3$ and $\Gamma_2$ has a symmetry of excited Fano resonance.
*Band overlap*. The overlap of $\fp$ and $\ff$ bands is observed in the band structure of the infinite square array of cylinders with radius $R=0.3a$ displayed in Fig. \[eps12-T\](a) As shown in the right panel, the overlap of the 3rd and 4th bands can be interpreted as a result of coupling of two bands shown by dot dashed and dashed lines, respectively. This overlap can be identified also from the complicated frequency dependence of the transmission coefficient [@sakoda-1997] shown in Fig. \[eps12-T\](b) for the slab of $N=24$ rows of cylinders. Similar analysis could be done for the overlap of the 5th and 6th bands in Fig. \[eps12-T\](a).[@xx]
Fano resonance {#sect:fano}
==============
In previous Section, we have shown that Fano resonances excited in linear array of dielectric cylinders create the $\ff$ bands in spectra of photonic crystals. Now we will discuss physical origin of Fano resonances.
Fano resonances have been observed recently in the most simple dielectric structure – the single dielectric cylinder [@rybin]. If the frequency of incident electromagnetic wave coincides with the eigenfrequency of any cylinder leaky eigenmode, the last can be excited. Then, the electromagnetic field in the neighbor of the cylinder is given by a superposition of two fields with the same frequency: the incident plane wave and field radiated by excited resonance [@rybin]. The excitation of resonance manifests itself as a maximum of coefficient $\beta$ shown in Fig. 6(a). The width of the resonance is proportional to inverse of its lifetime. Similarly, excitation of resonant guided mode in linear array of cylinders can be identified from sharp maxima in frequency dependence of corresponding coefficient $\beta$ (Figs. 2(b), 6(b)). The interference of two modes is the responsible for narrow maxima and minima in the transmission coefficient of incident electromagnetic wave displayed for instance in Figs. 1(b) and 2(a).
![(Color online) (a) Coefficients $\beta$ for (a) single dielectric cylinder with radius $R=0.4a$. (b) linear row of cylinders. (c) Coefficients $\beta_2^+$ (red) and an inverse determinant $\Delta^{-1}$ (black) for the clusters of $M$ dielectric cylinders ($M=1,2,4,24, 48,\infty$) along the $x$ axis. Thick blue line shows the position of the $\ff$ band $0.4067 < a/\lambda < 0.4254$ displayed in Fig. 1(a) by red dashed line. Multiple points for a given frequency corresponds with different intensity of excited resonance at individual cylinders as shown in (d) for the cluster with $M=48$ cylinders. []{data-label="fig:mie"}](pm-fig6ab.pdf "fig:"){width="0.99\linewidth"}\
![(Color online) (a) Coefficients $\beta$ for (a) single dielectric cylinder with radius $R=0.4a$. (b) linear row of cylinders. (c) Coefficients $\beta_2^+$ (red) and an inverse determinant $\Delta^{-1}$ (black) for the clusters of $M$ dielectric cylinders ($M=1,2,4,24, 48,\infty$) along the $x$ axis. Thick blue line shows the position of the $\ff$ band $0.4067 < a/\lambda < 0.4254$ displayed in Fig. 1(a) by red dashed line. Multiple points for a given frequency corresponds with different intensity of excited resonance at individual cylinders as shown in (d) for the cluster with $M=48$ cylinders. []{data-label="fig:mie"}](pm-fig6d.pdf "fig:"){width="0.89\linewidth"}
![(Color online) (a) Coefficients $\beta$ for (a) single dielectric cylinder with radius $R=0.4a$. (b) linear row of cylinders. (c) Coefficients $\beta_2^+$ (red) and an inverse determinant $\Delta^{-1}$ (black) for the clusters of $M$ dielectric cylinders ($M=1,2,4,24, 48,\infty$) along the $x$ axis. Thick blue line shows the position of the $\ff$ band $0.4067 < a/\lambda < 0.4254$ displayed in Fig. 1(a) by red dashed line. Multiple points for a given frequency corresponds with different intensity of excited resonance at individual cylinders as shown in (d) for the cluster with $M=48$ cylinders. []{data-label="fig:mie"}](pm-fig6c.pdf){width="0.99\linewidth"}
We start with the comparison of the frequency dependence of coefficients $\beta$ for isolated dielectric cylinder (Fig. \[fig:mie\](a) ) and for an infinite periodic array of cylinders (Figs. \[r40\](b) and \[fig:mie\](b)). For single cylinder and $E_z$ polarization, $\beta_k$ is a solution of system of linear equations [@stratton] $$\left(
\begin{array}{cr}
{\cal J}_k & - {\cal H}_k(R) \\
{\cal J'}_k & - \zeta
\end{array}
\right)
\left(
\begin{array}{c}
\alpha_k\\
\beta_k
\end{array}
\right)
=
\left(
\begin{array}{r}
J_k \\
\zeta J'_k
\end{array}
\right)$$ where $J_k=J_k(2\pi R/\lambda)$, $\zeta=\sqrt{\mu/\varepsilon}$ is an impedance, and the r.h.s is given by the expansion of incident plane wave into Bessel functions [@as] $$e^{i 2\pi R/\lambda \sin\theta} = \sum_k J_k(2\pi R/\lambda)e^{ik\theta}$$ As shown in Fig. \[fig:mie\](a), Fano resonances of cylinder lye very close to those of an array of cylinders (Figs. 2(b) and Fig. \[fig:mie\](b)). The first two resonances ($k=0$ and $k=1$ are relatively broad, especially for an infinite number of cylinders.
On the other hand, Fano resonances with $k\ge 2$ are significantly narrower when excited in an infinite linear chain of cylinders than in individual cylinder. To explore how the shape of the resonance depends on the number of cylinders, we analyze the scattering of incident electromagnetic wave on finite cluster consisting from $M$ cylinders along the $x$ direction (Fig. 1(c)). Coefficients $\beta_k^\pm(n_x)$ were calculated as a solution of system of linear equations $$\label{eq:lin}
\textbf{A} \vec{\beta} = \vec{a}$$ where $\textbf{A}$ is a matrix of the size $M\times (2N_B+1)$ and vector $\vec{a}$ represents incident electromagnetic wave. Figure \[fig:mie\](c) presents coefficients $\beta_2^+$ for finite cluster of $M$ cylinders excited by perpendicularly incident plane wave. The resonance indeed becomes narrower when number of cylinders increases. Since the spatial distribution of electromagnetic field along the finite cluster is not homogeneous, $\beta_2^+$ acquires various values for individual cylinders inside the cluster. As an example, we plot in Fig. \[fig:mie\](d) spatial distribution of $|\beta_2^+|$ as well as its real and imaginary part calculated in cluster of 48 cylinders.
Figure \[fig:mie\](c) shows also the frequency dependence of an inverse of the determinant of the matrix $\textbf{A}$ (Eq. \[eq:lin\]) which determines the eigenfrequencies and lifetimes of leaky guided modes [@economou] exited in the cluster. Comparison of the frequency dependence of an inverse determinant and $\beta_2$ confirms Fano character of observed excitation and enables to estimate the sign of the Fano parameter $q<0$ [@miro; @p2].
Absorption
==========
![(Color online) (a) Absorption of the electromagnetic wave in the photonic structure discussed in Fig. \[eps12-T\] but with small imaginary part of the cylinder permittivity. Absorption is large at band edges, where the group velocity is small, and in the region of the Fano bands. (b) Detailed frequency dependence of the absorption in the resonant frequency region for Imag $\varepsilon= 0.001$ and 0.01. []{data-label="abs"}](pm-fig7a.pdf "fig:"){width="0.55\linewidth"}\
\
![(Color online) (a) Absorption of the electromagnetic wave in the photonic structure discussed in Fig. \[eps12-T\] but with small imaginary part of the cylinder permittivity. Absorption is large at band edges, where the group velocity is small, and in the region of the Fano bands. (b) Detailed frequency dependence of the absorption in the resonant frequency region for Imag $\varepsilon= 0.001$ and 0.01. []{data-label="abs"}](pm-fig7b.pdf "fig:"){width="0.57\linewidth"}
Finally, we note an another difference between $\fp$ and $\ff$ bands: we expect that the $\ff$ bands are much more sensitive to the absorption loses than the $\fp$ bands. One reason is that typical $\ff$ band is narrow, therefore the group velocity of transmitted wave is small. However, more important is that $\ff$ bands are associated with the resonance which led to higher intensity of propagating electric field. Figure \[abs\] presents the absorption of electromagnetic field in the array of $R=0.3a$ cylinders with small imaginary part of the permittivity. Fano bands could be identified from the position of large maxima of the absorption. Note that very small imaginary part of the permittivity (Imag $\varepsilon$/Real $\varepsilon= 8.3\times 10^{-5}$) causes the absorption of 20% of energy when wave propagates through an array of 24 rows of cylinders.
Conclusion
==========
In conclusion, we showed that the band structure of square arrays of cylinders can be completely described in terms of two kinds of frequency bands. The $\fp$ bands originates from the reduction of the dispersion relation to the first reduced zone. The $\ff$ bands have origin in Fano resonances observed in the linear chain of cylinders. The field distribution in $\ff$ bands is determined by the symmetry of Fano resonance. Two bands, $\fp$ and $\ff$ might overlap, which complicates resulting band structure. Since resonance is typically accompanied by strong electric field, we expect that absorption is stronger in $\ff$ bands than in the $\fp$ bands.
This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0108-11 and by the Agency VEGA under the contract No. 1/0372/13.
[99]{}
K. Sakoda, *Optical Properties of Photonic Crystals*, Berlin, Heidelberg: Springer (2005).
C. M. Soukoulis (Editor), *Photonic Crystals and Light Localization in the 21st Century*, NATO Sci. Ser. C.: Mathematical and Physical Sciences **563** Kluwer Acad. Publ. (2001).
K. Inoue and K. Ohtaka (Editors), *Photonic Crystals: Physics, Fabrication and Application*, Springer (2010).
J. D. Joannopoulos, S. G. Johnson, J. N. Winnand R. G. Meade, *Photonic Crystals: Molding the Flow of Light* 2nd edition. Princeton: Princeton University Press (2008).
U. Fano, Phys. Rev. **124**, 1866 (1961).
A. E. Miroshnichenko, S. Flach andf Y. S. Kivshar, Rev. Mod. Phys. **82**, 2257 (2010).
S. Fan, W. Suh and J. D. Joannopoulos, J. Opt. Soc. Amm. A **20**, 569 (2003).
S. Fan and J. D. Joannopoulos, Phys. Rev. B **65**, 235112 (2002).
V. N. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. S. Skolnick, T. F. Krauss and R. M. De La Rue, J. Light. Technol. **17**, 2050 (1999).
G. Mie, Ann. Physik **25**, 377 (1908). H. C. van de Hulst, *Light scattering by small particles* Dover Publ. Inc, NY (1981)
M.I. Tribelsky, S. Flach, A. E. Miroshnichenko, A.V. Gorbach and Y. S. Kivshar, Phys. Rev. Lett. **100**, 04903 (2008).
M. V. Rybin, K.B. Samusev, I. S. Sinev, G. Semouchkin, E. Semouchjina, Y. S. Kivshar, M. F. Limonov, Optics Express **21**, 30107 (2013);
M. V. Rybin, D. S. Filonov, P.A. Belov, Y. S. Kivshar and M. F. Limonov, Sci. Rep. **5**, 8774 (2015).
B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen and Ch. T. Chong, Nature Materials **9**, 707 (2010).
A. N. Poddubny, M. V. Rybin, M. F. Limonov and Y. S. Kivshar, Nature Communications **3**, 914 (2012).
K. Inoue and K Ohtaka, in [@iok] p. 9.
M. V. Rybin, A. B. Khanikaev, M. Inoue, K. B. Samusev, M. J. Steel, G. Yushin and M. F. Limonov, Phys. Rev. Lett. **103**, 023901 (2009).
J. B. Pendry and A. MacKinnon, Phys. Rev. Lett. **69**, 2772 (1992).
P. Markoš and C. M. Soukoulis, Phys. Rev. E **65**, 036622 (2002).
J. A. Stratton, *Electromagnetic Theory* (New York: Mc Graw-Hill Comp. 1941).
K. Ohtaka, H. Numata, Phys. Lett. **73A**, 411 (1979).
K. Ohtaka, T.Ueda and K. Amemiya, Phys. Rev. B **57**, 2550 (1998).
To assure the numerical stability, Hankel functions are normalized by the frequency-dependent function $H'_k(2\pi R/\lambda)$. Consequently, this function scales coefficients $\beta_k$ shown in Figures throughout the paper.
P. Markoš, ArXiv:1501.05125 (unpublished).
W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe and J. D. Joannopoulos, Phys. Rev. Lett. **68**, 2023 (1992). K. Sakoda, Phys. Rev. B **52**, 7982 (1995).
K. Sakoda, J. Opt. Soc. A. B **14**, 1961 (1997).
Note that the overlap frequency interval, found by the the transfer matrix method in Fig. \[eps12-T\](c) differs slightly from that calculated by the plane wave expansion (Fig. \[eps12-T\](a). The origin of this discrepancy lies in the numerical limitation of the transfer matrix method due to the finite discretization of space.
M. Abramowitz and I. A. Stegun, *Handbook of Mathematical Functions* Dover Publ. (1965).
C. A. Pfeiffer, E. N. Economou and K. L. Ngai, [Phys. Rev. B]{} **10**, 3038 (1974).
Note that $\beta_2$ equals to scattering efficiency $Q_{\rm sca,2}$ [@rr] multiplied by analytic contiguous function of the frequency.
|
---
abstract: 'A solution to the existence problem of $G$-designs with given subdesigns is known when $G$ is a triangle with $p=0,1,$ or $2$ disjoint pendent edges: for $p=0$, it is due to Doyen and Wilson, the first to pose such a problem for Steiner triple systems; for $p=1$ and $p=2$, the corresponding designs are kite systems and bull designs, respectively. Here, a complete solution to the problem is given in the remaining case where $G$ is a $3$-sun, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor.'
author:
- |
[**Giovanni Lo Faro**]{}\
Dipartimento di Scienze Matematiche e Informatiche,\
Scienze Fisiche e Scienze della Terra\
Università di Messina, Messina, Italia\
email: lofaro@unime.it
- |
[**Antoinette Tripodi**]{}\
Dipartimento di Scienze Matematiche e Informatiche,\
Scienze Fisiche e Scienze della Terra\
Università di Messina, Messina, Italia\
email: atripodi@unime.it
title: '**The Doyen-Wilson theorem for $3$-sun systems[^1]**'
---
\[section\] \[section\] \[section\] \[section\] \[section\]
[**Keywords:**]{} $3$-sun systems; embedding; difference set.
[**Mathematics Subject Classification(2000):**]{} 05B05, 05B30.
Introduction
============
If $G$ is a graph, then let $V(G)$ and $E(G)$ be the vertex-set and edge-set of $G$, respectively. The graph $K_n$ denotes the complete graph on $n$ vertices. The graph $K_m\setminus K_n$ has vertex-set $V(K_m)$ containing a distinguished subset $H$ of size $n$; the edge-set of $K_m\setminus K_n$ is $E(K_m)$ but with the $n \choose 2$ edges between the $n$ distinguished vertices of $H$ removed. This graph is sometimes referred to as a complete graph of order $m$ with a *hole* of size $n$.
Let $G$ and $\Gamma$ be finite graphs. A [*$G$-design*]{} of $\Gamma$ is a pair $(X, {\cal B})$ where $X=V(\Gamma)$ and ${\cal B}$ is a collection of isomorphic copies of $G$ ([*blocks*]{}), whose edges partition $E(\Gamma)$. If $\Gamma=K_n$, then we refer to such a design as a *$G$-design of order $n$*.
A $G$-design $(X_1, \cB_1)$ of order $n$ is said to be *embedded* in a $G$-design $(X_2, \cB_2)$ of order $m$ provided $X_1\subseteq X_2$ and $
\cB_1\subseteq \cB_2$ (we also say that $(X_1, \cB_1)$ is a *subdesign* (or *subsystem*) of $(X_2, \cB_2)$ or $(X_2,
\cB_2)$ contains $(X_1, \cB_1)$ as subdesign). Let $N(G)$ denote the set of integers $n$ such that there exists a $G$-design of order $n$. A natural question to ask is: given $n,m\in N(G)$, with $m>n$, and a $G$-design $(X, \cB)$ of order $n$, does exists a $G$-design of order $m$ containing $(X, \cB)$ as subdesign? Doyen and Wilson were the first to pose this problem for $G=K_3$ (Steiner triple systems) and in 1973 they showed that given $n,m\in N(K_3)=\{ v\equiv 1,3\!\!\!\!\pmod {6}\}$, then *any Steiner triple system of order $n$ can be embedded in a Steiner triple system of order $m$ if and only if $m\geq 2n+1$ or $m=n$* (see [@dw]). Over the years, any such problem has come to be called a “Doyen-Wilson problem” and any solution a “Doyen-Wilson type theorem”. The work along these lines is extensive ([@br2], [@fllhh]-[@hy], [@lt], [@lt2], [@w]) and the interested reader is referred to [@br] for a history of this problem.
In particular, taking into consideration the case where $G$ is a triangle with $p=0,1,2,$ or $3$ mutually disjoint pendent edges, a solution to the Doyen-Wilson problem is known when $p=0$ (Steiner triple systems, [@dw]), $p=1$ (kite systems, [@lt; @lt2]) and $p=2$ (bull designs, [@fllhh]). Here, we deal with the remaining case ($p=3$) where $G$ is a $3$-*sun*, i.e. a graph on six vertices consisting of a triangle with three pendent edges which form a 1-factor, by giving a complete solution to the Doyen-Wilson problem for $G$-designs where $G$ is a $3$-sun (*$3$-sun systems*).
Notation and basic lemmas
=========================
The *$3$-sun* consisting of the triangle $(a, b, c)$ and the three disjoint pendent edges $\{{a, d\}, \{b, e\}, \{c, f\}}$ is denoted by $(a,b,c;d,e,f)$. A $3$-sun system of order $n$ (briefly, 3SS$(n)$) exsits if and only if $n\equiv 0,1,4,9\!\!\!\!\pmod {12}$ and if $(X, \cS)$ is a 3SS$(n)$, then $|\cS|=\frac {n(n-1)}{12}$ (see [@yg]).
Let $n,m \equiv 0,1,4,9\!\!\!\!\pmod {12}$, with $m=u+n$, $u\geq 0$. The Doyen-Wilson problem for $3$-sun systems is equivalent to the existence problem of decompositions of $K_{u+n}\setminus K_n$ into $3$-suns.
Let $r$ and $s$ be integers with $r<s$, define $[r, s] = \{r, r + 1, . . . , s\}$ and $[s, r] = \emptyset$. Let $Z_u=[0, u-1] $ and $H=\{\infty _1, \infty _2,\ldots ,\infty _t\}$, $H\cap Z_u=\emptyset$. If $S=(a,b,c; d,e,f)$ is a 3-sun whose vertices belong to $Z_u \cup H$ and $i\in Z_u$, let $S+i=(a+i,b+i,c+i; d+i,e+i,f+i)$, where the sums are modulo $u$ and $\infty +i=\infty$, for every $\infty \in H$. The set $(S)= \{S+i : i\in Z_u\}$ is called the *orbit of $S$ under $Z_u$* and $S$ is a *base block* of $(S)$.
To solve the Doyen-Wilson problem for $3$-sun systems we use the *difference method* (see [@ls], [@p]). For every pair of distinct elements $i,j \in Z_u$, define $|i-j|_u $= min$\{|i-j|, u-|i-j|\}$ and set $D_u =\{|i-j|_u : i,j\in Z_u\}=\{1, 2, \ldots, \lfloor \frac{u}{2}\rfloor\}$. The elements of $D_u$ are called *differences* of $Z_u$. For any $d\in D_u$, $d\neq \frac u2$, we can form a single 2-factor $\{\{i, d+i\}\, :\, i\in Z_u\}$, while if $u$ is even and $d=
\frac u2$, then we can form a 1-factor $\{\{i, i+\frac u2\}\, :\,
0 \leq i\leq \frac u2 -1 \}$. It is also worth remarking that 2-factors obtained from distinct differences are disjoint from each other and from the 1-factor.
If $D\subseteq D_u$, denote by $\langle Z_u\cup H, D\rangle$ the graph with vertex-set $V=Z_u\cup H$ and the edge-set $
E=\{\{i, j\}\, :\,|i-j|_u=d ,\ d\in
D \}\cup \{\{\infty , i\}\, :\, \infty \in H,\ i\in Z_u\}$. The graph $\langle Z_u\cup H, D_u\rangle$ is the complete graph $K_{u+t}\setminus K_t$ based on $Z_u\cup H $ and having $H$ as hole. The elements of $H$ are called *infinity points*.
Let $X$ be a set of size $n \equiv 0,1,4,9\!\!\!\!\pmod {12}$. The aim of the paper is to decompose the graph $\langle Z_u\cup X, D_u\rangle$ into $3$-suns. To obtain our main result the $\langle Z_u\cup X, D_u\rangle$ will be regarded as a union of suitable edge-disjoint subgraphs of type $\langle Z_u\cup H, D\rangle$ (where $H\subseteq X$ may be empty, while $D\subseteq D_u$ is always non empty) and then each subgraph will be decomposed into 3-suns by using the lemmas given in this section. From here on suppose $u\equiv 0,1,3,4,5,7,8,$ $9,11\!\!\!\!\pmod {12}$.
Lemmas \[2-1\] - \[8-2\] give decompositions of subgraphs of type $\langle Z_u\cup H, D\rangle$ where $D$ contains particular differences, more precisely, $D=\{ 2\}$, $D=\{ 2,4\}$ or $D=\{ 1, $ $\frac u3\}$.
\[2-1\] Let $u\equiv 0 \pmod {4}$, $u\geq 8$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2\}, $ $\{ 2\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the $3$-suns $$\begin{array}{l}
(\infty _1,{2+4i},{4i}; {3+4i},4+4i,\infty _2), \\
(\infty _2,{3+4i},{1+4i}; {2+4i},5+4i,\infty _1) ,
\end{array}$$ for $i=0,1, \ldots, \frac{u}{4}-1$. $\Box$
\[4-1\] Let $u\equiv 0 \pmod {12}$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2, \infty _3, $ $\infty _4\}, \{ 2\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the $3$-suns $$\begin{array}{l}
(\infty _1,{12i},{2+12i}; {7+12i},,\infty _3,\infty _4), \\
(\infty _1,{4+12i},{6+12i}; {9+12i}, \infty _3,\infty _4), \\
(\infty _1,{8+12i},{10+12i}; {11+12i}, \infty _3,\infty _4), \\
(\infty _2,{2+12i},{4+12i}; {1+12i}, \infty _3,\infty _4), \\
(\infty _2,{6+12i},{8+12i}; {7+12i}, \infty _3,\infty _4), \\
(\infty _2,{10+12i},{12+12i}; {11+12i}, \infty _3,\infty _4),\\
(\infty _3,{1+12i},{3+12i}; {9+12i}, \infty _1,\infty _2),\\
(\infty _3,{5+12i},{7+12i}; {11+12i}, \infty _1,9+12i),\\
(\infty _4,{3+12i},{5+12i}; {1+12i}, \infty _1,\infty _2),\\
(\infty _4,{9+12i},{11+12i}; {7+12i}, \infty _2,13+12i),\end{array}$$ for $i=0,1, \ldots, \frac{u}{12}-1$. $\Box$
\[4-2\] The graph $\langle Z_u\cup \{\infty _1, \infty _2,\infty _3, $ $\infty _4\}, \{2,4\}\rangle$, $u\geq 7$, $u\neq 8$, can be decomposed into $3$-suns.
[*Proof.*]{} Let $u=4k+r$, with $r=0,1,3$, and consider the $3$-suns $$\begin{array}{l}
(\infty _1,{4+4i},{6+4i}; {5+4i},8+4i,\infty _4),\\
(\infty _2,{5+4i},{7+4i}; {6+4i},9+4i,\infty _1),\\
(\infty _3,{6+4i},{8+4i}; {7+4i},10+4i,\infty _2),\\
(\infty _4,{7+4i},{9+4i}; {8+4i},11+4i,\infty _3),
\end{array}$$ for $i=0,1, \ldots, k-3$, $k\geq3$, plus the following blocks as the case may be.\
If $r=0$, $$\begin{array}{l}
(\infty _1,{0},{2}; {1},4,\infty _4),\\
(\infty _2,{1},{3}; {2},5,\infty _1),\\
(\infty _3,{2},{4}; {3},6,\infty _2),\\
(\infty _4,{3},{5}; {4},7,\infty _3),\\
(\infty _1,{4k-4},{4k-2}; {4k-3},0,\infty _4),\\
(\infty _2,{4k-3},{4k-1}; {4k-2},1,\infty _1),\\
(\infty _3,{4k-2},{0}; {4k-1},2,\infty _2),\\
(\infty _4,{4k-1},{1}; {0},3,\infty _3).
\end{array}$$ If $r=1$, $$\begin{array}{l}
(\infty _1,{0},{2}; {1},4,\infty _2),\\
(\infty _2,{1},{3}; {0},5,\infty _1),\\
(\infty _3,{2},{4}; {3},6,\infty _2),\\
(\infty _4,{3},{5}; {4},7,\infty _3),\\
(\infty _1,{4k-4},{4k-2}; {4k-3},4k,\infty _2),\\
(\infty _2,{4k-3},{4k-1}; {4k},0,\infty _1),\\
(\infty _3,{4k-2},{4k}; {4k-1},1,\infty _1),\\
(\infty _4,{4k-1},{0}; {4k-2},2,\infty _3),\\
(\infty _4,{4k},{1}; {2},3,\infty _3).
\end{array}$$ If $r=3$, $$\begin{array}{l}
(\infty _1,{0},{2}; {1},4,\infty _4),\\
(\infty _2,{1},{3}; {2},5,\infty _1),\\
(\infty _3,{2},{4}; {3},6,\infty _2),\\
(\infty _4,{3},{5}; {4},7,\infty _3),\\
(\infty _1,{4k-4},{4k-2}; {4k-3},4k,\infty _4),\\
(\infty _2,{4k-3},{4k-1}; {4k-2},4k+1,\infty _1),\\
(\infty _3,{4k-2},{4k}; {4k-1},4k+2,\infty _2),\\
(\infty _4,{4k-1},{4k+1}; {4k},0,\infty _3),\\
(\infty _1,{4k},{4k+2}; {4k+1},1,\infty _4),\\
(\infty _2,{4k+1},{0}; {4k+2},2,\infty _4),\\
(\infty _3,{4k+2},{1}; {0},3,\infty _4).
\end{array}$$ With regard to the difference $4$ in $Z_7$, note that $|4|_7=3$ and the seven distinct blocks obtained for $k=1$ and $r=3$ gives a decomposition of $\langle Z_7\cup \{\infty _1, \infty _2,\infty _3, $ $\infty _4\}, \{2,3\}\rangle$ into $3$-suns. $\Box$
\[8-2\] Let $u\equiv 0 \pmod {3}$, $u\geq 12$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2,$ $\ldots, \infty _8\}, \{ 1, $ $\frac u3\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} If $u\equiv 0 \pmod {6}$ consider the $3$-suns: $$\begin{array}{l}
(\infty _1,{2i},{\frac u3+2i}; {2\frac u3+2i},\infty _5,\infty _6), \ i=0,1, \ldots, \frac u6-1, \\
(\infty _1,{1+2i},{\frac u3+1+2i}; {2\frac u3+1+2i},\infty _6,\infty _5), i=0,1,\ \ldots, \frac u6-1, \\
(\infty _2,{2\frac u3+2i},{\frac u3+2i}; {2+2i},2i,\infty _5), \ i=0,1, \ldots, \frac u6-2, \\
(\infty _2,{2\frac u3+1+2i},{\frac u3+1+2i}; {3+2i},1+2i,\infty _6), i=0,1,\ \ldots, \frac u6-2, \\
(\infty _2,{u-2},{2\frac u3-2}; {0},\frac u3-2,\infty _5), \
(\infty _2,{u-1},{2\frac u3-1}; {1},\frac u3-1,\infty _6), \\
(\infty _3,{2i},{1+2i}; {2\frac u3+2i},\infty _7,\infty _8), \ i=0,1, \ldots, \frac u6-1, \\
(\infty _3,{\frac u3+2i},{\frac u3+1+2i}; {2\frac u3+1+2i},\infty _7,\infty _8), i=0,1,\ \ldots, \frac u6-1, \\
(\infty _4,{1+2i},{2+2i}; {2\frac u3+2+2i},\infty _7,\infty _8), \ i=0,1, \ldots, \frac u6-1, \\
(\infty _4,{\frac u3+1+2i},{\frac u3+2+2i}; {2\frac u3+1+2i},\infty _7,\infty _8), i=0,1,\ \ldots, \frac u6-1, \\
(\infty _5,{2\frac u3+2i},{2\frac u3+1+2i}; {1+2i},\infty _7,\infty _8), \ i=0,1, \ldots, \frac u6-1, \\
(\infty _6,{2\frac u3+3+2i},{2\frac u3+4+2i}; {2+2i},\infty _7,\infty _8), i=0,1,\ \ldots, \frac u6-2, \\
(\infty _6,{2\frac u3+1},{2\frac u3+2}; {2\frac u3},\infty _7,\infty _8).
\end{array}$$ If $u\equiv 3 \pmod {6}$ consider the $3$-suns: $$\begin{array}{l}
(\infty _1,{2i},{\frac u3+2i}; {2\frac u3+2i},\infty _5,\infty _6), \ i=0,1, \ldots, \frac{u-3}{6}, \\
(\infty _1,{1+2i},{\frac u3+1+2i}; {2\frac u3+1+2i},\infty _6,\infty _5), i=0,1,\ \ldots, \frac{u-9}{6}, \\
(\infty _2,{2\frac u3+2i},{\frac u3+2i}; {2+2i},2i,\infty _5), \ i=0,1, \ldots, \frac{u-9}{6}, \\
(\infty _2,{u-1},{2\frac u3-1}; {0},\frac u3-1,\infty _5), \\
(\infty _2,{2\frac u3+1+2i},{\frac u3+1+2i}; {3+2i},1+2i,\infty _6), i=0,1,\ \ldots, \frac{u-15}{6}, \\
(\infty _2,{u-2},{2\frac u3-2}; {1},\frac u3-2,\infty _6), \\
(\infty _3,{2i},{1+2i}; {2\frac u3+2i},\infty _7,\infty _8), \ i= 2,3,\ldots, \frac{u-3}{6}, \\
(\infty _3,{0},{1}; {2\frac u3}, \infty _6,\infty _8),
(\infty _3,{2},{3}; {2\frac u3+2}, \infty _6,\infty _8), \\
(\infty _3,{\frac u3+1+2i},{\frac u3+2+2i}; {2\frac u3+1+2i},\infty _7,\infty _8), i=0,1,\ \ldots, \frac{u-9}{6}, \\
(\infty _4,{1+2i},{2+2i}; {2\frac u3+2+2i},\infty _7,\infty _8), \ i=0,1, \ldots, \frac{u-9}{6}, \\
(\infty _4,{\frac u3+2i},{\frac u3+1+2i}; {2\frac u3+1+2i},\infty _7,\infty _8), i=0,1,\ \ldots, \frac{u-3}{6}, \\
(\infty _5,{2\frac u3+2i},{2\frac u3+1+2i}; {1+2i},\infty _7,\infty _8), \ i=0,1, \ldots, \frac{u-9}{6}, \\
(\infty _6,{2\frac u3+1+2i},{2\frac u3+2+2i}; {4+2i},\infty _7,\infty _8), \ i=0,1,\ \ldots, \frac{u-15}{6},\\
(\infty _6,{u-2},{u-1}; {2\frac u3},\infty _7,\infty _8), \
(\infty _7,{u-1},{0}; {2},\infty _5,\infty _8).
\end{array}$$ $\Box$
Lemmas \[3-2u2\] - \[3-3u2\] allow to decompose $\langle Z_u\cup H, D\rangle$ where $u$ is even and $ D$ contains the difference $\frac u2$.
\[3-2u2\] Let $u$ be even, $u\geq 8$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2, \infty _3\}, \{1,$ $\frac u2\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the $3$-suns $$\begin{array}{l}
(\infty _1,{2i},{1+2i}; {\frac u2+2+2i},\frac u2+2i,\infty _3), \ i=0,1, \ldots, \frac{u}{4}-2, \\
(\infty _1,{\frac u2-2},{\frac u2-1}; {\frac u2},u-2,\infty _3),\\
(\infty _2,{1+2i},{\frac u2+1+2i}; {2i},2+2i,\infty _1), \ i=0,1, \ldots, \frac{u}{4}-1, \\
(\infty _3,{\frac u2+1+2i},{\frac u2+2i}; {2i},\frac u2+2+2i,\infty _2), \ i=0,1, \ldots, \frac{u}{4}-1.
\end{array}$$ $\Box$
\[4-2u2\] Let $u\equiv 0 \pmod {12}$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2, \infty _3, \infty _4\}, $ $\{1,\frac u2\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the $3$-suns $$\begin{array}{l}
(\infty _1,{6i},{\frac u2+6i}; {4+6i},\infty _3 ,\infty _2), \\
(\infty _1,{1+6i},{\frac u2+1+6i}; {5+6i},\infty _4 ,\infty _2),\\
(\infty _1,{2+6i},{\frac u2+2+6i}; {\frac u2+3+6i},\infty _4,\infty _3), \\
(\infty _2,{1+6i},{ 6i}; {\frac u2+3+6i},\infty _3 ,\infty _4),\\
(\infty _2,{2+6i},{3+6i}; {\frac u2+4+6i},1+6i,\infty _4), \\
(\infty _2,{5+6i},{ 4+6i}; {\frac u2+5+6i},6+6i ,3+6i), \\ (\infty _3,{3+6i},{\frac u2+3+6i}; {2+6i},\infty _1 ,\frac u2+2+6i), \\
(\infty _3,{4+6i},{\frac u2+4+6i}; {\frac u2+6i},\infty _4 ,\infty _1),\\
(\infty _3,{5+6i},{\frac u2+5+6i}; {\frac u2+1+6i},\infty _4 , \infty _1), \\
(\infty _4,{\frac u2+1+6i},{\frac u2+2+6i}; {\frac u2+3+6i},\frac u2+6i ,\infty _2), \\
(\infty _4,{\frac u2+4+6i},{\frac u2+5+6i}; {\frac u2+6i},\frac u2+3+6i ,\frac u2+6+6i), \end{array}$$ for $i=0,1, \ldots, \frac{u}{12}-1$. $\Box$
\[6-2u2\] Let $u$ be even, $u\geq 8$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2, \dots, \infty _6\}, \{1,\frac u2\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the $3$-suns $$\begin{array}{l}
(\infty _1,{2i},{1+2i}; {\frac u2+2+2i},\frac u2+2i,\infty _3), i=0,1, \ldots, \frac{u}{4}-2,\\
(\infty _1,{\frac u2-2},{\frac u2-1}; {\frac u2},u-2,\infty _3), \\
(\infty _2,{1+2i},{\frac u2+1+2i}; {2i},\infty _6,\infty _1), i=0,1, \ldots, \frac{u}{4}-1,\\
(\infty _3,{\frac u2+1+2i},{\frac u2+2i}; {2i},\infty _6,\infty _2), i=0,1, \ldots, \frac{u}{4}-1,\\
(\infty _4,{1+2i},{2+2i}; {\frac u2+2+2i},\infty _5,\infty _6), i=0,1, \ldots, \frac{u}{4}-1,\\
(\infty _5,{\frac u2+1+2i},{\frac u2+2+2i}; {2+2i},\infty _4,\infty _6), i=0,1, \ldots, \frac{u}{4}-1.
\end{array}$$ $\Box$
\[7-2u2\] Let $u\equiv 0 \pmod {12}$. Then the graph $\langle Z_u\cup \{\infty _1, $ $\infty _2, \dots, \infty _7\}, \{1,\frac u2\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the $3$-suns $$\begin{array}{l}
(\infty _1,{6i},{\frac u2+6i}; {4+6i},\infty _7 ,\infty _2), \\
(\infty _1,{1+6i},{\frac u2+1+6i}; {\frac u2+3+6i},\infty _7 ,\infty _4),\\
(\infty _1,{2+6i},{\frac u2+2+6i}; {\frac u2+5+6i},\infty _5,\infty _2), \\
(\infty _2,{3+6i},{ \frac u2+3+6i}; {6i},\infty _1 ,\infty _4),\\
(\infty _2,{4+6i},{\frac u2+4+6i}; {2+6i}, \infty _7,\infty _1), \\
(\infty _2,{5+6i},{ \frac u2+5+6i}; {\frac u2+1+6i}, \infty _1,\infty _7), \\
(\infty _3,{ 6i},{1+6i}; {\frac u2+6i},\infty _5 ,\infty _6), \\
(\infty _3,{2+6i},{3+6i}; {\frac u2+2+6i},\infty _7 ,\infty _6),\\
(\infty _3,{4+6i},{ 5+6i}; {\frac u2+5+6i},\infty _5 , \infty _6), \\
(\infty _4,{ 1+6i},{\ 2+6i}; {\frac u2+6+6i},\infty _2,\infty _6), \\
(\infty _4,{3+6i},{4+6i}; {\frac u2+4+6i},\infty _7 ,\infty _6),\\
(\infty _4,{5+6i},{6+6i}; {\frac u2+5+6i},\infty _7 ,\infty _6),\\
(\infty _5,{\frac u2+ 6i},{ \frac u2+1+6i}; { 1+6i}, \infty _7,\infty _3), \\
(\infty _5,{\frac u2+2+6i},{ \frac u2+3+6i}; { 3+6i}, \infty _7,\infty _3), \\
(\infty _5, {\frac u2+4+6i},{ \frac u2+5+6i}; { 5+6i}, \infty _7,\frac u2+6+6i),\\
(\infty _6,{\frac u2+1+6i},{ \frac u2+2+6i}; { \frac u2+5+6i}, \infty _7,\infty _4), \\
(\infty _6,{\frac u2+3+6i},{ \frac u2+4+6i}; { \frac u2+6+6i}, \infty _7,\infty _3),
\end{array}$$ for $i=0,1, \ldots, \frac{u}{12}-1$. $\Box$
\[3-3u2\] Let $u\equiv 0 \pmod {12}$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2, \infty _3\}, $ $\{1,2,\frac u2\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the $3$-suns $$\begin{array}{l}
(\infty _1,{6i},{1+6i}; {\frac u2+1+6i},\frac u2+6i ,3+6i), \\
(\infty _1,{2+6i},{3+6i}; {\frac u2+5+6i},\frac u2+2+6i ,5+6i), \\
(\infty _1,{4+6i},{5+6i}; {\frac u2+2+6i},\frac u2+4+6i ,7+6i), \\
(\infty _1,{\frac u2+3+6i},{\frac u2+4+6i}; {\frac u2+6i},\frac u2+2+6i , \infty _2), \\
(\infty _2,{1+6i},{ \frac u2+1+6i}; {\frac u2+3+6i}, 2+6i, \frac u2+2+6i),\\
(\infty _2,{3+6i},{4+6i}; {2+6i},\frac u2+3+6i, 6+6i), \\
(\infty _2,{5+6i},{ \frac u2+5+6i}; {\frac u2+2+6i},6+6i ,\frac u2+6+6i), \\
(\infty _3,{2+6i},{ 6i}; {1+6i}, 4+6i, \infty _2 ), \\
(\infty _3,{\frac u2+2+6i},{\frac u2 +6i}; {4+6i},\frac u2+4+6i ,\infty _2),\\
(\infty _3,{\frac u2+1+6i},{\frac u2+3+6i}; {3+6i},\frac u2+6i, \frac u2+5+6i), \\
(\infty _3,{\frac u2+5+6i},{\frac u2+4+6i}; {5+6i},\frac u2+7+6i, \frac u2+6+6i), \end{array}$$ for $i=0,1, \ldots, \frac{u}{12}-1$. $\Box$
The following lemma “combines” one infinity point with one difference $d\neq \frac u2, \frac u3$ such that $\frac u{gcd(u,d)}\equiv 0 \pmod {3}$ (therefore, $u\equiv 0 \pmod {3}$).
\[1-1\] Let $u\equiv 0 \pmod {3}$ and $d\in D_u\setminus \{\frac u2, \frac u3\}$ such that $p=\frac u{gcd(u,d)} \equiv 0 \pmod {3}$. Then the graph $\langle Z_u\cup \{\infty\}, \{d\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} The subgraph $\langle Z_u, \{d\}\rangle$ can be decomposed into $\frac{u}{p}$ cycles of length $p=3q$, $q \geq 2$.\
If $q>2$, let $(x_1,x_2, \ldots, x_{3q})$ be a such cycle and consider the $3$-suns $$(\infty ,x_{2+3i},x_{3+3i}; x_{7+3i},x_{1+3i},x_{4+3i}),$$ for $i=0,1, \ldots, q-1$ (where the sum is modulo $3q$).\
If $q=2$, let $(x_1^{(j)},x_2^{(j)},x_3^{(j)},x_4^{(j)},x_5^{(j)} , x_{6}^{(j)})$, $j=0,1,\ldots,\frac{u}{6}-1$, be the $6$-cycles decomposing $\langle Z_u, \{d\}\rangle$ and consider the $3$-suns $$(\infty ,x_{2}^{(j)},x_{3}^{(j)}; x_{1}^{( j+1)},x_{1}^{(j)},x_{4}^{(j)}),\ (\infty ,x_{5}^{(j)},x_{6}^{(j)}; x_{4}^{(j+1)},x_{4}^{(j)},x_{1}^{(j)}),$$ for $j=0,1, \ldots, \frac{u}{6}-1$ (where the sums are modulo $\frac{u}{6}$). $\Box$
Subsequent Lemmas \[5-1\] - \[1-5\] allow to decompose $\langle Z_u\cup H, D\rangle$, where $|H|=1,2,3,5$, $|D|=6-|H|$ and $\frac u2 \not \in D$; here, $u$ and $D$ are any with the unique condition that if $D$ contains at least three differences $d_1,d_2,d_3$, then $d_3=d_2-d_1 $ or $d_1+d_2+ d_3=u$.
\[1-5\] Let $d_1,d_2,d_3,d_4,d_5 \in D_u\setminus \{\frac u2\}$ such that $d_3=d_2-d_1 $ or $d_1+d_2+ d_3=u$. Then the graph $\langle Z_u\cup \{\infty \},
\{d_1,d_2,d_3,d_4,d_5\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} If $d_3=d_2-d_1 $, consider the orbit of $(d_1,d_2, 0;\infty , d_2+ d_5 ,d_4) $ (or $(d_1,d_2, 0;\infty , d_2+ d_5 ,-d_4) $, if $d_2+d_5=d_4$) under $Z_u$. If $d_1+d_2+ d_3=u$, consider the orbit of $(-d_1,d_2, 0;\infty , d_2+ d_5 ,d_4) $ (or $(-d_1,d_2, 0;\infty , d_2+ d_5 ,-d_4) $, if $d_2+d_5=d_4$) under $Z_u$. $\Box$
\[2-4\] Let $d_1,d_2,d_3,d_4 \in D_u\setminus \{\frac u2\}$ such that $d_3=d_2-d_1 $ or $d_1+d_2+ d_3=u$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2\},
\{d_1,d_2,d_3,d_4\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the orbit of $(d_1,d_2, 0;\infty_1 , \infty_2 ,d_4) $ or $(-d_1,d_2, 0;\infty_1 , $ $\infty_2 ,d_4) $ under $Z_u$ when, respectively, $d_3=d_2-d_1 $ or $d_1+d_2+ d_3=u$. $\Box$
\[3-3\] Let $d_1,d_2,d_3\in D_u\setminus \{\frac u2\}$ such that $d_3=d_2-d_1 $ or $d_1+d_2+ d_3=u$. Then the graph $\langle Z_u\cup \{\infty _1, \infty _2,\infty _3\},
\{d_1,d_2,d_3\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} Consider the orbit of $(d_1,d_2, 0;\infty_1 , \infty_2 ,\infty_3) $ or $(-d_1,d_2, 0;\infty_1 , $ $\infty_2 ,\infty_3) $ under $Z_u$ when, respectively, $d_3=d_2-d_1 $ or $d_1+d_2+ d_3=u$.$\Box$
\[5-1\] Let $d\in D_u\setminus \{\frac u2\}$, the graph $\langle Z_u\cup
\{\infty _1, \infty _2,$ $\infty _3, \infty _4, \infty _5\}, \{d\}\rangle$ can be decomposed into $3$-suns.
[*Proof.*]{} The subgraph $\langle Z_u, \{d\}\rangle$ is regular of degree 2 and so can be decomposed into $l$-cycles, $l\geq 3$. Let $(x_1,x_2, \ldots, x_{l})$ be a such cycle. Put $l=3q+r$, with $r=0,1,2$, and consider the $3$-suns with the sums modulo $l$ $$\begin{array}{l}
(\infty _1,x_{1+3i},x_{2+3i}; x_{3+3i},\infty _4,\infty _5),\\ (\infty _2,x_{2+3i},x_{3+3i}; x_{4+3i},\infty _4,\infty _5), \\ (\infty _3,x_{3+3i},x_{4+3i}; x_{5+3i},\infty _4,\infty _5),
\end{array}$$ for $i=0,1, \ldots, q-2$, $q \geq 2$, plus the following blocks as the case may be.\
If $r=0$, $$\begin{array}{l}
(\infty _1,x_{3q-2},x_{3q-1}; x_{3q},\infty _4,\infty _5),\\
(\infty _2,x_{3q-1},x_{3q}; x_{1},\infty _4,\infty _5),\\
(\infty _3,x_{3q},x_{1}; x_{2},\infty _4,\infty _5).
\end{array}$$ If $r=1$, $$\begin{array}{l}
(\infty _1,x_{3q-2},x_{3q-1}; x_{3q+1},\infty _4,\infty _5),\\
(\infty _2,x_{3q-1},x_{3q}; x_{1},\infty _4,\infty _1),\\
(\infty _3,x_{3q},x_{3q+1}; x_{2},\infty _4,\infty _2),\\
(\infty _5,x_{3q+1},x_{1}; x_{3q},\infty _4,\infty _3).
\end{array}$$ If $r=2$, $$\begin{array}{l}
(\infty _1,x_{3q-2},x_{3q-1}; x_{3q+2},\infty _4,\infty _5),\\
(\infty _2,x_{3q-1},x_{3q}; x_{1},\infty _4,\infty _5),\\
(\infty _3,x_{3q},x_{3q+1}; x_{2},\infty _1,\infty _2),\\
(\infty _4,x_{3q+1},x_{3q+2}; x_{3q},\infty _1,\infty _3),\\
(\infty _5,x_{3q+2},x_{1}; x_{3q+1},\infty _2,\infty _3).
\end{array}$$ $\Box$
Finally, after settling the infinity points by using the above lemmas, if $u$ is large we need to decompose the subgraph $ \langle Z_u, L \rangle$, where $L$ is the set of the differences unused (*difference leave*). Since by applying Lemmas \[2-1\]-\[3-3\] it could be necessary to use the differences $1$, $2$ or $4$, while Lemma \[5-1\] does not impose any restriction, it is possible to combine infinity points and differences in such a way that the difference leave $L$ is the set of the “small” differences, where $1$, $2$ or $4$ could possibly be avoided.
\[55\] Let $\alpha \in \{0,4,8\}$ and $u$, $s$ be positive integers such that $u> 12s+\alpha$. Then there exists a decomposition of $ \langle Z_u, L \rangle$ into $3$-suns, where:
- $\alpha=0$ and $L= [1,6s]$;
- $\alpha=4$ and $L= [3,6s+2]$;
- $\alpha=8$ and $L= [3,6s+4]\setminus \{4,6s+3\}$.
[*Proof.*]{}
- Consider the orbits $(S_j)$ under $Z_u$, where $ S_j=(5s+1+j,\ 5s-j,\ 0; 3s,\ s, u-2-2j)$, $j=0,1,\ldots,s-1$.
- Consider the orbits in $i)$, where $(S_0)$ is replaced with the orbit of $(6s+1,\ 4s,\ 0; s,$ $\ 9s, 6s+2)$.
- Consider the orbits in $i)$, where the orbits $(S_0)$ and $(S_1)$ are replaced with the orbits of $(6s+1,\ 4s,\ 0; s,\ 9s, 6s+4)$ and $(5s+2,\ 5s-1,\ 0; 3s,\ s, 6s+2)$.$\Box$
The main result
===============
Let $(X, \cS)$ be a $3$-sun system of order $n$ and $m\equiv
0,1,4,9 \pmod {12}$.
\[NC\] If $(X, \cS)$ is embedded in a $3$-sun system of order $m>n$, then $m\geq \frac 75 n+1$.
[*Proof.*]{} Suppose $(X, \cS)$ embedded in $(X', \cS')$, with $|X'|=m=n+u$ ($u$ positive integer). Let $c_i$ be the number of 3-suns of $\cS'$ each of which contains exactly $i$ edges in $X'\setminus X$. Then $\sum_{i=1}^6i\times c_i=
{u\choose 2 }$ and $\sum_{i=1}^5(6-i)c_i= u\times n$, from which it follows $6c_2+12c_3+18c_4+24c_5+30c_6= \frac {u(5u-2n-5)}2$ and so $u\geq \frac 25
n+1$ and $m\geq \frac 75 n+1$. $\Box$
By previous Lemma:
1. if $n= 60k+5r$, $r=0,5,8,9$, then $m\geq 84k+7r+1$;
2. if $n= 60k+5r+1$, $r=0,3,4,7$, then $m\geq 84k+7r+3$;
3. if $n= 60k+5r+2$, $r=2,7,10,11$, then $m\geq 84k+7r+4$;
4. if $n= 60k+5r+3$, $r=2,5,6,9$, then $m\geq 84k+7r+6$;
5. if $n= 60k+5r+4$, $r=0,1,4,9$, then $m\geq 84k+7r+7$.
In order to prove that the necessary conditions for embedding a $3$-sun system $(X, \cS)$ of order $n$ in a $3$-sun system of order $m=n+u$, $u>0$ are also sufficient, the graph $\langle Z_u\cup X, D_u \rangle$ will be expressed as a union of edge-disjoint subgraphs $\langle Z_u\cup X, D_u \rangle=\langle Z_u\cup X, D \rangle \cup \langle Z_u, L \rangle$, where $L=D_u\setminus D$ is the difference leave, and $\langle Z_u\cup X, D \rangle$ (if necessary, expressed itself as a union of subgrapphs) will be decomposed by using Lemmas \[2-1\]-\[5-1\], while if $L\neq \emptyset$, $ \langle Z_u, L \rangle$ will be decomposed by Lemma \[55\]. To obtain our main result we will distiguish the five cases $1.-5.$ listed before by giving a general proof for any $k\geq 0$ with the exception of a few cases for $k=0$, which will be indicated by a star $\star$ and solved in Appendix. Finally, note that:
- $u\equiv 0,1,4,$ or $9 \pmod {12}$, if $n\equiv 0 \pmod {12}$;
- $u\equiv 0,3,8,$ or $11\pmod {12}$, if $n\equiv 1 \pmod {12}$;
- $u\equiv 0,5,8,$ or $9\pmod {12}$, $n\equiv 4\pmod {12}$;
- $u\equiv 0,3,4,$ or $7\pmod {12}$, if $n\equiv 9\pmod {12}$.
\[60\] For any $n=60k+5r$, $r=0,5,8,9$, there exists a decomposition of $K_{n+u}\setminus K_n$ into $3$-suns for every admissible $u\geq 24k+2r+1$.
[*Proof.*]{} Let $X=\{\infty _1,\infty _2, \ldots,$ $ \infty _{60k+5r}\}$, $r=0,5,8,9$, and $u=24k+2r+1+h$, with $h\geq 0$. Set $h=12s+l$, $0\leq l\leq11$ ($l$ depends on $r$), and distinguish the following cases.
*Case $1$*: $r=0,5,8,9$ and $l=0$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$\langle Z_u\cup X, D \rangle \cup \langle Z_u, L \rangle$, where $D= [6s+1, 12k+r+6s]$, $|D|=12k+r$, and $L= [1,6s]$, and apply Lemmas \[5-1\] and \[55\].\
*Case $2$*: $r=0,9$ and $l=8$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2,\infty _3\}, \{2,6s+3,6s+5\} \rangle \cup
\langle Z_u\cup \{\infty _4\}, \{1\} \rangle \cup
\langle Z_u\cup \{\infty _5\}, \{6s+4\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _3,\infty _4,\infty _5\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [ 6s+6,12k+r+6s+4]$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[3-3\], \[1-1\], \[5-1\] and \[55\].\
*Case $3$*: $r=5,8$ and $l=4$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3,\infty _4\}, \{2,4\} \rangle \cup
\langle Z_u\cup \{\infty _5\}, \{1\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _3,\infty _4,\infty _5\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [ 6s+3,12k+r+6s+2]\setminus \{6s+4\}$, $|D'|=12k+r-1$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[4-2\], \[1-1\], \[5-1\] and \[55\].\
*Case $4$*: $r=0,8$ and $l=3$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5\}, \{2\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\infty _3,\infty _4 ,\infty _{5}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3, 12k+r+6s+1]$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[3-2u2\], \[2-1\], \[5-1\] and \[55\].\
*Case $5$*: $r=0$ and $l=11$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,2,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5\}, \{4,6s+3,6s+5,6s+7\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _3 , \infty _4, \infty _{5}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'=[ 6s+6,12k+6s+5]\setminus \{ 6s+7\}$, $|D'|=12k-1$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[3-3u2\], \[2-4\], \[5-1\] and \[55\].\
*Case $6$*: $r=5$ and $l=1$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots,\infty _6\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _7,\infty _8, \infty _9,$ $\infty _{10}\}, \{2\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots,\infty _{10}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3, 12k+6s+5]$, $|D'|=12k+3$, and $L=[3,6s+2]$, and apply Lemmas \[6-2u2\], \[4-1\], \[5-1\] and \[55\].\
*Case $7$*: $r=5,9$ and $l=9$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5\}, \{2,6s+3,6s+4,6s+5\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\infty _3,\infty _4, \infty _{5}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+6, 12k+r+6s+4]$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[3-2u2\], \[2-4\], \[5-1\] and \[55\].\
*Case $8$*: $r=8$ and $l=7$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,2,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4\}, \{4\} \rangle \cup
\langle Z_u\cup \{\infty _5\} , \{6s+5\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _3 , \infty _4, \infty _{5}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3, 12k+6s+11]\setminus \{6s+4, 6s+5\}$, $|D'|=12k+7$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[3-3u2\], \[1-1\], \[5-1\] and \[55\].\
*Case $9$*: $r=9$ and $l=5$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4\}, \{2\} \rangle \cup
\langle Z_u\cup \{\infty _5\} , \{4\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _3 , \infty _4, \infty _{5}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3,12k+6s+11]\setminus \{6s+4\}$, $|D'|=12k+8$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[3-2u2\], \[1-1\], \[5-1\] and \[55\]. $\Box$
\[70\] For any $n=60k+5r+1$, $r=0,3,4,7$, there exists a decomposition of $K_{n+u}\setminus K_n$ into $3$-suns for every admissible $u\geq 24k+2r+2$.
[*Proof.*]{} Let $X=\{\infty _1,\infty _2, \ldots,$ $ \infty _{60k+5r+1}\}$, $r=0,3,4,7$, and $u=24k+2r+2+h$, with $h\geq 0$. Set $h=12s+l$, $0\leq l\leq11$, and distinguish the following cases.
*Case $1$*: $r=0,3$ and $l=1$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty \}, \{6s+2\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty \}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+1,12k+r+6s+1]\setminus \{6s+2\}$, $|D'|=12k+r$, and $L= [1,6s]$, and apply Lemmas \[1-1\], \[5-1\] and \[55\].\
*Case $2$*: $r=0,3,4,7$ and $l=9$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,6s+3,6s+4\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5,$ $\infty _6\}, \{2,6s+5,6s+7\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots,\infty _6\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'=[6s+6,12k+r+6s+5]\setminus \{6s+7\}$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[3-3\], \[5-1\] and \[55\].\
*Case $3$*: $r=4^{\star},7$ and $l=5$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3,\infty _4\}, \{2,4\} \rangle \cup
\langle Z_u\cup \{\infty _5\}, \{1\} \rangle \cup
\langle Z_u\cup \{\infty _6\}, \{6s+8\} \rangle\cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots,\infty _6\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3,12k+r+6s+3]\setminus \{6s+4,6s+8\}$, $|D'|=12k+r-1$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[4-2\], \[1-1\], \[5-1\] and \[55\].\
*Case $4$*: $r=0,4$ and $l=6$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5,\infty _6\}, \{2,6s+3, 6s+5\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\ldots,\infty _6\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+4,12k+r+6s+3]\setminus \{6s+5\}$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[3-2u2\], \[3-3\], \[5-1\] and \[55\].\
*Case $5$*: $r=0$ and $l=10$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots,\infty _6\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _7,\infty _8, \infty _9,$ $\infty _{10}\}, \{2\} \rangle \cup
\langle Z_u\cup \{\infty _{11}\}, \{4, 6s+3,6s+5,6s+6, 6s+7\} \rangle\cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots,\infty _{11}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+8, 12k+6s+5]$, $|D'|=12k-2$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[6-2u2\], \[4-1\], \[1-5\], \[5-1\] and \[55\].\
*Case $6$*: $r=3,7$ and $l=0$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots, \infty _6\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\ldots,$ $\infty _6\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= \{2\}\cup [6s+3,12k+r+6s]$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[6-2u2\], \[5-1\] and \[55\].\
*Case $7$*: $r=3$ and $l=4$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3, \infty _4\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _5\} , \{2\} \rangle \cup
\langle Z_u\cup \{\infty _6\}, \{6s+5\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots, \infty _{6}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3, 12k+6s+5]\setminus \{6s+5\}$, $|D'|=12k+2$, and $L=[3,6s+2]$, and apply Lemmas \[4-2u2\], \[1-1\], \[5-1\] and \[55\].\
*Case $8$*: $r=4$ and $l=2$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3, \infty _4\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _5,\infty _6\}, \{2\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\ldots,\infty _{6}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3, 12k+6s+5]$, $|D'|=12k+3$, and $L=[3,6s+2]$, and apply Lemmas \[4-2u2\], \[2-1\], \[5-1\] and \[55\].\
*Case $9$*: $r=7$ and $l=8$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,2,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5,\infty _6\}, \{4,$ $ 6s+3,6s+7\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\ldots,\infty _{6}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+5, 12k+6s+11]\setminus \{6s+7\}$, $|D'|=12k+6$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[3-3u2\], \[3-3\], \[5-1\] and \[55\]. $\Box$
\[80\] For any $n=60k+5r+2$, $r=2,7,10,11$, there exists a decomposition of $K_{n+u}\setminus K_n$ into $3$-suns for every admissible $u\geq 24k+2r+2$.
[*Proof.*]{} Let $X=\{\infty _1,\infty _2, \ldots,$ $ \infty _{60k+5r+2}\}$, $r=2,7,10,11$, and $u=24k+2r+2+h$, with $h\geq 0$. Set $h=12s+l$, $0\leq l\leq11$, and distinguish the following cases.
*Case $1$*: $r=2,11$ and $l=3$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty_1 \}, \{6s+2\} \rangle \cup \langle Z_u\cup \{\infty_2 \}, \{6s+4\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty_1,\infty_2 \}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+1, 12k+r+6s+2]\setminus \{6s+2, 6s+4\}$, $|D'|=12k+r$, and $L= [1,6s]$, and apply Lemmas \[1-1\], \[5-1\] and \[55\].\
*Case $2$*: $r=2,7,10,11$ and $l=7$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2\}, \{1,2,6s+3,6s+4\} \rangle
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2\}), $ $ D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+5,12k+r+6s+4]$, $|D'|=12k+r$, and $L=[3,6s+2]$, and apply Lemmas \[2-4\], \[5-1\] and \[55\].\
*Case $3$*: $r=7,10$ and $l=11$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,6s+3,6s+4\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5,$ $\infty _6\}, \{2,6s+5,6s+7\} \rangle \cup
\langle Z_u\cup \{\infty _7\}, \{6s+8\} \rangle\cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots,\infty _7\}), $ $D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+6,12k+r+6s+6]\setminus \{6s+7,6s+8\}$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[3-3\], \[1-1\], \[5-1\] and \[55\].\
*Case $4$*: $r=2$ and $l=6$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3, \infty _4\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _5,\infty _6,\infty _7\}, \{2,$ $ 6s+3,6s+5\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots, \infty _{7}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+4,12k+6s+5]\setminus \{6s+5\}$, $|D'|=12k+1$, and $L=[3,6s+2]$, and apply Lemmas \[4-2u2\], \[3-3\], \[5-1\] and \[55\].\
*Case $5$*: $r=2,10$ and $l=10$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots, \infty _6\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _7\}, \{2, 6s+3,6s+4,6s+5,6s+6\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots, \infty _{7}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+7, 12k+r+6s+5]$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[6-2u2\], \[1-5\], \[5-1\] and \[55\].\
*Case $6$*: $r=7,11$ and $l=4$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5, \infty _6, \infty _{7}\}, $ $ \{2,4\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots,\infty _{7}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3,12k+r+6s+2]\setminus \{6s+4\}$, $|D'|=12k+r-1$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[3-2u2\], \[4-2\], \[5-1\] and \[55\].\
*Case $7$*: $r=7$ and $l=8$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2,\infty _3\}, \{1,\frac{u}{2}\} \rangle \cup \{\infty _4,\infty _5,\infty _6\}, \{2, 6s+3,6s+5\} \rangle
\langle Z_u\cup \{\infty _7\}, \{6s+7\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots, \infty _{7}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+4,12k+6s+11]\setminus \{6s+5,6s+7\}$, $|D'|=12k+6$, and $L=[3,6s+2]$, and apply Lemmas \[3-2u2\], \[3-3\], \[1-1\], \[5-1\] and \[55\].\
*Case $8$*: $r=10$ and $l=2$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots, \infty _6\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _7\}, \{2\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots, \infty _{7}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3,12k+6s+11]$, $|D'|=12k+9$, and $L=[3,6s+2]$, and apply Lemmas \[6-2u2\], \[1-1\], \[5-1\] and \[55\].\
*Case $9$*: $r=11$ and $l=0$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots, \infty _7\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, $ $ \ldots, \infty _{7}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= \{2\}\cup [6s+3,12k+6s+11]$, $|D'|=12k+10$, and $L=[3,6s+2]$, and apply Lemmas \[7-2u2\], \[5-1\] and \[55\]. $\Box$
\[90\] For any $n=60k+5r+3$, $r=2,5,6,9$, there exists a decomposition of $K_{n+u}\setminus K_n$ into $3$-suns for every admissible $u\geq 24k+2r+3$.
[*Proof.*]{} Let $X=\{\infty _1,\infty _2, \ldots,$ $ \infty _{60k+5r+3}\}$, $r=2,5,6,9$, and $u=24k+2r+3+h$, with $h\geq 0$. Set $h=12s+l$, $0\leq l\leq11$, and distinguish the following cases.
*Case $1$*: $r=2,5,6,9$ and $l=4$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,6s+3,6s+4\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, $ $\infty _{3}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= \{2\}\cup [6s+5,12k+r+6s+3]$, $|D'|=12k+r$, and $L=[3,6s+2]$, and apply Lemmas \[3-3\], \[5-1\] and \[55\].\
*Case $2$*: $r=2,5$ and $l=8$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2\}, \{1,6s+3,6s+4,6s+5\} \rangle \cup
\langle Z_u\cup \{\infty _3\}, \{2\} \rangle \cup (X\setminus \{\infty _1,\infty _2, $ $\infty _{3}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+6, 12k+r+6s+5]$, $|D'|=12k+r$, and $L=[3,6s+2]$, and apply Lemmas \[2-4\], \[1-1\], \[5-1\] and \[55\].\
*Case $3$*: $r=6,9$ and $l=0$ (odd $u$).\
If $s=0$, then write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2,\ldots, \infty _8\}, \{1,\frac{u}{3}\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots, \infty _8\}), D' \rangle $, where $D'= [2, 12k+r+1]\setminus \{\frac{u}{3}\}$, $|D'|=12k+r-1$, and apply Lemmas \[8-2\] and \[5-1\]. If $s>0$, then write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,5s,5s+1\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5, \infty _6\}, \{2,6s+1,6s+3\} \rangle \cup
\langle Z_u\cup \{\infty _7\}, \{6s+2\} \rangle \cup
\langle Z_u\cup \{\infty _8\}, \{6s+4\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots,\infty _{8}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= \{2s+1,4s\}\cup [6s+5,12k+r+6s+1]$, $|D'|=12k+r-1$, and $L= [3,6s]\setminus \{2s+1,4s,5s,5s+1\}$, and apply Lemmas \[3-3\], \[1-1\] and \[5-1\] to decompose the first five subgraphs, while to decompose the last one apply Lemma \[55\] $i)$ and delete the orbit $(S_0)$.\
*Case $4$*: $r=2,6$ and $l=1$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _{3}\}), $ $ D' \rangle \cup \langle Z_u, L \rangle$, where $D'= \{2\}\cup [6s+3,12k+r+6s+1]$, $|D'|=12k+r$, and $L=[3,6s+2]$, and apply Lemmas \[3-2u2\], \[5-1\] and \[55\].\
*Case $5$*: $r=2^\star$ and $l=5$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots, \infty _6\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _7,\infty _8, \infty _9,$ $ \infty _{10}\}, \{2\} \rangle \cup
\langle Z_u\cup \{\infty _{11},\infty _{12},\infty _{13}\}, \{4,6s+3,6s+7\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, $ $ \ldots,\infty _{13}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+5, 12k+6s+5]\setminus \{6s+7\}$, $|D'|=12k$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[6-2u2\], \[4-1\], \[3-3\], \[5-1\] and \[55\].\
*Case $6$*: $r=5,9$ and $l=7$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2,\ldots, \infty _6\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _7,\infty _8\}, \{2,6s+3,6s+4, 6s+5\} \rangle \cup \langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\ldots, \infty _{8}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+6,12k+r+6s+4]$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[6-2u2\], \[2-4\], \[5-1\] and \[55\].\
*Case $7$*: $r=5$ and $l=11$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3, \infty _4\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _5,\infty _6\}, \{2,6s+3, 6s+5,6s+6\} \rangle \cup
\langle Z_u\cup \{\infty _7\}, \{4\} \rangle
\cup
\langle Z_u\cup \{\infty _8\}, \{6s+7\} \rangle
\cup \langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\ldots, \infty _{8}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'=[6s+8,12k+6s+11]$, $|D'|=12k+4$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[4-2u2\], \[2-4\], \[1-1\], \[5-1\] and \[55\].\
*Case $8$*: $r=6$ and $l=9$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _4,\infty _5,\infty _6\}, \{2,6s+3, 6s+5\} \rangle \cup
\langle Z_u\cup \{\infty _7\}, \{4\} \rangle
\cup
\langle Z_u\cup \{\infty _8\}, \{6s+7\} \rangle
\cup \langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\ldots,$ $ \infty _{8}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'=[6s+6,12k+6s+11]\setminus \{ 6s+7\}$, $|D'|=12k+5$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[3-2u2\], \[3-3\], \[1-1\], \[5-1\] and \[55\].\
*Case $9$*: $r=9$ and $l=3$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,2,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _{3}\}), $ $ D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3,12k+6s+11]$, $|D'|=12k+9$, and $L=[3,6s+2]$, and apply Lemmas \[3-3u2\], \[5-1\] and \[55\]. $\Box$
\[100\] For any $n=60k+5r+4$, $r=0,1,4,9$, there exists a decomposition of $K_{n+u}\setminus K_n$ into $3$-suns for every admissible $u\geq 24k+2r+3$.
[*Proof.*]{} Let $X=\{\infty _1,\infty _2, \ldots,$ $ \infty _{60k+5r+4}\}$, $r=0,1,4,9$, and $u=24k+2r+3+h$, with $h\geq 0$. Set $h=12s+l$, $0\leq l\leq11$, and distinguish the following cases.
*Case $1$*: $r=0,1^\star,4,9$ and $l=2$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3, \infty _4\}, \{2,4\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _3, $ $ \infty _4\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= \{1,6s+3\}\cup [6s+5,12k+r+6s+2]$, $|D'|=12k+r$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[4-2\], \[5-1\] and \[55\].\
*Case $2$*: $r=0,9$ and $l=6$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,6s+3,6s+4\} \rangle \cup \langle Z_u\cup \{\infty _4\}, \{2\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _{3} , \infty _4\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+5,12k+r+6s+4]$, $|D'|=12k+r$, and $L=[3,6s+2]$, and apply Lemmas \[3-3\], \[1-1\], \[5-1\] and \[55\].\
*Case $3$*: $r=1,4$ and $l=10$ (odd $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2\}, \{1,6s+3,6s+5,6s+6\} \rangle \cup
\langle Z_u\cup \{\infty _3\}, \{2\} \rangle \cup
\langle Z_u\cup \{\infty _4\}, \{6s+4\} \rangle
\cup \langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _{3}, \infty_4\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+7, 12k+r+6s+6]$, $|D'|=12k+r$, and $L=[3,6s+2]$, and apply Lemmas \[2-4\], \[1-1\], \[5-1\] and \[55\].\
*Case $4$*: $r=0,4$ and $l=5$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2,\ldots, \infty _6\}, \{1,\frac{u}{2}\} \rangle \cup \langle Z_u\cup \{\infty _7,\infty _8, \infty _9\}, \{2, $ $6s+3,6s+5\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots, \infty_{9}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+4,12k+r+6s+3]\setminus\{6s+5\}$, $|D'|=12k+r-1$, and $L=[3,6s+2]$, and apply Lemmas \[6-2u2\], \[3-3\], \[5-1\] and \[55\].\
*Case $5$*: $r=0$ and $l=9$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3, \infty _4\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _5,\infty _6,\infty _7\}, $ $\{2,6s+3, 6s+5\} \rangle \cup
\langle Z_u\cup \{\infty _8\}, \{4\} \rangle
\cup
\langle Z_u\cup \{\infty _9\}, \{6s+7\} \rangle
\cup \langle Z_u\cup (X\setminus \{\infty _1,\infty _2,\ldots, \infty _{9}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+6,12k+6s+5]\setminus\{6s+7\}$, $|D'|=12k-1$, and $L=[3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[4-2u2\] , \[3-3\], \[1-1\], \[5-1\] and \[55\].\
*Case $6$*: $r=1$ and $l=7$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots, \infty _7\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _8,\infty _9\}, $ $\{2,4,$ $6s+3,6s+5\} \rangle
\cup \langle Z_u\cup (X\setminus \{\infty _1,\infty _2,$ $\ldots, \infty _{9}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [ 6s+6,12k+6s+5]$, $|D'|=12k$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[7-2u2\], \[2-4\], \[5-1\] and \[55\].\
*Case $7$*: $r=1,9$ and $l=11$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup \{\infty _{4}\}, \{2,4,6s+3,6s+5,6s+6\} \rangle
\cup \langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \infty _{3}, \infty _{4}\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [ 6s+7,12k+r+6s+6]$, $|D'|=12k+r$, and $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[3-2u2\], \[1-5\], \[5-1\] and \[55\].\
*Case $8$*: $r=4$ and $l=1$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3, \infty _4\}, \{1,\frac{u}{2}\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, $ $\infty _{3}, \infty _4\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= \{2\}\cup [6s+3,12k+6s+5]$, $|D'|=12k+4$, and $L=[3,6s+2]$, and apply Lemmas \[4-2u2\], \[5-1\] and \[55\].\
*Case $9$*: $r=9$ and $l=3$ (even $u$).\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3\}, \{1,\frac{u}{2}\} \rangle \cup \langle Z_u\cup \{\infty _4\}, \{2\} \rangle \cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, $ $\infty _{3}, \infty _4\}), D' \rangle \cup \langle Z_u, L \rangle$, where $D'= [6s+3,12k+6s+11]$, $|D'|=12k+9$, and $L= [3,6s+2]$, and apply Lemmas \[3-2u2\], \[1-1\], \[5-1\] and \[55\]. $\Box$
Combining Lemma \[NC\] and Propositions \[60\]–\[100\] gives our main theorem.
Any 3SS$(n)$ can be embedded in a 3SS$(m)$ if and only if $m\geq
\frac 75 n+1$ or $m=n$.
[**Appendix**]{}
- $n=21$, $u=12s+15$\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \infty _3,\infty _4\}, \{2,4\} \rangle \cup
\langle Z_u\cup \{\infty _5\}, \{1\} \rangle \cup
\langle Z_u\cup \{\infty _6\}, \{6s+7\} \rangle\cup
\langle Z_u\cup (X\setminus \{\infty _1,\infty _2, \ldots,\infty _6\}), \{6s+3,6s+5,6s+6\} \rangle \cup \langle Z_u, L \rangle$, where $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[4-2\], \[1-1\], \[5-1\] and \[55\].
- $n=13$, $u=12s+12$\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,\infty _2, \ldots, \infty _6\}, \{1,6s+6\} \rangle \cup
\langle Z_u\cup \{\infty _7,$ $ \infty _8, \infty _9,\infty _{10}\}, \{2\} \rangle \cup
\langle Z_u\cup \{\infty _{11},\infty _{12},\infty _{13}\}, \{4,6s+3,6s+5\} \rangle \cup \langle Z_u, L \rangle$, where $L= [3,6s+4]\setminus \{4,6s+3\}$, and apply Lemmas \[6-2u2\], \[4-1\], \[3-3\] and \[55\].
- $n=9$, $u=12s+7$\
Write $\langle Z_u\cup X, D_u \rangle$=$
\langle Z_u\cup \{\infty _1,$ $\infty _2, \infty _3, \infty _4\}, \{2,4\} \rangle \cup
\langle Z_u\cup \{\infty _5,\infty _6, $ $ \infty _7, \infty _8, \infty _9\}, \{1\}\rangle \cup \langle Z_u, L \rangle$, where $L= [3,6s+3]\setminus \{4\}$, and apply Lemmas \[4-2\], \[5-1\] and decompose $ \langle Z_u, L \rangle$ as in Lemma \[55\] $iii)$, taking in account that $|6s+4|_{12s+7}=6s+3$.
[66]{}
[br2]{} D.E. Bryant and C.A. Rodger, [*The Doyen-Wilson theorem extended to $5$-cycles*]{}, J. Comb. Theory A 68 (1994), 218-225.
[br]{} D.E. Bryant and C.A. Rodger, [*On the Doyen-Wilson theorem for $m$-cycle systems*]{}, J. Comb. Des. 4 (1994), 253-271.
[dw]{} J. Doyen and R.M. Wilson, [*Embeddings of Steiner triple systems*]{}, Discrete Math. 5 (1973), 229-239.
[fllhh]{} C.-M. Fu, Y.-L. Lin, S.-W. Lo, Y.-F. Hsu, and W.-C. Huang, [*The Doyen–Wilson theorem for bull designs*]{}, Discrete Math. 313 (2013), 498-507.
[fl]{} H.-L. Fu and C.C. Lindner, [*The Doyen-Wilson theorem for maximum packings of $K_n$ with $4$-cycles*]{}, Discrete Math. 183 (1998), 103-117.
[flr]{} H.-L. Fu, C.C. Lindner, and C.A. Rodger, [*Two Doyen-Wilson theorems for maximum packings of $K_n$ with triples*]{}, Discrete Math. 178 (1998), 63-71.
[gw]{} H. Gao and J.-H. Wang, [*Doyen-Wilson theorem for perfect hexagon triple systems*]{}, Discrete Math. 311 (2011), 1006-1014.
[hy]{} W.-C. Huang and W.-C. Yang, [*The Doyen-Wilson theorem for extended directed triple systems*]{}, Ars Combinatoria 84 (2007), 77–83.
[ls]{} H. Lenz and G. Stern, [*Steiner triple systems with given subspaces: another proof of the Doyen-Wilson-theorem*]{}, B. Unione Mat. Ital. 5, 17-A (1980), 109-114.
[lt]{} G. Lo Faro and A. Tripodi, [*Embeddings of $\lambda$-fold kite systems, $\lambda \geq 2$*]{}, Australas. J. Combin. 36 (2006), 143-150.
[lt2]{} G. Lo Faro and A. Tripodi, [*The Doyen-Wilson theorem for kite systems*]{}, Discrete Math. 306 (2006), 2695-2701.
[p]{} R. Pelteshon, [*Eine Lösung der beiden Heffterschen Differenzenprobleme*]{}, Compos. Math. 6 (1938), 251-257.
[w]{} J.-H. Wang, [*Perfect hexagon triple systems with given subsystems*]{}, Discrete Math. 309 (2009), 2930-2933.
[yg]{} J. X. Yin and B.-S. Gong, [*Existence of $G$-designs with $|V (G)| = 6$*]{}, In: Combinatorial designs and applications (Huangshan, 1988), Lecture Notes in Pure and Appl. Math., Vol. 126, Dekker, New York, 1990, 201-218.
[^1]: Supported by P.R.I.N. and I.N.D.A.M. (G.N.S.A.G.A.)
|
---
abstract: 'In this paper we provide detailed proofs for some of the uniqueness results presented in Ref. [@Nima]. We show that: (1) Yang-Mills and General Relativity amplitudes are completely determined by gauge invariance in $n-1$ particles, with minimal assumptions on the singularity structure; (2) scalar non-linear sigma model and Dirac-Born-Infeld amplitudes are fixed by imposing full locality and the Adler zero condition (vanishing in the single soft limit) on $n-1$ particles. We complete the proofs by showing uniqueness order by order in the single soft expansion for Yang-Mills and General Relativity, and the double soft expansion for NLSM and DBI. We further present evidence for a greater conjecture regarding Yang-Mills amplitudes, that a maximally constrained gauge invariance alone leads to both locality and unitarity, without any assumptions on the existence of singularities. In this case the solution is not unique, but a linear combination of amplitude numerators.'
author:
- Laurentiu Rodina
title: '**Uniqueness from gauge invariance and the Adler zero**'
---
Introduction
============
Recently, in Ref. [@Nima] it was conjectured that after only fixing the number and form of possible singularities, gauge invariance uniquely determines the Yang-Mills and gravity scattering amplitudes. It was also stated that the same is true for scalar theories like the non-linear sigma model (NLSM) and Dirac-Born-Infeld (DBI), when gauge invariance is replaced by vanishing in the single soft limit. Crucially, in all of these cases locality and unitarity are never assumed, and so arise automatically as a consequence of uniqueness. Here by locality we mean that poles correspond to propagators of cubic diagrams (for YM and GR), and quartic diagrams (for NLSM and DBI), while by unitarity we mean factorization on those poles. In some sense then we see the emergence of spacetime and local quantum interactions purely from gauge invariance. A similar result was presented in Ref. [@bcfwl], that locality and vanishing under large BCFW shifts are also sufficient to completely fix the Yang-Mills amplitude. Beyond their conceptual implications, these uniqueness results have a very practical application: if a given expression can be verified to be gauge invariant and contain the correct singularity structure, it is now guaranteed to match the corresponding amplitude. This has many implications for a wide variety of recently developed formalisms, like BCFW recursion relations [@BCFW], the BCJ duality [@bcj], or the CHY scattering equations [@CHY1]-[@CHY3], among others.
The proof used in this article also demonstrates a new powerful application of soft limits, as well as novel derivations of the well-known leading theorems [@Weinberg][@doublesoftstuff]. Leading and subleading soft theorems have already proven very useful in a number of very surprising ways. Originally, they showed that charge conservation or the equivalence principle can be derived from S-matrix arguments [@Weinberg]. More recently, the theorems were interpreted as consequences of new symmetries [@strom1; @strom2], with further implications for black-hole information [@BH]. They were also used for recursion relations for effective field theories [@congkao].
The goal of this paper is three-fold. First, we present the full details of the argument used in Ref. [@Nima] to prove the uniqueness claims for Yang-Mills, gravity, NLSM, and DBI, when locality is assumed. Second, we extend the argument to prove the conjecture that uniqueness still holds without assuming locality. And third, we make an even larger conjecture, that gauge invariance alone, with no assumptions on the presence of any singularities, is sufficient to imply both locality and unitarity.[^1]
Assumptions and results
-----------------------
In all four theories, Yang-Mills, gravity, NLSM, and DBI, we start with an ansatz $B_n(p^k)$, based on our assumptions of the singularity structure and mass dimension. In general, this ansatz contains functions of the form: $$\begin{aligned}
\label{nl}
B_n(p^k)\equiv \sum_{i} \frac{ N_i(p^k)}{P_i}\, ,\end{aligned}$$ where the numerators $N(p^k)$ are general polynomials with $k$ powers of momenta, and linear in some number of polarization vectors/tensors (for YM/GR). The denominators $P_i$ can be any polynomial of $p_i.p_j$ factors. If we assume locality, it means we must restrict each $P_i$ to be a product of simple poles, which can be associated to propagators of cubic (for YM and GR) or quartic (for NLSM and DBI) diagrams. That is, a local ansatz has a form: $$\begin{aligned}
\label{ll}
B_n(p^k)\equiv \sum_{\textrm{diags. } i} \frac{ N_i(p^k)}{\prod_{\alpha_i}P_{\alpha_i}^2}\, ,\end{aligned}$$ with $\alpha_i$ corresponding to the channels of each diagram. As discussed in [@bcj], it is always possible to put amplitudes in this cubic diagram form, by adding artificial propagators to the higher point vertex interactions. For YM and NLSM the diagrams are ordered, while for gravity and DBI they are not. We can relax locality by dropping the underlying diagram structure, allowing each term to have some number $s$ of any singularities $P_\mathcal{S}^2=(\sum_i p_i)^2$, with the $p_i$ consecutive for YM and NLSM: $$\begin{aligned}
\label{lll}
B_n(p^k)\equiv \sum_{ i} \frac{ N_i(p^k)}{P_{\mathcal{S}_1}^2\ldots P_{\mathcal{S}_s}^2 }\, ,\end{aligned}$$
Then the claim is that for the smallest $s$ and $k$ which admit solutions, the ansatz (\[lll\]) is uniquely fixed by gauge invariance/vanishing in the single soft limit in $n-1$ particles. Concretely, these smallest values for $s$ and $k$ are:
- Yang-Mills: $s=n-3$, $k=n-2$
- Gravity: $s=n-3$, $k=2n-4$
- NLSM: $s=n/2-2$, $k=n-2$
- DBI: $s=n/2-2$, $k=2n-4$
In this article, we prove the following results:
1. Local Singularities + Gauge Invariance $\Rightarrow$ Locality + Unitarity
2. Locality + Adler zero $\Rightarrow$ Unitarity
The stronger version of claim number 2 for NLSM and DBI (that Local Singularities + Adler zero $\Rightarrow$ Locality + Unitarity) is less susceptible to the argument we use in this article, but a more direct approach was already presented in [@Nima]. We also prove a stronger result for Yang-Mills, by allowing non-local singularities $(\sum_i a_i p_i)^2$, with some mild restrictions. Further, we conjecture that completely ignoring the singularity structure, gauge invariance alone forces general polynomials to be linear combinations of amplitude numerators.
Surprisingly, imposing gauge invariance/vanishing in the soft limit for the $n^{\rm th}$ particle is not required, and is automatic once the other $n-1$ constraints have been imposed. Without loss of generality we can take particle 3 to be this $n^{\textrm{th}}$ particle, and we will always impose momentum conservation by expressing $p_3$ in terms of the other momenta. Tying the unneeded $n^{\rm th}$ constraint to momentum conservation ensures that we always avoid checks of the form $e_3\rightarrow p_3=-p_1-p_2-p_4-\ldots$, which would complicate the analysis.
To begin, the above statements can be easily tested explicitly for a small number of particles. For Yang-Mills, at four points, we can only have terms with one pole, either $(p_1+p_2)^2$ or $(p_1+p_4)^2$. Then the most general term we can write down is a linear combination of 60 terms, of the form $$\begin{aligned}
M_4(p^2)=a_1\frac{e_1.e_2 \,e_3.e_4\, p_1.p_4}{p_1.p_2}+a_2\frac{e_1.p_2\, e_3.p_2\, e_2.e_4}{p_1.p_2}+\ldots\, .\end{aligned}$$ Imposing gauge invariance in particles 1, 2, and 4 forces the coefficients $a_i$ to satisfy some linear equations with a unique solution, which turns out to be precisely the scattering amplitude of four gluons.
At five points, the most general non-local ansatz, where we only assume two cyclic singularities per term, contains some 7500 terms, and it can be checked that gauge invariance in four particles leads to the five point amplitude. It is indeed quite remarkable that gauge invariance is so constraining to produce a unique solution. Actually, it is even more remarkable that any solution exists at all! The amplitudes are the result of a striking conspiracy between the propagator structure and momentum conservation.
It is easy to make a gauge invariant in $n$ particles by taking different contractions of $\prod_{i=1}^n (e_i^{\mu_i} p_i^{\nu_i}-e_i^{\nu_i} p_i^{\mu_i})$, but this requires a mass dimension of $[n]$. No single diagram has enough momenta in the numerator to accommodate this product, so different diagrams must cancel each other. But less obviously, without momentum conservation, such contractions will always contain terms with at least $n$ factors of $e_i.p_j$. This is impossible to achieve even with several diagrams, since each diagram numerator can have at most $n-2$ such factors per term, while the denominators only contains $p_i.p_j$ factors. But with momentum conservation, together with a cubic propagator structure, it turns out that $n-2$ factors are sufficient, creating an object which satisfies more constraints than expected by simple counting. In fact, we will show that this structure is indeed very special. There is no non-trivial way of deforming or adding things to produce different solutions. Identical facts hold for the other theories as well. For example, the NLSM requires vanishing under $n-1$ particles, which naively would require $A\propto \mathcal{O}(p_i)$ for $n-1$ particles. Again, however, only at $k=n-2$, with momentum conservation and quartic diagrams we find an exception, which is the amplitude itself. The absence of solutions below this critical mass dimension is at the heart of the proof.
The basic strategy for the proof is the following. We start with an appropriate ansatz (local, non-local, etc), and we take single/double soft expansions. Then order by order we show that gauge invariance/the Adler zero condition uniquely fix the corresponding amplitudes. In this process we do not assume any form of the soft theorems, but we end up re-deriving the well known leading terms [@Weinberg; @doublesoftstuff]. Our approach does not directly provide the subleading terms (which for gravity have a particularly nice form [@strom3]), but only proves their uniqueness.[^2] The whole proof rests on showing that after the first non-vanishing order is fixed, none of the higher orders can produce independent solutions. The reason for this is that the subleading orders must have a growing number of soft momenta in the numerator, leaving fewer momenta to satisfy the necessary requirements.
Organization of the article
---------------------------
In section \[gaugeinv\], we begin by exploring the notion of constrained gauge invariance. We find a very simple proof that functions with at most $k<n-2$ factors of momenta in the numerator can be gauge invariant in at most $k$ particles, while the same is true for tensors with $k\le n-2$.
In section \[unique1\], we first prove a weaker version of our statement for YM and GR, by assuming locality. In section \[unique2\] the same argument is applied to the NLSM and DBI amplitudes, with gauge invariance replaced by the Adler zero condition.
In section \[locality\], relaxing our assumptions on the underlying cubic diagrams, we instead consider a more general singularity structure. We only keep the requirement of (local) singularities of the form $(\sum_i p_i)^2$, and recover the unique amplitude, as long as the number of such singularities per term is $s= n-3$. For fewer singularities there are no solutions, while for more the answer can always be factorized as $(\sum \textrm{poles})\times \textrm{(amplitude)}$. This proves the conjecture originally made in Ref [@Nima].
Finally, in section \[l2\], we investigate the extent to which more general singularities can be used to fix the Yang-Mills amplitude. Completely non-local singularities of the form $(\sum_i a_i p_i)^2$, with some minor restrictions, are also shown to provide a unique solution. Trying to find a less arbitrary ansatz, we are lead to consider polynomials again, with no singularities at all. We conjecture that yet an even stronger statement can be made, namely that the smallest mass dimension polynomial that admits a solution is fixed to a linear combination of amplitude numerators, when gauge invariance in [*all*]{} $n$ particles is imposed. The usual argument can be used to provide leading order evidence for this fact.
Constrained gauge invariance {#gaugeinv}
============================
Polynomials
-----------
Let $B(k)$ be a polynomial linear in polarization vectors, with at most $k$ factors of dot products of the type $e_i.p_j$ in any given term. Let $g$ be the total number of gauge invariance requirements, and $\Delta=g-k$ be the “excess" of gauge invariance requirements compared to the maximum number of $e.p$ factors. When appropriate, we will use the notation $\overline{e}_i$ to distinguish polarization vectors which are not used for gauge invariance. We wish to prove that, without momentum conservation, $B(k)$ can be gauge invariant in at most $k$ particles, ie. satisfy at most $\Delta=0$ constraints. Then we will prove that with momentum conservation the statement is still true, but only for $k<n-2$.
### No momentum conservation
Having no momentum conservation implies gauge invariants in particle $i$ must be proportional to $G_i^{\mu\nu}=e_i^\mu p_i^\nu-e_i^\nu p_i^\nu$. Therefore the only way to obtain gauge invariants in $k$ particles is with linear combinations of different contractions of products $\prod_i G_i^{\mu_i\nu_i}$. We will show that such expressions always contain at least one term with $k$ factors of $e.p$.
By assumption there are always more $e$’s than $e.p$’s, so at least one of the polarization vectors needed for gauge invariance will be in a factor $e.e$ or $e.\overline{e}$. Consider first as an example the following term in a polynomial with $k=2$: $$\begin{aligned}
\label{eq0}e_1.e_2\ e_3.p\ \overline{e}_4.p\ p.p\, .\end{aligned}$$ We wish to show that such a term cannot be gauge invariant in three particles, say particles 1, 2 and 3. We start with a polarization vector sitting in a $e.e$ factor, for example $e_1$. To make a gauge invariant in particle 1, at least one of the $p's$ above must be a $p_1$, and a pair term must exist with $e_1$ and a $p_1$ switched. We can use either the $p$ in the $e_3.p$ (or $\overline{e}_4$) factor, or one in the $p.p$ factor, to make the gauge invariants: $$\begin{aligned}
\label{eq1}
G_1&= (e_1.e_2)\ (e_3.p_1)\ \overline{e}_4.p\ p.p - (p_1.e_2)\ (e_3.e_1)\ \overline{e}_4.p\ p.p\, ,\\
\nonumber &\textrm{or}\\
\label{eq2}G_1'&=\ (e_1.e_2)\ e_3.p\ \overline{e}_4.p\ (p_1.p)- (p_1.e_2)\ e_3.p\ \overline{e}_4.p \ (e_1.p)\, .\end{aligned}$$ However, the second option leads to a term with four $e.p$ factors, contradicting our claim that just two factors are sufficient. The first option is fine, and so we can only use $p$’s in $e.p$ factors for this mark-and-switch procedure.
Now consider the second term in $G_1$ above, and note that we ended up with another $e.e$ factor, namely $e_3.e_1$. Applying the same reasoning for gauge invariance in 3 forces us to fix the factor $\overline{e}_4.p$ to $\overline{e}_4.p_3$. Therefore the second piece of $G_1$ can form a gauge invariant in particle 3 in the pair: $$\begin{aligned}
G_3=(p_1.e_2)\ (e_3.e_1)\ \overline{e}_4.p_3\ p.p-p_1.e_2\ p_3.e_1\ \overline{e}_4.e_3\ p.p\, .\end{aligned}$$ Now we do not need gauge invariance in 4, so this chain $1\rightarrow 3\rightarrow \overline{4}$ is finished. Note we do not care about making the first piece of $G_1$ gauge invariant in particle 3, we are only interested in finding some minimal constraints. Instead, we go back to (\[eq0\]) to check gauge invariance in the remaining particle 2. But the choices we made so far by imposing gauge invariance in 1 and 3 fixed both $e.p$ factors in this initial term to $$\begin{aligned}
e_1.e_2\ e_3.p_1\ \overline{e}_4.p_3\ p.p\, ,\end{aligned}$$ so now there is no way to make it gauge invariant in 2, as all the allowed $p$’s have been used up. Therefore the term (\[eq0\]) is not compatible with gauge invariance in {1,2,3}.
The general strategy is the same. New chains always start in the original term from $e.\overline{e}$ or $e.e$ factors, which are aways present by assumption. Next, for each jump we fix a $p$ in an $e.p$ or $\overline{e}.p$ factor, which becomes unavailable for other gauge invariants. The chain ends when reaching an $\overline{e}.p$ factor, and a new chain is started from the original term, and so on. The process ends when all the chains have ended on $\overline{e}.p$ factors, which means gauge invariance is compatible with this counting argument, or when all the $e.p$’s have been marked and a chain is unable to continue, which means the term is ruled out.
In general, we said we need gauge invariance in $k+1=\#[\overline{e}.e]+2\#[e.e]+\#[e.p]$ particles, but have only $k=\#[\overline{e}.p]+\#[e.p]$ factors. This means the difference between how many chains must start and how many can end is: $\left(\#[\overline{e}.e]+2\#[e.e]\right)-\#[\overline{e}.p]=1$. Therefore there is always at least one chain which cannot end, so all possible starting terms are ruled out. Then there is no way to make a polynomial $B(k)$ gauge invariant in $k+1$ particles.
### With momentum conservation
Now we consider the same type of polynomials from above, but on the support of momentum conservation, $B_n(k)\equiv B(k)\delta_n$. To impose momentum conservation explicitly, we can choose three particles (for example 2, 3, and 4), and use the following relations: $$\begin{aligned}
p_3&=-\sum_{i\neq 3} p_i\, ,\\
e_3.p_4&=-\sum_{i\neq 3,4} e_3.p_i\, ,\\
p_2.p_4&=-\sum_{i,j\neq 3} p_i.p_j\equiv P_{24}\, .\end{aligned}$$ This allows other ways of forming gauge invariance in particles 2, 3 and 4. For example, for particle 2 we can now have a new gauge invariant of the form $$\begin{aligned}
\label{alt}
G_2^\mu =e_2.p_4 p_2^\mu-P_{24} e_2^\mu\, .\end{aligned}$$ Such an expression avoids our previous argument: now both particles 2 and 4 can share the same $e_2.p_4$ factor above. When checking gauge invariance in at most $n-3$ particles this is not a problem, as we can always impose momentum conservation in such a way as to avoid these three special particles. However, for more than $n-3$ particles, at least one particle has to be affected by momentum conservation.
The worst case that we will need to prove is that $B_n(n-3)$ cannot be gauge invariant in $n-2$ particles. Without loss of generality, since we can change how we impose momentum conservation, we can leave out 3 and 4, and assume that these $n-2$ particles are $\{1,2,5,6,\ldots,n\}$. Then $n-3$ of the particles can only form gauge invariants of the form $ e_i^\mu p_i^\nu-e_i^\nu p_i^\nu$, while particle 2 allows gauge invariants of the form (\[alt\]). Since we only have $k=n-3$ factors of $e.p$, the first $n-3$ constraints already fix all such factors. However, we are not checking gauge invariance in particle 4, so none of the factors will be fixed to $e.p_4$, more specifically to the $e_2.p_4$ needed in eq. (\[alt\]). Therefore there is still no room to form the gauge invariant for particle 2.
This proves that a polynomial $B_n(k)$ with $k<n-2$ can be gauge invariant in at most $k$ particles. It is easy to see that the last step of the argument fails for $k=n-2$. In that case the counting allows for gauge invariance in not just $n-1$ particles, but all the way to $n$ particles. This is of course what we should expect, since a polynomial $B_n(n-2)$ corresponds to the full amplitude numerator, and is gauge invariant in $n$ particles.
Functions and tensors with singularities
----------------------------------------
The previous results for polynomials can be extended to functions with poles, such as those we initially introduced in eq. (\[nl\]): $$\begin{aligned}
\label{nonlocal}
B_n(p^k)\equiv \sum_{i} \frac{ N_i(p^k)}{P_i}\, .\end{aligned}$$ Because the $P_i$ are only functions of $p.p$ factors, a function with at most $k$ momenta in the numerators can be expressed in terms of a polynomial with at most $k$ factors of $e.p$: $$\begin{aligned}
B_n(p^k)=\frac{B_n(k)}{\prod_i P_i}\, .\end{aligned}$$ This implies that a function $B_n(p^k)$ cannot be gauge invariant in $k+1$ particles, if $k< n-2$. This statement can be generalized to tensors $B^{\mu\nu}_n(p^k)$. We can write out the components of such a tensor: $$\begin{aligned}
\label{qq1}B_n^{\mu\nu}(p^k)&=\sum_{i, j}p_i^\mu p_j^\nu B_{ij}(p^{k-2})+\sum_{i,j} p_i^\mu e^\nu_j C_{ij}(p^{k-1})+\sum_{i,j} e_i^\mu p^\nu_j C'_{ij}(p^{k-1})+\sum_{i\neq j} e^\mu_i e^\nu_j D_{ij}(p^{k})\, ,\end{aligned}$$ and determine what constraints each of the functions above must satisfy in order for $B^{\mu\nu}_n(p^k)$ to be gauge invariant in $k+1$ particles. We can treat each different type of function in order:
- $p_i^\mu p_j^\nu B_{ij}(p^{k-2})$: if we check gauge invariance in some particle $m$, with $m\neq i,j$, the prefactor remains unique, and none of the other terms in (\[qq1\]) may cancel against $B_{ij}$. This implies $B_{ij}(p^{k-2})$ itself must be gauge invariant in at least $k+1-|\{i,j\}|=k-1$ particles so is ruled out, if $k-2< n-2$.
- $e_i^\mu p_i^\nu C_{ii}(p^{k-1})$, or $\overline{e}_i^\mu p_j^\nu C_{ij}(p^{k-1})$: the same logic as before implies $C(p^{k-1})$ must be gauge invariant in $k$ particles, so is also ruled out if $k-1<n-2$.
- $e_i^\mu p_j^\nu C_{ij}$: can only form a gauge invariant in $i$ together with a term $p_i^\mu p_j^\nu B_{ij}$, which was ruled out.
- $e_i^\mu e_j^\nu D_{ij}(p^k)$, or $e_i^\mu \overline{e}_j^\nu D_{ij}(p^k)$: under $e_i\rightarrow p_i$, $D_{ij}$ can only form a gauge invariant with $C_{ij}$, which vanished, so this case is ruled out.
- $\overline{e}_i^\mu \overline{e}_j^\nu D_{ij}(p^k)$: is only ruled out for $k<n-2$
To summarize, we obtain just three types of cases: $$\begin{aligned}
&B_n(p^{k-2}), \textrm{ gauge invariant in $k-1$}\, ,\\
&C_n(p^{k-1}), \textrm{ gauge invariant in $k$}\, ,\\
\label{3rd}&D_n(p^{k}), \textrm{ gauge invariant in $k+1$}\,\end{aligned}$$ all of which vanish for $k< n-2$. For $k=n-2$, the first two also vanish, but for the third we have an apparent contradiction, since we know that functions $D_n(p^{n-2})$ have sufficient momenta to satisfy gauge invariance in $n-1$ particles. However, when $k=n-2$, case (\[3rd\]) does not exist. The only tensor $B_n^{\mu\nu}(p^{n-2})$ that will show up in the actual proof has $\overline{G}=\{e_3\}$, so it does not contain a component $\overline{e}_i\overline{e}_j D_{ij}$.
Therefore, the tensors we are interested in cannot be gauge invariant in $k+1$ particles for $k\le n-2$. In conclusion, so far we have shown that:
- functions $B_n(p^k)$ cannot satisfy $\Delta=1$ constraints for $k<n-2$
- tensors $B^{\mu\nu}_n(p^k)$ cannot satisfy $\Delta=1$ constraints for $k\le n-2$
It turns out that these results can be generalized even further: we can take linear combinations of the above functions/tensors, with factors of $p_i.p_j$ as coefficients, and still the above statements hold. These results will be necessary for the following sections.
Unitarity from locality and gauge invariance {#unique1}
============================================
In this section we will consider local functions, as in eq. (\[ll\]): $$\begin{aligned}
\label{local}
B_n(p^{k})= \sum_{\textrm{diags. } i} \frac{ N_i(p^{k})}{\prod_{\alpha_i}P^2_{\alpha_i}}\, .\end{aligned}$$ In the above notation the actual gluon amplitude $A_n(p^{n-2})$ is a subset of $B_n(p^{n-2})$, with $G=\{1,2,4,\ldots,n\}$ and $\overline{G}=\{3\}$, so $g=n-1$ and $\Delta=1$. Now we wish to prove that $A_{n+1}$ is uniquely fixed by gauge invariance in $n$ particles, under the assumption that $A_n$ is fixed by gauge invariance in $n-1$ particles. Consider the most general $(n+1)$-point local function $M_{n+1}$, and let $p_{n+1}=z q$. The Taylor series expansion around $z=0$ is: $$\begin{aligned}
\nonumber M_{n+1}\delta_{n+1}=&(z^{-1} M^{-1}_{n+1}+z^0 M_{n+1}^0+\ldots)(\delta_n+z q.\delta_n'+\ldots)\\
\nonumber \label{Taylor}=&z^{-1}M_{n+1}^{-1}\delta_n+z^0\left(M_{n+1}^{-1}q.\delta'_n+M^0_{n+1}\delta_n\right)+\ldots\\
=&z^{-1}\mathcal{M}_{n+1}^{-1}+z^0\mathcal{M}_{n+1}^0+\ldots\, .\end{aligned}$$ First we must investigate the pole structure of a local function in this limit. There are two types of poles that can show up. First, there are $q$-poles which are singular. These correspond to diagrams of the type:
and can be written as: $$\begin{aligned}
\frac{N}{q.p_i \mathcal{P}_n(q)}=D_{n+1}\end{aligned}$$ with $i=1$ or $i=n$ for Yang-Mills, because of ordering. In the limit $q\rightarrow 0$, we can factor out the $q.p_i$ pole and incorporate the remaining propagator structure into the lower point local function: $$\begin{aligned}
\label{d1}
D_{n+1}\rightarrow \frac{1}{q.p_i} \frac{N}{\mathcal{P}_n(0)}=\frac{1}{q.p_i} B_n\, .\end{aligned}$$ Then there are non-singular poles, which appear when two propagators become equal in the $q\rightarrow 0$ limit:
and can be written as: $$\begin{aligned}
D_{n+1}=\frac{N}{P_L(q)(p_1+p_2+\ldots+p_i)^2(q+p_1+p_2+\ldots+p_i)^2 P_R(q)}\, ,\end{aligned}$$ where for Yang-Mills $P_i^2=(p_1+p_2+\ldots+p_i)^2$ contains only consecutive momenta up to particle $i$, $i=\overline{2,n-2}$. We will factor out one of the $P_i^2$’s, and incorporate the other into the lower point local function $B_n$: $$\begin{aligned}
\label{d2}
D_{n+1}\rightarrow \frac{N}{(p_1+p_2+\ldots+p_i)^2\mathcal{P}_n(q)}=\frac{1}{(p_1+p_2+\ldots+p_i)^2}B_n\, .\end{aligned}$$ The argument by induction can be used precisely because of this factorization into the lower point local functions.
Yang-Mills
----------
#### Leading order
The leading $z^{-1}$ piece of the soft limit (\[Taylor\]) can only come from $q$-pole terms. Using linearity in $e_{n+1}=e$, it can be written as: $$\begin{aligned}
M_{n+1}^{-1}(p^{n-1})= \frac{e^\mu B^\mu_{n}(p^{n-1})}{q.p_1}+\frac{e^\mu C^\mu_{n}(p^{n-1})}{q.p_n}\, ,\end{aligned}$$ where $B_n^\mu$ and $C_n^\mu$ are local (vectors) at $n$-points. Next, gauge invariance in $q$ requires $B^\mu_n=p_1^\mu B_n$, and $C^\mu_n=-p_n^\mu B_n$, where $B_n$ is a local function at $n$ points. Then the leading piece is: $$\begin{aligned}
M_{n+1}^{-1}(p^{n-1})= \left(\frac{e.p_1}{q.p_1}-\frac{e.p_n }{q.p_n}\right)B_n(p^{n-2})\, .\end{aligned}$$ By assumption, gauge invariance in the remaining $(n-1)$ particles uniquely fixes $B_n=A_n$, reproducing the well known Weinberg soft factor [@Weinberg]. Note that unlike other methods of obtaining the soft term, we have not used any information on factorization, but only gauge invariance.
Now that the leading order is fixed, consider instead the function $B_{n+1}=M_{n+1}-A_{n+1}$. $B_{n+1}$ is also local, and must be gauge invariant in $n$ particles, but has a vanishing leading order. We will show that all higher orders in the soft expansion of $B_{n+1}$ also vanish, implying that $M_{n+1}=A_{n+1}$. This procedure will be identical for gravity, NLSM and DBI.
#### Sub-leading order
Because the leading order vanishes, the sub-leading piece is given only by: $$\begin{aligned}
\label{sub1}
B_{n+1}^0(p^{n-1})=\sum_{i=1;n} \frac{e^\mu q^\nu B^{\mu\nu}_{n;i}(p^{n-2}) }{q.p_i}+\sum_{i=2}^{n-2} \frac{e^\mu C_{n;i}^\mu(p^{n-1})}{P_i^2}\, ,\end{aligned}$$ which includes both singular and non-singular pole parts. The non-singular pole terms are ruled out by gauge invariance in $q$, while the $q$-pole terms must be proportional to $e^{[\mu}q^{\nu]}$. Bringing everything under a common denominator we can write $$\begin{aligned}
B_{n+1}^0(p^{n-1})\propto e^{[\mu}q^{\nu]} \left( q.p_n B^{\mu\nu}_{n;1}(p^{n-2})+q.p_1 B^{\mu\nu}_{n;n}(p^{n-2})\right)\equiv e^{[\mu}q^{\nu]}{B'_n}^{\mu\nu}(p^{n-2})\, ,\end{aligned}$$ where ${B'_n}^{\mu\nu}(p^{n-2})$ is a linear combination of tensors with $k=n-2$, so is ruled out by requiring gauge invariance in $n-1$ particles. Therefore $B_{n+1}^0=0$.
#### Sub-sub-leading order
The sub-sub-leading piece is given by: $$\begin{aligned}
B_{n+1}^1(p^{n-1})=e^{[\mu}q^{\nu]}\left( \sum_{i=1,n}\frac{q^\rho B^{\mu\nu\rho}_n(p^{n-3})}{q.p_i}+\sum_{i=2}^{n-2}\frac{C^{\mu\nu}_n(p^{n-2}) }{P_i^2}\right)\, .\end{aligned}$$ This time the non-singular pole terms are not ruled out just by gauge invariance in $q$. We can still write: $$\begin{aligned}
B_{n+1}^1(p^{n-1})\propto e^{[\mu}q^{\nu]} {B'_n}^{\mu\nu}(p^{n-2})\, .\end{aligned}$$ We obtain similar constraints as in the subleading case, which imply $B_{n+1}^1=0$.
#### Sub$^{s\ge 3}$-leading order
At arbitrary order $s\ge 3$ we can write: $$\begin{aligned}
B_{n+1}^{s-1}(p^{n-1})\propto e^{[\mu}q^{\nu]}q^{\rho_1}\ldots q^{\rho_{s-2}} B^{\mu\nu\rho_1\ldots \rho_{s-2}}_n(p^{n-s})\, .\end{aligned}$$ And all constraints will have $k\le n-3$, with $\Delta=s-1\ge 1$, so $B^{s-1}_{n+1}=0$ to all orders up to $s=n$, where the soft expansion terminates. Therefore $M_{n+1}=A_{n+1}$, proving uniqueness.
Gravity
-------
For gravity, we can simply write polarization tensors in terms of polarization vectors as $e_i^{\mu\nu}=e_i^\mu f_{i}^\nu$. Then gauge invariance in one graviton becomes equivalent to gauge invariance in two “gluons". The polynomial statement from section 2 still applies, so ignoring momentum conservation, no polynomial with at most $k$ $e.p$ factors can be gauge invariant in $k+1$ “gluons”. With momentum conservation, in the case of gravity this is true for $k< 2n-4$. This implies that a tensor $B_n^{\mu\nu}$ with $k$ powers of momenta in the numerator cannot be gauge invariant in $k+1$ particles for $k\le 2n-4$.
One other difference is that for gravity we are no longer restricted only to cyclic poles, since there is no ordering. In the end, the proof is almost identical to that for Yang-Mills. We assume that $A_n(p^{2n-4})$ is unique, and prove the same is true for $A_{n+1}(p^{2n-2})$.
#### Leading order
The leading piece has a form: $$\begin{aligned}
M_{n+1}^{-1}=\sum_i \frac{e^\mu f^\nu B_{n;i}^{\mu\nu}}{q.p_i}\, .\end{aligned}$$ Gauge invariance in $e$ and $f$ can only be satisfied on the support of momentum conservation, by $B^{\mu\nu}_{n;i}=p_i^\mu p_i^\nu B_n$, where $B_n$ is a local function at $n$ points. Then the leading piece is: $$\begin{aligned}
M_{n+1}^{-1}(p^{2n-2})=\sum_i \frac{e.p_i f.p_i}{q.p_i}B_n(p^{2n-4})\, ,\end{aligned}$$ and now by assumption gauge invariance in the other particles fixes $B_n=A_n$. Using the same trick as before, we consider instead the function $B_{n+1}=M_{n+1}-A_{n+1}$.
#### Sub-leading order
The subleading piece is given by: $$\begin{aligned}
B_{n+1}^0=\sum_{i} \frac{e^\mu f^\nu q^\rho B^{\mu\nu\rho}_{n;i}}{q.p_i}+\sum_{i} \frac{e^\mu f^\nu C_{n;i}^{\mu\nu}}{P_i^2}\, .\end{aligned}$$ Gauge invariance in $e$ and $f$ rules out the non-singular pole contributions, and fixes the first term to: $$\begin{aligned}
\nonumber B_{n+1}^0(p^{2n-2})&=\sum_{i} \frac{e^{[\mu} q^{\nu]} f.p_i B^{\mu\nu}_{n;i}(p^{2n-4})}{q.p_i}\\
&=e^{[\mu} q^{\nu]} B^{\mu\nu}_n(p^{2n-4})\, ,\end{aligned}$$ which is ruled out by gauge invariance in the remaining particles.
#### Sub-sub-leading order
For higher orders, which go up to $s=2n-1$, the same argument rules out any other solutions, so $A_{n+1}$ is uniquely fixed by gauge invariance.
Unitarity from locality and the Adler zero {#unique2}
==========================================
For the NLSM and DBI we will also deal with local functions $B_n(p^k)$, with $k$ powers of momenta in the numerator. However, the poles are now associated to propagators of quartic diagrams, ordered for the NLSM, un-ordered for DBI. The Adler zero condition [@adler] states that the amplitude $A_n$ must vanish when a particle is taken soft. Exactly how rapidly it must vanish sets the difference between the NLSM and DBI [@trnka1]-[@trnka4]. The limit $p_i\rightarrow 0$ is taken as $p_i=w_i p_i$, $w_i\rightarrow 0$. Then for the NLSM we require the amplitude to vanish as $\mathcal{O}(w_i)$, while for DBI we require $\mathcal{O}(w_i^2)$, $\forall i\neq 3$.
As for gauge invariance, it will be useful to quantify the difference between available momenta and total constraints. In general, if we require a function $B_n(p^k)$ to vanish as $\mathcal{O}(w_i^{g_i})$ for some particle $i$, let the corresponding constraint be $g_i$. Then $g=\sum_i g_i$ will the total constraints $B_n(p^k)$ must satisfy, and define $\Delta=g-k$ as before. This time we wish to show that the NLSM amplitude $A_n(p^{n-2})$ is the unique object satisfying $\Delta=1$ constraints, while the DBI amplitude $A_n(p^{2n-4})$ uniquely satisfies $\Delta=2$ constraints.
We will also show this by counting possible solutions, order by order in the double soft expansion $q= z q$, $\tilde{q}= z \tilde{q}$, $z\rightarrow 0$. The double soft expansion is chosen now, since the functions simply vanish in the single soft limit. The proof will again be almost identical with the ones for YM and GR, with one important difference. In the first two cases, the simple polynomial statement of section 2 significantly streamlined the argument. Remember that in the soft limit, we encountered tensors $B^{\mu\nu}(p^k)$, with $k\le n-2$, which were immediately ruled out. The key in that case was that we could always associate a polynomial with $k$ $e.p$ factors to a function with $k$ momenta in the numerator. For the NLSM, there is no such (simple) distinction to be made, since all we have are $p_i.p_j$ factors, both in the numerator and denominator.
Therefore, for scalars we do not have a direct proof for the following fact: a function $B_n(p^k)$ cannot satisfy $k+1$ constraints, if $k<n-2$. Instead, this statement must be proven by induction. We will write the proof only for the uniqueness statement, ie $k=n-2$, under the assumption that the non-existence statement is true. The proof for the latter case is identical, only with $k<n-2$.
The Taylor series expansion is identical to eq. (\[Taylor\]). We have the singular $q$-pole terms:
of the form: $$\begin{aligned}
D_{n+2}=\frac{N}{(q+\tilde{q}+p_i)^2 \mathcal{P}_n(q,\tilde{q})}\end{aligned}$$ where $i=1$ or $i=n$ for NLSM due to ordering. In the soft limit we can also write this in terms of the lower point local function: $$\begin{aligned}
D_{n+2}\rightarrow \frac{1}{(q+\tilde{q}).p_i} \frac{N}{\mathcal{P}_n(0,0)}=\frac{1}{(q+\tilde{q}).p_i}B_n\, .\end{aligned}$$ Next there are non-singular poles, which are more varied than in the cubic diagram case. There is still the equivalent of the double poles from before:
which we can write as: $$\begin{aligned}
D_{n+2}=\frac{N}{P_L(q,\tilde{q})(p_1+p_2+\ldots+p_i)^2(q+\tilde{q}+p_1+p_2+\ldots+p_i)^2 P_R(q,\tilde{q})}\, ,\end{aligned}$$ In the soft limit this becomes: $$\begin{aligned}
D_{n+2}\rightarrow \frac{1}{(p_1+p_2+\ldots+p_i)^2}B_n\, .\end{aligned}$$ There are also more complicated non-singular poles, when the $q$ and $\tilde{q}$ legs are separated. However, even in such cases it is easy to write the terms in a form: $$\begin{aligned}
D_{n+2}\rightarrow \frac{1}{P_i^2}B_n\, .\end{aligned}$$
NLSM
----
#### Leading order
The leading $1/z$ term can only come from $q$-pole terms: $$\begin{aligned}
M_{n+2}^{-1}=\frac{N_1(0,0)}{p_1.(q+\tilde{q})}+\frac{N_2(0,0)}{p_n.(q+\tilde{q})}\, ,\end{aligned}$$ imposing vanishing under $\tilde{q}\rightarrow 0$ implies: $$\begin{aligned}
\left(\frac{N_1}{q.p_1}+\frac{N_2}{q.p_n}\right)=0\, ,\end{aligned}$$ so $N_1=N_2=0$, and $M_{n+2}^{-1}=0$.
#### Sub-leading order
At this level both types of poles can contribute. The $q$-pole piece is: $$\begin{aligned}
M_{n+2}^0=\frac{q^\mu B^\mu_{n}+{\tilde{q}}^\mu C^\mu_{n}}{(q+\tilde{q}).p_1}+\frac{q^\mu D^\mu_{n}+{\tilde{q}}^\mu E^\mu_{n}}{(q+\tilde{q}).p_n}\, .\end{aligned}$$ Vanishing under $\tilde{q}\rightarrow 0$ implies $B^\mu=p_1^\mu B_n$, and $D_n^\mu=-p_n^\mu B_n$, and similarly $q\rightarrow 0$ leads to $C_n^\mu=p_1^\mu C_n$ and $E_n^\mu=-p_n^\mu C_n$. The subleading term becomes: $$\begin{aligned}
M^0_{n+2}(p^n)=\left(\frac{q.p_1}{(q+\tilde{q}).p_1}-\frac{q.p_n}{(q+\tilde{q}).p_n}\right)(B_n-C_n)\, .\end{aligned}$$ Now $(B_n-C_n)\equiv B_n(p^{n-2})$ is also a general local function at $n$-points, so by assumption vanishing in the other soft limits fixes $B_n=A_n$.
Terms with non-singular poles have a form: $$\begin{aligned}
\sum_i \frac{N_i}{P_i}\, ,\end{aligned}$$ but are quickly ruled out by requiring vanishing under $q$ or $\tilde{q}$.
#### Sub-sub-leading order
The most general $q$-pole sub-sub-leading term is: $$\begin{aligned}
\nonumber M_{n+2}^1=&\frac{1}{(q+\tilde{q}).p_1}(q^\mu q^\nu B^{\mu\nu}+q^\mu \tilde{q}^\nu C^{\mu\nu}+\tilde{q}^\mu \tilde{q}^\nu D^{\mu\nu}+ q.\tilde{q} E)\\
&-\frac{1}{(q+\tilde{q}).p_n}(q^\mu q^\nu F^{\mu\nu}+q^\mu \tilde{q}^\nu G^{\mu\nu}+\tilde{q}^\mu \tilde{q}^\nu H^{\mu\nu}+ q.\tilde{q} I)\, .\end{aligned}$$ Now we expand the remaining $p_i=w_i p_i$ and require $M_{n+2}^1\propto \mathcal{O}(w_i^{1})$ for each of the ($n-1$) particles left. All of the above functions can be treated as independent because of their unique prefactors. Taking the $p$’s in the denominator into account, we obtain the following constraints for all functions: $$\begin{aligned}
B^{\mu\nu},\, C^{\mu\nu}\, D^{\mu\nu},\, E&\propto \mathcal{O}(w_1^2), \mathcal{O}(w_{i\neq 1}^1)\, ,\\
F^{\mu\nu},\, G^{\mu\nu}\, H^{\mu\nu},\, I&\propto \mathcal{O}(w_n^2), \mathcal{O}(w_{i\neq n}^1)\, ,\end{aligned}$$ while component-wise there will be two types of constraints. First: $$\begin{aligned}
\label{nlsm1}
B(p^{n-4}),C(p^{{n-4}}),D(p^{n-4}),G(p^{n-4}) \propto n-2\, .\end{aligned}$$ These are $\Delta=2$ with $k<n-2$ so are ruled out. The other constraints are: $$\begin{aligned}
E(p^{n-2}),\ I(p^{n-2})\propto n\, .\end{aligned}$$ First, we can use $n-1$ of the usual $\mathcal{O}(w_i^1)$ constraints to fix $E=I=A_n$. But then $A_n$ cannot satisfy the extra requirement of $\mathcal{O}(w_1^2)$ or $\mathcal{O}(w_n^2)$, so it must mean that $I=E=0$.
Non-singular poles are still ruled out by vanishing under $q$ and $\tilde{q}$, and so $M_{n+2}^1=0$.
#### Sub$^{s\ge3}$-leading
With each extra $q^\mu$ or $\tilde{q}^\mu$ being added, $k$ decreases by 1, so $\Delta$ can only increase by at least 1, leading to $\Delta_s=s\ge 2$ constraints. Therefore any sub$^s$-leading order vanishes, and so $M_{n+2}=0$, proving our statement, with the caveat below.
##### Neutral poles
At the sub$^{3}$-leading order some special combinations of non-singular pole terms are not directly ruled out. Consider for example the two diagrams, which we take to have equal numerators:
given by: $$\begin{aligned}
\nonumber D_{n+2}(p^n)=&\frac{N(p^{n})}{P_L^2 (P_L+i+j)^2 (P_L+i+j+q+\tilde{q})^2P_R^2(q,\tilde{q})}\\
&-\frac{N(p^n)}{P_L^2 (P_L+i+\tilde{q})^2 (P_L+i+j+q+\tilde{q})^2P_R^2(q,\tilde{q})}\, .\end{aligned}$$ At sub$^{s\ge3}$-leading order their contribution is: $$\begin{aligned}
\label{extranlsm}
D_{n+2}^{2}(p^n)=q^\mu \tilde{q}^\nu N^{\mu\nu}(p^{n-2})\left(\frac{1}{P_L^2 (P_L+p_i+p_j)^2P_R^2}-\frac{1}{P_L^2 (P_L+p_i)^2P_R^2}\right)\, .\end{aligned}$$ Now $N^{\mu\nu}(p^{n-2})$ has enough momenta to trivially satisfy $n-2$ of the remaining $n-1$ constraints. But vanishing in particle $j$ is automatic because the two denominators in (\[extranlsm\]) become equal when $p_j\rightarrow 0$. Therefore $D^2_{n+2}$ is not ruled out by our usual argument. Instead, such terms can be ruled out by taking different soft limits. Specifically, it must be soft limits which lead to soft-singularities in $P_L^2$ or $P_R^2$. This ensures that $D^2_{n+2}$ above avoids non-singular pole terms in the new soft limit.
Dirac-Born-Infeld
-----------------
For DBI we can use the same notation from the previous section. In this case, the Adler zero condition is stronger, as we require $A_n\propto \mathcal{O}(w_i^2)$ under $p_i=w_i p_i\rightarrow 0$. The proof is identical to the one for the NLSM, with the minor difference that now non-cyclic poles are allowed. Also, in all cases non-singular poles can be ruled out easily - the issue appearing in the NLSM is not present, since the vanishing under $p_j\rightarrow 0 $ ensured by eq. (\[extranlsm\]) can not provide the full $\mathcal{O}(w_i^2)$ needed. Instead, there is a different issue appearing at the same order, which can be resolved by demanding permutation invariance.
#### Leading order
The leading piece is given by: $$\begin{aligned}
B^{-1}_{n+2}=\sum_{i=1}^n \frac{N_i}{(q+\tilde{q}).p_i}\, ,\end{aligned}$$ but is ruled out by requiring $B_{n+2}^{-1}\propto \mathcal{O}(w^1)$ in $q$ or $\tilde{q}$ .
#### Sub-leading order
Regular $q$-pole terms have the form: $$\begin{aligned}
\sum_{i=1}^n\frac{1}{(q+\tilde{q}).p_i}(q^\mu B_i^\mu+\tilde{q}^\mu C_i^\mu)\, ,\end{aligned}$$ but are ruled out by requiring $\mathcal{O}(w^2)$ under $q$ and $\tilde{q}$.
#### Sub-sub-leading order
Have the form: $$\begin{aligned}
M^1_{n+2}&=\sum_i\frac{1}{(q+\tilde{q}).p_i}(q^\mu q^\nu\, B_i^{\mu\nu}+\tilde{q}^\mu \tilde{q}^\nu\, C_i^{\mu\nu}+q^\mu \tilde{q}^\nu D_i^{\mu\nu}+q.\tilde{q} E_i )\, .\end{aligned}$$ Requiring $\mathcal{O}(w^2)$ in $q, \tilde{q}$ we end up with: $$\begin{aligned}
\nonumber M^1_{n+2}(p^{2n})&=\sum_i\frac{1}{(q+\tilde{q}).p_i}((q.p_i)^2 B+(\tilde{q}.p_i)^2 C+q.p_i \tilde{q}.p_i D )\\
&=\sum_i\frac{q.p_i \tilde{q}.p_i}{(q+\tilde{q}).p_i} (-B-C+ D )\, .\end{aligned}$$ Now $(-B-C+D)\equiv B_n(p^{2n-4})$ is a general local function at $n$-points, so imposing the remaining $2n-2$ constraints fixes $B_n=A_n$ by assumption.
#### Sub$^3$-leading order
Like for the NLSM, at this order extra care is required. The usual arguments rule out all terms except: $$\begin{aligned}
D_{n+2}^2(p^{2n})= q.\tilde{q}\sum_i \frac{ q.p_i B_{n;i}+\tilde{q}.p_i C_{n;i}}{(q+\tilde{q}).p_i}\, ,\end{aligned}$$ under the condition that $\sum_i B_{n;i}=\sum_i C_{n;i}=0$. The functions $B_i(p^{2n-4})$ and $C_i(p^{2n-4})$ must satisfy $\Delta=2$ constraints and are fixed (up to some coefficient) to $A_n$ by assumption. Then the extra conditions become $\sum_i B_{n;i}=\sum_i b_i A_n=0$, and similarly $\sum C_i=\sum_i c_i A_n=0$. The sub$^3$-leading term becomes: $$\begin{aligned}
D_{n+2}^2= q.\tilde{q} A_n \sum_i\frac{b_i \,q.p_i+c_i\, \tilde{q}.p_i}{(q+\tilde{q}).p_i}\, .\end{aligned}$$ But now if we require symmetry in $q\leftrightarrow \tilde{q}$, then $b_i=c_i$, so $D_{n+2}^2=0$, and this order vanishes.
#### Sub$^{s>3}$leading
All such terms are ruled out, so $M_{n+2}=0$ to all orders, and $A_{n+2}$ is unique.
Locality and unitarity from singularities and gauge invariance {#locality}
==============================================================
The general argument we used in the previous sections can be easily extended when we relax our cubic graph assumptions, and instead consider a more general singularity structure, as long as the singularities themselves have a form $\left(\sum_i p_i\right)^2$, with consecutive momenta in the case of Yang-Mills. This means we allow double poles as well as overlaps. There are three cases to consider depending on how many singularities $s$ we allow. We will show that:
- for $s<n-3$ there is no solution
- for $s=n-3$ there is a unique solution, $A_n$
- for $s> n-3$ solutions can be factorized in a form $\left(\sum \textrm{poles} \right)\times A_n$
We prove these three results for five points Yang-Mills. It is easy to extend the proof for general $n$, including for gravity.
In the following, we will call a function with $s$ singularities $B_{n;s}$, and to simplify notation, let: $$\begin{aligned}
S_0=\frac{e.p_1}{q.p_4}-\frac{e.p_4}{q.p_4}\, .\end{aligned}$$
Case 1. $s<n-3$
---------------
This case is easy to prove by induction. Assume that $B_{4;0}(p^{2})$ is ruled out by gauge invariance. Then the five point function with just one singularity has a leading order: $$\begin{aligned}
M_{5;1}^{-1}(p^3)=S_0 B_{4;0}(p^2)\, ,\end{aligned}$$ which by assumption is ruled out. Higher order terms are again ruled out as usual, so there are no solutions for $s< n-3$.
Case 2. $s=n-3$
---------------
At five points, in this case we have two (cyclic) poles per term, and now we also allow double poles and overlaps.
#### Order $z^{-2}$
The lowest order is now $z^{-2}$, coming from three possible terms, which were not present before: $$\begin{aligned}
M_{5;2}^{-2}(p^3)=\frac{N_a}{(q.p_1)^2}+\frac{N_b}{(q.p_1)(q.p_4)}+\frac{N_c}{(q.p_4)^2}\, .\end{aligned}$$ Gauge invariance in $q$ requires the forms: $$\begin{aligned}
M_{5;2}^{-2}(p^3)=\frac{1}{q.p_1}S_0 B_{4;0}(p^2)+\frac{1}{q.p_4}S_0 C_{4;0}(p^2)\, ,\end{aligned}$$ so both $B_{4;0}$ and $C_{4;0}$ vanish by the previous argument.
#### Order $z^{-1}$
At this order we have the usual leading piece, but also terms with the non-local poles from above: $$\begin{aligned}
\nonumber M_{5}^{-1}=& S_0 B_4(p^{2})+e^{[\mu}q^{\nu]} \left(\frac{N_a^{\mu\nu}}{(q.p_1)^2}+\frac{N_b^{\mu\nu}}{(q.p_1)(q.p_4)}+\frac{N_c^{\mu\nu}}{(q.p_4)^2}\right)\, .\end{aligned}$$ For the second piece we need tensors $N_4^{\mu\nu}(p^2)$, gauge invariant in three particles, which is not possible. Therefore the leading piece is just $$\begin{aligned}
\label{un}
M_{5}^{-1}=S_0A_4\, ,\end{aligned}$$ and so far we get the same answer as usual. However, we must deal with a subtle issue that was not present before. We have shown that at the leading order, all possible functions must map onto the unique expression (\[un\]). But when we allow a non-local singularity structure, it is possible for two different $n+1$ point functions to have an identical leading order piece. Consider for example the actual amplitude, which contains a local term such as: $$\begin{aligned}
\label{good}
A_5=\ldots \frac{e.p_1 N}{q.p_1 (q+p_1+p_2)^2}+\ldots \, ,\end{aligned}$$ and a similar function $M_5$, but where we replace the term from above with a non-local one: $$\begin{aligned}
\label{bad}
M_5=\ldots \frac{e.p_1 N}{q.p_1 (p_1+p_2)^2}+\ldots\, .\end{aligned}$$ In the soft limit $q\rightarrow 0$ both functions are equal at the leading order, so apparently we have two different solutions, contradicting our statement. The issue can still be resolved by considering all orders of the soft expansion of $B_5=M_5-A_5$. The subleading order is now different than the usual (\[sub1\]), because $B_5$ now has a contribution originating from the Taylor series expansion of the denominator in eq. (\[good\]), which was absent before. We obtain: $$\begin{aligned}
B_5^{0}=\frac{e^\mu q^\nu B^{\mu\nu}_4}{q.p_i}+e^\mu C_4^\mu -\frac{e.p_iq.(p_1+p_2) N}{q.p_i(p_1.p_2)^2}+\ldots\end{aligned}$$ where the third term is new. But using our previous arguments $B_5^{0}$ is still ruled out by gauge invariance. Higher order terms can be treated in a similar manner, so $B_5=0$ to all orders. Therefore the five point Yang-Mills amplitude is completely fixed even if we start with these non-local assumptions.
Case 3. $s> n-3$
----------------
For this case, where we are not expecting to obtain a unique answer, but the same soft limit argument can be used to count the maximum total number of independent solutions, order by order. First, at four points it is easy to check that with $s=2$ poles, there are two solutions: $$\begin{aligned}
M_{4;2}=\left(\frac{a_1}{p_1.p_2}+\frac{a_2}{p_1.p_4}\right)A_4\, .\end{aligned}$$ Now at five points, with three poles, we want to show there are five solutions, corresponding to the five different cyclic poles: $$\begin{aligned}
\label{five}
M_{5;3}=\left(\frac{a_1}{p_1.p_2}+\frac{a_2}{p_2.p_3}+\ldots+\frac{a_5}{p_5.p_1}\right)A_5\, .\end{aligned}$$
Again taking a soft limit, and imposing gauge invariance in $p_5=q$, we obtain:
#### Order $\mathcal{O}(z^{-3})$
$$\begin{aligned}
M_{5;3}^{-3}= \frac{1}{(q.p_1)^2}S_0 B_{4;0}+\frac{1}{(q.p_4)^2}S_0 C_{4;0}\, ,\end{aligned}$$
which was shown to vanish, so no solutions at this level.
#### Order $\mathcal{O}(z^{-2})$
$$\begin{aligned}
M_{5;3}^{-2}=\frac{1}{q.p_1}S_0 B_{4;1}+\frac{1}{q.p_4}S_0 C_{4;1}\, ,\end{aligned}$$
which is fixed by gauge invariance to $$\begin{aligned}
M_{5;3}^{-2}=\left(\frac{a_5}{q.p_1}+\frac{a_4}{q.p_4}\right)S_0 A_4\, .\end{aligned}$$ Therefore from this order we obtain two possible solutions.
#### Order $\mathcal{O}(z^{-1})$
Because we are only counting [*independent*]{} solutions, we can simply ignore the contributions from the lower order above. Therefore we are only interested in the term: $$\begin{aligned}
M_{5;3}^{-2}=S_0 B_{4;2}\, .\end{aligned}$$ By assumption this gives two independent solutions corresponding to the poles $p_1.p_2$ and $p_1.p_4$, but starting from five points there are three poles which map onto these two in the soft limit: $$\begin{aligned}
&p_1.p_2\rightarrow p_1.p_2\, ,\\
&p_2.p_3=(p_4+p_5+p_1)^2\rightarrow p_1.p_4\, ,\\
&p_3.p_4=(p_5+p_1+p_2)^2\rightarrow p_1.p_2\, .\end{aligned}$$ And so we obtain three independent solutions at this order. For higher orders, the usual arguments rule out other solutions, and so we end up with at most five possible solutions. We have not derived what these must be, but since we can just write down the five terms of Eq. (\[five\]), this must be all of them. The result is easy to generalize to an arbitrary number of extra poles.
The argument can also easily be extended to general $n$-point amplitudes, as well as gravity. Once it is shown that functions with $s<n-3$ singularities are ruled out, for $s=n-3$ the only non-vanishing contribution will be the Weinberg term at order $1/z$, which by the usual argument implies uniqueness. We suspect the same type of argument can be used for NLSM and DBI, although some extra complications might appear at the sub$^3$-leading orders, which were already troublesome. Regardless, a more direct argument ruling out the non-local terms was already presented in Ref. [@Nima] for these theories.
Generalizing singularities {#l2}
==========================
Non-local singularities
-----------------------
In the previous sections we have assumed that the denominators are always products of singularities $P_\mathcal{S}^2=\left(\sum_i p_i\right)^2$. An obvious next step is to relax even this assumption, and allow completely non-local singularities of the form $(\sum_i a_i p_i)^2$. In full generality, this doesn’t work out. Even at four points, allowing a singularity of the form $a\, p_1.p_2+b\, p_1.p_4$ no longer provides a unique local solution. We can write the four point numerator as $N_4= (t N_s+ s N_t)=(t,s)\cdot(N_s,N_t)$, with $A_4=N_4/(st)$. Now we can do any 2D rotation to obtain $N_4=(t',s')\cdot(N'_s,N'_t)$, where $s'=s\, \textrm{cos}\, \theta - t\, \textrm{sin}\,\theta $ and $t'=t\, \textrm{cos}\, \theta+s\, \textrm{sin}\,\theta$. But now diving by $(s't')$ we obtain a (non-unique) gauge invariant with the non-local poles $s'$ and $t'$, so our claim is invalidated if we allow such poles.
However, there exists a special set of non-local “cyclic” poles from which full locality can still be derived, if we are careful about momentum conservation. To obtain this set, we must start from a local cyclic pole $P_{jk}^2=(\sum_{i=j}^k p_i)^2$. Now only [*after*]{} using momentum conservation $p_3=-\sum p_i$, we can add arbitrary coefficients $(\sum_i p_i)^2\rightarrow (\sum_i a_i p_i)^2$. For example, from a six point local pole like $(p_1+p_2+p_3)^2$, we can obtain $(a\, p_4+b\, p_5+c\, p_6)^2$. Note how this rule doesn’t allow the four point pole $a\, p_1.p_2+b\, p_1.p_4$ from above. It can only come from the pole $p_1.p_3=p_1.p_2+p_1.p_4$, which is not cyclic. At five points, the most general set of singularities that can be used is: $$\begin{aligned}
\nonumber (p_1+p_2)^2&=p_1.p_2\\
\nonumber(p_2+p_3)^2&=(p_1+p_4+p_5)^2\rightarrow a_1\, p_1.p_4+a_2\, p_1.p_5+a_3\, p_4.p_5\\
\nonumber(p_3+p_4)^2&=(p_1+p_2+p_5)^2\rightarrow a_4\, p_1.p_2+a_5\, p_1.p_5+a_6 \,p_2.p_5\\
\nonumber(p_4+p_5)^2&=p_4.p_5\\
(p_5+p_1)^2&=p_1.p_5\end{aligned}$$ For an $n$ point amplitude, $n-2$ of the singularities keep the form $p_i.p_{i+1}$, while the others are promoted to carry these extra coefficients. Now, the usual proof by induction will work, as long as we avoid taking soft the particles adjacent to 3, which is of course always possible from four points and higher. This procedure ensures that the soft-singularities $q.p_i$, critical for the leading term, are not affected in any way. Then the leading term is as usual $$\begin{aligned}
B_{n+1}^\textrm{non-local}\rightarrow \left(\frac{e.p_1}{q.p_1}-\frac{e.p_n}{q.p_n}\right)B_n^{\textrm{non-local}}\, ,\end{aligned}$$ where now $B_n^{\textrm{non-local}}$ also contains the non-local singularities described above. If by assumption even this non-local $B_n$ is uniquely fixed by gauge invariance, ultimately so will $B_{n+1}$. The claim is in fact trivial at four points, where none of the poles may be modified, so $B_4^{\textrm{non-local}}=B_4^{\textrm{local}}$.
With a few extra restrictions, a similar result can be shown for gravity as well, though the procedure is somewhat more complicated because for gravity soft-singularities involving $p_3$ are not so easily avoided. The solution is to require several extra poles to keep their usual local form, in such a way to ensure that even after taking multiple soft limits, there always exists a particle which forms no soft-singularities with $p_3$.
No singularities {#nonlocality}
----------------
So far, we have mostly looked at functions with singularities of the form $(\sum_i p_i)^2$, and in some cases we showed that singularities of the type $(\sum_i a_i p_i)^2$ also lead to uniqueness. But what about allowing the denominators to be polynomials of some degree $s^2$, instead of $s$ products of singularities? In general, this is a very difficult question to systematically analyze, and given the four point counter-example from the previous section, it might simply be an ill-posed question. But instead of trying to understand all such completely general poles, there is an even more general alternative to pursue. We can completely disregard singularities, and investigate gauge invariance directly at the level of the total numerator, by considering general polynomials instead of functions with poles. Clearly, given sufficient mass dimension, a general polynomial can always be thought of as originating from the most general singularity structure possible. We can start with the minimal polynomial which admits any solution, which has $n-2$ $e.p$ factors, and $(n-2)^2$ total mass dimension, the same as an actual amplitude numerator. It turns out that imposing our usual $n-1$ gauge invariance constraints does not provide a meaningful solution, but imposing the full $n$ constraints does: we obtain a linear combination of amplitude numerators! The $n^{\rm th}$ extra constraint essentially is required to replace the information we lost by not considering denominators which are products of singularities. From this perspective, the singularities do no play any crucial or physical role, but only provide a useful method of organizing terms in the polynomial. While we do not have a proof for this fact for $n>4$, it is easily testable at five points. There we obtain six solutions, which are linear combinations of five point amplitude numerators, corresponding to different orderings. Below we provide leading order evidence for this fact.
We can again use our usual soft argument to count the solutions at leading order. First, it is easy to check that imposing all four gauge invariance conditions on the four point polynomial $N_4((e.p)^2,p^4)$ gives a unique solution. This corresponds to the fact that all four point amplitudes have the same numerator. That is, any amplitude can be obtained by dividing the same numerator by the desired propagator structure: $$\begin{aligned}
A(1,2,3,4)=\frac{N}{p_1.p_2 \,p_1.p_4}\, ,\\
A(1,3,2,4)=\frac{N}{p_1.p_3 \,p_1.p_4}\, .\end{aligned}$$ At five points, the leading piece of the general polynomial must have a form: $$\begin{aligned}
N_5((e.p)^3,p^9)=e^{[\mu} q^{\nu]}N^{\mu\nu}(p^8)\, .\end{aligned}$$ After imposing the other four constraints all possible components are ruled out, except the following: $$\begin{aligned}
N_5((e.p)^3,p^9)=S_{12}N_{a}+S_{14}N_{b}+S_{24}N_{c}\, ,\end{aligned}$$ where $S_{ij}=e.p_i q.p_j-e.p_jq.p_i$. Now the $N_i((e.p)^2,p^6)$ must also satisfy the four constraints. First, we can rewrite $N((e.p)^2,p^6)=N((e.p)^2,p^4)\sum_{i,j} a_{ij} p_i.p_j$, after a reshuffling of the coefficients. Then, the constraints imposed on $N((e.p)^2,p^6)$ instead act on $N((e.p)^2,p^4)$, which by assumption is fixed uniquely to the four point numerator. Finally, there are two independent $p_i.p_j$ factors at four points. Therefore we obtain $$\begin{aligned}
N((e.p)^3,p^9)=(a_1 p_1.p_2+a_2 p_1.p_4 )(b_1S_{12}+b_2S_{14}+b_3S_{24}) N_4\, ,\end{aligned}$$ ie. six independent solutions, which are related to the leading pieces of amplitude numerators. Unfortunately, the subleading order is not ruled out so quickly. The $N$ still have enough momenta to provide gauge invariant contributions even at this order: $$\begin{aligned}
N((e.p)^3,p^9)=(a_1.q.p_1+a_2 q.p_2 +a_3 q.p_4)(b_1S_{12}+b_2S_{14}+b_3S_{24}) N_4\, ,\end{aligned}$$ and so the usual argument fails in its simplest form. However, considering all orders, eventually these extra solutions become tied to the original six, and in the end just six solutions are left. The argument becomes even less well suited for higher points, so clearly a better strategy is required.
Summary of the results and future directions {#ending}
============================================
In this note we have presented the full proofs for some of the uniqueness claims originally made in [@Nima]. We summarize these results below. Let $s$ be the number of poles of the form $\left(\sum_i p_i\right)^2$, and $k$ the mass dimension of the numerators.
#### Yang-Mills and General Relativity:
- Unique solution for $s=n-3$, with $k_{\textrm{YM}}=n-2$, $k_{\textrm{GR}}=2n-4$
- No solutions for $s$ or $k$ smaller than above
- Factorized solutions $(\sum \textrm{poles})\times A_n$ for $s$ larger than above
#### NLSM and DBI:
- Uniqueness assuming quartic diagrams, with $k_{\textrm{NLSM}}=n-2$, $k_{\textrm{DBI}}=2n-4$
- No solutions for $k$ smaller than above
For Yang-Mills, we also proved that uniqueness holds when allowing specific types of non-local singularities $\left(\sum_i a_i p_i\right)^2$. Finally, we conjectured that general polynomials of minimal mass dimension lead to linear combinations of amplitude numerators, and so to both locality and unitarity.
The next step is understanding how to approach such polynomials with absolutely no singularity structure. It would be very interesting to see if the soft limit argument can be extended even further, or if an even more powerful argument is required. Meanwhile, for NLSM and DBI, it is not even clear what the equivalent claim should be, if it exists. For YM, the number of $e.p$ factors always helped distinguish what $p$’s come from numerators and which come from propagators. For scalar theories, there is no distinction to be made: all the $p$’s are equal. We should note that an equivalent claim for gravity does not exist. It is trivial to obtain many different solutions by gluing together Yang-Mills amplitudes, while there is a unique gravity numerator. Nevertheless, even if the numerator statement is less fundamental than the other results, it is a very useful exercise. After all, thinking about polynomials lead to the crucial results of section \[gaugeinv\], so perhaps there is more to be learned from this perspective.
A more important issue to be understood is that of the gram determinant relations. When working in some fixed dimension $D$, at most $D-1$ vectors can be linearly independent ($-1$ because of momentum conservation). For example, if we restrict to 4D, starting at six points, we can express $p_6$ in terms of the other four independent momenta: $$\begin{aligned}
\label{lin}
p_6=a\, p_1+b\, p_2+ c\, p_4+d\, p_5\end{aligned}$$ This could allow for different solutions to our requirements. The linear dependence (\[lin\]) can be viewed as another form of momentum conservation: $$\begin{aligned}
\label{mln}
p_3=-(p_1+ p_2+ p_4+ p_5+p_6)\end{aligned}$$ We already saw that adding momentum conservation limited the applicability of our initial polynomial argument to $k< n-2$: at $k=n-2$ momentum conservation allowed for some “free” gauge invariants to be formed. Luckily, this was still sufficiently constraining for our purposes. It is not inconceivable, though would be very surprising, that the gram determinant relations could allow such free gauge invariants starting at $k=n-3$ for example.
Ultimately, these results strongly suggest that scattering amplitudes might have a different definition, perhaps geometric, in line with the amplituhedron program [@amplituhedron]. A formulation where both this minimal singularity structure and gauge invariance/vanishing in the soft limit are manifest could potentially uncover yet more unknown features of these theories.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank Nima Arkani-Hamed and Jaroslav Trnka for the insights and many discussions which lead to this work, and Song He for valuable discussions.
[99]{} N. Arkani-Hamed, L. Rodina and J. Trnka, “Locality and Unitarity from Singularities and Gauge Invariance,” arXiv:1612.02797 \[hep-th\]. L. Rodina, “Uniqueness from locality and BCFW shifts,” arXiv:1612.03885 \[hep-th\].
R. Britto, F. Cachazo, and Bo Feng, Nucl. Phys. B [**715**]{}, 499 (2005) \[arXiv:hep-th/0412308\]. R. Britto, F. Cachazo, B. Feng, and E. Witten, Phys. Rev. Lett. [**94**]{},181602 (2005) \[arXiv:hep-th/0501052\]. Z. Bern, J. J. M. Carrasco and H. Johansson, “New Relations for Gauge-Theory Amplitudes,” Phys. Rev. D [**78**]{}, 085011 (2008) doi:10.1103/PhysRevD.78.085011 \[arXiv:0805.3993 \[hep-ph\]\]. F. Cachazo, S. He and E. Y. Yuan, “Scattering of Massless Particles in Arbitrary Dimensions,” Phys. Rev. Lett. 113, no. 17, 171601 (2014) \[arXiv:1307.2199 \[hep-th\]\]. F. Cachazo, S. He and E. Y. Yuan, “Scattering of Massless Particles: Scalars, Gluons and Gravitons,” JHEP 1407, 033 (2014) \[arXiv:1309.0885 \[hep-th\]\] F. Cachazo, S. He and E. Y. Yuan, “Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM,” JHEP 1507, 149 (2015) \[arXiv:1412.3479 \[hep-th\]\]
S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. [**140**]{}, B516 (1965). F. Cachazo, S. He and E. Y. Yuan, “New Double Soft Emission Theorems,” Phys. Rev. D [**92**]{}, no. 6, 065030 (2015) doi:10.1103/PhysRevD.92.065030 \[arXiv:1503.04816 \[hep-th\]\].
A. Strominger, “On BMS Invariance of Gravitational Scattering,” arXiv:1312.2229 \[hepth\]. T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” arXiv:1401.7026 \[hep-th\]. S. W. Hawking, M. J. Perry and A. Strominger, “Soft Hair on Black Holes,” Phys. Rev. Lett. [**116**]{}, no. 23, 231301 (2016) doi:10.1103/PhysRevLett.116.231301 \[arXiv:1601.00921 \[hep-th\]\].
H. Luo and C. Wen,“Recursion relations from soft theorems,”JHEP [**1603**]{}, 088 (2016)\[arXiv:1512.06801 \[hep-th\]\].
L. A. Barreiro and R. Medina, “RNS derivation of N-point disk amplitudes from the revisited S-matrix approach,” Nucl. Phys. B [**886**]{}, 870 (2014) doi:10.1016/j.nuclphysb.2014.07.015 \[arXiv:1310.5942 \[hep-th\]\]. R. H. Boels and R. Medina, “Graviton and gluon scattering from first principles,” arXiv:1607.08246 \[hep-th\].
F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” arXiv:1404.4091 \[hep-th\]. J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, “Constraining subleading soft gluon and graviton theorems,” Phys. Rev. D [**90**]{}, no. 6, 065024 (2014) \[arXiv:1406.6574 \[hep-th\]\].
S. L. Adler, “Consistency conditions on the strong interactions implied by a partially conserved axial vector current,” Phys. Rev. [**137**]{}, B1022 (1965). doi:10.1103/PhysRev.137.B1022
K. Kampf, J. Novotny and J. Trnka, “Tree-level Amplitudes in the Nonlinear Sigma Model,” JHEP [**1305**]{}, 032 (2013) doi:10.1007/JHEP05(2013)032 \[arXiv:1304.3048 \[hep-th\]\].
C. Cheung, K. Kampf, J. Novotny and J. Trnka, “Effective Field Theories from Soft Limits of Scattering Amplitudes,” Phys. Rev. Lett. [**114**]{}, no. 22, 221602 (2015) doi:10.1103/PhysRevLett.114.221602 \[arXiv:1412.4095 \[hep-th\]\].
C. Cheung, K. Kampf, J. Novotny, C. H. Shen and J. Trnka, “On-Shell Recursion Relations for Effective Field Theories,” Phys. Rev. Lett. [**116**]{}, no. 4, 041601 (2016) doi:10.1103/PhysRevLett.116.041601 \[arXiv:1509.03309 \[hep-th\]\].
C. Cheung, K. Kampf, J. Novotny, C. H. Shen and J. Trnka, “A Periodic Table of Effective Field Theories,” arXiv:1611.03137 \[hep-th\].
N. Arkani-Hamed and J. Trnka, “The Amplituhedron,” JHEP [**1410**]{}, 030 (2014) \[arXiv:1312.2007 \[hep-th\]\].
[^1]: This is similar to the results of [@boel1; @boel2], where it was found that imposing full gauge invariance, while also allowing extra kinematical invariants as coefficients, leads to $(n-3)!$ independent solutions, forming the BCJ basis of Yang-Mills amplitudes.
[^2]: See also [@delta] for a very illuminating discussion on fixing the subleading terms.
|
---
abstract: |
-0.04in Physics of chaos in a localized phase-space region is exploited to produce a longitudinally uniformly distributed beam. Theoretical study and simulations are used to study its origin and applicability in phase-space dilution of beam bunch. Through phase modulation to a double-rf system, a central region of localized chaos bounded by invariant tori are generated by overlapping parametric resonances. Condition and stability of the chaos will be analyzed. Applications include high-power beam, beam distribution uniformization, and industrial beam irradiation.\
author:
- |
S.Y. Lee, Indiana University, Bloomington, IN 47405, US\
K.Y. Ng, Fermilab, Batavia, IL 60510, US
title: 'APPLICATION OF A LOCALIZED CHAOS GENERATED BY RF-PHASE MODULATIONS IN PHASE-SPACE DILUTION[^1]'
---
INTRODUCTION
============
ALPHA, under construction at IU CEEM, is a 20-m electron storage ring. [@alpha] The project calls for storing a tiny synchrotron-radiation-damped bunch to be extended to about 40 ns with uniform longitudinal distribution. RF barriers should be the best candidate for bunch lengthening. Unfortunately, this ring is only 66.6 ns in length, and the widths of the barriers must be of the order of 10 ns or less. The risetime of the barrier voltage will therefore be a few ns, or the rf generating the barrier voltage will be in the frequency range of a few hundred MHz. Ferrite is very lossy at such high frequencies and is therefore unsuitable for the job. Even if another material could be substituted, the barriers of such narrow widths would require very high rf peak voltage; the rf system would be very costly.
Another way to achieve bunch lengthening is to perform phase modulation of the rf wave so as to produce a large chaotic region at the center of the rf bucket, but bounded by well-behaved tori. The beam at the bucket center will be blown up to the much larger chaotic region. If true chaoticity is achieved, the particle distribution will be uniform. Such an idea has been demonstrated experimentally at the IUCF Cooler ring in 1997, [@jeon] where a double-rf system was used and the diffusion was found rather sensitive to the phase difference $\Delta\phi_0$ between the two rf waves. In this paper, the modulation method is further investigated by first determining the choice of $\Delta\phi_0$, and next analyzing the condition and stability of the localized chaotic region.
THE MODEL
=========
The model to be studied is described by the Hamiltonian $H=H_0+H_1$, where [@liu] H\_0=12\_s\^2 +\_s{1--}, H\_1=a\_s(\_m+). \[h1\] Here $r$ is the ratio of the two rf voltages, $h$ is the ratio of the two rf harmonics, $\nu_s$ is the small-amplitude synchrotron tune in the absence of the second rf, $\nu_m$ is the phase modulation tune, $\eta$ is the modulation phase, $a$ is the modulation amplitude, $\phi$ is the rf phase, $\delta$ is the canonical momentum offset, and $\theta$ advances by $2\pi$ per revolution turn. This model entails a number of parameters. In this paper, however, we restrict ourselves to the special case of $r=1/2$, $h=2$, $\nu_m/\nu_s=2$, and $\eta=0$, thus leaving behind only the phase offset $\Delta\phi_0$ and the modulation amplitude $a$.
Choice of $\Delta\phi_0$
------------------------
The action at the bottom of an rf potential well $V(\phi)$ is zero and so are the [*resonance strengths*]{} generated by phase modulation (see Fig. \[strength-30deg-fig\] below). Since the bunch will be tiny, it will be difficult to be driven into parametric resonances if it sits at the bottom of the rf potential. This explains why a two-rf system is necessary. The phase difference $\Delta\phi_0$ between the two rf’s shifts the potential-well bottom away from the center of the longitudinal phase space, where the tiny bunch is located. The action at the bunch is now finite. Thus the farther the potential-well bottom is shifted, the larger the resonant strengths.
To generate a large region of chaoticity, the eventual modulation amplitude $a$ will be large. However, a perturbative approach is taken in the analysis so as to get a ball-park understanding of the mechanism. For the unperturbed Hamiltonian, the position of the potential-well bottom $\phi_0$ is given by $V'(\phi_0)/\nu_s=
\sin\phi_0-r\sin(h\phi_0+\Delta\phi_0)=0,$ and is at a maximum when $V''(\phi_0)=0$. This leads to the solution $\phi_0=\pm\sin^{-1}r$. The corresponding phase difference between the two rf’s is therefore (see Fig. \[minimum-fig\]) \_0=-h\^[-1]{}r. When $h\!=\!1/r\!=\!2$, the largest offset of well bottom is $\phi_0\!=\!\pm30^\circ$ and the corresponding phase difference between the two rf’s is $\Delta\phi_0=\pm30^\circ$.
Simulations show that diffusion of a tiny bunch at the phase-space center is possible when $20^\circ\lesssim|\Delta\phi_0|\lesssim50^\circ$. In below, we first study the case of $\Delta\phi_0=30^\circ$, and attempt another value later.
Synchrotron Tune and Resonance Strengths
----------------------------------------
For the unperturbed Hamiltonian, the action and angle of an oscillatory orbit are given by J=()d, =\_[\_1]{}\^, \[psi\] where $\phi_1$ is smaller end of the phase excursion. When $\phi$ reaches the larger end of the phase excursion $\phi=\phi_2$ while $\psi$ advances by $\pi$, the synchrotron tune $Q_s$ of the oscillatory torus can be extracted, and is depicted in Fig. \[syntune-30deg-fig\], where the asymmetric rf potential is also plotted in dot-dashes.
With the modulation tune $\nu_m/\nu_s\!=\!2$ (horizontal line), we see that the tiny bunch at the phase-space center will be driven into 3:1 and 5:2 resonances. The synchrotron tune can also be computed as a function of the action $J$, and this is depicted in Fig. \[syntune-J-30deg-fig\]. It is important to point out that we require the synchrotron tune to be large at the central part than the edges of the phase space, because we wish to have a central chaotic region bounded by well-behaved tori. To express the perturbative Hamiltonian $H_1$ of Eq. (\[h1\]) in terms of action-angle variables, the reduced momentum offset $\delta$ is expanded into Fourier series, =\_[n=-]{}\^ g\_n(J)e\^[in]{}, g\_n(J)=\_[-]{}\^e\^[-in]{}d. Keeping only the first-order perturbative terms, the Hamiltonian can now be expressed as $$\begin{aligned}
H\!=&H_0(J)+a\nu_s\bigg\{\!\sum_{n>0}|g_n(J)|
\Big[\sin(\nu_m\theta\!+\!\eta\!+\!\chi_n\!+\!n\psi)~~~~~~\nonumber\\
&\!\!\!\!+\sin(\nu_m\theta\!+\!\eta\!+\!\chi_n\!-\!n\psi)\Big]
\!+g_0(J)\sin(\nu_m\theta\!+\!\eta)\!\bigg\},\!\!\!\!\!
\label{first-order}\end{aligned}$$ with $\chi_n$ the phase of $g_n$, showing all the first-order parametric resonances. The resonance strength function $|g_n|$ is a measure of its ability to drive the $n\!:\!1$ parametric resonance, and is depicted in Fig. \[strength-30deg-fig\] for $n\!=\!1$ to 4. Note that all the strength functions vanish at $J=0$, which is the reason why we need to shift the potential-well bottom away from the center of the phase space. Because of the asymmetry of $\delta(\phi)$, $g_n(J)$ no longer vanishes when $n$ is even, implying that both 2:1 and 3:1 resonances can be excited depending on the choice of the modulation tune $\nu_m$.
.
SIMULATIONS WITH $\Delta\phi_0=30^\circ$
========================================
Simulation is performed by tracking each macro-particle from the $k$-th turn to the $(k\!+\!1)$-th turn according to the Hamiltonian of Eq. (\[h1\]): $$\begin{aligned}
\!\!\!\phi_{k+1}&\!=\!\phi_k\!+\!2\pi\nu_s\big[\delta_k
\!+\!a\sin(2\pi k\nu_m )\big], \nonumber\\
\!\!\!\delta_{k+1}&\!=\!\delta_k\!-\!2\pi\nu_s\big[\sin\phi_{k+1}
\!-\!r\sin(h\phi_{k+1}+\Delta\phi_0)
\big],\end{aligned}$$ where the assumption of slowly varying particle position within a turn has been applied. The modulation period has been chosen to be exactly 515 turns with $\nu_m/\nu_s=2$. The modulation phase has been taken to be $\eta=0$. Other values of $\eta$ will lead to different rotated stroboscopic views of the tracking results. However, all the conclusions on diffusion and beam enlargement will not be affected.
We first study the structure of the phase space at modulation amplitude $a=8^\circ$ and $\Delta\phi_0=30^\circ$, as depicted in Fig. \[a8-30deg-8-fig\].
This is the cumulative stroboscopic modulation-period views in half million turns. The central stable region is bounded by the 5:2 resonance. After that we see the remnant of the 8:3 resonance which merges partially with the 3:1 resonance. Then comes the 25:8 resonance and the well-behaved tori. In order for the bunch, initially at the phase-space center, to be enlarged via diffusion, we require the 5:2 resonance to collapse and the central stable region to shrink so that the bunch is inside the chaotic region initially. This occurs when the modulation amplitude increases to $a=46^\circ$.
Next a Gaussian distributed bunch of rms spread $\sigma_\phi=0.001$ rad consisting of 10000 macro-particles at the center of the longitudinal phase space is tracked. The particle distribution with modulation amplitude $a=58^\circ$ is shown in Fig. \[a58-30deg-82-fig\] at the last modulation period in 1.2 million turns. Essentially, the tiny bunch at the phase-space center is driven into the thick stochastic layers surrounding the separatrices of the 3:1 resonance.
The thick stochastic layers, on the other hand, come from the overlapping of the 5:2, 8:3, and possibly many other higher-order resonances, which are not included in the first order perturbation of the modulation presented in Eq. (\[first-order\]). The distribution appears to be uniform except for the four big empty space, where the four stable fixed points of the 3:1 resonance are located. The rms beam size is computed turn by turn and is depicted in Fig. \[a58-30deg-93-fig\].
We see that the rms-bunch-size squared, $\sigma_\phi^2$ or $\sigma_\delta^2$, grows linearly with turn number, signaling that the bunch enlargement is indeed a diffusion process. The growth levels off after about $1\times10^5$ to $2\times10^5$ turns. The rms phase and momentum spreads increase to $\sigma_\phi=0.81\pm0.03$ rad and $\sigma_\delta=0.57\pm0.01$, respectively.
We next continue the tracking by doubling the number of turns; the distribution remains bounded and the pattern does not change. We also track a particle initially at $\phi\!=\!0.94$ rad and $\delta\!=\!0$ for $1\!\times\!10^5$ turns. Its positions at every modulation period are shown as red dots in Fig. \[a58-30deg-82-fig\]. These red dots constitute a well-behaved chain of islands, confirming that the diffused bunch will be well-bounded.
Variation of Modulation Amplitude
---------------------------------
When the modulation amplitude is varied from $a=46^\circ$ to $70^\circ$, there is not much difference in the shape of the final diffused bunch distribution. The only significant change is the gradual left-shifting of the chaotic pattern in Fig. \[a58-30deg-82-fig\], which is a consequence of the detuning of the synchrotron tune $Q_s$ as the modulation amplitude $a$ increases. The diffused rms phase and momentum spreads are still roughly at $\sigma_\phi\approx0.8$ rad and $\sigma_\delta\approx0.58$, respectively, as depicted in Fig. \[bunchspread-30deg-fig\].
When the modulation strength is increased past $a=70^\circ$, suddenly no diffusion is observed independent of how long the simulations are performed. This turns out to be the moment when the rightmost empty region in Fig. \[a58-30deg-82-fig\] has been left-shifted to include the phase-space origin. The bunch is now inside the rightmost island of the 3:1 resonance making small tori around the stable fixed points of the 3:1 resonance.
The diffusion of the bunch does return when the modulation strength increases up to $a\geq85^\circ$. Now the three outside islands of the 3:1 resonance are completely filled up by higher-order resonances so that the tiny bunch initially at the phase-space center can diffuse outward again.
As the modulation amplitude continues to increase, the phase-space structure tends to contract and shift further to the left. As $a>94^\circ$, the bunch original position moves out of the chaotic region and no diffusion occurs. A typical phase-space pattern at $a=104^\circ$ is shown in Fig. \[a104-30deg-8-fig\].
As $a$ increases past $110^\circ$, beam loss occurs.
SIMULATIONS WITH $\Delta\phi_0=45^\circ$
========================================
Figures \[syntune-30deg-fig\] and \[syntune-J-30deg-fig\] shows that the approximate intercepts of horizontal line $\nu_m/\nu_s=2$ with the $3\times Q_s/\nu_s$ curve are far away from the initial location of the particle bunch. In other words, the bunch is initially far from the unstable fixed points of the 3:1 resonance. This limits the bunch from falling inside the stochastic layers surrounding the separatrices unless the modulation amplitude is sufficiently large. Figure \[strength-30deg-fig\] shows that the strength function of the 3:1 resonance is much smaller than that of the 2:1 resonance and as a result very large modulation amplitude has to be employed. All these reasons educe us to a deviation from the maximum potential-well-bottom offset. Here, we try the rf phase difference $\Delta\phi_0\!=\!45^\circ$, which leads to a well-bottom offset of $\phi_0\!=\!29.12^\circ$, which is only 3% less than the maximum value of $30^\circ$. This explains why the range of $\Delta\phi_0$ that can produce bunch lengthening is not too narrow.
Corresponding to Fig. \[syntune-J-30deg-fig\], the synchrotron tune as a function of action for $\Delta\phi_0=45^\circ$ is shown in Fig. \[syntune-J-45deg-fig\].
Observe that where the horizontal line $\nu_m/\nu_s\!=\!2$ cuts the $2\times Q_s/\nu_s$ curve is extremely close to $J\!=\!0.053$, the initial location of bunch, implying close proximity of the bunch from an unstable fixed point of the 2:1 resonance. We should expect diffusion to occur at relatively smaller modulation amplitudes. The resonance strength functions for $\Delta\phi_0\!=\!45^\circ$ do not differ much in value from those for $\Delta\phi_0\!=\!30^\circ$ in Fig. \[strength-30deg-fig\].
The phase-space structure when $\Delta\phi_0=45^\circ$ and $a=7^\circ$ is illustrated in Fig. \[a7-45deg-8-fig\].
We first notice that the phase-space center is very close to the 2:1 resonance as speculated. However, the tiny bunch there can only spread out inside the thin stochastic layers of the 2:1 resonance, and cannot reach the larger chaotic region between the island chains of the 8:5 and 7:3 resonances. The bunch can diffuse into this region only when the chains of higher-order islands enclosing the 2:1 resonance collapse. This happens when $a\approx9^\circ$, which explains the rapid jump of simulated bunch-spread results from $a=8^\circ$ to $10^\circ$ in Fig. \[bunchspread-45deg-fig\].
A typical diffused bunch distribution is shown in Fig. \[a14-45deg-82-fig\], corresponding to modulation amplitude $a\!=\!28^\circ$ at the last modulation period after roughly 0.5 million turns. Compared with Fig. \[a58-30deg-82-fig\] at $\Delta\phi_0\!=\!30^\circ$, it is evident that the chaotic area is much larger and the empty space inside is very much smaller. It also looks much more rectangular, and will provide a more uniform linear density. At the same time the modulation amplitude $a=28^\circ$ is about one-half smaller. The red dots are stroboscopic loci of one particle initially located at $\phi\!=\!1.46$ rad and $\delta\!=\!0$. These loci provide a well-behaved torus bounding the diffused bunch.
The rms phase and momentum-offset spreads shown in Fig. \[a28-45deg-93-fig\] reveal linear growths of $\sigma_\phi^2$ and $\sigma_\delta^2$, demonstrating the occurrence of diffusion. Compared with Fig. \[a58-30deg-93-fig\] at $\Delta\phi_0\!=\!30^\circ$, equilibrium is reached much earlier. This is understandable because the bunch initially is much closer to the separatrices of the 2:1 resonance, and obviously, will take less time to diffuse.
As illustrated in Fig. \[bunchspread-45deg-fig\], there is another jump of beam size around $a\approx 20^\circ$. This can be explained by the hump of the synchrotron frequency around action $J\approx0.7$ in Fig. \[syntune-J-45deg-fig\]. As a result, there are two sets of 8:3 resonances, one going out and one coming in as the modulation amplitude $a$ increases. The one going out has already broken at $a\approx7$. The incoming set encircling the chaotic region starts collapsing around $a=20^\circ$, and the chaotic region is increased after that. This is illustrated in Fig. \[a196-45deg-82-fig\].
CONCLUSION
==========
We have devised a method of phase modulation of the rf wave to create a large chaotic region in the central longitudinal phase space bounded by well-behaved tori. To accomplish this, we require (1) large modulation amplitude so that higher-order parametric resonances collapse to form a large chaotic area, and (2) the initial position of the tiny bunch inside this chaotic region. Since the bunch is initially located at the phase-space center, we must offset the relative phase of the two-rf system so that the potential-well bottom is shifted away from the phase-space center. The maximum well-bottom offset and the corresponding relative phase difference between the two rf’s are computed.
[9]{}
J. Doskow, etal, “The ALPHA Project at IU CEEM,” IPAC’10, 268.
D. Jeon, etal, “A Mechanism of Anomalous Diffusion in Particle Beams,” Phys. Rev. Lett. [**80**]{}, (1998) 2314. C.M. Chu, etal, “Effects of Overlapping Parametric Resonances on the Particle Diffusion Process,” Phy. Ref. [**E60**]{}, (1999) 6051. K.Y. Ng, “Particle Diffusion in Overlapping Resonances,” Advanced IFCA Workshop on Beam Dynamics Issues for $e^+e^-$ Factories, Frascati, Oct. 20-25, 1997 (Fermilab-Conf-98/001).
J.Y. Liu, etal, “Analytic Solution of Particle Motion in a Double RF System,” Particle Accelerators [**49**]{}, (1995) 221.
[^1]: Work supported by the US DOE under contracts DE-FG02-92ER40747, DE-AC02-76CH030000, and the NSF under contract NSF PHY-0852368.
|
---
abstract: 'In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric framework. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.'
author:
- 'T.O. Gallouët, M. Laborde, L. Monsaingeon'
bibliography:
- 'Laborde\_bib.bib'
title: 'An unbalanced Optimal Transport splitting scheme for general advection-reaction-diffusion problems'
---
Introduction
============
Since the seminal works of Jordan-Kinderlehrer-Otto [@JKO], it is well known that certain diffusion equations can be interpreted as gradient flows in the space of probability measures, endowed with the quadratic Wasserstein distance ${\mathtt{W}}$. The well-known JKO scheme (a.k.a. minimizing movement), which is a natural implicit Euler scheme for such gradient flows, naturally leads to constructive proofs of existence for weak solutions to equations or systems with mass conservation such as, for instance, Fokker-Planck equations [@JKO], Porous Media Equations [@O1], aggregation equation [@CDFLS], double degenerate diffusion equations [@O], general degenerate parabolic equation [@A] etc. We refer to the classical textbooks of Ambrosio, Gigli and Savaré [@AGS] and to the books of Villani [@V1; @V2] for a detailed account of the theory and extended bibliography. Recently, this theory has been extended to study the evolution of interacting species with mass-conservation, see for examples [@DFF; @Zinsl; @L; @KMX; @CL1].
Nevertheless in biology, for example for diffusive prey-predator models, the conservation of mass may not hold, and the classical optimal transport theory does not apply. An unbalanced optimal transport theory was recently introduced simultaneously in [@CPSV; @CPSV1; @KMV; @LMS_big_15; @LMS_small_15], and the resulting Wasserstein-Fisher-Rao (${\mathtt{WFR}}$) metrics (also referred to as the Hellinger-Kantorovich distance ${\mathtt{HK}}$) allows to compute distances between measures with variable masses while retaining a convenient Riemannian structure. See section \[part4-section2WFR\] for the definition and a short discussions on this ${\mathtt{WFR}}$ metric. We also refer to [@piccoli_rossi_generalized_14; @figalli2010new] for earlier attempts to account for mass variations within the framework of optimal transport.
The ${\mathtt{WFR}}$ metrics can be seen as an *inf-convolution* between Wasserstein/transport and Fisher-Rao/reaction processes, and is therefore extremely convenient to control both in a unified metric setting. This allows to deal with non-conservative models of population dynamics, see e.g. [@KMV; @Kondratyev20162784]. In [@GM], the first and third authors proposed a variant of the JKO scheme for ${\mathtt{WFR}}$-gradient flows corresponding to some particular class of reaction-diffusion PDEs: roughly speaking, the reaction and diffusion were handled separately in two separate ${\mathtt{FR}},{\mathtt{W}}$ metrics, and then patched together using a particular uncoupling of the inf-convolution, namely ${\mathtt{WFR}}^2\approx {\mathtt{W}}^2+{\mathtt{FR}}^2$ in some sense (see [@GM section 3] for a thorough discussion). However, the analysis was restricted to very particular structures for the PDE, corresponding to pure ${\mathtt{WFR}}$ gradient-flows.
In this work we aim at extending this splitting scheme in order to handle more general reaction-diffusion problems, not necessarily corresponding to gradient flows. Roughly speaking, the structure of our splitting scheme is the following: the transport/diffusion part of the PDE is treated by a single Wasserstein JKO step $$\rho^k\xrightarrow[\mbox{transport}]{{\mathtt{W}}}\rho^{k+1/2},$$ and the next Fisher-Rao JKO step $$\rho^{k+1/2}\xrightarrow[\mbox{reaction}]{{\mathtt{FR}}}\rho^{k+1}$$ handles the reaction part of the evolution. As already mentioned, the ${\mathtt{WFR}}$ metric will allow to suitable control both steps in a unified metric framework. We will first state a general convergence result for scalar reaction-diffusion equations, and then illustrate on a few particular examples how the general idea can be adapted to treat e.g. prey-predator systems or very degenerate Hele-Shaw diffusion problems. In this work we do not focus on optimal results and do not seek full generality, but rather wish to illustrate the efficiency of the general approach.
Another advantage of the splitting scheme is that is well adapted to existing Monge/Kantorovich/Wasserstein numerical solvers, and the Fisher-Rao step turns out to be a simple pointwise convex problem which can be implemented in a very simple way. See also [@simone_lenaic; @chizat2016scaling] for a more direct numerical approach by entropic regularization. Throughout the paper we will illustrate the theoretical results with a few numerical tests. All the numerical simulations were implemented with the augmented Lagrangian [ALG2-JKO]{} scheme from [@BCL] for the Wasserstein step, and we used a classical Newton algorithm for the Fisher-Rao step.\
The paper is organized as follows. In section \[part4-section2WFR\] we recall the basic definitions and useful properties of the Wasserstein-Fisher-Rao distance ${\mathtt{WFR}}$. Section \[sec:KFRsplitting\] contains the precise description of the splitting scheme and a detailed convergence analysis for a broad class of reaction-diffusion equations. In section \[section:systems\] we present an extension to some prey-predator multicomponent systems with nonlocal interactions. In section \[part4-section2HS\] we extend the general result from section \[sec:KFRsplitting\] to a very degenerate tumor growth model studied in [@PQV], corresponding to a pure ${\mathtt{WFR}}$ gradient flow: we show that the splitting scheme captures fine properties of the model, particularly the $\Gamma$-convergence of discrete gradient flows as the degenerate diffusion parameter of Porous Medium type $m\to\infty$. The last section \[part4-section2HSN\] contains an extension to a tumor-growth model coupled with an evolution equation for the nutrients.
Preliminaries {#part4-section2WFR}
=============
Let us first fix some notations. Throughout the whole paper, $\Omega$ denotes a possibly unbounded convex subset of $\operatorname*{\mathbb{R}}^d$, $Q_T$ represents the product space $[0,T] \times \Omega$, for $T>0$, and we write ${\mathcal{M}}^+={\mathcal{M}}^+(\Omega)$ for the set of nonnegative finite Radon measures on $\Omega$. We say that a curve of measures $t\mapsto \rho_t\in\mathcal C_w([0,1];{\mathcal{M}}^+)$ is narrowly continuous if it is continuous with respect to the narrow convergence of measures, namely for the duality with ${\mathcal{C}}_b(\Omega)$ test-functions.
\[def:FR\] The Fisher-Rao distance between $\rho_0,\rho_1\in {\mathcal{M}}^+$ is $${\mathtt{FR}}(\rho_0,\rho_1):= \min_{(\rho_t,r_t) \in {\mathcal{A}}_{{\mathtt{FR}}}[\rho_0,\rho_1]} \int_0^1\int_\Omega |r_t|^2 \,d\rho_t(x)dt,$$ where the admissible set $A_{{\mathtt{FR}}}[\rho_0,\rho_1]$ consists in curves $[0,1]\ni t\mapsto (\rho_t,r_t)\in{\mathcal{M}}^+\times{\mathcal{M}}$ such that $t\mapsto \rho_t\in\mathcal C_w([0,1];{\mathcal{M}}^+)$ is narrowly continuous with endpoints $\rho_t(0)=\rho_0,\rho_t(1)=\rho_1$, and $$\partial_t\rho_t=\rho_tr_t$$ in the sense of distributions $\mathcal D'((0,1)\times\Omega)$.
The Monge-Kantorovich-Wasserstein admits several equivalent definitions and formulations, and we refer e.g. to [@V1; @V2; @AGS; @S] for a complete description. For our purpose we shall only need the dynamical Benamou-Brenier formula:
\[theo:benamou\_brenier\] There holds $$\label{eq:benamou_brenier_formula}
{\mathtt{W}}^2(\rho_0,\rho_1)=\min\limits_{(\rho,{\mathbf{v}})\in\mathcal A_{\mathtt{W}}[\rho_0,\rho_1]}\int_0^1\int_\Omega|{\mathbf{v}}_t|^2{\mathrm{d}}\rho_t{\mathrm{d}}t,$$ where the admissible set $\mathcal A_{\mathtt{W}}[\rho_0,\rho_1]$ consists in curves $(0,1)\ni t\mapsto (\rho_t,{\mathbf{v}}_t)\in {\mathcal{M}}^+\times {\mathcal{M}}(\Omega;\operatorname*{\mathbb{R}}^d)$ such that $t\mapsto\rho_t$ is narrowly continuous with endpoints $\rho_t(0)=\rho_0$, $\rho_t(1)=\rho_1$ and solving the continuity equation $$\partial_t\rho_t+\operatorname*{div}(\rho_t{\mathbf{v}}_t)=0$$ in the sense of distributions $\mathcal D'((0,1)\times\Omega)$.
According to the original definition in [@CPSV] we have
\[def:KFR\] The Wasserstein-Fisher-Rao distance between $\rho_0, \rho_1 \in {\mathcal{M}}^+(\Omega)$ is $$\label{eq:def_KFR}
{\mathtt{WFR}}^2(\rho_0,\rho_1):=\inf_{(\rho,{\mathbf{v}},r) \in {\mathcal{A}}_{{\mathtt{WFR}}}[\rho_0,\rho_1]} \int_0^1 \int_\Omega (|{\mathbf{v}}_t(x)|^2+|r_t|^2)\,d\rho_t(x)dt,$$ where the admissible set ${\mathcal{A}}_{{\mathtt{WFR}}}[\rho_0,\rho_1]$ is the set of curves $t \in [0,1] \mapsto (\rho_t,v_t,r_t) \in {\mathcal{M}}^+\times {\mathcal{M}}(\Omega;\operatorname*{\mathbb{R}}^d) \times {\mathcal{M}}$ such that $t\mapsto \rho_t \in \mathcal{C}_w([0,1],{\mathcal{M}}^+)$ is narrowly continuous with endpoints $\rho_{|t=0}=\rho_0$, $\rho_{|t=1}=\rho_1$ and solves the continuity equation with source $$\partial_t \rho_t +\operatorname*{div}(\rho_t v_t) = \rho_t r_t.$$
Comparing definition \[def:KFR\] with definition \[def:FR\] and Theorem \[theo:benamou\_brenier\], this dynamical formulation [*à la Benamou-Brenier*]{} shows that the ${\mathtt{WFR}}$ distance can be viewed as an inf-convolution of the Wasserstein and Fisher-Rao distances ${\mathtt{W}},{\mathtt{FR}}$. From [@CPSV; @CPSV1; @KMV; @LMS_big_15] the infimum in is always a minimum, and the corresponding minimizing curves $t\mapsto\rho_t$ are of course constant-speed geodesics ${\mathtt{WFR}}(\rho_t,\rho_s)=|t-s|{\mathtt{WFR}}(\rho_0,\rho_1)$. Then $({\mathcal{M}}^+,{\mathtt{WFR}})$ is a complete metric space, and ${\mathtt{WFR}}$ metrizes the narrow convergences of measures (see again [@CPSV; @CPSV1; @KMV; @LMS_big_15]). Interestingly, there are other possible formulations of the distance in terms of static unbalanced optimal transportation, primal-dual characterizations with relaxed marginals, lifting to probability measures on a cone over $\Omega$, duality with subsolutions of Hamilton-Jacobi equations, and we refer to [@CPSV; @CPSV1; @KMV; @LMS_small_15; @LMS_big_15] for more details.
As a first useful interplay between the distances ${\mathtt{WFR}},{\mathtt{W}},{\mathtt{FR}}$ we have
\[prop:comparison\_d\_W\_H\] Let $\rho_0,\rho_1\in {\mathcal{M}}^+_2$ such that $|\rho_0|=|\rho_1|$. Then $${\mathtt{WFR}}^2(\rho_0,\rho_1)\leqslant {\mathtt{W}}^2(\rho_0,\rho_1).$$ Similarly for all $\mu_0,\mu_1\in {\mathcal{M}}^+$ (with possibly different masses) there holds $${\mathtt{WFR}}^2(\mu_0,\mu_1)\leqslant {\mathtt{FR}}^2(\mu_0,\mu_1).$$ Finally, for all $\nu_0,\nu_1\in {\mathcal{M}}^+_2$ such that $|\nu_0|=|\nu_1|$ and all $\nu\in {\mathcal{M}}^+$, there holds $${\mathtt{WFR}}^2(\nu_0,\nu)\leqslant 2({\mathtt{W}}^2(\nu_0,\nu_1)+{\mathtt{FR}}^2(\nu_1,\nu)).$$
Moreover, we have the following link between the reaction and the velocity in , which was the original definition in [@KMV]:
\[prop:coupling=uncoupling\_d\] The definition of the ${\mathtt{WFR}}$ distance can be restricted to the subclass of admissible paths $({\mathbf{v}}_t,r_t)=(\nabla u_t,u_t)$ for potentials $u_t\in H^1({\mathrm{d}}\rho_t)$ and continuity equations $$\partial_t\rho_t+\operatorname*{div}(\rho_t\nabla u_t)=\rho_t u_t.$$
This shows that $({\mathcal{M}}^+,{\mathtt{WFR}})$ can be endowed with the formal Riemannian structure constructed as follow: any two tangent vectors $\xi^1=\partial_t\rho^1,\xi^2=\partial_t\rho^2$ can be uniquely identified with potentials $u^i$ by solving the elliptic equations $$\xi^i=-\operatorname*{div}(\rho\nabla u^i)+\rho u^i.$$ Then the Riemaniann tensor is naturally constructed on the $H^1({\mathrm{d}}\rho)$ scalar product, i-e $$g_\rho(\xi^1,\xi^2):=\langle u^1,u^2\rangle_{H^1({\mathrm{d}}\rho)}=\int_{\Omega}(\nabla u^1\cdot\nabla u^2+ u^1 u^2){\mathrm{d}}\rho.$$ This is purely formal, and we refer again to [@GM] for discussions. Given a functional $${\mathcal{F}}(\rho):=\int_\Omega F(\rho)+\int_\Omega \rho V +\frac{1}{2}\int_{\Omega}(K\ast \rho)\rho,$$ this Riemannian structure also allows to compute ${\mathtt{WFR}}$ gradients as $${\operatorname{grad}}_{{\mathtt{WFR}}}{\mathcal{F}}(\rho)
=-\operatorname*{div}\left(\rho\nabla \frac{\delta{\mathcal{F}}}{\delta\rho}\right)+\rho \frac{\delta{\mathcal{F}}}{\delta\rho}
={\operatorname{grad}}_{{\mathtt{W}}}{\mathcal{F}}(\rho)+{\operatorname{grad}}_{\mathtt{FR}}{\mathcal{F}}(\rho),$$ where $\frac{\delta{\mathcal{F}}}{\delta\rho}=F'(\rho)+V+K\ast\rho$ denotes the Euclidean first variation of ${\mathcal{F}}$ with respect to $\rho$. In other words, the Riemannian tangent vector ${\operatorname{grad}}_{\mathtt{WFR}}{\mathcal{F}}(\rho)$ is represented in the previous $H^1({\mathrm{d}}\rho)$ duality by the scalar potential $u=\frac{\delta{\mathcal{F}}}{\delta\rho}$.
An existence result for general parabolic equations {#sec:KFRsplitting}
===================================================
In this section, we propose to solve scalar parabolic equations of the form $$\label{eq:KFR-general equation}
\left\{
\begin{array}{l}
\partial_t \rho = \operatorname*{div}(\rho \nabla (F'_1(\rho) +V_1)) -\rho( F'_2(\rho)+V_2) \\
\rho|_{t=0}=\rho_0\\
\left.{}_{}\rho \nabla (F'_1(\rho) +V_1)\right|_{\partial\Omega}\cdot \nu =0
\end{array}
\right.$$ in a bounded domain $\Omega \subset {\mathbb{R}^d}$ with Neumann boundary condition and suitable initial conditions. Our goal is to extend to the case $F_1\neq F_2,V_1\neq V_2$ the method initially introduced in [@GM] for variational ${\mathtt{WFR}}$-gradient flows, i-e with $F_1=F_2$ and $V_1=V_2$.
We assume for simplicity that $F_1 \, : \, \operatorname*{\mathbb{R}}\rightarrow \operatorname*{\mathbb{R}}$ is given by $$\begin{aligned}
\label{assumption Diffusion}
F_1(z)=\left\{
\begin{array}{ll}
z\log z -z & \mbox{(linear diffusion)}\\
\mbox{or}\\
\frac{1}{m_1-1}z^{m_1} & \mbox{(Porous Media diffusion)}
\end{array}
\right.,\end{aligned}$$
and $F_2\, : \, \operatorname*{\mathbb{R}}\rightarrow \operatorname*{\mathbb{R}}$ is given by $$\begin{aligned}
\label{assumption reaction}
F_2(z) = \frac{1}{m_2-1}z^{m_2}, \qquad\mbox{for some } m_2>1.\end{aligned}$$ Note that we cannot take $F_2(z)=z\log z-z$ because the Boltzmann entropy is not well behaved (neither regular nor convex) with respect to the Fisher-Rao metric in the reaction step, see [@GM; @LMS_small_15; @LMS_big_15] for discussions. In addition, we assume that $$V_1 \in W^{1,\infty}(\Omega)
\qquad\mbox{and}\qquad
V_2 \in L^\infty(\Omega).$$ We denote ${\mathcal{E}}_1, {\mathcal{E}}_2 \, :\, {\mathcal{M}}^+ \rightarrow \operatorname*{\mathbb{R}}$ the energy functionals $${\mathcal{E}}_i(\rho) := {\mathcal{F}}_i(\rho) + {\mathcal{V}}_i(\rho),$$ where $${\mathcal{F}}_i(\rho):= \left\{ \begin{array}{ll}
\int_\Omega F_i(\rho) & \text{ if } \rho \ll \mathcal{L}_{|\Omega}\\
+\infty & \text{ otherwise, }
\end{array}\right.
\qquad
\text{and}
\qquad
{\mathcal{V}}_i(\rho) := \int_\Omega V_i \rho.$$ Although more general statements with suitable structural assumptions could certainly be proved, we do not seek full generality here and choose to restrict from the beginning to the above simple (but nontrivial) setting for the sake of exposition.
\[def:weak\_solutions\] A weak solution of is a curve $ [0, +\infty) \ni t\mapsto \rho(t,\cdot) \in L^1_+\cap L^\infty(\Omega)$ such that for all $T<\infty$ the pressure $P_1(\rho):=\rho F_1'(\rho)-F_1(\rho)$ satisfies $\nabla P_1(\rho) \in L^2([0,T] \times \Omega)$, and $$\int_0^{+\infty}\left( \int_\Omega (\rho \partial_t \phi -\nabla V_1 \cdot \nabla \phi \rho - \nabla P_1(\rho) \cdot \nabla \phi - \rho (F_2'(\rho) +V_2) \phi) \,dx \right)\, dt = -\int_\Omega \phi(0,x) \rho_0(x) \,dx$$ for every $\phi \in {\mathcal{C}}^\infty_c([0,+\infty) \times {\mathbb{R}^d})$.
Note that the pressure $P_1$ is defined so that the diffusion term $\operatorname*{div}(\rho\nabla F_1'(\rho))=\Delta P_1(\rho)$, at least for smooth solutions.\
The starting point of our analysis is that can be written, at least formally as, $$\partial_t \rho = \operatorname*{div}(\rho \nabla (F'_1(\rho) +V_1)) -\rho( F'_2(\rho)+V_2)
\quad \leftrightarrow\quad
\partial_t\rho=-{\operatorname{grad}}_{{\mathtt{W}}}\mathcal E_1(\rho)-{\operatorname{grad}}_{{\mathtt{FR}}}\mathcal E_2(\rho).$$ Our splitting scheme is a variant of that originally introduced in [@GM], and can be viewed as an operator splitting method: each part of the PDE above is discretized (in time) in its own ${\mathtt{W}},{\mathtt{FR}}$ metric, and corresponds respectively to a ${\mathtt{W}}$/transport/diffusion step and to a ${\mathtt{FR}}$/reaction step. More precisely, let $h>0$ be a small time step. Starting from the initial datum $ \rho_h^0 :=\rho_0$, we construct two recursive sequences $(\rho_h^k)_k$ and $(\rho_h^{k+1/2})_k$ such that $$\begin{aligned}
\label{splitting scheme}
\left\{\begin{array}{l}
\rho_h^{k+1/2} \in \operatorname*{argmin}\limits_{\rho \in {\mathcal{M}}^+, |\rho|=|\rho_h^k|} \left\{ \frac{1}{2h} {\mathtt{W}}^2(\rho, \rho_h^k) + {\mathcal{E}}_1(\rho ) \right\},\\
\\
\rho_h^{k+1} \in \operatorname*{argmin}\limits_{\rho \in {\mathcal{M}}^+} \left\{ \frac{1}{2h} {\mathtt{FR}}_2^2(\rho, \rho_h^{k+1/2}) + {\mathcal{E}}_2(\rho ) \right\}.
\end{array}\right.\end{aligned}$$
With our structural assumptions on $F_i,V_i$ and arguing as in [@GM], the direct method shows that this scheme is well-posed, i-e that each minimizing problem in admits a unique minimizer. We construct next two piecewise-constant interpolating curves $$\begin{aligned}
\label{piecewise-constant curves}
\left\{\begin{array}{l}
\rho_h(t) = \rho_h^{k+1},\\
{\tilde{\rho}}_h(t) = \rho_h^{k + 1/2},
\end{array}\right.
\text{ for all }t\in (kh, (k+1)h].\end{aligned}$$
Our main results in this section is the constructive existence of weak solutions to :
\[theo:existence differente energies\] Assume that $\rho_0 \in L^1_+\cap L^\infty(\Omega)$. Then, up to a discrete subsequence (still denoted $h\to 0$ and not relabeled here), $\rho_h$ and ${\tilde{\rho}}_h$ converge strongly in $L^1((0,T)\times \Omega)$ to a weak solution $\rho$ of .
Note that any uniqueness for would imply convergence of the whole (continuous) sequence $\rho_h,{\tilde{\rho}}_h\to \rho$ as $h\to 0$, but for the sake of simplicity we shall not address this issue here.
The main technical obstacle in the proof of Theorem \[theo:existence differente energies\] is to retrieve compactness in time. For the classical minimizing scheme of any energy ${\mathcal{E}}$ on any metric space $(X,d)$, suitable time compactness is usually retrieved in the form of the *total-square distance estimate* $\frac{1}{2h}\sum\limits_{k\geq 0}d^2(x^k,x^{k+1})\leqslant {\mathcal{E}}(x_0)-\inf {\mathcal{E}}$. This usually works because only one functional is involved, and ${\mathcal{E}}(x_0)-\inf{\mathcal{E}}$ is obtained as a telescopic sum of one-step energy dissipations ${\mathcal{E}}(x^{k+1})-{\mathcal{E}}(x^k)$. Here each of our elementary step in involves one of the ${\mathtt{W}},{\mathtt{FR}}$ metrics, and we will use the ${\mathtt{WFR}}$ distance to control both simultaneously: this strongly leverages the inf-convolution structure, the ${\mathtt{WFR}}$ distance being precisely built on a compromise between ${\mathtt{W}}$/transport and ${\mathtt{FR}}$/reaction. On the other hand we also have two different functionals ${\mathcal{E}}_1,{\mathcal{E}}_2$, and we will have to carefully estimate the dissipation of ${\mathcal{E}}_1$ during the ${\mathtt{FR}}$ reaction step (driven by ${\mathcal{E}}_2$) as well as the dissipation of ${\mathcal{E}}_2$ during the ${\mathtt{W}}$ transport/diffusion step (driven by ${\mathcal{E}}_1$).
We start by collecting one-step estimates, exploiting the optimality conditions for each elementary minimization procedure, and postpone the proof of Theorem \[theo:existence differente energies\] to the end of the section.
Optimality conditions and pointwise $L^\infty$ estimates
--------------------------------------------------------
The optimality conditions for the first Wasserstein step $\rho^k\to\rho^{k+1/2}$ in are by now classical, and can be written for example
$$\begin{aligned}
\label{opt Wasserstein}
\frac{-\nabla \varphi_h^{k+1/2}}{h} \rho_h^{k+1/2} = \nabla P_1(\rho_h^{k+1/2}) + \rho_h^{k+1/2} \nabla V_1\qquad \mbox{a.e.}\end{aligned}$$
Here $\varphi_h^{k+1/2}$ is an optimal (backward) Kantorovich potential from $\rho_h^{k+1/2}$ to $\rho_h^{k}$.
\[lem:maximum\_principle\_W\_step\] For all $k \geqslant 0$, $$\label{eq: mass-conservation W}
\|\rho_h^{k+1/2}\|_{L^1}=\|\rho_h^k\|_{L^1}
$$ and for all constant $C$ such that $V_1 \leqslant C$, $$\label{eq:PMAX_W}
\rho_h^k (x) \leqslant (F_1')^{-1}(C -V_1(x))\, \mbox{a.e.}
\qquad
\Rightarrow \qquad
\rho_h^{k+1/2} (x) \leqslant (F_1')^{-1}(C -V_1(x)) \, \mbox{a.e.}$$
The Wasserstein step is mass conservative by construction, so the first part is obvious. The second part is a direct consequence of a generalization [@PT lemma 2] of Otto’s maximum principle [@O1].
Note that if $\rho_h^k \leqslant M$, we may take $C=F_1'(M)+\|V_1\|_{L^\infty}$ in . Formally, this corresponds to taking $\overline{\rho}(x):=(F_1')^{-1}(C -V_1(x))$ as a stationary Barenblatt supersolution for $\partial_t\rho =\operatorname*{div}(\rho\nabla (F_1'(\rho)+V_1))$ at the continuous level. In addition, if $V_1\equiv 0$ we recover Otto’s maximum principle [@O1] in the form $\|\rho^{k+1/2}\|_{L^\infty}\leqslant \|\rho^k\|_{L^\infty}$.
For the second Fisher-Rao reaction step, the optimality condition has been obtained in [@GM section 4.2] in the form $$\begin{aligned}
\label{opt Fisher-Rao}
\left( \sqrt{\rho_h^{k+1}} - \sqrt{\rho_h^{k+1/2}}\right) \sqrt{\rho_h^{k+1}}= - \frac{h}{2}\rho_h^{k+1} \left( F_2'(\rho_h^{k+1}) + V_2 \right) \qquad \mbox{a.e.}\end{aligned}$$ As a consequence we have
\[lem encadrement\] There is $C\equiv C(V_2)>0$ such that for $h\leqslant h_0(V_2)$ small enough we have
$$\label{encadrement FR_sup}
\rho_h^{k+1}(x) \leqslant (1+Ch)\rho_h^{k+1/2}(x) \qquad \mbox{a.e.},$$
and for all $M>0$ there is $c\equiv c(M,V_2)$ such that if $\|\rho_h^{k+1/2}\|_{\infty}\leqslant M$ then $$\label{encadrement FR_inf}
(1-ch)\rho_h^{k+1/2}(x) \leqslant \rho_h^{k+1}(x) \qquad \mbox{a.e.}$$
Note in particular that this immediately implies $$\label{eq:supports}
{\mathop{\rm supp}}\,\rho_h^{k+1} = {\mathop{\rm supp}}\,\rho_h^{k+1/2},$$ which was to be expected since the reaction part $\partial_t\rho =-\rho (F_2'(\rho)+V_2)$ of the PDE preserves strict positivity.
We start with the upper bound: inside ${\mathop{\rm supp}}\rho_h^{k+1}$, and $F_2'\geqslant 0$ give $$\begin{aligned}
\sqrt{\rho_h^{k+1}(x)} - \sqrt{\rho_h^{k+1/2}(x)} &=& -h \sqrt{\rho_h^{k+1}(x)}(F_2'(\rho_h^{k+1}(x)) +V_2(x))\\
&\leqslant & -hV_2(x) \sqrt{\rho_h^{k+1}(x)}
\leqslant h\|V_2\|_\infty\sqrt{\rho_h^{k+1}(x)}\end{aligned}$$ whence $$\sqrt{\rho_h^{k+1}(x)} \leqslant \frac{1}{1-h\|V_2\|_\infty}\sqrt{\rho_h^{k+1/2}(x)}.$$ Taking squares and using $$\frac{1}{(1-h\|V_2\|_\infty)^2} =1+2\|V_2\|_{L^\infty}h+\mathcal O(h^2)\leqslant 1+3\|V_2\|_{L^\infty}h
$$ for small $h$ gives the desired inequality.
For the lower bound , we first observe that since $F_2''\geqslant 0$ and from we have $F_2'(\rho^{k+1}_h)\leqslant F_2'((1+Ch)\rho^{k+1/2}_h)\leqslant F_2'(2M)$ if $h$ is small enough. Then gives inside ${\mathop{\rm supp}}\rho^{k+1}$ $$\begin{aligned}
\sqrt{\rho_h^{k+1}(x)} - \sqrt{\rho_h^{k+1/2}(x))}&=& -h\sqrt{\rho_h^{k+1}(x)} (F_2'(\rho_h^{k+1}(x)) +V_2(x))\\
&\geqslant & -h(F_2'(2M) +\|V_2\|_\infty)\sqrt{\rho_h^{k+1}(x)},\end{aligned}$$ hence $$\rho_h^{k+1}(x)\geqslant \frac{1}{(1+h(F_2'(2M) +\|V_2\|_\infty))^2}\rho_h^{k+1/2}(x)\geqslant (1-ch)\rho_h^{k+1/2}(x)$$ for small $h$.
Combining Lemma \[lem:maximum\_principle\_W\_step\] and Lemma \[lem encadrement\], we obtain at the continuous level
\[prop:maximum principle\] For all $T>0$ there exist constants $M_T,M_T'$ such that for all $t \in [0,T]$, $$\|\rho_h(t)\|_{L^1\cap L^\infty} , \|{\tilde{\rho}}_h(t)\|_{L^1\cap L^\infty} \leqslant M_T$$ and $$\|\rho_h(t)-{\tilde{\rho}}_h(t)\|_{L^1}\leqslant h M_T'$$ uniformly in $h\geqslant 0$.
Note from the second estimate that strong $L^1((0,T)\times \Omega)$ convergence of $\rho_h$ will immediately imply convergence of ${\tilde{\rho}}_h$ to the same limit.
By induction combining and , we obtain, for all $t \in [0,T]$, $$\|\rho_h(t)\|_{L^\infty} , \|{\tilde{\rho}}_h(t)\|_{L^\infty} \leqslant C_T,$$ where $C_T$ is a constant depending on $\| V_1 \|_{L^\infty}$, see [@PT lemma 2]. The $L^1$ bound is even easier: since the Wasserstein step is mass preserving, we can integrate in space to get $$\|\rho^{k+1}_h\|_{L^1}\leqslant (1+Ch)\|\rho_h^{k+1/2}\|_{L^1}=(1+Ch)\|\rho_h^{k+1}\|_{L^1}.$$ For $t\leqslant T\Leftrightarrow k\leqslant \lfloor T/h\rfloor$ the $L^{1}$ bounds immediately follow by induction, with $(1+Ch)^{\lfloor T/h\rfloor}\lesssim e^{CT}$. and we conclude again by induction.
In order to compare now $\rho_h$ and ${\tilde{\rho}}_h$, we take advantage of the above upper bound to write $\rho^{k+1/2}_h\leqslant M_T$ as long as $kh\leqslant T$. Taking $c=c(M_T)$ in and combining with , we have $$-ch \rho^{k+1/2}_h\leqslant \rho^{k+1/2}_h-\rho^{k+1}_h\leqslant Ch \rho^{k+1/2}_h\qquad \mbox{a.e.}$$ Integrating in $\Omega$ we conclude that $$\|\rho_h(t)-{\tilde{\rho}}_h(t)\|_1=\|\rho^{k+1}_h-\rho_h^{k+1/2}\|_1\leqslant h\max\{c,C\}\|\rho_h^{k+1/2}\|_1
\leqslant h\max\{c,C\}M_T=hM'_T$$ and the proof is complete.
Energy dissipation
------------------
Our goal is here to estimate the crossed dissipation along each elementary ${\mathtt{W}},{\mathtt{FR}}$ step.
Testing $\rho=\rho_h^k$ in the first Wasserstein step in , we get as usual $$\begin{aligned}
\label{sum telescopique MK 1}
\frac{1}{2h}{\mathtt{W}}^2(\rho_h^{k+1/2}, \rho_h^k) \leqslant {\mathcal{F}}_1(\rho_h^k)-{\mathcal{F}}_1(\rho_h^{k+1/2})+\int_\Omega V_1(\rho_h^{k}-\rho_h^{k+1/2}).\end{aligned}$$ Since $V_1$ is Globally Lipschitz we can first use standard methods from [@DFF; @L] to control $\int_\Omega V_1(\rho_h^{k}-\rho_h^{k+1/2})$ in terms of ${\mathtt{W}}^2(\rho_h^{k+1/2},\rho_h^{k})$, and suitably reabsorb in the left-hand side to obtain $$\begin{aligned}
\label{sum telescopique MK 2}
\frac{1}{4h}{\mathtt{W}}^2(\rho_h^{k+1/2}, \rho_h^k) \leqslant {\mathcal{F}}_1(\rho_h^k)-{\mathcal{F}}_1(\rho_h^{k+1/2})+C_Th.\end{aligned}$$
The dissipation of ${\mathcal{F}}_1$ along the Fisher-Rao step is controlled as
\[prop:decr F1\] For all $T>0$ there exists a constant $C_T>0$ such that, for all $k \geqslant 0$ and $k\leq \lfloor T/h\rfloor$, $$\label{décroissance F1}
{\mathcal{F}}_1(\rho_h^{k+1}) \leqslant {\mathcal{F}}_1(\rho_h^{k+1/2}) +C_Th.$$
We first treat the case of $F_1(z) = \frac{1}{m_1 -1}z^{m_1}$ with $m_1>1$. Since $F_1$ is increasing, we use to obtain $$\begin{aligned}
{\mathcal{F}}_1(\rho_h^{k+1}) - {\mathcal{F}}_1(\rho_h^{k+1/2}) & \leqslant & \frac{( (1+Ch)^{m_1} -1)}{m_1 -1}\int_\Omega (\rho_h^{k+1/2})^{m_1}\\
&\leqslant & Ch \|\rho^{k+1/2}\|_{L^\infty}^{m_1-1}\,\|\rho^{k+1/2}\|_{L^1},\end{aligned}$$ and we conclude from Proposition \[prop:maximum principle\].
In the second case $F_1(z) = z\log(z)-z$, we have $${\mathcal{F}}_1(\rho_h^{k+1}) = \int_{\{\rho_h^{k+1} \leqslant e^{-1}\}} \rho_h^{k+1}\log(\rho_h^{n+1}) +\int_{\{\rho_h^{k+1} \geqslant e^{-1}\}} \rho_h^{k+1}\log(\rho_h^{k+1}) -\int_\Omega \rho_h^{k+1}.$$ Note from Proposition \[prop:maximum principle\] that the $z$ contribution in $F_1(z)=z\log z-z$ is immediately controlled by $|\int \rho_h^{k+1}-\int\rho^{k+1/2}_h|\leqslant \|\rho^{k+1}_h-\rho_h^{k+1/2}\|_{L^1}\leqslant hM_T'$, so we only have to estimate the $z\log z$ contribution. Since $z\mapsto z\log z$ is increasing on $\{z\geqslant e^{-1}\}$ and using , the second term in the right hand side becomes
$$\begin{aligned}
\int_{\{\rho_h^{k+1} \geqslant e^{-1}\}} \rho_h^{k+1}\log(\rho_h^{k+1}) &\leqslant & \int_{\{\rho_h^{k+1} \geqslant e^{-1}\}} (1+Ch)\rho_h^{k+1/2}\log((1+Ch)\rho_h^{k+1/2})\\
&\leqslant & \int_{\{\rho_h^{k+1} \geqslant e^{-1}\}} \rho_h^{k+1/2}\log(\rho_h^{k+1/2})
+ Ch\int_{\{\rho_h^{k+1} \geqslant e^{-1}\}} \rho_h^{k+1/2}\log(\rho_h^{k+1/2}) \\
& & \hspace{1.5cm} + (1+Ch)\int_{\{\rho_h^{k+1}\geqslant e^{-1}\}}\rho_h^{k+1/2}\log(1+Ch)\\
&\leqslant & \int_{\{\rho_h^{k+1} \geqslant e^{-1}\}} \rho_h^{k+1/2}\log(\rho_h^{k+1/2}) +C_Th,\end{aligned}$$
where we used $\|\rho_h^{k+1/2}\|_{L^1} \leqslant M_T$ from Proposition \[prop:maximum principle\] as well as $\log(1+Ch) \leqslant Ch$ in the last inequality. Using the same method with the bound from below on $\{\rho^{k+1}_h\leqslant e^{-1}\}$ (where $z\mapsto z\log z$ is now decreasing), we obtain similarly $$\int_{\{\rho_h^{k+1} \leqslant e^{-1}\}} \rho_h^{k+1}\log(\rho_h^{k+1}) \leqslant \int_{\{\rho_h^{k+1} \leqslant e^{-1}\}} \rho_h^{k+1/2}\log(\rho_h^{k+1/2}) +C_Th.$$ Combining both inequalities gives $$\int_\Omega \rho_h^{k+1}\log(\rho_h^{k+1}) \leqslant \int_\Omega \rho_h^{k+1/2}\log(\rho_h^{k+1/2}) +C_Th$$ and the proof is complete.
Summing and over $k$ we obtain $$\label{eq:sum telescopic MK}
\frac{1}{2h}\sum_{k=0}^{N-1}{\mathtt{W}}^2(\rho_h^{k+1/2}, \rho_h^k) \leqslant {\mathcal{F}}_1(\rho_0)-{\mathcal{F}}_1(\rho_h^{N})+C_T,$$ where $N=\lfloor \frac{T}{h}\rfloor$.\
In the above estimate we just controlled the dissipation of ${\mathcal{F}}_1$ along the ${\mathtt{FR}}$/reaction steps, and the goal is now to similarly estimate the dissipation of ${\mathcal{F}}_2$ along the Wasserstein step. Testing $\rho=\rho_h^{k+1/2}$ in the second Fisher-Rao step in , we obtain $$\begin{aligned}
\label{sum telescopique FR 1}
\frac{1}{2h}{\mathtt{FR}}_2(\rho_h^{k+1}, \rho_h^{k+1/2}) \leqslant {\mathcal{F}}_2(\rho_h^{k+1/2})-{\mathcal{F}}_2(\rho_h^{k+1})+\int_\Omega V_2(\rho_h^{k+1/2}-\rho_h^{k+1}).\end{aligned}$$
Since we assumed $V_2 \in L^\infty(\Omega)$ and because $\rho_h(t)=\rho^{k+1}_h$ remains close to ${\tilde{\rho}}_h(t)=\rho_h^{k+1/2}$ in $L^1$ uniformly in $t,h$ by Proposition \[prop:maximum principle\], we immediately control the potential part as $$\label{sum telescopique FR 2}
\int_\Omega V_2(\rho_h^{k+1/2}-\rho_h^{k+1}) \leqslant \|V_2\|_\infty C_Th.$$ For the internal energy we argue exactly as in the proof Proposition \[prop:decr F1\] (for the Porous Media part, since we chose here $F_2(z)=\frac{1}{m_2-1}z^{m_2}$), and obtain $$\label{décroissance F2}
{\mathcal{F}}_2(\rho_h^{k+1/2})-{\mathcal{F}}_2(\rho_h^{k+1}) \leqslant C_Th.$$
Combining , and , we immediately deduce that $$\begin{aligned}
\label{sum telescopic FR}
\frac{1}{2h}\sum_{k=0}^{N-1}{\mathtt{FR}}^2(\rho_h^{k+1/2}, \rho_h^{k+1}) \leqslant C_T,\end{aligned}$$ where $N=\lfloor \frac{T}{h}\rfloor$ as before.\
Finally, we recover an approximate compactness in time in the form
\[prop:1/2\_holder\] There exists a constant $C_T>0$ such that for all $h$ small enough and $k\leqslant N=\lfloor T/h\rfloor$, $$\label{estimate sum KFR}
\frac{1}{h}\sum_{k=0}^{N-1} {\mathtt{WFR}}^2(\rho_h^k,\rho_h^{k+1}) \leqslant 4{\mathcal{F}}_1(\rho_0) +C_T.$$
Adding and gives $$\frac{1}{h}\sum_{k=0}^{N-1} {\mathtt{W}}^2(\rho_h^k,\rho_h^{k+1/2})+{\mathtt{FR}}^2(\rho_h^{k+1/2},\rho_h^{k+1}) \leqslant 2\left( {\mathcal{F}}_1(\rho_0)-{\mathcal{F}}_1(\rho_h^{N})+C_T\right )+2C_T\leqslant 2{\mathcal{F}}_1(\rho_0) +C_T,$$ since in any case $F_1(z)=\frac{1}{m_1-1}z^{m_1}\geqslant 0$ and $F_{1}(z)=z\log z-z\geqslant -1$ is bounded from below on the bounded domain $\Omega$, hence ${\mathcal{F}}_1(\rho_h^{N})\geqslant -C_{\Omega}$ uniformly. It then follows from Proposition \[prop:comparison\_d\_W\_H\] that ${\mathtt{W}}^2(\rho_h^k,\rho_h^{k+1/2})+{\mathtt{FR}}^2(\rho_h^{k+1/2},\rho_h^{k+1})\geqslant \frac{1}{2}{\mathtt{WFR}}^2 \rho_h^k,\rho_h^{k+1}$ in the left-hand side, and the result immediately follows.
Estimates and convergences
--------------------------
From the total-square distance estimate we recover as usual the approximate $\frac{1}{2}$-Hölder estimate $$\begin{aligned}
\label{estimate holder}
{\mathtt{WFR}}(\rho_h(t),\rho_h(s))+{\mathtt{WFR}}({\tilde{\rho}}_h(t),{\tilde{\rho}}_h(s)) \leqslant C_T|t-s+h|^{1/2}\end{aligned}$$ for all fixed $T>0$ and $t,s\in[0,T]$. From and Proposition \[prop:comparison\_d\_W\_H\] we have moreover $$\begin{aligned}
\label{estimate diff}
{\mathtt{WFR}}(\rho_h(t),{\tilde{\rho}}_h(t)) \leqslant {\mathtt{FR}}(\rho_h(t),{\tilde{\rho}}_h(t)) \leqslant C\sqrt{h}.\end{aligned}$$ Using a refined version of Ascoli-Arzelà theorem, [@AGS prop. 3.3.1] and arguing exactly as in [@GM prop. 4.1], we see that for all $T>0$ and up to extraction of a discrete subsequence, $\rho_h$ and ${\tilde{\rho}}_h$ converge uniformly to the same ${\mathtt{WFR}}$-continuous curve $\rho \in \mathcal{C}^{1/2}([0,T], {\mathcal{M}}^+_{{\mathtt{WFR}}})$ as $$\sup_{t\in [0,T]} ({\mathtt{WFR}}(\rho_h(t),\rho(t))+ {\mathtt{WFR}}({\tilde{\rho}}_h(t),\rho(t)) ) \rightarrow 0.$$
In order to pass to the limit in the nonlinear terms, we first strengthen this ${\mathtt{WFR}}$-convergence into a more tractable $L^1$ convergence. The first step is to retrieve compactness in space:
For all $T>0$, $\rho_h$ and ${\tilde{\rho}}_h$ satisfies $$\label{estimate P1}
\|P_1({\tilde{\rho}}_h)\|_{L^2([0,T];H^1(\Omega))} \leqslant C_T.$$
From and the $L^1\cap L^\infty$ bounds from Proposition \[prop:maximum principle\] we see that $$\begin{aligned}
\int_\Omega |\nabla P_1(\rho_h^{k+1/2})|^2
&\leqslant & \frac{1}{2h^2}\int_\Omega|\nabla \varphi_h^{k+1/2}|^2 (\rho_h^{k+1/2})^2 + \frac{1}{2} \int_\Omega|\nabla V_1|^2(\rho_h^{k+1/2})^2\\
&\leqslant & \frac{C_T}{2h^2}\int_\Omega|\nabla \varphi_h^{k+1/2}|^2 \rho_h^{k+1/2} + \frac{1}{2} \|\nabla V_1\|_{\infty}^2\int _\Omega(\rho_h^{k+1/2})^2\\
&\leqslant & C_T\left(\frac{{\mathtt{W}}^2(\rho_h^{k+1/2},\rho_h^{k})}{h^2} + 1\right)\end{aligned}$$ since $\varphi^{k+1/2}_h$ is the optimal (backward) Kantorovich potential from $\rho_h^{k+1/2}$ to $\rho_h^{k}$. Multiplying by $h>0$, summing over $k$, and exploiting gives $$\|P_1({\tilde{\rho}}_h)\|^2_{L^2([0,T];H^1(\Omega))}\leqslant \sum _{k=0}^{N-1}h\|P_1(\rho^{k+1/2}_h)\|^2_{H^1}
\leqslant C_T ({\mathcal{F}}_1(\rho_0)-{\mathcal{F}}_1(\rho_h^{N})+1)\leqslant C_T,$$ where we used as before ${\mathcal{F}}_1(\rho_h^N)\geqslant -C_{\Omega}$ in the last inequality.
We are now in position of proving our main result:
Exploiting and , we can apply the extension of the Aubin-Lions lemma established by Rossi and Savaré in [@RS] to obtain that ${\tilde{\rho}}_h$ converges to $\rho$ strongly in $L^1(Q_T)$ (see [@L]). By diagonal extraction if needed, we can assume that the convergence holds in $L^1(Q_T)$ for all fixed $T>0$. Then by Proposition \[prop:maximum principle\] we have
$$\|\rho_h - \rho \|_{L^1(Q_T)} \leqslant \|\rho_h - {\tilde{\rho}}_h \|_{L^1(Q_T}+\|{\tilde{\rho}}_h - \rho \|_{L^1(Q_T)}\leqslant C_T h +\|{\tilde{\rho}}_h - \rho \|_{L^1(Q_T)}\to 0$$ hence $\rho_h\to \rho$ as well.
Moreover, since $P_1({\tilde{\rho}}_h)$ is bounded in $L^2((0,T),H^1(\Omega))$ we can assume that $\nabla P_1({\tilde{\rho}}_h) \rightharpoonup \nabla P_1(\rho)$ in $L^2((0,T),H^1(\Omega))$ for all $T>0$. Exploiting the Euler-Lagrange equations and arguing exactly as in [@GM Theorem 4], it is easy to pass to the limit to conclude that $$\int_\Omega \rho(t_2)\varphi -\rho(t_1)\varphi=-\int_{t_1}^{t_2}\int_\Omega \Big\{
\nabla P(\rho)\cdot \nabla\varphi
+\rho \nabla V_1\cdot \nabla\varphi
-\rho(F'_2(\rho)+V_2)\varphi\Big\}$$ for all $0<t_1<t_2$ and $\varphi\in \mathcal C^1_b(\Omega)$. Since $\rho\in\mathcal C([0,T];{\mathcal{M}}^+_{{\mathtt{WFR}}})$ takes the initial datum $\rho(0)=\rho_0$ and ${\mathtt{WFR}}$ metrizes the narrow convergence of measures, this is well-known to be equivalent to our weak formulation in Definition \[def:weak\_solutions\], and the proof is complete.
In the above proofs one can check that Theorem \[theo:existence differente energies\] extends in fact to all ${\mathcal{C}}^1$ nonlinearities $F_2$ such that $F_2' \geqslant C$ for some $C \in \operatorname*{\mathbb{R}}$. Likewise, we stated and proved our main result in bounded domains for convenience: all the above arguments immediately extend to $\Omega=\operatorname*{\mathbb{R}}^d$ at least for $F_1(z)=\frac{1}{m_1-1}z^{m_1}\geqslant 0$. The only place where we actually used the boundedness of $\Omega$ was in the proof of Proposition \[prop:1/2\_holder\], when we bounded from below $\mathcal F_1(\rho^N_h)\geqslant -C_\Omega$ in order to retrieve the total-square distance estimate. When $\Omega=\operatorname*{\mathbb{R}}^d$ and $F_1(z)=z\log z-z$ a lower bound $\mathcal F_1(\rho^N_h)\geqslant -C_T$ still holds, but the proof requires a tedious control of the second moments $\mathfrak m_2(\rho)=\int_{\operatorname*{\mathbb{R}}^d}|x|^2\rho$ hence we did not address this technical issue for the sake of brevity.
Application to systems {#section:systems}
======================
In this section we shall try to illustrate that the previous scheme is very tractable and allows to solve systems of the form
$$\begin{aligned}
\label{eq:KFR-general system}
\left\{\begin{array}{l}
\partial_t \rho_1 = \operatorname*{div}(\rho_1 \nabla(F_1'(\rho_1)+ V_1[\rho_1,\rho_2]))-\rho_1(G_1'(\rho_1) +U_1[\rho_1,\rho_2]),\\
\partial_t \rho_2 = \operatorname*{div}(\rho_2 \nabla(F_2'(\rho_2)+ V_2[\rho_1,\rho_2]))-\rho_2 (G_2'(\rho_2) +U_2[\rho_1,\rho_2]),\\
{\rho_1}_{|t=0}=\rho_{1,0}, \, {\rho_2}_{|t=0}=\rho_{2,0}.
\end{array}\right.\end{aligned}$$
For simplicity we assume again that $\Omega$ is a smooth, bounded subset of ${\mathbb{R}^d}$. Then the system is endowed with Neumann boundary conditions, $$\rho_1 \nabla(F_1'(\rho_1)+ V_1[\rho_1,\rho_2]) \cdot \nu =0 \text{ and } \rho_2 \nabla(F_2'(\rho_2)+ V_2[\rho_1,\rho_2]) \cdot \nu=0 \qquad \text{ on } \mathbb{R}^+ \times \partial \Omega,$$ where $\nu$ is the outward unit normal to $\partial \Omega$. In system of the form , we allow interactions between densities in the potential terms $V_i[\rho_1,\rho_2]$ and $U_i[\rho_1,\rho_2]$. In the mass-conservative case (without reaction terms), this system has already been studied in [@DFF; @L; @CL1], using a semi-implicit JKO scheme introduced by Di Francesco and Fagioli, [@DFF]. This section combines the splitting scheme introduced in the previous section and semi-implicit schemes both for the Wasserstein JKO step and for the Fisher-Rao JKO step.
For the ease of exposition we keep the same assumptions for $F_i$ and $G_i$ as in the previous section, i.e the diffusion terms $F_i$ satisfy and the reaction terms $G_i$ satisfy . Moreover, since the potentials depend now on the densities $\rho_1$ and $\rho_2$, we need stronger hypotheses: we assume that $V_i \, :\, L^1(\Omega;\operatorname*{\mathbb{R}}^+)^2 \rightarrow \mathcal{C}^{1}(\Omega)$ are continuous and verify, uniformly in $\rho_1,\rho_2 \in L^1(\Omega;\operatorname*{\mathbb{R}}^+)$, $$\begin{aligned}
\label{assumption V system}
\begin{array}{c}
\|V_i[\rho_1,\rho_2]\|_{W^{1,\infty(\Omega)}} \leqslant K (1+ \|\rho_1\|_{L^1(\Omega)} +\|\rho_2\|_{L^1(\Omega)}),\\
\|\nabla( V_i[\rho_1,\rho_2]) - \nabla (V_i[\mu_1,\mu_2]) \|_{L^\infty(\Omega)} \leqslant K(\|\rho_1 -\mu_1\|_{L^1(\Omega)} +\|\rho_2 -\mu_2\|_{L^1(\Omega)}).
\end{array}\end{aligned}$$
The interacting potentials we have in mind are of the form $V_i[\rho_1,\rho_2]= K_{i,1} \ast \rho_1 + K_{i,2}\ast \rho_2$, where $K_{i,1},K_{i,2} \in W^{1,\infty}(\Omega)$ and then $V_i$ satisfies . For the reaction, we assume that the potentials $U_i$ are continuous from $L^1(\Omega)_+^2$ to $L^1$ with moreover
$$\label{assumption U}
U_i[\rho_1,\rho_2] \geqslant -K, \qquad \forall\,\rho_1,\rho_2 \in L^1(\Omega;{\operatorname*{\mathbb{R}}}^+)$$
for some $K\in\operatorname*{\mathbb{R}}$, and $$\label{assumption U max}
\| U_i[\rho_1,\rho_2] \|_{L^\infty(\Omega)} \leqslant K_M,
\qquad
\forall \|\rho_1\|_{L^1(\Omega)},\|\rho_2\|_{L^1(\Omega)} \leqslant M$$ for some nondecreasimg function $K_M\geqslant 0$ of $M$. The examples we have in mind are of the form $$U_1[\rho_1,\rho_2]=C_1\frac{\rho_2}{1+\rho_1},
\quad
U_2[\rho_1,\rho_2]=-C_2\frac{\rho_1}{1+\rho_1}$$ for some constants $C_i\geq 0$, or nonlocal reactions $$U_i[\rho_1, \rho_2](x) = \int_\Omega K_{i,1}(x,y) \rho_1(y) \,dy + \int_\Omega K_{i,2}(x,y)\rho_2(y) \,dy$$ for some nonnegative kernels $K_{i,j}\in L^1\cap L^\infty$. Such reaction models appear for example in biological adaptive dynamics [@Per].
\[def:weak\_solutions system\] We say that $(\rho_1,\rho_2)\, : \, \operatorname*{\mathbb{R}}^+ \rightarrow L^1_+ \cap L^\infty_+(\Omega)$ is a weak solution of if, for $i\in \{1,2\}$ and all $T< +\infty$, the pressure $P_i(\rho_i):=\rho_i F_i'(\rho_i)-F_i(\rho_i)$ satisfies $\nabla P_i(\rho_i) \in L^2([0,T] \times \Omega)$, and $$\begin{gathered}
\int_0^{+\infty}\left( \int_\Omega (\rho \partial_t \phi_i -\rho_i\nabla V_i[\rho_1,\rho_2] \cdot \nabla \phi_i - \nabla P_i(\rho_i) \cdot \nabla \phi_i - \rho_i (G_i'(\rho_i) +U_i[\rho_1,\rho_2]) \phi_i) \,dx \right)\, dt\\
= -\int_\Omega \phi_i(0,x) \rho_{i,0}(x) \,dx,\end{gathered}$$ for all $\phi_i \in {\mathcal{C}}^\infty_c([0,+\infty) \times {\mathbb{R}^d})$.
Then, the following result holds,
\[theo:existence system\] Assume that $\rho_{1,0},\rho_{2,0} \in L^1\cap L^\infty_+(\Omega)$ and that $V_i,U_i$ satisfy . Then admits at least one weak solution.
Note that this result can be easily adapted to systems with an arbitrary number of species $N \geqslant 2$, coupled by nonlocal terms $V_i[\rho_1,\dots,\rho_N]$ and $U_i[\rho_1,\dots,\rho_N]$.
A refined analysis shows that our approach would allow to handle systems of the form $$\left\{\begin{array}{l}
\partial_t \rho_1 - \operatorname*{div}(\rho_1 \nabla(F_1'(\rho_1)+ V_1))=-\rho_1 h_1(\rho_1,\rho_2),\\
\partial_t \rho_2 - \operatorname*{div}(\rho_2 \nabla(F_2'(\rho_2)+ V_2))=+\rho_2 h_2(\rho_1),
\end{array}\right.$$ where $h_1$ is a nonnegative continuous function and $h_2$ is a continuous functions.
Indeed since $h_1\geq 0$ the reaction term is the first equation is nonpositive, hence $
\|\rho_1(t) \|_{L^\infty(\Omega)} \leqslant C_T$. Then it follows that $-h_2(\rho_1)$ satisfies assumptions and . A classical example is $ h_2(\rho_1)= \rho_1^\alpha$ and $h_1(\rho_1,\rho_2)= \rho_1^{\alpha-1} \rho_2$, where $\alpha \geqslant 1$, see for example [@MPsurvey] for more discussions.
As already mentioned, the proof of theorem \[theo:existence system\] is based on a semi-implicit splitting scheme. More precisely, we construct four sequences $\rho_{1,h}^{k+1/2},\rho_{1,h}^{k+1},\rho_{2,h}^{k+1/2},\rho_{2,h}^{k+1}$ defined recursively as $$\begin{aligned}
\label{eq:splitting scheme Wass-FR syst}
\left\{\begin{array}{l}
\rho_{i,h}^{k+1/2} \in \operatorname*{argmin}\limits_{\rho \in {\mathcal{M}}^+, |\rho|=|\rho_{i,h}^k|} \left\{ \frac{1}{2h} {\mathtt{W}}^2(\rho, \rho_{i,h}^k) + {\mathcal{F}}_i(\rho) +{\mathcal{V}}_i(\rho | \rho_{1,h}^k, \rho_{2,h}^k) \right\}\\
\\
\rho_{i,h}^{k+1} \in \operatorname*{argmin}\limits_{\rho \in {\mathcal{M}}^+} \left\{ \frac{1}{2h} {\mathtt{FR}}^2(\rho, \rho_{i,h}^{k+1/2}) + {\mathcal{G}}_i(\rho) +\mathcal{U}_i(\rho | \rho_{1,h}^k, \rho_{2,h}^k) \right\}
\end{array}\right.,\end{aligned}$$ where the fully implicit terms $${\mathcal{F}}_i(\rho):= \left\{ \begin{array}{ll}
\int_\Omega F_i(\rho) & \text{ if } \rho \ll \mathcal{L}_{|\Omega}\\
+\infty & \text{ otherwise }
\end{array}\right.
\quad\text{and}\quad
{\mathcal{G}}_i(\rho):= \left\{ \begin{array}{ll}
\int_\Omega G_i(\rho) & \text{ if } \rho \ll \mathcal{L}_{|\Omega}\\
+\infty & \text{ otherwise }
\end{array}\right.,$$ and the semi-implicit terms $${\mathcal{V}}_i(\rho | \mu_1, \mu_2):= \int_\Omega V_i[\mu_1,\mu_2] \rho
\quad\text{and}\quad
\mathcal{U}_i(\rho | \mu_1, \mu_2):= \int_\Omega U_i[\mu_1,\mu_2] \rho.$$
In the previous section, the proof of theorem \[theo:existence differente energies\] for scalar equations strongly leveraged the uniform $L^\infty(\Omega)$-bounds on the discrete solutions. Here an additional difficulty arises due to the nonlocal terms $\nabla V_i[\rho_1,\rho_2]$ and $U_i[\rho_1,\rho_2]$, which are a priori not uniformly bounded in $L^\infty(\Omega)$. Using assumption we will first obtain a uniform $L^1(\Omega)$-bound on $\rho_1,\rho_2$, and then extend proposition \[prop:maximum principle\] to the system . This in turn will give a uniform $W^{1,\infty}$ control on $V_i[\rho_1,\rho_2]$ and $L^\infty$ control on $U_i[\rho_1,\rho_2]$ through our assumptions --, which will finally allow to argue as in the previous section and give $L^\infty$ control on $\rho_1,\rho_2$.
Numerical simulations for a diffusive prey-predator system are presented at the end of this section.
Properties of discrete solutions
--------------------------------
Arguing as in the case of one equation, the optimality conditions for the Wasserstein step and for the Fisher-Rao step first give
\[lem:L1 bound\] For all $k \geqslant 0$ and $i\in \{1,2\}$, we have $$\label{eq:PMAX_W system}
\|\rho_{i,h}^{k+1/2} \|_{L^1} = \|\rho_{i,h}^k\|_{L^1}.$$ Moreover, there exists $C_i\equiv C(U_i) >0$ (uniform in $k$) such that $$\label{encadrement FR_sup system}
\rho_{i,h}^{k+1}(x) \leqslant (1+ C_i h) \rho_{i,h}^{k+1/2}(x) \qquad a.e.$$
The first part is simply the mass conservation in the Wasserstein step, and the second part follows the lines of the proof of in Lemma \[lem encadrement\] using assumption .
As a direct consequence we have uniform control on the $L^1$-norms:
\[lem:L1\_bound\_systems\] For all $T>0$ there exist constants $C_T,C_T'>0$ such that, for all $t\in [0,T]$, $$\| \rho_{i,h}(t)\|_{L^1}, \| {\tilde{\rho}}_{i,h}(t)\|_{L^1} \leqslant C_T$$ and $$\label{encadrement potential V system}
\| V_i[\rho_{1,h}(t), \rho_{2,h}(t)] \|_{W^{1,\infty}},\| V_i[{\tilde{\rho}}_{1,h}(t), {\tilde{\rho}}_{2,h}(t)] \|_{W^{1,\infty}} \leqslant C'_T.$$
Integrating and iterating with , we obtain for all $t\leqslant T$ and $k\leqslant \lfloor T/h\rfloor$ $$\| \rho_{i,h}^{k+1} \|_{L^1} \leqslant (1+C_ih)\| \rho_{i,h}^{k} \|_{L^1} \leqslant (1+C_ih)^k\| \rho_{i,0}\|_{L^1} \leqslant e^{C_iT}\| \rho_{i,0}\|_{L^1}
.$$ Then follows from our assumption on the interactions.
Combining and , we deduce
\[prop:encadrment L1\] For all $T>0$, there exists $M_T$ such that for all $t\in [0,T]$, $$\| \rho_{i,h}(t)\|_{L^\infty}, \| {\tilde{\rho}}_{i,h}(t)\|_{L^\infty} \leqslant M_T.$$ Then, there exists $c_i \equiv c(M_T,U_i) \geq 0$, such that, for all $k \leqslant \lfloor T/h\rfloor$ and $h \leqslant h_0(U_1,U_2)$, $$(1-c_ih) \rho_{i,h}^{k+1/2} \leqslant \rho_{i,h}^{k+1}.$$ In particular, there exist $M_T'>0$ such that for all $t\in [0,T]$, $$\| \rho_{i,h}(t) - {\tilde{\rho}}_{i,h}(t) \|_{L^1} \leqslant hM_T'.$$
The first $L^\infty$ estimate can be found in [@PT Lemma 2], and the rest of our statement can be proved exactly as in Lemma \[lem encadrement\] and Proposition \[prop:maximum principle\].
Estimates and convergences
--------------------------
Since we proved that $V_1[\rho_{1,h},\rho_{2,h}]$ and $V_2[\rho_{1,h},\rho_{2,h}]$ are bounded in $L^\infty([0,T],W^{1,\infty}(\Omega))$, we can argue exactly as in the previous section for the Wasserstein step and obtain $$\begin{aligned}
\label{eq:sum telescopic MK system}
\frac{1}{4h}{\mathtt{W}}^2(\rho_{i,h}^{k+1/2}, \rho_{i,h}^k) \leqslant {\mathcal{F}}_i(\rho_{i,h}^k)-{\mathcal{F}}_i(\rho_{i,h}^{k+1/2})+C_Th,\end{aligned}$$ see - for details. Since ${\tilde{\rho}}_{1,h}$ and ${\tilde{\rho}}_{2,h}$ are uniformly bounded in $L^1(\Omega)$ (Lemma \[lem:L1\_bound\_systems\]), our assumption ensures that $U_1[\rho_{1,h}^{k+1/2},\rho_{2,h}^{k+1/2}]$ and $U_2[\rho_{1,h}^{k+1/2},\rho_{2,h}^{k+1/2}]$ are uniformly bounded in $L^\infty(\Omega)$. Proposition \[prop:encadrment L1\] then allows to argue exactly as in -- for the Fisher-Rao step, and we get $$\begin{aligned}
\label{eq:sum telescopic FR system}
\frac{1}{2h}{\mathtt{FR}}^2(\rho_h^{k+1}, \rho_h^{k+1/2}) \leqslant {\mathcal{G}}_i(\rho_{i,h}^{k+1/2})-{\mathcal{G}}_i(\rho_{i,h}^{k+1}) +C_Th.\end{aligned}$$
The dissipation of ${\mathcal{F}}_i$ along the Fisher-Rao step is obtained in the same way as Proposition \[prop:decr F1\] and we omit the details:
For all $T>0$ and $i\in\{1,2\}$, there exist constants $C_T,C'_T>0$ such that, for all $k\geqslant 0$ with $hk \leqslant T$, $$\begin{array}{c}
{\mathcal{F}}_i(\rho_{i,h}^{k+1}) \leqslant {\mathcal{F}}_i(\rho_{i,h}^{k+1/2}) +C_Th,\\
{\mathcal{G}}_i(\rho_{i,h}^{k+1/2}) \leqslant {\mathcal{G}}_i(\rho_{i,h}^{k+1}) +C'_Th.
\end{array}$$
From and this immediately gives a telescopic sum $$\frac{1}{2h}\left({\mathtt{W}}^2(\rho_{i,h}^{k}, \rho_{i,h}^{k+1/2})+ {\mathtt{FR}}^2(\rho_h^{k+1/2}, \rho_h^{k})\right)
\leqslant 2[{\mathcal{F}}_i(\rho_{i,h}^k)-{\mathcal{F}}_i(\rho_{i,h}^{k+1})] + C_Th$$ which in turn yields an approximate $\frac 12$-Hölder estimate (with respect to the ${\mathtt{WFR}}$ distance) as in Proposition \[prop:1/2\_holder\]. The rest of the proof of Theorem \[theo:existence system\] is then identical to section \[sec:KFRsplitting\] and we omit the details.
Numerical application: prey-predator systems
--------------------------------------------
Our constructive scheme can be implemented numerically, by simply discretizing in space. We use the augmented Lagrangian method ALG-JKO from [@BCL] to solve the Wasserstein step, and the Fisher-Rao step is just a convex pointwise minimization problem. Indeed, it is known [@GM; @LMS] that ${\mathtt{FR}}^2(\rho,\mu)=4\|\sqrt{\rho}-\sqrt{\mu}\|^2_{L^2}$, hence the Fisher-Rao step in is a mere convex pointwise minimization problem of the form: for all $x \in \Omega$ (and omitting all indexes $\rho_{i,h}$), $$\label{eq:pointwise_convex_FR}
\rho^{k+1}(x) = \operatorname*{argmin}\limits_{\rho \geq 0} \left\{ 4\left| \sqrt{\rho} - \sqrt{\rho^{k+1/2}(x)} \right|^2 + 2h F(\rho)\right\}.$$ This is easily solved using any simple Newton procedure.
Figure shows the numerical solution of the following diffusive prey-predator system $$\left\{\begin{array}{l}
\partial_t \rho_1 - \Delta \rho_1 - \operatorname*{div}(\rho_1 \nabla V_1[\rho_1,\rho_2])=A\rho_1\left( 1-\rho_1 \right) - B\frac{\rho_1 \rho_2}{1+\rho_1},\\
\partial_t \rho_2 - \Delta \rho_2 - \operatorname*{div}(\rho_2 \nabla V_2[\rho_1,\rho_2])=\frac{B\rho_1 \rho_2}{1+\rho_1}-C\rho_2,
\end{array}\right..$$ Here the $\rho_1$ species are preys and $\rho_2$ are predators, see for example [@M], the parameters $A=10,C=5,B=70$, and the interactions are chosen as $$V_{1}[\rho_1,\rho_2]=|x|^2 \ast \rho_1 - |x|^2 \ast \rho_2,
\quad
V_2[\rho_1,\rho_2]=|x|^2 \ast \rho_1 + |x|^2 \ast \rho_2.$$ In this corresponds to $$G_1(\rho_1) =A\frac{\rho_1^2}{2}, \quad G_2(\rho_2) =0,
\quad
U_1[\rho_1,\rho_2]=\frac{B\rho_2}{1+\rho_1}-A,
\quad
U_2[\rho_1,\rho_2]=-\frac{B\rho_1}{1+\rho_1}+C.$$ Of course, $U_1$ and $U_2$ satisfy assumptions and , and then Theorem \[theo:existence system\] gives a solution of the prey-predator system. As before, we shall disregard the uniqueness issue for the sake of simplicity. Figure depicts the mass evolution of the prey and predator species: we observe the usual oscillations in time with phase opposition, a characteristic behaviour for Lotka-Volterra types of systems.
------- ---------- ---------- --------- ---------- ---------- ------- --
$t=0$ $t=0.15$ $t=0.35$ $t=0.5$ $t=0.65$ $t=0.85$ $t=1$
------- ---------- ---------- --------- ---------- ---------- ------- --
Application to a tumor growth model with very degenerate enery {#part4-section2HS}
==============================================================
In this section we take interest in the equation $$\begin{aligned}
\label{equation HS}
\left\{\begin{array}{l}
\partial_t \rho =\operatorname*{div}( \rho \nabla p)+ \rho (1-p),\\
p \geqslant 0\quad\mbox{and}\quad p(1-\rho)=0\\
0\leqslant \rho \leqslant 1,\\
\rho_{|t=0}=\rho_0.
\end{array} \right.\end{aligned}$$ This equation is motivated by tumor growth models [@PQV; @PTV] and exhibits a Hele-Shaw patch dynamics: if $\rho_0=\chi_{\Omega_0}$ then the solution remains an indicator $\rho(t)=\chi_{\Omega(t)}$ and the boundary moves with normal velocity $V=-\nabla p|_{\partial\Omega(t)}$, see [@alexander2014quasi] for a rigorous analysis in the framework of viscosity solutions.
At least formally, we remark that is the Wasserstein-Fisher-Rao gradient flow of the singular functional $${\mathcal{F}}(\rho) :={\mathcal{F}}_\infty(\rho) -\int_\Omega \rho,$$ where $${\mathcal{F}}_\infty(\rho):= \left\{\begin{array}{ll}
0 & \text{ if } \rho \leqslant 1 \,\mbox{ a.e},\\
+\infty & \text{ otherwise.}
\end{array}\right.$$ Indeed, the compatibility conditions $p\geqslant 0$ and $p(1-\rho)=0$ in really mean that the pressure $p$ belongs to the subdifferential $\partial {\mathcal{F}}_\infty(\rho)$, and thus reads as the gradient flow $$\partial_t\rho =\operatorname*{div}(\rho \nabla u)-\rho u,\qquad u=p-1\in -\partial {\mathcal{F}}(\rho).$$ However, this functional is too singular for the previous splitting scheme to correctly capture the very degenerate diffusion. Indeed, the naive and direct approach from section \[sec:KFRsplitting\] would lead to $$\left\{\begin{array}{l}
\rho_h^{k+1/2} \in \operatorname*{argmin}\limits_{\rho \leqslant1, \, |\rho|=|\rho_h^k|}\left\{ \frac{1}{2h} {\mathtt{W}}^2(\rho, \rho_{h}^k) -\int_\Omega \rho \right\},\\
\\
\rho_h^{k+1} \in \operatorname*{argmin}\limits_{\rho \leqslant 1} \left\{ \frac{1}{2h} {\mathtt{FR}}^2(\rho, \rho_h^{k+1/2}) -\int_{\Omega} \rho \right\}.
\end{array}\right.$$ Since the Wasserstein step is mass-conservative by definition, the $\int\rho$ term has no effect in the first step and the latter reads as “project $\rho_{h}^k$ on $\{\rho\leqslant 1\}$ w.r.t to the ${\mathtt{W}}$ distance”. Since the output of the reaction step $\rho^{k+1}_h\leqslant 1$, the Wasserstein step will never actually project anything, and the diffusion is completly shut down. As an example, it is easy to see that if the initial datum is an indicator $\rho_0=\chi_{\Omega_0}$ then the above naive scheme leads to a stationary solution $\rho^{k+1}_h=\rho^{k+1/2}_h=\rho_0$ for all $k\geqslant 0$, while the real solution should evolve according to the aforementioned Hele-Shaw dynamics $\rho(t)=\chi_{\Omega(t)}$ [@alexander2014quasi; @PQV]. One could otherwise try to write a semi-implicit scheme as follows: 1) keep the projection on $\{\rho\leqslant 1 \}$ in the first Wasserstein step. As in [@MRCSV] a pressure term $p^{k+1/2}_h$ appears as a Lagrange multiplier in the Wasserstein projection. 2) in the ${\mathtt{FR}}$/reaction step, relax the constraint $\rho\leqslant 1$ and minimize instead $\rho^{k+1}\in\operatorname*{argmin}\left\{\frac{1}{2h}{\mathtt{FR}}^2(\rho)+\int\rho p^{k+1/2}-\int\rho\right\}$, and keep iterating. This seems to correctly capture the diffusion at least numerically speaking, but raises technical issues in the rigorous proof of convergence and most importantly destroys the variational structure at the discrete level (due to the fact that the reaction step becomes semi-explicit). We shall use instead an approximation procedure, which preserves the variational structure at the discrete level: it is well-known that the Porous-Medium functional $${\mathcal{F}}_m(\rho):=\left\{\begin{array}{ll}
\int_\Omega \frac{\rho^m}{m-1} & \text{ if } \rho^m \in L^1(\Omega)\\
+\infty& \text{ otherwise}
\end{array}\right.$$ $\Gamma$-converges to ${\mathcal{F}}_\infty$ as $m\to\infty$, see [@Braides]. In the spirit of [@serfaty_sandier_gamma], one should therefore expect that the gradient flow $\rho_m$ of ${\mathcal{F}}_m(\rho)-\int\rho$ converges to the gradient flow $\rho_\infty$ of the limiting functional ${\mathcal{F}}(\rho)={\mathcal{F}}_\infty(\rho)-\int\rho$. Implementing the splitting scheme for the regular energy functional ${\mathcal{F}}_m(\rho)-\int\rho$ gives a sequence $\rho_{h,m}$, and we shall prove below that $\rho_{h,m}$ converges to a solution of the limiting gradient flow as $m\to\infty$ and $h\to 0$. However, it is known [@fleissner2016gamma] that the limit depends in general on the interplay between the time-step $h$ and the regularization parameter ($m\to\infty$ here), and for technical reasons we shall enforce the condition $$mh\to 0
\qquad \mbox{as }m\to\infty\mbox{ and }h\to 0.$$ Note that [@PQV] already contained a similar approximation $m\to\infty$ but without exploiting the variational structure of the $m$- gradient flow, and our approach is thus different. The above gradient-flow structure was already noticed and fully exploited in the ongoing work [@simone_lenaic], where existence and uniqueness of weak solutions is proved and numerical simulations are performed needless of any splitting an using directly the ${\mathtt{WFR}}$ structure. Here we rather emphasize the fact that the splitting does capture delicate $\Gamma$-convergence phenomena.\
In order to make this rigorous, we fix a time step $h>0$ and construct two sequences $(\rho_{h,m}^{k+1/2})_k$ and $(\rho_{h,m}^{k})_k$, with $\rho_{h,m}^0=\rho_{0}$, defined recursively as $$\begin{aligned}
\label{splitting scheme implicit}
\left\{\begin{array}{l}
\rho_h^{k+1/2} \in \operatorname*{argmin}\limits_{ \rho\in {\mathcal{M}}^+,\,|\rho|=|\rho_h^k|} \left\{ \frac{1}{2h} {\mathtt{W}}^2(\rho, \rho_{h,m}^k) + {\mathcal{F}}_m(\rho) -\int_\Omega\rho\right\} ,\\
\\
\rho_h^{k+1} \in \operatorname*{argmin}\limits_{\rho \in {\mathcal{M}}^+} \left\{ \frac{1}{2h} {\mathtt{FR}}^2(\rho, \rho_h^{k+1/2}) +{\mathcal{F}}_m(\rho) -\int_{\Omega} \rho \right\}.
\end{array}\right.\end{aligned}$$
As is common in the classical theory of Porous Media Equations [@vazquez2007porous], we define the pressure as the first variation $$p_m:=F_m'(\rho)=\frac{m}{m-1}\rho^{m-1}.$$ We accordingly write $$p_{h,m}^{k+1/2}:=\frac{m}{m-1}(\rho_{h,m}^{k+1/2})^{m-1}
\qquad \text{and}\qquad
p_{h,m}^{k+1}:=\frac{m}{m-1}(\rho_{h,m}^{k+1})^{m-1}$$ for the discrete pressures. As in section \[sec:KFRsplitting\] we denote by $\rho_{h,m}(t), p_{h,m}(t)$ and ${\tilde{\rho}}_{h,m}(t),\tilde{p}_{h,m}(t)$ the piecewise constant interpolations of $\rho_{h,m}^{k+1}, p_{h,m}^{k+1}$ and $\rho_{h,m}^{k+1/2}, p_{h,m}^{k+1/2}$, respectively.
Our main result is
\[existence HS implicit\] Assume that $\rho_0 \in BV(\Omega)$, $\rho_0 \leqslant 1$, and $mh\to 0$ as $h\to 0$ and $m\to \infty$. Then for all $T>0$, $\rho_{h,m},{\tilde{\rho}}_{h,m}$ both converge to some $\rho$ strongly in $L^1((0,T)\times \Omega)$, the pressures $p_{h,m},\tilde{p}_{h,m}$ both converge to some $p$ weakly in $L^2((0,T) , H^1(\Omega))$, and $(\rho,p)$ is the unique weak solution of .
Since we have a ${\mathtt{WFR}}$ gradient-flow structure, uniqueness should formally follows from the $-1$ geodesic convexity of the driving functional ${\mathcal{E}}_\infty(\rho)-\int_\Omega\rho$ with respect to the ${\mathtt{WFR}}$ distance [@liero2013gradient; @LMS_small_15] and the resulting contractivity estimate ${\mathtt{WFR}}(\rho^1(t),\rho^2(t))\leq e^t{\mathtt{WFR}}(\rho^1_0,\rho^2_0)$. This is proved rigorously in [@simone_lenaic], and therefore we retrieve convergence of the whole sequence $\rho_{h,m}\to \rho$ in Theorem \[existence HS implicit\] (and not only for subsequences). Given this uniqueness, it is clearly enough to prove convergence along any discrete (sub)sequence, and this is exactly what we show below.
The strategy of proof for Theorem \[existence HS implicit\] is exactly as in section \[sec:KFRsplitting\], except that we need now the estimates to be uniform in both in $h\to 0$ and $m\to\infty$.
Estimates and convergences
--------------------------
In this section, we improve the previous estimates from section \[sec:KFRsplitting\]. We start with an explicit $L^\infty$-bound:
\[lem:borne sup inf 1\] Assume that $\rho_0 \leqslant 1$, then for all $t \in \operatorname*{\mathbb{R}}^+$, $$\| \rho_{h,m}(t,\cdot) \|_\infty,\| {\tilde{\rho}}_{h,m}(t,\cdot) \|_\infty \leqslant 1 .$$
We argue by induction at the discrete level, starting from $\rho_0=\rho^0_{h,m}\leqslant 1$ by assumption. If $\| \rho_{h,m}^k \|_\infty \leqslant 1$, Otto’s maximum principle [@O] implies that $\| \rho_{h,m}^{k+1/2} \|_\infty \leqslant\| \rho_{h,m}^{k} \|_\infty\leqslant 1$ in the Wasserstein step.
Assume now by contradiction that $E:=\{\rho_{h,m}^{k+1} >1\}$ has positive Lebesgue measure. The optimality condition for the Fisher-Rao minimization step gives, dividing by $\sqrt{\rho_{h,m}^{k+1}}>0$ in $E$, $$\sqrt{\rho_{h,m}^{k+1}}- \sqrt{\rho_{h,m}^{k+1/2}}= \frac{h}{2}\sqrt{\rho_{h,m}^{k+1}} \left(1- \frac{m}{m-1} (\rho_{h,m}^{k+1})^{m-1} \right)$$ Then $1- \frac{m}{m-1} (\rho_{h,m}^{k+1})^{m-1} \leqslant 1-\frac{m}{m-1}< 0$ in the right-hand side, hence the desired contradiction $\rho_{h,m}^{k+1} < \rho_{h,m}^{k+1/2} \leqslant 1$.
Noticing that the functional $\frac{1}{m-1}\int \rho^m-\int\rho$ corresponds to taking explicitly $F_2(z)=z^{m}/m-1$ and $V_2(x)\equiv-1$ in section \[sec:KFRsplitting\], it is easy to reproduce the computations from the proof of Lemma \[lem encadrement\] and carefully track the dependence of the constants w.r.t $m>1$ to obtain
There exists $c>0$ such that, for all $m>m_0$ large enough and all $h\leq h_0$ small enough, $$\label{eq:encadrement_FR_HS}
(1-ch)\rho^{k+1/2}_{h,m}(x)\leqslant \rho^{k+1}_{h,m}(x)\leqslant (1+h)\rho^{k+1/2}_{h,m}(x)\qquad
\mbox{a.e.}$$
Note that this holds regardless of any compatibility such as $hm\to 0$. The key point is here that the lower bound $c$ previously depended on an upper bound $M$ on $\rho^{k+1/2}$ in Lemma \[lem encadrement\], but since we just obtained in Lemma \[lem:borne sup inf 1\] the universal upper bound $\rho^{k+1/2}\leqslant 1$ we end up with a lower bound which is also uniform in $h,m$. The proof is identical to that of Lemma \[lem encadrement\] and we omit the details for simplicity.
Recalling that the Wasserstein step is mass-preserving, we obtain by immediate induction and for all $0\leq t\leq T$ $$\|\rho_{h,m}(t)\|_{L^1},\,\|{\tilde{\rho}}_{h,m}(t)\|_{L^1}\leqslant e^T\|\rho_0\|_{L^1}$$ as well as $$\label{eq:comparison_p_ptilda_HS}
\|\rho_{h,m}(t)-{\tilde{\rho}}_{h,m}(t)\|_{L^1}\leqslant C_Th.$$
Testing successively $\rho=\rho_{h,m}^k$ and $\rho = \rho_{h,m}^{k+1/2}$ in , we get $$\frac{1}{2h}\left({\mathtt{W}}^2(\rho_{h,m}^k,\rho_{h,m}^{k+1/2}) +{\mathtt{FR}}^2( \rho_{h,m}^{k+1/2},\rho_{h,m}^{k+1}) \right) \leqslant {\mathcal{F}}_m(\rho_{h,m}^k) -{\mathcal{F}}_m(\rho_{h,m}^{k+1}) +\int_{\Omega}(\rho_{h,m}^{k+1/2}-\rho_{h,m}^{k+1}).$$ Using Proposition \[prop:comparison\_d\_W\_H\] to control ${\mathtt{WFR}}^2\lesssim 2({\mathtt{W}}^2+{\mathtt{FR}}^2)$ and the lower bound in yields $$\begin{aligned}
\frac{1}{4h}{\mathtt{WFR}}^2(\rho^{k+1}_{h,m},\rho^k_{h,m})
&\leqslant & \frac{1}{2h}\left({\mathtt{W}}^2(\rho_{h,m}^k,\rho_{h,m}^{k+1/2}) +{\mathtt{FR}}^2( \rho_{h,m}^{k+1/2},\rho_{h,m}^{k+1}) \right)\\
&\leqslant & {\mathcal{F}}_m(\rho_{h,m}^k) -{\mathcal{F}}_m(\rho_{h,m}^{k+1}) +\int_{\Omega}(\rho_{h,m}^{k+1/2}-\rho_{h,m}^{k+1})\\
&\leqslant & {\mathcal{F}}_m(\rho_{h,m}^k) -{\mathcal{F}}_m(\rho_{h,m}^{k+1}) +c h\int_{\Omega}\rho_{h,m}^{k+1/2}\\
&\leqslant &{\mathcal{F}}_m(\rho_{h,m}^k) -{\mathcal{F}}_m(\rho_{h,m}^{k+1}) +c he^T\end{aligned}$$ for all $k\leqslant N:= \lfloor T/h \rfloor$.
Summing over $k$ we get $$\begin{aligned}
\frac{1}{4h}\sum_{k=0}^{N-1} {\mathtt{WFR}}^2(\rho_{h,m}^k, \rho_{h,m}^{k+1}) &\leqslant & {\mathcal{F}}_m(\rho_0) - {\mathcal{F}}_m(\rho_{h,m}^N) +C_T\\
&\leqslant &\frac{1}{m-1}\int_\Omega\rho_0^m + C_T \leqslant \frac{1}{m-1}\int_\Omega\rho_0 + C_T\leqslant C_T,\end{aligned}$$
where we used successively $F_m\geq 0$ to get rid of ${\mathcal{F}}_m(\rho^N_{h,m})$, and $\rho_0^m\leq \rho_0$ for $\rho_0\leq 1$ and $m>1$.
Consequently, for all fixed $T>0$ and any $t,s\in[0,T]$ we obtain the classical $\frac{1}{2}$-Hölder estimate $$\begin{aligned}
\label{part42-holder estimates}
\left\{ \begin{array}{l}
{\mathtt{WFR}}(\rho_{h,m}(t),\rho_{h,m}(s)) \leqslant C_T|t-s+h|^{1/2},\\
{\mathtt{WFR}}({\tilde{\rho}}_{h,m}(t),{\tilde{\rho}}_{h,m}(s)) \leqslant C_T|t-s+h|^{1/2}.
\end{array}\right.\end{aligned}$$\
Exploiting the explicit algebraic structure of $F_m(z)=\frac{1}{m-1}z^m$, compactness in space will be given here by
\[prop:BV\_estimate\] If $\rho_0 \in BV(\Omega)$ then $$\sup_{t \in [0,T]} \left\{\| \rho_{h,m}(t,\cdot) \|_{BV(\Omega)}, \| {\tilde{\rho}}_{h,m}(t,\cdot) \|_{BV(\Omega)} \right\} \leqslant e^T \| \rho_0 \|_{BV(\Omega)}.$$
The argument closely follows the lines of [@GM prop. 5.1]. We first note from [@DPMSV thm. 1.1] that the $BV$-norm is nonincreasing during the Wasserstein step, $$\| \rho_{h,m}^{k+1/2} \|_{BV(\Omega)} \leqslant \| \rho_{h,m}^{k} \|_{BV(\Omega)} .$$ Using as before the implicit function theorem, we show below that $\rho_{h,m}^{k+1} = R(\rho_{h,m}^{k+1/2})$ for some suitable $(1+h)$-Lispchitz function $R$. By standard $Lip\circ BV$ composition [@AFP] this will prove that $$\| \rho_{h,m}^{k+1} \|_{BV(\Omega)} \leqslant (1+h)\| \rho_{h,m}^{k+1/2} \|_{BV(\Omega)}$$ and will conclude the proof by immediate induction.
Indeed, we already know from that $\rho_{h,m}^{k+1/2}$ and $\rho_{h,m}^{k+1}$ share the same support. In this support and from it is easy to see that $\rho=\rho_{h,m}^{k+1}(x)$ is the unique positive solution of $f(\rho,\rho_{h,m}^{k+1/2}(x))=0$ with $$f(\rho,\mu)= \sqrt{\rho}\left(1- \frac{h}{2} \left(1- \frac{m}{m-1}\rho^{m-1} \right) \right) - \sqrt{\mu}.$$ For $\mu >0$, the implicit function theorem gives the existence of a $\mathcal{C}^1$ map $R$ such that $f(\rho, \mu)=0\Leftrightarrow \rho =R(\mu)$, with $R(0)=0$. An algebraic computation shows moreover that $0<\frac{d R}{d\mu}= - {\frac{\partial_\mu f}{\partial_\rho f}}_{|\rho =R(\mu)}\leqslant (1+h)$ uniformly in $m>1$, hence $R$ is $(1+h)$-Lipschitz as claimed and the proof is complete.
\[prop:CV\_phm\_rhohm\] Up to extraction of a discrete sequence $h\to 0,m\to\infty$, there holds $$\rho_{h,m},\,{\tilde{\rho}}_{h,m}\to \rho \qquad \mbox{strongly in }L^1(Q_T)$$ $$p_{h,m}\rightharpoonup p \quad\mbox{and}\quad\tilde p_{h,m}\rightharpoonup \tilde p \qquad\mbox{weakly in all }L^q(Q_T)$$ for all $T>0$. If in addition $mh\to 0$ then $p=\tilde p$.
The first part of the statement follows exactly as in section \[sec:KFRsplitting\], exploiting the $\frac{1}{2}$-Hölder estimates and the space compactness from Proposition \[prop:BV\_estimate\] in order to apply the Rossi-Savaré theorem [@RS]. The fact that $\rho_{h,m},{\tilde{\rho}}_{h,m}$ have the same limit comes from .
For the pressures, we simply note from $\rho_{h,m}\leqslant 1$ and $m\gg 1$ that $p_{h,m}=\frac{m}{m-1}\rho_{h,m}^{m-1}\leqslant 2\rho_{h,m}$ is bounded in $L^1\cap L^\infty(Q_T)$ uniformly in $h,m$ in any finite time interval $[0,T]$. Thus up to extraction of a further sequence we have $p_{h,m}\rightharpoonup p$ in all $L^q(Q_T)$, and likewise for $\tilde p_{h,m}\rightharpoonup \tilde p$.
Finally, we only have to check that $p=\tilde p$ if $hm\to 0$. Because $\rho_{h,m},{\tilde{\rho}}_{h,m}\leqslant 1$ and $z\mapsto z^{m-1}$ is $(m-1)$-Lipschitz on $[0,1]$ we have for all fixed $t\geqslant 0$ that $$\begin{aligned}
\int_\Omega |p_{m,h}(t,\cdot) - \tilde{p}_{m,h}(t,\cdot) | &= & \int_\Omega \frac{m}{m-1}|\rho_{h,m}^{m-1}(t,\cdot)-{\tilde{\rho}}_{h,m}^{m-1}(t,\cdot) |\\
&\leqslant & m\int_\Omega |{\rho_{h,m}(t)}-{{\tilde{\rho}}_{h}(t)} |
\leqslant C_Thm \longrightarrow 0,\end{aligned}$$ where we used in the last inequality. Hence $p=\tilde p$ and the proof is complete.
In order to pass to the limit in the diffusion term $\operatorname*{div}(\rho\nabla p)$ we first improve the convergence of $\tilde{p}_{h,m}$:
\[borne pression porous media\] There exists a constant $C_T$, independent of $h$ and $m$, such that $$\| \tilde{p}_{h,m} \|_{L^2((0,T),H^1(\Omega))} \leqslant C_T$$ for all $T>0$. Consequently, up to a subsequence, $\tilde{p}_{h,m}$ converges weakly in $L^2((0,T),H^1(\Omega))$ to $p$.
The proof is based on the flow interchange technique developed by Matthes, McCann and Savaré in [@MMCS]. Let $\eta$ be the (smooth) solution of $$\left\{ \begin{array}{l}
\partial_t \eta = \Delta \eta^{m-1} + {\varepsilon}\Delta \eta,\\
\eta|_{t=0}=\rho^{k+1/2}_{h,m}.
\end{array}\right.$$ It is well known [@AGS] that $\eta$ is the Wasserstein gradient flow of $${\mathcal{G}}(\rho):= \int_\Omega \frac{\rho^{m-1}}{m-2} + {\varepsilon}\int_\Omega \rho \log(\rho).$$ Since ${\mathcal{G}}$ is geodesically $0$-convex, $\eta$ satisfies the Evolution Variational Inequality (EVI) $$\left.\frac{1}{2}{\frac{d^+}{dt}}\right|_{t=s} {\mathtt{W}}^2(\eta(s), \rho) \leqslant {\mathcal{G}}(\rho) -{\mathcal{G}}(\eta(s)),$$ for all $s>0$ and for all $\rho \in \operatorname*{\mathcal{P}^{ac}}(\Omega)$, where $\frac{d^+}{dt}f(t):=\limsup\limits_{s\rightarrow 0^+} \frac{f(t+s) -f(t)}{s}$. By optimality of $\rho^{k+1/2}_{h,m}$ in , we obtain that $$\left.\frac{1}{2}{\frac{d^+}{dt}}\right|_{t=s} {\mathtt{W}}^2(\eta(s),\rho_{h,m}^{k})\geqslant -h \left.{\frac{d^+}{dt}}\right|_{t=s} {\mathcal{F}}_m(\eta(s)).$$ Since $\eta$ is smooth due to the regularizing ${\varepsilon}\Delta$ term, we can legitimately integrate by parts for all $s>0$ $$\begin{aligned}
\frac{d}{ds} {\mathcal{F}}_m(\eta(s))&=& \int_\Omega \frac{m}{m-1} \eta(s)^{m-1}(\Delta\eta(s)^{m-1} + {\varepsilon}\Delta \eta(s) )\\
& = & -\int_\Omega \frac{m}{m-1} |\nabla \eta(s)^{m-1} |^2 -{\varepsilon}\int_\Omega m\eta(s)^{m-2}|\nabla \eta(s) |^2\\
& \leqslant & -\int_\Omega \frac{m}{m-1} |\nabla \eta(s)^{m-1} |^2
=-\frac{m-1}{m} \int_\Omega \left|\nabla \left(\frac{m}{m-1}\eta(s)^{m-1}\right) \right|^2\end{aligned}$$
Remarking that $\frac{m}{m-1}\eta(s)^{m-1}\to\frac{m}{m-2}\rho_{h,m}^{k+1/2}=p_{h,m}^{k+1/2}$ as $s\to 0$, an easy lower semi-continuity argument gives that $$\int_\Omega \frac{m-1}{m} |\nabla p_{h,m}^{k+1/2}|^2= \int_\Omega \frac{m}{m-1} |\nabla (\rho_{h,m}^{k+1/2})^{m-1} |^2 \leqslant \liminf_{s\searrow 0}\left.{\frac{d^+}{dt}}\right|_{t=s} {\mathcal{F}}_m(\eta(s)).$$
Then we have $$\begin{aligned}
h\int_\Omega \frac{m-1}{m} |\nabla p_{h,m}^{k+1/2}|^2 & \leqslant {\mathcal{F}}_{m-1}(\rho_{h,m}^k) - {\mathcal{F}}_{m-1}(\rho_{h,m}^{k+1/2})\\
& + {\varepsilon}\left( \int_\Omega \rho_{h,m}^k \log(\rho_{h,m}^k) -\int_\Omega \rho_{h,m}^{k+1/2} \log(\rho_{h,m}^{k+1/2}) \right).\end{aligned}$$ First arguing as in Proposition \[prop:decr F1\] to control $${\mathcal{F}}_{m-1}(\rho_{h,m}^{k+1}) \leqslant {\mathcal{F}}_{m-1}(\rho_{h,m}^{k+1/2}) +C_Th,$$ and then passing to the limit ${\varepsilon}\searrow 0$, we obtain $$h\int_\Omega \frac{m-1}{m} |\nabla p_{h,m}^{k+1/2}|^2 \leqslant {\mathcal{F}}_{m-1}(\rho_{h,m}^k) - {\mathcal{F}}_{m-1}(\rho_{h,m}^{k+1}) +C_Th.$$ Summing over $k$ gives $$\int_0^T \int_\Omega |\nabla \tilde{p}_{h,m}(t,x)|^2 \,dxdt \leqslant \frac{m}{m-1}( {\mathcal{F}}_{m-1}(\rho_0)- {\mathcal{F}}_{m-1}(\rho^N_{h,m})+C_T)
\leqslant 2 {\mathcal{F}}_{m-1}(\rho_0)+C_T$$ for all $T<+\infty$. Due to $\rho_0\leqslant 1$ and $m\gg1 $ we can bound ${\mathcal{F}}_{m-1}(\rho_0)=\frac{1}{m-2}\int\rho_0^{m-1}\leqslant \frac{1}{m-2}\int\rho_0\leqslant \|\rho_0 \|_{L^1(\Omega)}$ and the result finally follows.
Properties of the pressure $p$ and conclusion {#subsec:prop_pressure_HS}
---------------------------------------------
We start by showing that the limits $\rho,p$ satisfy the compatibility conditions in .
\[lem:properties pressure\] There holds $$0\leqslant \rho,p\leqslant 1 \quad \text{and} \quad p(1-\rho) =0 \, \text{ a.e. in }Q_T.$$
By Lemma \[lem:borne sup inf 1\] it is obvious that $ 0\leqslant \rho \leqslant 1$ and $0\leqslant p \leqslant 1$ are inherited from $0\leqslant \rho_{h,m}\leqslant 1$ and $0\leqslant p_{h,m}=\frac{m}{m-1}\rho^{m-1}_{h,m}\leqslant \frac{m}{m-1}$. In order to prove that $p(1-\rho)=0$, we first observe that $$p_{h,m}(1-\rho_{h,m})\to 0\qquad \mbox{a.e. in }Q_T.$$ Indeed, since $\rho_{h,m}\to \rho$ strongly in $L^1(Q_T)$ we have $\rho_{h,m}(t,x)\to \rho(t,x)$ a.e. If the limit $\rho(t,x)<1$ then $\rho_{h,m}(t,x)\leqslant (1-{\varepsilon})$ for small $h$ and large $m$. Hence $p_{h,m}(t,x)=\frac{m}{m-1}\rho_{h,m}^{m-1}\leqslant \frac{m}{m-1}(1-{\varepsilon})^{m-1}\to 0$ while $1-\rho_{h,m}$ remains bounded, and therefore the product $p_{h,m}(1-\rho_{h,m})\to 0$. Now if the limit $\rho(t,x)=1$ then the pressure $p_{h,m}=\frac{m}{m-1}\rho_{h,m}^{m-1}\leqslant \frac{m}{m-1}$ remains bounded, while $1-\rho_{h,m}(t,x)\to 0$ hence the product goes to zero in this case too.
Thanks to the uniform $L^\infty$ bounds $\rho_{h,m}\leqslant 1$ and $p_{h,m}\leqslant \frac{m}{m-1}\leqslant 2$ we can apply Lebesgue’s convergence theorem to deduce from this pointwise a.e. convergence that, for all fixed nonnegative $\varphi\in\mathcal \mathcal C^{\infty}_c(Q_T)$, there holds $$\lim \int_{Q_T}p_{h,m}(1-\rho_{h,m})\varphi =0.$$ On the other hand since $\rho_{h,m}\to \rho$ strongly in $L^1(Q_T)$ hence a.e, and because $0\leqslant \rho_{h,m}\leqslant 1$, we see that $(1-\rho_{h,m})\varphi\to (1-\rho)\varphi$ in all $L^q(Q_T)$. From Proposition \[prop:CV\_phm\_rhohm\] we also had that $p_{h,m}\rightharpoonup p$ in all $L^q(Q_T)$, hence by strong-weak convergence we have that $$\int_{Q_T}p (1-\rho)\varphi=\lim \int_{Q_T}p_{h,m}(1-\rho_{h,m})\varphi=0$$ for all $\varphi\geqslant 0$. Because $p(1-\rho)\geqslant 0$ we conclude that $p(1-\rho)=0$ a.e. in $Q_T$ and the proof is achieved.
We end this section with
We only sketch the argument and refer to [@GM] for the details. Fix any $0<t_1<t_2$ and $\varphi\in \mathcal C^2_c(\operatorname*{\mathbb{R}}^d)$. Exploiting the Euler-Lagrange equations and summing from $k=k_1=\lfloor t_1/h \rfloor$ to $k=k_2-1=\lfloor t_2/h\rfloor-1$, we first obtain $$\int_{\operatorname*{\mathbb{R}}^d}\rho_{h,m}(t_2)\varphi-\rho_{h,m}(t_1)\varphi
+\int_{k_1h}^{k_2h}\int_{\operatorname*{\mathbb{R}}^d}{\tilde{\rho}}_{h,m}\nabla \tilde p_{h,m}\cdot\nabla\varphi
=-\int_{k_1h}^{k_2h}\int_{\operatorname*{\mathbb{R}}^d}\rho_{h,m}(1-p_{h,m})\varphi + R(h,m),$$ where the remainder $R(h,m)\to 0$ for fixed $\varphi$. The strong convergence $\rho_{h,m},\tilde\rho_{h,m}\to \rho$ and the weak convergences $\nabla\tilde p_{h,m}\rightharpoonup \nabla \tilde p=\nabla p$ and $p_{h,m}\rightharpoonup p$ are then enough pass to the limit to get the corresponding weak formulation for all $0<t_1<t_2$. Moreover since the limit $\rho\in {\mathcal{C}}([0,T];{\mathcal{M}}^+_{{\mathtt{WFR}}})$ the initial datum $\rho(0)=\rho_0$ is taken at least in the sense of measures. This gives an admissible weak formulation of , and the proof is complete.
Numerical simulation
--------------------
The constructive scheme naturally leads to a fully discrete algorithm, simply discretizing the minimization problem in space for each ${\mathtt{W}},{\mathtt{FR}}$ step. We use again the [ALG2-JKO]{} scheme [@BCL] for the Wasserstein steps. As already mentioned the Fisher-Rao step is a mere convex pointwise minimization problem, here explicitly given by: for all $x \in \Omega$, $$\rho_{h,m}^{k+1}(x) = \operatorname*{argmin}_{\rho \geq 0}
\left\{ 4\left| \sqrt{\rho} - \sqrt{\rho_{h,m}^{k+1/2}(x)} \right|^2 + 2h \left( \frac{\rho^{m}}{m-1} -1 \right)\right\}$$ and poses no difficulty in the practical implementation using a standard Newton method.
Figure \[figure m=100 \] depicts the evolution of the numerical solution $\rho_{h,m}$ for $m=100$ and with a time step $h=0.005$. We remark that the tumor first saturates the constraint ($\rho\nearrow 1$) in its initial support, and then starts diffusing outwards. This is consistent with the qualitative behaviour described in [@PQV].
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![*Snapshot of the approximate solution $\rho_{h,m}(t,.)$ to , with $m=100$, $h=0.005$.*[]{data-label="figure m=100 "}](dens-jkofr100-00-eps-converted-to.pdf "fig:") ![*Snapshot of the approximate solution $\rho_{h,m}(t,.)$ to , with $m=100$, $h=0.005$.*[]{data-label="figure m=100 "}](dens-jkoCFL-60.pdf "fig:") ![*Snapshot of the approximate solution $\rho_{h,m}(t,.)$ to , with $m=100$, $h=0.005$.*[]{data-label="figure m=100 "}](dens-jkoCFL-100.pdf "fig:") ![*Snapshot of the approximate solution $\rho_{h,m}(t,.)$ to , with $m=100$, $h=0.005$.*[]{data-label="figure m=100 "}](dens-jkoCFL-140.pdf "fig:") ![*Snapshot of the approximate solution $\rho_{h,m}(t,.)$ to , with $m=100$, $h=0.005$.*[]{data-label="figure m=100 "}](dens-jkoCFL-200.pdf "fig:")
$t=0$ $t=0.3$ $t=0.5$ $t=0.7$ $t=1$
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A tumor growth model with nutrient {#part4-section2HSN}
==================================
In this section we use the same approach for the following tumor growth model with nutrients, appearing e.g. in [@PQV]
$$\begin{aligned}
\label{system HS-nut}
\left\{\begin{array}{l}
\partial_t \rho -\operatorname*{div}( \rho \nabla p)= \rho \left((1-p)(c + c_1) -c_2 \right),\\
\partial_t c - \Delta c = -\rho c,\\
0\leqslant \rho \leqslant 1,\\
p \geqslant 0 \mbox{ and } p(1-\rho)=0, \\
\rho_{|t=0}=\rho_0, \, c_{|t=0}= c_0.
\end{array} \right.\end{aligned}$$
Here $c_1$ and $c_2$ are two positive constants, and the nutrient $c$ is now diffusing in $\Omega$ in addition to begin simply consumed by the tumor $\rho$, according to the second equation. For technical convenience we work here on a convex bounded domain $\Omega\subset{\mathbb{R}^d}$, endowed with natural Neumann boundary conditions for both $\rho$ and $c$.
Contrarily to section \[part4-section2HS\] this is not a ${\mathtt{WFR}}$ gradient flow anymore, and we therefore introduce a semi-implicit splitting scheme. Starting from the initial datum $\rho^0_{h,m}:=\rho_0,c_{h,m}^0:=c_0$ we construct four sequences $\rho_{h,m}^{k+1/2},\rho_{h,m}^k,c_{h,m}^{k+1/2},c_{h,m}^k$, defined recursively as $$\begin{aligned}
\label{splitting scheme w}
\left\{\begin{array}{l}
\rho_{h,m}^{k+1/2} \in \operatorname*{argmin}\limits_{\rho \in {\mathcal{M}}^+, |\rho|=|\rho_{h,m}^k|} \left\{ \frac{1}{2h} {\mathtt{W}}^2(\rho, \rho_{h,m}^k) + {\mathcal{F}}_m(\rho ) \right\},\\
\\
c_{h,m}^{k+1/2} \in \operatorname*{argmin}\limits_{c \in {\mathcal{M}}^+,|c|=|c_{h,m}^k|} \left\{ \frac{1}{2h} {\mathtt{W}}^2(c, c_{h,m}^{k}) + {\mathcal{E}}(\rho ) \right\},
\end{array}\right.\end{aligned}$$ and $$\begin{aligned}
\label{splitting scheme fr}
\left\{\begin{array}{l}
\rho_{h,m}^{k+1} \in \operatorname*{argmin}\limits_{\rho \in {\mathcal{M}}^+} \left\{ \frac{1}{2h} {\mathtt{FR}}^2(\rho, \rho_{h,m}^{k+1/2}) + {\mathcal{E}}_{1,m}(\rho | c_{h,m}^{k+1/2}) \right\},\\
\\
c_{h,m}^{k+1} \in \operatorname*{argmin}\limits_{c\in {\mathcal{M}}^+} \left\{ \frac{1}{2h} {\mathtt{FR}}^2(c, c_{h,m}^{k+1/2}) + {\mathcal{E}}_2(c |\rho_{h,m}^{k+1/2}) \right\},
\end{array}\right.\end{aligned}$$ where $${\mathcal{E}}(\rho):= \int_\Omega \rho \log(\rho),$$ $${\mathcal{E}}_{1,m}(\rho |c):= \int_\Omega \left(c +c_1\right) \frac{\rho^m}{m-1} + \int_\Omega (c_2 - c -c_1) \rho ,$$ and $${\mathcal{E}}_2(c|\rho):= \int_\Omega \rho c.$$
As earlier it is easy to see that these sequences are well-defined (i-e there exists a unique minimizer for each step), and the pressures are defined as before as $$p_{h,m}^{k+1/2}:=\frac{m}{m-1}(\rho_{h,m}^{k+1/2})^{m-1}
\quad\mbox{and}\quad
p_{h,m}^{k+1}:=\frac{m}{m-1}(\rho_{h,m}^{k+1})^{m-1}.$$ We denote again by $a_{h,m}(t),\tilde a_{h,m}(t)$ the piecewise constant interpolation of any discrete quantity $a^{k+1}_{h,m},a^{k+1/2}_{h,m}$ respectively. Our main result reads:
\[theo:existence system nutrient\] Assume $\rho_0 \in BV(\Omega)$ with $\rho_0 \leqslant 1$ and $c_0 \in L^\infty(\Omega) \cap BV(\Omega)$. Then $\rho_{h,m}$ and ${\tilde{\rho}}_{h,m}$ strongly converge to $\rho$ in $L^1((0,T)\times \Omega)$ and $c_{h,m}$ and $\tilde{c}_{h,m}$ strongly converge to $c$ in $L^1((0,T)\times \Omega)$ when $h\searrow 0$ and $m\nearrow+\infty$. Moreover, if $mh\rightarrow 0$, then $p_{h,m},\tilde{p}_{h,m}$ converge weakly in $L^2((0,T) , H^1(\Omega))$ to a unique $p$, and $(\rho,p,c)$ is a solution of .
Note that uniqueness of solutions would result in convergence of the whole sequence. Uniqueness was proved in [@PQV thm. 4.2] for slightly more regular weak solutions, but we did not push in this direction for the sake of simplicity. The method of proof is almost identical to section \[part4-section2HS\] so we only sketch the argument and emphasize the main differences.
We start by recalling the optimality conditions for the scheme -. The Euler-Lagrange equations for the tumor densities in the Wasserstein and Fisher-Rao steps are $$\begin{aligned}
\label{eq:optimality_rho_nutrients}
\left\{ \begin{array}{l}
\rho_{h,m}^{k+1/2} \nabla p_{h,m}^{k+1/2} = \frac{\nabla \varphi}{h}\rho_{h,m}^{k+1/2},\\
\sqrt{\rho_{h,m}^{k+1}} - \sqrt{\rho_{h,m}^{k+1/2}} = \frac{h}{2}\sqrt{\rho_{h,m}^{k+1}} \left( (1 - p_{h,m}^{k+1})(c_{h,m}^{k+1/2} + c_1) -c_2 \right),
\end{array}\right.\end{aligned}$$ where $\varphi$ is a (backward) Kantorovich potential for ${\mathtt{W}}(\rho_{h,m}^{k+1/2},\rho_{h,m}^{k})$. For the nutrient, the Euler-Lagrange equations are $$\begin{aligned}
\label{eq:optimality_c_FR_nutrients}
\left\{ \begin{array}{l}
\nabla c_{h,m}^{k+1/2} = \frac{\nabla \psi}{h}c_{h,m}^{k+1/2},\\
\sqrt{c_{h,m}^{k+1}} - \sqrt{c_{h,m}^{k+1/2}} =- \frac{h}{2}\sqrt{c_{h,m}^{k+1}} \rho_{h,m}^{k+1/2},
\end{array}\right.\end{aligned}$$ with $\psi$ a Kantorovich potential for ${\mathtt{W}}(c_{h,m}^{k+1/2},c_{h,m}^{k})$.
Using the optimality conditions for the Fischer-Rao steps, we obtain directly the following $L^\infty$ bounds:
\[borne sup inf 1 and nut\] For all $k \geqslant 0$ $$\|c_{h,m}^{k+1} \|_{L^\infty(\Omega)} \leqslant \|c_{h,m}^{k+1/2} \|_{L^\infty(\Omega)} \leqslant \|c_{h,m}^{k} \|_{L^\infty(\Omega)},$$ and at the continuous level $$\|c_{h,m}(t,\cdot) \|_{L^\infty(\Omega)}, \|\tilde{c}_{h,m}(t,\cdot) \|_{L^\infty(\Omega)} \leqslant \|c_0\|_{L^\infty(\Omega)}
\qquad \forall\, t\geq 0.$$ Moreover, $$\| \rho_{h,m}(t,\cdot) \|_\infty,\| {\tilde{\rho}}_{h,m}(t,\cdot) \|_\infty \leqslant 1$$ and there exists $c_T \equiv c_T(\|c_0\|_{L^\infty}),C_T \equiv C_T(\|c_0\|_{L^\infty}) >0$ such that $$\label{eq:encadrement nutrient}
\begin{array}{c}
(1-c_Th)\rho_{h,m}^{k+1/2}(x) \leqslant \rho_{h,m}^{k+1}(x) \leqslant (1+C_Th)\rho_{h,m}^{k+1/2}(x)
\qquad\mbox{a.e. in } \Omega.\\
(1-h) c_{h,m}^{k+1/2}(x) \leqslant c_{h,m}^{k+1}(x) \leqslant c_{h,m}^{k+1/2}(x) \qquad\mbox{a.e. in } \Omega.
\end{array}$$
The proof of the estimates on $c_{h,m}$ and $\tilde{c}_{h,m}$ is obvious because one step of Wasserstein gradient flow with the Boltzmann entropy decreases the $L^\infty$-norm in (see [@O1; @A]), and, because the product $\sqrt{c_{h,m}^{k+1}} \rho_{h,m}^{k+1/2}$ is nonnegative in , the $L^\infty$-norm is also nonincreasing during the Fischer-Rao step. The proof for $\rho_{h,m}$ and ${\tilde{\rho}}_{h,m}$ is the same as in lemma \[lem:borne sup inf 1\]. Using the fact that $\| {\tilde{\rho}}_{h,m}(t,\cdot) \|_\infty \leqslant 1$, we see that the term $\Phi(p_{h,m}^{k+1},c_{h,m}^{k+1/2}):=(1 - p_{h,m}^{k+1})(c_{h,m}^{k+1/2} + c_1) -c_2 $ in is bounded in $L^\infty$ uniformly in $k$. This allows to argue exactly as in Lemma \[lem encadrement\] to retrieve the estimate and concludes the proof.
With these bounds it is easy to prove as in proposition \[décroissance F1\] that $$\begin{array}{c}
{\mathcal{F}}_m(\rho_{h,m}^{k+1}) \leqslant {\mathcal{F}}_m(\rho_{h,m}^{k+1/2}) +C_Th,\\
{\mathcal{E}}_{1,m}(\rho_{h,m}^{k+1/2} |c_{h,m}^{k+1/2}) - {\mathcal{E}}_{1,m}(\rho_{h,m}^{k+1} |c_{h,m}^{k+1/2}) \leqslant C_Th,\\
{\mathcal{E}}(c_{h,m}^{k+1}) \leqslant {\mathcal{E}}(c_{h,m}^{k+1/2}) +C_Th,\\
{\mathcal{E}}_2(c_{h,m}^{k+1/2} |\rho_{h,m}^{k+1/2}) - {\mathcal{E}}_2(c_{h,m}^{k+1} |\rho_{h,m}^{k+1/2}) \leqslant C_Th,
\end{array}.$$ for some $C_T$ independent of $m$. Then we obtain the usual $\frac 12$-Hölder estimates in time with respect to the ${\mathtt{WFR}}$ distance, which in turn implies that $\rho_{h,m},{\tilde{\rho}}_{h,m}$ converge to some $\rho \in L^\infty([0,T],L^1(\Omega))$ and $c_{h,m},\tilde{c}_{h,m}$ converge to some $c\in L^\infty([0,T],L^1(\Omega))$ pointwise in time with respect to ${\mathtt{WFR}}$, see , Proposition \[prop:1/2\_holder\], and for details.
As before we need to improve the convergence in order to pass to the limit in the nonlinear terms. For $\rho_{h,m}$ and ${\tilde{\rho}}_{h,m}$, this follows from
For all $T>0$, if $\rho_0,c_0 \in BV(\Omega)$, $$\begin{array}{c}
\sup\limits_{t \in [0,T]} \left\{\| \rho_{h,m}(t,\cdot) \|_{BV(\Omega)}+\| c_{h,m}(t,\cdot) \|_{BV(\Omega)} \right\} \leqslant e^{C_TT} (\| \rho_0 \|_{BV(\Omega)} +\|c_0 \|_{BV(\Omega)})\\
\sup\limits_{t \in [0,T]} \left\{\| {\tilde{\rho}}_{h,m}(t,\cdot) \|_{BV(\Omega)}+\| \tilde{c}_{h,m}(t,\cdot) \|_{BV(\Omega)}\right\} \leqslant e^{C_TT} (\| \rho_0 \|_{BV(\Omega)} +\|c_0 \|_{BV(\Omega)}).
\end{array}$$
The argument is a generalization of Lemma \[prop:BV\_estimate\], see [@GM remark 5.1]. First, the $BV$-norm is nonincreasing during the Wasserstein step, [@DPMSV thm. 1.1], $$\| \rho_{h,m}^{k+1/2} \|_{BV(\Omega)} \leqslant \| \rho_{h,m}^{k} \|_{BV(\Omega)} \text{ and } \| c_{h,m}^{k+1/2} \|_{BV(\Omega)} \leqslant \| c_{h,m}^{k} \|_{BV(\Omega)}.$$ Arguing as in Lemma \[prop:BV\_estimate\], we observe that, inside ${\mathop{\rm supp}}\rho_{h,m}^{k+1/2}={\mathop{\rm supp}}\rho_{h,m}^{k+1}$, the minimizer $\rho=\rho_{h,m}^{k+1}(x)$ is the unique positive solution of $f(\rho, \rho_{h,m}^{k+1/2}(x),c_{h,m}^{k+1/2}(x))=0$, with $$f(\rho, \mu , c) = \sqrt{\rho}\left( 1-\frac{h}{2}\left( \left( 1- \frac{m}{m-1}\rho^{m-1}\right)(c+c_1) -c_2 \right)\right) -\sqrt{\mu}.$$ For $\mu >0$ the implicit function theorem gives as before a $\mathcal{C}^1$ map $R$ such that $f(\rho,\mu,c)=0 \Leftrightarrow \rho=R(\mu,c)$. An easy algebraic computation and then gives $0 <\partial_\mu R(\mu,c) \leqslant (1+C_Th)$ and $|\partial_c R(\mu,c) | \leqslant C_Th$ for some constant $C_T>0$ independent of $h,m,k$. This implies that $$\begin{aligned}
\| \rho_{h,m}^{k+1} \|_{BV(\Omega)} & \leqslant & (1+C_Th)\| \rho_{h,m}^{k+1/2} \|_{BV(\Omega)} + C_Th\| c_{h,m}^{k+1/2} \|_{BV(\Omega)}\\
&\leqslant & (1+C_Th)\| \rho_{h,m}^{k} \|_{BV(\Omega)} + C_Th\| c_{h,m}^{k} \|_{BV(\Omega)}.\end{aligned}$$
The same argument shows that $$\| c_{h,m}^{k+1} \|_{BV(\Omega)} \leqslant (1+C_Th)\| c_{h,m}^{k} \|_{BV(\Omega)} + C_Th\| \rho_{h,m}^{k} \|_{BV(\Omega)},$$ and a simple induction allows to conclude.
Up to extraction of a discrete sequence $h \to 0, m\to +\infty$, $$\rho_{h,m},\,{\tilde{\rho}}_{h,m}\to \rho \qquad \mbox{strongly in }L^1(Q_T)$$ $$p_{h,m}\rightharpoonup p \mbox{ and }\tilde p_{h,m}\rightharpoonup \tilde p \qquad\mbox{weakly in all }L^q(Q_T)$$ for all $T>0$. If in addition $mh\to 0$ then $p=\tilde p \in L^2((0,T),H^1(\Omega))$ and $(\rho,p)$ satisfies $$0 \leqslant \rho, p \leqslant 1 \quad \text{and} \quad p(1-\rho)=0 \qquad \text{a.e. in } Q_T.$$
The proof is the same as Proposition \[prop:CV\_phm\_rhohm\], Lemma \[borne pression porous media\], and Lemma \[lem:properties pressure\].
In order to conclude the proof of Theorem \[theo:existence system nutrient\] we only need to check that $\rho,p,c$ satisfy the weak formulation of : the strong convergence of $\rho_{h,m},c_{h,m}$ and the weak convergence of $p_{h,m}$ are enough to take the limit in the nonlinear terms as in section \[subsec:prop\_pressure\_HS\], and we omit the details.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We warmly thank G. Carlier for fruitful discussions and suggesting us the problem in section \[sec:KFRsplitting\]
|
---
abstract: 'Molecular communications (MC), where molecules are used to encode, transmit, and receive information, is a promising means of enabling the coordination of nanoscale devices. The paradigm has been extensively studied from various aspects, including channel modeling and noise analysis. Comparatively little attention has been given to the physical design of molecular receiver and transmitter, envisioning biological synthetic cells with intrinsic molecular reception and transmission capabilities as the future nanomachines. However, this assumption leads to a discrepancy between the envisaged applications requiring complex communication interfaces and protocols, and the very limited computational capacities of the envisioned biological nanomachines. In this paper, we examine the feasibility of designing a molecular receiver, in a physical domain other than synthetic biology, meeting the basic requirements of nanonetwork applications. We first review the state-of-the-art biosensing approaches to determine whether they can inspire a receiver design. We reveal that nanoscale field effect transistor based electrical biosensor technology (bioFET) is a particularly useful starting point for designing a molecular receiver. Focusing on bioFET-based molecular receivers with a conceptual approach, we provide a guideline elaborating on their operation principles, performance metrics and design parameters. We then provide a simple model for signal flow in silicon nanowire (SiNW) FET-based molecular receiver. Lastly, we discuss the practical challenges of implementing the receiver and present the future research avenues from a communication theoretical perspective.'
author:
- 'Murat Kuscu, and Ozgur B. Akan, [^1] [^2]'
title: |
On the Physical Design of\
Molecular Communication Receiver\
Based on Nanoscale Biosensors
---
Molecular communications, receiver, nanoscale biosensor, bioFET, affinity-based biorecognition, electrical biosensing, sensitivity, selectivity, limit of detection, SNR
Introduction {#Introduction}
============
fully functional nanoscale devices with sensing, computing and actuating capabilities has been a long-standing goal of science and engineering community. Recent advances in nanotechnology including the inventions of novel nanomaterials like carbon nanotube and graphene have enabled the miniaturization of various functional devices like computing and memories into nanoscale dimensions. This exciting progress has led to a thorough investigation on the feasibility of enabling communication between nanomachines, forming *nanonetworks*, to realize more complex tasks for ground breaking applications such as collaborative in-body drug delivery and continuous health monitoring [@Akyildiz2008] [@Kuscu2015-2].
Since implementing conventional electromagnetic communications among nanomachines are obstructed by the limitations of antenna sizes, researches have started a quest for finding alternative communication methods. Among the several paradigms proposed for use in nanonetworks, MC is the most promising one because it already exists in nature as the main communication mechanism of living cells and other microorganisms, and thus, its feasibility in nanoscale domain is readily proven. MC uses molecules to encode, transmit and receive information. One of the physical characteristics of molecules, such as type or concentration, is modulated to be transferred either through active propagation or diffusion-based passive propagation in a fluid medium [@Nakano2012] [@Kuran2011].
A substantial body of work provides signal propagation and noise models for both active and passive propagation with various modulation schemes [@Akyildiz2013]. Many aspects of MC like addressing and routing have been widely investigated; and advanced communication protocols for transport, network and link layers have just begun to be designed [@Nakano2014]. However, so far, little attention has been given to the physical design of transmitter and receiver devices compatible with MC.
A limited number of works have focused on the signal processing aspects of molecular receiver with the aim of developing optimal detection schemes generally being adapted from the conventional communication theoretical tools [@Kilinc2013]. However, these studies ignore the design of the receiver antenna and the characteristics of the antenna’s output signal which is to be processed. They implicitly assume that the output signal is already suitable, in terms of physical form, amplitude and resolution, for the proposed detection schemes, all of which require fast and complex signal processing with high computational resources. On the other hand, the current predictions on the design of nanomachines with MC ability are based on the synthetic biology approaches, and a widespread consensus is established on this completely bio-inspired vision [@Nakano2012] [@Nakano2014]. However, synthetic biology is currently far away from taking the full control over the functionality of a living cell or designing artificial cells that are capable of operating in a nanonetwork application. Moreover, relying only on synthetic cells limits the application range of MC to *in vivo* operations. It is clear that novel approaches are required to remove the discrepancy between the envisioned applications and the device architectures. At this point, this paper aims to pave the way for the physical design of a nanoscale MC receiver compatible with the proposed communication schemes requiring high computational capacity. An MC receiver should be capable of *in-situ*, continuous, label-free and selective detection of molecular messages that arrive in the close vicinity of the nanomachine on which it is implemented. It can be considered to consist of a biorecognition unit selectively recognizing the information molecules; a transducer unit converting recognition events to a processable signal; and a processor unit analyzing the transducer output to extract the transmitted information based on a preset scheme. Functionality of an MC receiver is very similar to the one of biosensors, which are also designed for the aim of quantifying analyte concentrations in a solution [@Schellera2001]. Therefore, in this paper, we first review the state-of-the-art biosensing approaches to determine whether and in what extent they satisfy the design requirements of an MC receiver.
A careful examination of biosensing options, ranging from optical to mechanical and electrical methods, reveals that field-effect transistor-based electrical biosensors (bioFETs) are quite promising for the design of molecular receiver [@Poghossian2014]. Hence, in the second part of the paper, we focus on a conceptual FET-based molecular receiver. We provide a comprehensive design guideline comprising of the operation principles and the performance metrics of the device. We elaborate on the design options of recognition and transducer units for the optimization of the overall performance from the communication theoretical perspective. Moreover, we present a simple model for a SiNW FET-based MC receiver, and evaluate the receiver performance in terms of sensitivity and Signal-to-Noise Ratio (SNR). We examine integrability of the proposed receiver design into more advanced communication schemes including molecular division multiple access (MDMA), molecular shift keying (MoSK), spatial diversity combining. Lastly, we discuss the challenges before the implementation of the device from a practical aspect, and state the future research directions.
Molecular Communications Receiver {#MCreceiver}
=================================
MC between a single pair of nanomachines can be depicted as in Fig. \[fig:MolCom\], where the molecular messages encoded into concentration propagate through diffusion from the transmitting nanomachine to the receiving nanomachine. Receiver of a nanomachine is responsible for detecting the incoming molecular messages, converting them into a processable signal, and processing the signal for extracting the encoded information. Information may then be used by the nanomachine to realize a prespecified operation. Therefore, the performance of the receiver is critical for the proper functioning of the nanomachine, and thus of the nanonetwork application.
![MC between a pair of nanomachines.[]{data-label="fig:MolCom"}](Kuscu_fig1){width="7.5cm"}
MC receiver slightly differs from an EM communication receiver. Fig. \[fig:Receiver\] demonstrates the block diagram of its envisioned architecture. It basically consists of two subunits, namely, a molecular antenna, and a processing unit which is coupled to the output of the antenna. Molecular antenna is comprised of a recognition unit and a transducer. Recognition unit is the interface between the communication channel and the receiver. Its function is to establish the selectivity only to the molecules that carry information, and thus to minimize the interference from other molecules in the communication environment. Transducer works intimately with the recognition unit, and converts the molecular recognition events into a processable form by means of electrical, optical or chemical signals. Transduced signals, i.e., output signals of the antenna, are simultaneously collected by the processing unit which then may amplify and demodulate the signal to recover the transmitted information based on a preset modulation scheme.
To be operable in an MC application, the receiver should basically satisfy the following requirements:
- ***In situ* operation and in-device processing:** Nanomachines are required to operate in a physically independent manner. If nanomachines are controlled through an external agent, the interface between them should not interrupt the system operation. In the same manner, an MC receiver integrated into a nanomachine should be capable of *in situ* operation. Receiver should detect molecular messages through in-device processing, i.e., without requiring a post-processing of the transduced signals by an external macroscale device or a human controller.
- **Label-free detection:** Receiver should be able to recognize the information carrying molecules based on their intrinsic characteristics, i.e., without requiring a molecular labelling procedure or any other preparation stage.
- **Continuous operation:** Detection should be continuous. This requires the recognition and transduction to be reversibly responsive, i.e., they should return to the initial state after detection to accept succeeding molecular messages without causing a communication error.
- **Nanoscale dimensions:** The components of the receiver should have nanoscale dimensions for the integration of the overall receiver into a nanomachine.
![Functional units of an MC receiver.[]{data-label="fig:Receiver"}](Kuscu_fig2){width="7.5cm"}
In addition to these requirements, for *in vivo* applications, the receiver should also be biocompatible. In other words, the operation of the receiver should not have any detrimental effect on the biological environment, or its performance should not be degraded by the environmental conditions.
Overview of Biosensing Mechanisms {#Overview}
=================================
Developing a biosensor capable of detecting the existence of a target molecule in an environment or quantifying the concentration of a molecular specie has been a long-standing goal of analytical chemistry. For this aim, many different biosensing approaches have been proposed for a plethora of target molecular species ranging from proteins to oligonucleotides, as well as small molecules like glucose [@Perumal2014].
A biosensor, in general, consists of a biorecognition layer and a transducer which converts the recognition events into a processable form. It is not surprising that a biosensor and a molecular receiver have common design principles, since they both aim to analytically recognize the molecules. However, there are also fundamental differences between them. First, it is not sufficient for an MC receiver just to detect the presence of a molecule in an equilibrium state as for the case of a biosensor. The receiver should be capable of detecting the information, which is encoded into a physical property related to the molecules, such as concentration, type, or arrival time, by continuously observing the environment. This requires communication theoretical tools to be applied. Another critical difference arises from the application domain. The biosensor literature is mostly directed to the design of biosensing approaches suitable for laboratory applications that necessitate the operation of macroscale readout devices and human observers due to the lack of an integrated processor. Moreover, biosensors have not yet been completely scaled down to nanoscale dimensions to be integrated into an envisioned nanomachine. Nevertheless, MC receiver design can be inspired from a biosensing method.
Biosensors proposed so far, in general, can be classified into three categories based on the transduction mechanism: electrical, optical, and mechanical biosensors. Electrical biosensing is based on the alterations in the current, voltage, or conductance in the transducer resulting from a chemical reaction or a binding event realized in the recognition unit [@Grieshaber2008]. Similarly, optical sensing exploits the changes in one of the optical characteristics of the transducer upon the recognition. Optical transduction and recognition are generally realized on the same unit. Several optical sensing mechanisms are developed based on surface plasmon resonance (SPR), chemiluminescence, fluorescence and FRET [@Borisov2008]. On the other hand, mechanical sensing benefits from modulation of mechanical resonance with binding of molecules to the transducer surface [@Tamayo2013].
In addition to these engineered sensing methods, a fourth mechanism which is called biochemical or molecular transduction is prevalent in nature as part of the cellular communications [@Cantrell1996]. The mechanism converts the biorecognition events on the cellular membrane into an ion flux through the membrane pores of the cell to trigger a function with the potential change, or the events activate the secondary messengers in the cytoplasm to realize cellular functions.
Considering the basic requirements of the MC receiver, we suggest focusing on electrical biosensing approaches. The reason for eliminating the optical and mechanical biosensing options is that they rely on macroscale detectors that cannot be accepted for an MC receiver considering the requirement of *in situ* operation. Although the biorecognition and transduction units can be downsized to nanoscale, these units should be excited by optical lasers, and the resultant signals, for both optical and mechanical sensing, should be observed through optical detectors to extract the sensing information. Even for bioluminescence-based methods which do not require excitation of the transducer, there is currently no approach towards realizing *in situ* processing of the resultant optical signals mainly due to the diffraction limit of light.
To compare the suitability of electrical and biochemical sensing for an MC receiver, we also need to consider the corresponding nanomachine architectures, which are in the nanobioelectronic and synthetic-cell domain, respectively. In this context, the advantages of electrical sensing are several. First, there is a clear trend towards the miniaturization of electrical processing units into nanoscale. Since electrical biosensors intrinsically provide processable inputs for these devices, they can be readily integrated in nanobioelectronic domain. On the other hand, the assumption of synthetic cell-based nanomachines, which is based on the combination of the biochemical transduction and high processing capabilities, seems very vague. Biochemical sensing implies also biochemical processing. Electrical operations is incomparably faster than biochemical processes, thus, we can assert that electrical sensing also promises for fast processing of the transduced signals. Moreover, electrical sensing can provide the receiver with much more diverse functionality which can be tuned and controlled electrically in a similar way with the conventional counterparts. Also, interfacing with macroscale equipments in a wireless manner is more feasible for the electrical domain. Although synthetic-cell based nanomachines intrinsically provide ultimate biocompatibility; with the advances in bioelectronics, it is not a difficult problem also for the nanobioelectronic devices. Therefore, comparing the state-of-the-art in the two research domains, obviously electrical nanomachines seem more suitable and feasible for the envisaged nanonetwork applications. This leads us to investigate, in more details, the electrical sensing approaches to be adapted for an MC receiver.
Electrical biosensors can be grouped into two classes according to their recognition method: biocatalytic and affinity-based sensors. Biocatalytic recognition employs specific enzymes immobilized on a layer to detect the targets based on enzymatic reactions [@Wang2008]. When a target is bound to an enzyme, an electroactive specie, e.g., hydrogen ion, is generated as a result of the reaction. If the specie achieves to diffuse into the working electrode of the transducer, it modulates one of the electrical characteristics of the device. Commercialized macroscale glucose sensors can be given as example to the biocatalytic electrical sensors. On the other hand, affinity-based sensing is based on the selective binding of certain receptor-ligand pairs on the recognition layer [@Rogers2000].
Biocatalytic techniques require two-step reaction, i.e., enzymatic reaction and consecutive detection of the produced electroactive species. This complicates the procedure as the device gets smaller. Also, generation of additional products may not be desired during the operation. Additionally, the method is limited to specific analytes that generate the electroactive species upon the reaction with the corresponding enzymes. On the other hand, affinity-based sensing uses the nature’s approach for recognition offering a more direct detection scheme. Also the method has proven feasible for a larger range of target molecules with higher sensitivity and selectivity. Moreover, affinity-based biorecognition is more convenient for the current molecular communication models which are mostly based on ligand-receptor binding, thus, provides an easier path to be adapted into the receiver models. Therefore, we focus our attention into affinity-based electrical sensing.
![Conceptual design of FET-based MC receiver.[]{data-label="fig:Design"}](Kuscu_fig3){width="9cm"}
Affinity-based sensing is also classified depending on the detection format in the transducer unit: amperometric, potentiometric, conductometric, and impedance-based [@Rogers2000]. Briefly, amperometric detection is based on the changes in the current; potentiometric detection relies on the change of the potential between two reference electrodes; conductometric detection is based on the modulation of the conductance by the recognition events; and impedance-based detection is a frequency-domain method benefitting from the impedance variations as the result of molecule binding events. These methods are developed primarily for *ex situ* analysis in laboratory conditions; thus, they do not promise for portable and *in situ* applications. Fortunately, advances in nanotechnology in the last decade have led to the design of a new method for affinity-based electrical sensing which is based on FET technology using nanowires, nanotubes, organic polymers, and graphene, as the transducer unit [@Yang2010] [@Curreli2008]. It is based on the modulation of transducer conductivity by the electrostatic effects resulting from the bindings on the recognition layer. FET-based biosensors (bioFETs) promises for label-free, continuous and *in situ* operation in nanoscale dimensions, thus, the design of a bioFET stands as a good starting point for the design of an MC receiver. Hence, in the remaining of the paper, we elaborate on the FET-based molecular receiver.
![Expected time domain response of a p-type doped FET-based MC receiver in terms of drain-to-source current in the transducer channel, when (a) no information molecule is present in the reception space, (b) a negatively charged information molecule binds to a surface receptor, (c) the bound molecule leaves the recognition unit.[]{data-label="fig:Response"}](Kuscu_fig4){width="7.5cm"}
FET-Based Molecular Receiver {#FETbased}
============================
In conventional FET type transistors, current flows between source and drain electrodes through a semiconductor channel whose conductance is controlled via a perpendicular electric field created by a gate electrode. Conductivity is proportional to the carrier density accumulated in the channel by the electric field, and reflected to the changes in the voltage-current characteristics. Replacing the gate electrode with a biofunctionalized surface through the immobilization of affinity-based receptors directly on top of the channel as depicted in Fig. \[fig:Design\], the device becomes a biosensor, namely bioFET.
Binding of charged analytes to the surface receptors results in accumulation or depletion of the carriers on the semiconductor channel, and modulates the channel conductivity; thus, the conductivity becomes a function of the analyte density and the amount of analyte charges. For example, if the channel is p-type doped, negatively charged analytes accumulate the positive hole carriers in the channel when they bind to the receptors; as a result, the conductivity increases depending on the density of the bound analytes as shown in Fig. \[fig:Response\]. The conductivity change is reflected to the current ($I_{DS}$) flowing through the channel if a constant source-to-drain potential ($V_{DS}$) is maintained. For applications in ionic solutions, usually, a reference electrode is immersed in the solution to stabilize the surface potential of the channel on its base level which may be highly fluctuated due to undesirable ionic adsorptions complicating the detection of target analytes [@Nair2007]. It is clear that FET-based biosensing enables direct, label-free, continuous and *in situ* sensing of the molecules by not requiring any complicated processes like labelling of the molecules or the use of any macroscale equipment for readout and processing operations.
Simple operation principles together with the extensive literature on FETs established through many years, electrical controllability of the main device parameters, high-level integrability, and plethora of optimization options for varying applications make FET-based biosensing technology also a quite promising approach for electrical MC receiver. Therefore, we direct our attention to devise a molecular receiver based on the principles of affinity-based bioFETs.
Signal Flow and Noise Sources {#Principles}
-----------------------------
To characterize the operation of a FET-based molecular receiver, we need to investigate the input-output signal relations together with the expected contributions of the noise sources, which result in random fluctuations in the electrical output signal. Noise sources for a FET-based MC receiver can be divided into three categories [@Hassibi2007]:
### Reception Noise
A molecule in the reception space undergoes random motion which is governed by Brownian dynamics, and stochastically binds to the receptors on the recognition layer. The uncertainty in the location and the binding state of the molecules results in random fluctuations of the transduced signal and may severely hamper the instant detection of the concentration, and thus, the molecular messages [@Hassibi2005]. This type of reception noise is well-studied from the MC perspective [@Pierobon2011].
### Transducing noise
Transducing operation also includes contributions from noise sources which can be classified into two types: thermal noise and $1/f$ (flicker) noise [@Rajan2013], [@Deen2006]. Thermal noise is resulting from thermal fluctuations of charge carriers on the bound ligands. On the other hand, $1/f$ noise is caused by the traps and defects in the semiconductor channel, and could be effective at low frequencies where MC receiver is expected to operate. Hence, $1/f$ noise may dominate over other noise sources, and needs a careful investigation.
### Biological Interference
For crowded environments, it is possible that molecules of other species, which show similar affinities for the receptors, exist in the reception space, thus, occasionally bind to the receptors and complicate the detection. This biological interference, which is also termed the background noise, could severely deteriorate the detection performance as the concentration and the charge of the interferers are comparably high [@Hassibi2005]. Biological interference should not be confused with the intersymbol interference (ISI) or co-channel interference which have been already investigated in [@Pierobon2012], [@Pierobon2014], from the aspect of MC. Note that the randomness of interferers’ motion and binding to the receptors also contributes to the total reception noise of the receiver.
Incorporating the contribution of noise sources, a simple signal flow diagram of the receiver can be given as in Fig. \[fig:SignalFlow\], assuming that the binding of interferer molecules and messenger molecules are uncorrelated. In fact, this is a loose assumption; because the receptor-ligand binding kinetics is expected to be affected by the number of bound interferer molecules. However, when the concentration of the receptors in the recognition layer is high enough compared to the total concentration of ligand and interferer molecules in the reception space, this assumption holds true. Then, the electrical output signal can be expressed as follows: $$s(t) = T \overline{X(t)} + T u_x(t) + u_T(t) + \sum_i^m T_{i} \left( \overline{X_{(i)}(t)} + u_{x(i)} \right), \label{OutputSignal}$$ where $X(t)$ is a stochastic state matrix which represents the locations of the ligands, including the conjugated ones, in the reception space [@Hassibi2007]. The matrix relates the molecular density of the reception space to the occupancy rate of the receptors on the recognition layer. $T$ is the transducing vector which operates on the state matrix, and $\overline{X(t)}$ is the ensemble average of the state matrix. The $u_x(t)$ and $u_T(t)$ denote the reception noise and the total transducing noise, respectively. $\overline{X_{(i)}(t)}$ is the ensemble average of the state matrix of the $i$th interferer molecule, and $u_{x(i)}$ is the corresponding reception noise. $T_{i}$ is the transducing vector for the $i$th interferer. Note that, the reception noise resulting from the stochastic binding events is amplified by the transducing operation.
![Signal flow diagram of an MC receiver antenna. The triangles and circles denote the information and interferer molecules, respectively.[]{data-label="fig:SignalFlow"}](Kuscu_fig5){width="9cm"}
Performance Metrics {#Performance}
-------------------
### Sensitivity
Sensitivity is the capability of the receiver to perceive the small differences in the molecular concentration [@Nair2007]. It is a critical performance metric, especially for the design of MC systems which encodes the information into molecular concentrations, e.g., MC with Concentration Shift Keying (CSK). It can be defined as the change in the transducer current per change in the molecular concentration in the reception space of the receiver for a concentration range where linear operation can be assumed. Linear operation corresponds to the operation range where the receptors on the receiver surface are not saturated by the information molecules.
### Selectivity
The capability of the receiver to uniquely detect the information molecules in the reception space, which is probably crowded with interferers, is the measure of its selectivity [@Nair2007]. It is clear that the recognition unit is directly in charge of the selectivity as it sets the affinity with messenger ligands and possible interferers. Higher selectivity implies that the probability of the interferer molecules to bind the receptors is lower than the one of the information carrying molecules. Selectivity is a critical design consideration especially for MoSK-based MC systems [@Kuran2011].
### Signal-to-Noise Ratio (SNR)
SNR at the antenna output determines how easy the receiver can recover the transmitted signal in the presence of noise sources. It is a crucial metric to evaluate the reliability of the detection. It can be simply calculated as the ratio of the received signal power to the total noise power, i.e., $SNR = P_{signal}/P_{noise}$, in a given operation bandwidth of the receiver.
### Limit of Detection
The minimum molecular concentration in the reception space required for the receiver to detect the existence of an information molecule is termed the limit of detection (LoD) [@Rajan2013-2]. It is not to be confused with the minimum concentration required for the exact determination of the concentration. Once the limit is determined, it can be exploited by the transmitter to save on the number of molecules transmitted, especially when the information is encoded into the molecule type as in MoSK [@Kuran2011]. The limit of detection is mainly set by the noise level of the overall reception process and the flow conditions of the environment.
### Temporal Resolution
Temporal resolution defines how fast the receiver is able to sample the molecular concentration in the reception space. Considering that electrical processes are very fast compared to molecular processes, the limiting factor for the temporal resolution is expected to be the diffusion and binding kinetics. In order for the receiver to be able to detect all the messages carried by molecules into reception space, biorecognition should be realized in transport-limited manner. In other words, the binding kinetics should not be a limiting factor on the sampling rate. We further discuss the effect of binding kinetics on the temporal resolution in Section \[Receptors\].
In addition to these metrics, there are some other measures that can be used to quantify the performance of the electrical MC receiver. For example, the amount of information about the received molecular signal contained in the electrical output of the receiver is a quite important metric to determine the detection performance. In the information theory, this is represented by the mutual information between the receiver output and the incoming molecular signals. Mutual information is determined based on the conditional entropy, i.e., equivocation, of the received signal given the receiver’s electrical output. These information theoretical metrics would be especially important for the design of optimal detection algorithms. They are needed for determining the end-to-end communication capacity of an MC system and developing encoding schemes that can achieve this capacity. However, they require a probabilistic model of the overall reception process combined with the random propagation of the information signals in MC channel. Likewise, if the receiver is implemented along with a differential amplifier, Common Mode Rejection Ratio (CMRR), which measures the capability of the device to reject the common modes in the incoming signals, will become an important metric to determine the receiver performance.
Design Options {#Design}
==============
The performance of the receiver should be optimized by the design parameters under different application scenarios. This section elaborates on the main parameters corresponding to the two main receiver components, i.e., the transducer and the biorecognition unit.
Semiconductor Channel {#Channel}
---------------------
Semiconductor channel between source and drain is the main transducing element of the receiver. Therefore, its electrical characteristics are critical for the performance of the receiver. $1/f$ noise is mainly resulting from the defects and traps on the channel. Moreover, the channel geometry is directly relevant to the surface coverage of the receiver, and affects its integrability to a nanomachine. Many nanomaterials have proven suitable for use in bioFET channel, such as silicon nanowires (SiNWs) [@Patolsky2005], single walled carbon nanotubes (SWCNTs), graphene [@Yang2010], molybdenum disulfide (MoS$_2$) [@Sarkar2014], and organic materials like conducting polymers [@Torsi2013]. These materials, some of which are demonstrated in Fig. \[fig:Channel\], are also candidates for the transducer channel of MC receiver.
First generation bioFETs rely on the use of one dimensional materials like SWCNT and SiNW as the channel in a bulk form. However, use in the form of single material or aligned arrays have proved to surpass the performance of the bulk channels in terms of sensitivity and reduced noise [@Curreli2008]. SWCNT-based bioFETs have attracted more attention due to the excellent electrical characteristics of carbon nanotubes which lead to higher sensitivity; however, clean fabrication of SWCNT without defects is the most challenging among all of the candidates, which may hamper its proper practice in an MC receiver [@Maehashi2009]. On the other hand, SiNWs can be fabricated in a relatively easy and controlled manner, with adjustable length, diameter, and doping levels which have direct influence on the material conductance [@Chena2011]. However, SiNWs still possess substantial production challenges due to their 1D nature.
![FET-based MC receiver with different type of transducer channels: (a) SiNW channel, (b) SWCNT chanel, (c) graphene channel.[]{data-label="fig:Channel"}](Kuscu_fig6){width="6cm"}
Graphene has emerged as an alternative with its 2D planar structure providing higher spatial coverage in a single device with the possibility of immobilizing higher number of receptors [@Park2012]. The same 2D structure also leads to a higher sensitivity compared to 1D materials, since all of its atoms are able to closely interact with the bound molecules on the recognition layer. Moreover, intrinsic flexibility of graphene provides higher chance of integration into devices with non-planar surfaces which can be more suitable for the design of nanomachines in an MC application [@Yang2010]. Another 2D alternative for the transduction channel is MoS$_2$ which has been very recently shown to be more sensitive than the graphene channel [@Sarkar2014]. The possession of a bandgap which is not the case for graphene has been pointed out as the reason for the higher sensitivity of MoS$_2$-based bioFETs.
Organic conductive polymers also have been shown to provide similar electronic performances with the inorganic counterparts when used as the semiconducting channel. They can be utilized in both 1D and 2D forms [@Lin2012]. In addition, they provide ultimate biocompatibility which make them more preferable for *in vivo* applications of MC.
The selection among these materials should be carefully made considering the requirements of the MC application [@Yang2010]. Also, the architecture of the overall nanomachine can mandate the use of flexible planar materials. On the other hand, some applications may require more complex receivers with more than one channels for multiplexing purposes which will be discussed in Section \[Advanced\]; and for such cases, 1D materials may be more suitable allowing dense channel deployments.
Bioreceptors and Ligands {#Receptors}
------------------------
Bioreceptor is another design fundamental which is the principal determinant of the receiver’s selectivity. The array of receptors constitutes the interface between the receiver and the MC channel, being the active components of the recognition unit. The design considerations on the receptor molecules cannot be isolated from the information molecules, i.e., ligands; since the modulation of the transducer current is realized as a result of the binding of the ligands and the receptors to each other. The binding reaction can be simply represented as $\ce{N_R + N_L <=>[k_{+}][k_{-}] N_{B}}$, where $N_R$, $N_L$, and $N_B$ denote the number of unbounded receptor, ligand, and receptor-ligand complexes, respectively; $k_+$ and $k_-$ are the association and dissociation reaction rates. The affinity between the ligands and the receptors is quantized by the dissociation constant which can be given by $K_D = k_{-}/k_{+}$. Affinity is inversely proportional to $K_D$. In order for the receiver to be selective only to the ligands that represent the information, the affinity of the receptors for the ligands should be significantly higher than their affinity for the interferer molecules that may exist in the reception space.
Mainly three types of recognition pairs used in affinity-based FET-type biosensors are feasible for an MC application: antibody/antigen (Ab/Ag), natural receptor/ligand, and aptamer/ligand [@Poghossian2014] [@Rogers2000]. These pairs are suitable because they bind to each other reversibly, and their sizes are small enough to be utilized in an MC application.
Antibodies are Y-shaped immune system proteins that are generated to bind to specific antigens which are perceived as the proteins of an harmful microorganism inside the body. Antibodies generally operate with lock-and-key principle, thus, provide high level selectivity for the corresponding antigens [@Curreli2008]. Antibodies are usually generated by introducing antigens into an animal. Therefore, their utilization may face production challenges and biocompatibility risks for *in vivo* applications.
Natural receptors, i.e., neuroreceptors, taste and olfactory receptors, are cellular proteins immobilized within the plasma membrane. They have proven to be successfully implemented on the recognition unit of bioFETs to detect neurotransmitters, taste molecules, and odorants [@Song2014]. Since these receptors naturally operate in living organisms, their utilization in *in vivo* applications do not bring up any biocompatibility problem. Moreover, their ability to detect odorants promises for long range MC applications reviewed in [@Gine2009].
Additionally, aptamers, which are artificial single-stranded DNAs and RNAs, have also been widely utilized as recognition elements in bioFETs to detect a wide range of targets, i.e., ligands, including small molecules, proteins, ions, aminoacids, or other oligonucleotides. An aptamer for a specific ligand is selected through the SELEX process which is based on scanning a large library of DNAs and RNAs to determine a convenient nucleic acid sequence [@Luzi2003]. Availability of almost infinite number of different aptamer-ligand combinations with varying affinities makes aptamer-based recognition promise for highly selective MC receivers.
The characteristics of the ligand-receptor pairs affect the performance of the receiver, especially its sensitivity. The range of distance from the surface of the receiver where the transducer can detect a recognition event is limited because of the screening of the ligand charges by the ions existing in the communication environment [@Nair2008]. The mean effective charge of a free electron on a ligand as observed by the transducer is degraded as the distance between the ligand and the transducer increases. The relation is given by $$q_{eff} = q e^{-\frac{r}{\lambda_D}}, \label{qeff}$$ where $q$ is the elementary charge, and $r$ is the average distance of the ligand electrons to the transducer’s surface [@Rajan2013]. $\lambda_D$ is the Debye length which quantizes the ionic strength of the solution according to the following relation $$\lambda_D = \sqrt{\frac{\epsilon_R k_B T}{2 N_A q^2 c_{ion}}}, \label{eq:debye}$$ where $\epsilon_R$ is the permittivity of the solvent, $k_B$ is the Boltzmann’s constant, $T$ is the temperature, and $N_A$ is Avogadro’s number, $c_{ion}$ is the ionic concentration of the medium [@Rajan2013]. Debye length is a key parameter in determining an appropriate receptor-ligand pair. The distance of the receptor part, where the charged ligand binds, from the transducer surface should not exceed the Debye length. Otherwise, charges of the bound ligands cannot be effectively reflected to transducer current. Therefore, receptors with lengths exceeding the Debye length should be avoided to develop a sensitive receiver. Aptamers and natural receptors with lengths typically smaller than 2nm are advantageous compared to antibodies [@Luzi2003]. On the other hand, antibodies can be shortened using only their fab fragments which are the main recognition elements [@Elnathan2012].
Intrinsic charge of ligands also should be high enough for a proper receiver operation. Most biomolecules like proteins and nucleic acids are highly (and negatively) charged in physiological conditions, but the amount of charge may significantly alter depending on the pH of the environment [@Nair2007].
Additionally, density of the receptors can remarkably affect the performance. It is obvious that dense deployment of the receptors on the receiver surface increases the chance of binding to a ligand. However, the receptor density should be optimized together with the concentration of ligands used for communication. On one hand, if the ligand concentration is very low compared to the receptor density, then, most of the receptors will be unbounded; and thus, prone to the binding of the interferers, which may result in higher background noise. On the other hand, if the information is represented by very high concentrations of ligands compared to the receptor density, then, the recognition unit may face saturation, and the linear operation range may be disturbed, which then complicates the detection.
A Modeling Approach for SiNW FET-Based MC Receiver {#Numerical}
==================================================
In Section \[Principles\], we gave an expression in to describe the signal flow for the operation of a general MC receiver. That expression includes a transducer vector and a stochastic state matrix denoting the random locations and binding/unbinding states of information molecules. Although it gives an intuition for the receiver operation, it is only useful for simulation-based approaches, and it does not lead to the derivation of analytical expressions for deterministic signal flow to evaluate the performance of the receiver.
In this section, based on the existing modeling efforts in the literature [@Berezhkovskii2013] [@Kuscu2015c], we present an analytical model of molecular signal reception for a SiNW FET-based MC receiver and evaluate its performance in terms of mean electrical response, sensitivity and SNR. For modulation of molecular signals, we assume that CSK is utilized such that the messages are represented by different levels of ligand concentration. The receiver detects the concentration-encoded messages by observing the transducer channel current.
Model of Biorecognition Unit
----------------------------
For the analytical model of biorecognition unit, following assumptions are made:
Diffusion of ligands are assumed to be fast enough such that the reception is not mass transport limited. Therefore, the ligands are uniformly distributed in the reception space, and all of the receptors are always exposed to the same concentration of ligands.
Biological interference of any chemically similar molecule is neglected.
These assumptions are prevalent in MC and biosensor studies [@Akyildiz2013] [@Pierobon2011], and lead to a pseudo-first order ligand-receptor dynamics, where the first time derivative of the mean number of bound receptors at time $t$ can be expressed by $$\frac{d \overline{N_B (t)}}{dt} = k_+ c_L^R(t)(N_R - \overline{N_B(t)}) - k_- \overline{N_B(t)}, \label{eq:dNb/dt}$$ where $k_+$ and $k_-$ are the intrinsic association and dissociation rate constants of the receptor-ligand complex, respectively. $N_R$ is the total number of receptors on the bioFET surface, and $c_L^R(t)$ is the ligand concentration in the vicinity of the recognition layer, i.e., in the reception space.
Since we assume reaction-limited operation for simplicity, we can neglect the transient phase between different levels of ligand concentration such that $c_L^R(t) = c_i$ for $t \in [t_i, t_i+1/B)]$, where $c_i$ is the ligand concentration corresponding to the $i$th message, $t_i$ is the transition time from the $(i-1)$th message to the $i$th message in reception space, and $1/B$ is the symbol duration with $B$ being the symbol transmission rate. In other words, biorecognition layer of the receiver is assumed to be exposed to a constant concentration $c_i$ for $t \in [t_i, t_i+1/B)]$.
Given the initial condition $\overline{N_B(t_i-\epsilon)} = N_{B,i-1}$ with $\epsilon \rightarrow 0$, the solution of can be given as [@Berezhkovskii2013] [@Kuscu2015c] $$\begin{gathered}
\overline{N_B(t)} = \overline{N_{B,i}^{ss}} + \left( N_{B,i-1} - \overline{N_{B,i}^{ss}} \right) e^{-(k_+ c_i + k_-) (t-t_i)} \\ \text{for}\ t \in [t_i, t_i+1/B),\end{gathered}$$ where $\overline{N_{B,i}^{ss}}$ is the mean number of bound receptors at steady-state, i.e., when $d\overline{N_B(t)}/dt = 0$. We can infer from this equation that although the ligand concentration level immediately changes in the reception space, it takes a certain time for the receptors to adapt the concentration level of the new message. The time to reach steady-state is governed by the reaction timescale $\tau_B = (k_+ c_i + k_-)^{-1}$; thus, for higher ligand concentrations, the adaptation time of the recognition layer decreases. The mean number of bound receptors at steady-state is given by $$N_{B,i}^{ss} = \frac{k_+ c_i}{k_+ c_i + k_-} N_R = \frac{c_i}{c_i + K_D} N_R, \label{ss}$$
Receiver is assumed to sample the state of receptors at steady-state; thus, the mean number of occupied receptors corresponding to the $i$th message can be given as $\overline{N_{B,i}} = \overline{N_{B,i}^{ss}}$. The occupation states of the receptors fluctuate around the mean value even in the steady-state. This phenomenon is referred to as receptor noise, which we discussed in Section \[Principles\]. As being a summation of Bernoulli random variables, the number of bound receptors at steady-state follows the well-known binomial distribution, variance of which is given by $$Var(N_{B,i}) = \frac{k_- k_+ c_i}{(k_+ c_i + k_-)^2} N_R. \label{eq:variance}$$ The autocorrelation function for the stationary fluctuations of the number of bound receptors at steady-state can be approximated with a single exponential [@Bialek2005]: $$R(\tau) = Var(N_{B,i}) e^{-\frac{\tau}{\tau_B}}, \label{eq:acf}$$ where the characteristic timescale of ligand-receptor binding $\tau_B$ is also being the correlation time of binding noise. The Fourier Transform of gives the Power Spectral Density (PSD) of the fluctuations: $$S_{\Delta N_B}(f) = Var(N_{B,i}) \frac{2 \tau_B}{1+(2 \pi f \tau_B)^2}. \label{eq:SNb}$$
Model of Transducer Unit
------------------------
The charged ligands bound to the surface receptors induce opposite charges on the NW surface to some extent depending on their intrinsic charge. The mean amount of charge generated by the bound ligands for the $i$th message is given by $Q_i = \overline{N_{B,i}} \, N_e \, q_{eff}$, where $N_e$ is the average number of free electrons per ligand molecule. $q_{eff}$ is the mean effective charge induced by a single electron on a ligand molecule which is given in , where the mean vertical distance of ligand electrons at the bound state to the transducer surface is assumed equal to the length of the receptor molecule $r = L_R$.
![Equivalent circuit model for the transducer of the SiNW FET-based MC receiver [@Deen2006] [@Kuscu2015c] [@Spathis2015]. *REF* denotes the reference electrode, which stabilizes the gate voltage $V_G$.[]{data-label="fig:Circuit"}](Kuscu_fig7){width="7cm"}
The induced charges on the NW surface are translated into the variation of the surface potential through the equivalent circuit of the transducer [@Deen2006] [@Kuscu2015c] [@Spathis2015], which is demonstrated in Fig. \[fig:Circuit\]. By neglecting the current through $R_{layer}$, i.e., resistance of the layer of bound ligands, which is on the order of tens of $G \Omega$s [@Spathis2015] [@Bang2008], the mean potential difference resulting from the bound ligands can be written as $$\overline{\Delta \Psi_{i}} = \frac{Q_i}{C_{eq,i}},$$ where $C_{eq,i}$ is the overall capacitance of the equivalent circuit: $$\begin{gathered}
C_{eq,i} = \left((C_{ox} W L)^{-1} + (C_{s} W L)^{-1}\right)^{-1}\\ + \left(C_{rec}^{-1} + C_{layer,i}^{-1} + (C_{dl} W L)^{-1}\right)^{-1}.\end{gathered}$$ Here, $C_{ox}$, $C_{s}$ and $C_{dl}$ are the silicon oxide (SiO$_2$), the semiconductor, i.e. SiNW, and the double layer capacitances per unit area, respectively; $C_{rec}$ and $C_{layer,i}$ are the capacitances of the receptor layer and the layer of bound ligands when the $i$th message is received, respectively; and $W$ and $L$ are the width and length of the transducer’s active region. $C_{ox} = \epsilon_{ox}/t_{ox}$ with $\epsilon_{ox}$ and $t_{ox}$ being the permittivity and the thickness of the oxide layer. $C_{rec} = N_R \times C_{mol,R}$, $C_{layer,i} = \overline{N_{B,i}} \times C_{mol,L}$ with $C_{mol,R}$ and $C_{mol,L}$ being the capacitance of a single receptor and a single ligand molecule, respectively.
Size of active region ($W \times L$) $0.1 \times 5$ ($\mu m$)
--------------------------------------------------------------------- -------------------------------------------
Temperature ($T$) $298$ ($K$)
Relative permittivity of SiO$_2$ layer ($\epsilon_{ox}/\epsilon_0$) $3.9$
Thickness of SiO$_2$ layer ($t_{ox}$) $17.5$ ($nm$)
Effective mobility ($\mu_{eff}$) $16 \times 10^{-3}$ ($m^2 V^{-1} s^{-1}$)
Drain-source voltage ($V_{DS}$) $0.1$ ($V$)
Relative permittivity of solvent ($\epsilon_R/\epsilon_0$) $78$
Ionic concentration of medium ($c_{ion}$) $70$ ($mM$)
Trap density ($N_t$) $2.3 \times 10^{24}$ ($eV^{-1} m^{-3}$)
Tunneling distance ($\lambda$) $0.05$ ($nm$)
Average net charge of ligands ($N_e$) $4$
Length of receptor ($L_R$) $4$ ($nm$)
Binding rate ($k_+$) $2 \times 10^{-18}$ ($m^3 s^{-1}$)
Unbinding rate ($k_-$) $10$ ($s^{-1}$)
Ligand concentration in reception space ($c_i$) 4$K_D$
Concentration of receptors on the surface ($c_R$) $2 \times 10^{16}$ ($m^{-2}$)
Molecular capacitance ($C_{mol,L}, C_{mol,R}$) $2 \times 10^{-20}$ ($F$)
Capacitance of dielectric layer ($C_{dl}$) $5 \times 10^{-2}$ ($F/m^2$)
Capacitance of silicon ($C_s$) $2 \times 10^{-3}$ ($F/m^2$)
: Simulation Parameters
\[table:parameters\]
The deviation of the surface potential is reflected into a change in the FET device threshold as $\overline{\Delta V_{TH,i}} = \overline{\Delta \Psi_i}$. We know that the drain-to-source current of FETs in the linear operation regime can be written as $$I_{DS} = \frac{W}{L} \mu_{eff} C_{ox} (V_{GS} - V_{TH}) V_{DS}, \label{eq:Is}$$ where $\mu_{eff}$ is the effective carrier mobility in the transducer channel, and $V_{DS}$ is the drain-to-source voltage, which is held constant [@Rajan2013-2], $V_{GS}$ is the gate-to-source voltage, which is stabilized by the reference electrode. In case the threshold voltage is shifted, the mean variation reflected to $I_{DS}$ can be given by $$\overline{\Delta I_{DS,i}} = \overline{\Delta V_{TH,i}} \frac{W}{L} \mu_{eff} C_{ox} V_{DS}, \label{eq:Is}$$ We can simply express the mean current variation by $\overline{\Delta I_{DS,i}} = g_m \overline{\Delta V_{TH,i}} = g_m \Delta \Psi_i$, where $g_m$ is the transconductance of the device, which can be given as $$g_m = \frac{W}{L} \mu_{eff} C_{ox} V_{DS}, \label{eq:conductance}$$ The processor unit of the receiver uses the signal $\Delta I_{DS,i}$, to infer the incoming ligand concentration $c_i$; and thus the molecular message $i$, based on a predefined CSK scheme.
As discussed in Section \[Principles\], random diffusion of free electrons on the layer of bound ligands results in thermal noise. The uncertainty in the location of electrons are reflected to fluctuations in the threshold voltage of bioFET antenna. Using the thermal noise model derived in [@Spathis2015], the PSD of voltage fluctuations on the layer is given by $S_{\Delta V_{TH,i}^{T,layer}} = 4 k_B T R_{layer,i}$, where $R_{layer,i}$ is the mean resistance of the layer when the $i$th message is received. The fluctuations are reflected to the threshold voltage through the RC filter given in Fig. \[fig:Circuit\]. Using the its transfer function, the PSD of the thermal noise contribution on $\Delta V_{TH,i}$ is written as [@Kuscu2015c] [@Spathis2015]
$$S_{\Delta V_{TH,i}^T}(f) = S_{\Delta V_{TH,i}}^{T,layer} \left( 1 + \left[2 \pi R_{layer,i} C_{eq}' f\right]^2 \right)^{-1}, \label{eq:thermalpsd}$$with $C_{eq}' = C_{layer,i}+\left[(C_{dl}^{-1}+C_{ox}^{-1}+C_s^{-1}) (WL)^{-1}+C_{rec}^{-1}\right]^{-1}.$ In addition to the receptor and thermal noise, the low-frequency operation of bioFET-based molecular antenna is also suffered from $1/f$ noise. Using the well-known number fluctuation model [@Kuscu2015c] [@Rajan2013-2], the PSD of $1/f$ noise can be written as follows $$S_{\Delta V_{TH}^F}(f) = \frac{\lambda k T q^2 N_t}{W L C_{ox}^2 |f|},$$ where $\lambda$ is the characteristic tunneling distance, and $N_t$ is the trap density of the NW channel. $1/f$ noise is independent of the received signals, and shows an additive behavior on the overall threshold voltage fluctuations [@Spathis2015].
Performance Analysis
--------------------
In this section, we analyze the receiver performance in terms of expected response, sensitivity and SNR. Table \[table:parameters\] lists the default parameter values used in the analysis. Considering that most of the nanonetwork applications are envisioned for *in vivo* scenarios, the parameter values pertaining to medium characteristics are selected based on the physiological conditions. For example, the default value of ionic concentration, $c_{ion}$, is selected according to the reported value for bovine serum (70mM) [@Okada1990]; but in the sensitivity and SNR analyses, we investigate the performance for ionic concentrations corresponding to a wide range of solutions including highly diluted solutions ($c_{ion}$=1mM) and human blood plasma ($c_{ion}>$100mM) [@Duan2012]. The solvent is water by default in physiological conditions, which implies the relative permittivity, $\epsilon_R/\epsilon_0 = 78$ [@Hediger2012]. The employed receptors on the FET surface are considered to be aptamers, the production process of which provides full control over the selection of length, binding and unbinding rates, as well as the type of corresponding ligand molecules (see Section \[Receptors\]). The default length of receptors is set to 4nm, which corresponds to 12 base pair-aptamers. Binding and unbinding rates, $k_+$ and $k_-$, are set, considering the accepted values in the MC literature [@Pierobon2011] and the range of rates that aptamers can provide [@Luzi2003]. Aptamers can bind to a large set of ligands, such as, aptamers, small proteins, RNA and DNA, and even non-organic molecules, which can attain a broad range of elementary charges, as will be discussed in Section \[ReceiverResponse\]. The capacitance of receptors and ligands, $C_{mol,R}$ and $C_{mol,L}$, are selected based on the reported values for oligonucleotides and small molecules [@Lu2008], which correspond to a mean value of $2 \times 10^{-20}$F. The relative permittivity of SiO$_2$ layer is reported as $\epsilon_{ox}/\epsilon_0 = 3.9$ [@Deen2006]. The thickness of the SiO$_2$ layer, $t_{ox}$, is a design parameter, for which we select a default value of $17.5$nm corresponding to the design in [@Deen2006]. However, we carry out the analyses for a wide range of $t_{ox}$ values considering that the thickness can be reduced to below $5$nm, as reported in [@Kobayashi1998]. Depending on the fabrication, the tunneling distance for SiO$_2$ is on the order of 0.01-0.1nm [@Rajan2011a]. We set $\lambda = 0.05$nm as reported in [@Deen2006], which also reports the effective mobility as $16 \times 10^{-3}$m$^2/$Vs. The capacitance of dielectric layer, $C_{dl}$, and silicon substrate, $C_s$ are design parameters for which we set the default values as reported in [@Deen2006]. The trap density $N_t$ depends on the quality of fabrication, and can attain a wide range of values [@Rajan2011a]. As default, we set a moderate value of $2.3 \times 10^{24}$eV$^{-1}$m$^{-3}$. In the analyses, we also discuss the experimental results obtained in [@Duan2012] and compare them with our numerical results.
### Receiver Response {#ReceiverResponse}
Fig. \[fig:response\_Ne\], presents the receiver response in terms of the expected variation of $\Delta V_T$, which is the mean variation of the channel current normalized by the device transconductance $g_m$, as a function of the average number of elementary charges that ligands possess. As is seen, the net charge of ligands critically affects the receiver response, since the transducer operation is based on the field effect generated by the ligand charges. As reviewed in Section \[Receptors\], oligonucleotides are negatively charged in physiological conditions, i.e., at pH 7.4, due to their highly charged phosphate backbone. For example, a DNA sequence with 4 base-pairs at pH 7.4 can attain a net charge of $-8e$. Likewise, small proteins and antigens, depending on the pH of the environment, can attain a net charge of up to $\pm 4$e [@Tsai2006].
Employing highly charged ligands as information molecules is not sufficient to obtain a proper receiver performance, since the ligand charges are substantially screened by the ions present in the surrounding medium. The major determinant of this so-called Debye screening is the ionic concentration of the medium, i.e., $c_{ion}$. In Fig. \[fig:response\_cion\], we present the receiver response for different values of $c_{ion}$. As it is clear, the shift in the threshold voltage significantly increases with decreasing ionic concentration. 20-fold increase in the threshold voltage is obtained when the ions in the solution is diluted to 1mM from 70mM concentration, which is the ionic concentration of bovine serum [@Okada1990]. Note that physiological conditions for human body generally imply ionic concentrations higher than 100mM [@Rajan2013].
Third analysis is performed for varying concentration of surface receptors. As is seen from Fig. \[fig:response\_cR\], deploying receptor molecules more densely on the surface of the semiconductor channel, increases the receiver response almost linearly. However, size restrictions and possible correlations among densely deployed receptors, which are not captured by this model, should be accounted for in a real world implementation. Lastly, we analyze the effect of oxide layer thickness $t_{ox}$, which is the main determinant of the oxide capacitance $C_{ox}$, on the receiver response. Fig. \[fig:response\_tox\] demonstrates the results for conventional values of $t_{ox}$. As can be inferred, higher $t_{ox}$ implies a better receiver performance.
The response of a typical SiNW bioFET, which is the base of the proposed receiver, to varying ligand concentration can also be observed from the experimental results presented in Fig. \[fig:exp\]. These results have been acquired from [@Duan2012], where the authors employed a SiNW bioFET to quantify the affinity of DNA sequences with HMGB1 proteins. HMBG1 proteins of an unspecified concentration were deployed on the SiNW surface, and varying concentration of DNAs as the charged ligands at pH 7.4 were introduced into the reception space of the bioFET through a microfluidic system. The association on the surface through the ligand-receptor binding dynamics as the DNAs were supplied at a constant rate with different concentrations, and the dissociation after the steady-state was reached, can be observed from Fig. \[fig:exp1\]. The binding dynamics of DNA and HMGB1 were confirmed to be reaction-limited; thus, it does not violate the well-mixed assumption made for the analytical model of biorecognition layer. Unfortunately, many of the critical system parameters, such as the concentration of the surface receptors, the average ligand charge, the average receptor length, each of which substantially affects the device response, were not provided in that study. Thus, we are not able to make a one-to-one comparison of the response given in Fig. \[fig:exp2\] with the numerical results obtained using the model provided in this paper. Nevertheless, the experimental results clearly show the same saturation trend observed for the device response and very well fall into the range of the numerical results obtained in our analysis.
### Sensitivity Analysis
Sensitivity measures the responsivity of the receiver to the varying concentration of ligands, thus, determines how successful the receiver is in discriminating the messages encoded into different ligand concentrations. It can be defined as the derivative of with respect to input concentration $c_i$:
$$S(c_i) = \frac{K_D N_e q_{eff} g_m (N_R C_{mol,L} C_{eq,i} C_{p,i}^2 -1)}{(K_D + c_i)^2 C_{mol,L} C_{eq,i}^2 C_{p,i}^2}, \label{eq:sensitivity}$$ where $C_{p,i} = C_{rec}^{-1} + C_{layer,i}^{-1} + (C_{dl} W L)^{-1}$. Note that $C_{eq,i}$ and $C_{p,i}$ are functions of input concentration $c_i$.
Fig. \[fig:SEN\] demonstrates the sensitivity with the input concentration normalized by the dissociation constant $K_D$, such that the normalized sensitivity in the results correspond to the amount of increase in the output current due to a unit increase in the normalized concentration $c_i/K_D$. As can be inferred from Fig. \[fig:SENc\], the sensitivity substantially decreases as the concentration level in the reception space increases. This is because the receptors on the biorecognition layer are saturated at higher ligand concentrations, thus become insensitive to the changes in the input. The results presented in Fig. \[fig:SENcion\] and Fig. \[fig:SENLr\] reveal the effect of Debye screening such that the sensitivity decreases with increasing ion concentration and with increasing receptor length. Since it is not always possible to control the ionic concentration of the medium, it is critical to employ surface receptors with lengths smaller than the Debye length to obtain a highly sensitive receiver antenna. Physiological conditions generally imply ionic concentrations higher than $100mol/m^3$, which decrease the Debye length below $1nm$. As discussed in Section \[Receptors\], aptamers and natural receptors with smaller sizes are advantageous compared to antibodies to be employed as receptors in *in vivo* applications. The last analysis in Fig. \[fig:SENtox\] demonstrates that the thickness of the oxide layer $t_{ox}$ has also a significant effect on the sensitivity. As $t_{ox}$ increases, the oxide capacitance $C_{ox}$ decreases, which directly reduces the transconductance $g_m$ as given in . Fortunately, control over the oxide layer thickness is possible, and $t_{ox}$ values lower than $5nm$ are reported in the literature [@Kobayashi1998].
### SNR Analysis
The receptor noise, which is the fluctuations in the number of bound receptors at steady-state, is reflected to the threshold voltage fluctuations by the relation $S_{\Delta V_{TH}^B}(f) = S_{\Delta N_B}(f) V_m^2$, where $V_m = (N_e q_{eff})/C_{eq,i}$ is the mean deviation in the threshold voltage resulting from the binding of a single ligand. Receptor and thermal noise can be assumed uncorrelated, since they are largely separated in frequency domain [@Spathis2015]. Including additive $1/f$ noise, overall PSD of the $\Delta V_{TH}$-referred noise can be given by [@Kuscu2015c] $$S_{\Delta V_{TH}}(f) = S_{\Delta V_{TH}^B}(f) + S_{\Delta V_{TH}^T}(f) + S_{\Delta V_{TH}^F}(f).$$ Additive noise effective on $\Delta V_{TH}$ is reflected to channel current noise via $S_{\Delta I_{DS}}(f) = S_{\Delta V_G}(f) g_m^2$. Assuming a resistance of 1$\Omega$ for the channel, we formulate the receiver SNR as follows $$SNR = \frac{I_{DS}^2}{\int_{-\infty}^{\infty} S_{\Delta I_{DS}}(f) df}. \label{eq:SNRI}$$ Using , we investigate the effect of different system parameters on the SNR of the receiver’s electrical output. SNR for varying ligand concentration corresponding to different symbols is plotted in Fig. \[fig:SNRc\], which clearly shows that SNR is significantly improved with increasing concentration. However, it begins to saturate at around $25$dB due to the saturation of the surface receptors for high ligand concentrations. The effect of ionic strength of the fluidic medium on the output SNR is given in Fig. \[fig:SNRcion\]. When the ionic concentration increases above 100mol/m$^3$, the Debye length decreases below 1nm resulting in substantial screening of ligand charge. Therefore, SNR significantly decreases with increasing ionic strength. We also investigate the effect of receptor length on the SNR when the ionic strength is 70mM which makes the Debye length equal to 1.15nm. As seen in Fig. \[fig:SNRLr\], SNR in dB decreases linearly as the receptor length increases. Lastly, we analyze the SNR for varying trap density which is inversely proportional to the purity of the transducer channel. Trap density increases the $1/f$ noise, which is very effective in the frequency range of the antenna’s operation. As is shown in Fig. \[fig:SNRNt\], the effect of trap density on the $1/f$ noise, and thus, on the SNR, is evident especially for $N_t > 10^{24}$eV$^{-1}$m$^{-3}$. Fortunately, experimentally reported trap densities for SiNW bioFETs are on the order of $10^{22}$eV$^{-1}$m$^{-3}$ [@Rajan2011a].
Advanced Design Issues {#Advanced}
======================
Molecular Division Multiple Access (MDMA) and MoSK
--------------------------------------------------
Use of different types of molecules brings up several opportunities to improve the performance of MC. MoSK, where messages are encoded into the type of molecules, is one of the two robust modulation techniques envisioned for MC [@Kuran2011]. Combining MoSK scheme with CSK, i.e., representing messages by both molecular concentration and type, multiplies the alphabet size, thus, can boost the communication rate. Moreover, employing various types of molecules for encoding can enable a particular code division multi-access scheme termed Molecular Division Multiple Access (MDMA) [@Gine2009].
All of these modulation and multiple access schemes require for different messages to be discriminated by the receiver. The receiver architecture based on bioFETs can be adapted to each of these schemes through one of the following modifications:
- Employing multiple antennas with different types of receptors each corresponding to a different ligand as shown in Fig. \[fig:MDMA\](a), receiver can separately process the outputs of different antennas, and thus, easily decode different messages. Hence, it does not require a complex signal processing for detection; however, it implies a larger area of deployment, and more energy consumption which multiplies with the number of antennas.
- Employing a single antenna is convenient if a significant diversity exists in efficient charges, $N_e q_{eff}$, among different ligands. The receptors can be of the same type if the ligands have similar affinity to the same receptor; otherwise, for each of the ligand type, different receptors should be deployed on the recognition layer as shown in Fig. \[fig:MDMA\](b). It is clear that employing a single antenna is more energy-efficient and space-saving compared to the other architecture. However, for the case of MDMA where different signals received at the same time are superposed at the output, this architecture requires more complex signal processing on the electrical signal to discriminate the contributions of different messages [@Horesh2011].
![Receiver antennas for MDMA and MoSK schemes.[]{data-label="fig:MDMA"}](Kuscu_fig12){width="7.5cm"}
Diversity Combining
-------------------
The size of nanomachines could be significantly larger compared to a single receiver or transmitter antenna, and they are generally envisioned as motile undergoing both translational and rotational motion, which means that the distance between the transmitter and the receiver antennas of communicating nanomachines may alter notably during the operation. The distance between transmitter and receiver antennas has strong effect on the attenuation of molecular signals [@Akyildiz2013]. Hence, employing multiple receiver antennas with spatial diversity on a single nanomachine can lead to important diversity gains and reductions of the transmission delays. Several diversity combining schemes like selective combining, maximum ratio combining are applicable for the electrical receiver [@Meng2012].
Energy Efficiency Schemes
-------------------------
Nanomachines are envisioned to have limited power; thus, energy consumption of the receiver should be minimized. This objective can be realized by optimizing the hardware, or employing energy-efficient scheduling algorithms. Since the receiver is electrically powered, a nanomachine has the control on its operation times and sampling periods. Therefore, most of the conventional energy-efficient schemes in the literature can also be incorporated to the molecular receiver [@Akyildiz2002]. However, power saving modes of receiver’s operation like sleep, idle, standby, and active modes, need to be defined in the context of the nanocommunications application.
EM-Molecular Hybrid Operation
-----------------------------
Graphene and CNT, with extraordinary electromagnetic (EM) properties, have also proven to be promising candidates for the design of transceiver antennas in nanoscale EM communications [@Jornet2013a]. Designing the MC receiver with graphene or CNT-based transducer channels can allow a hybrid receiver architecture that can operate both in MC and EM nanocommunications. This hybrid design could enable novel applications. For example, the hybrid receiver can act as a nanoscale gateway that connects an *in vivo* molecular nanonetwork to an external macroscale network, e.g., Internet, through an EM-based communication channel to enable Internet of Nano Things (IoNT) [@Akyildiz2015] [@Kuscu2015]. In a similar manner, two distant molecular nanonetwork can be connected with an EM link. However, additional challenges should be addressed. For example if the intention is to design a hybrid receiver antenna, the effect of biofunctionalization on the electromagnetic characteristics of the transducer channel should be analyzed.
Challenges {#Challenges}
==========
Fabrication Challenges
----------------------
Nanoscale semiconductors, which constitute the transducer channel, can be fabricated following any of the two fundamental approaches: bottop-up or top-down [@Lu2013]. Both approaches impose unique challenges to the proper fabrication of semiconductors. Although top-down methods are more costly compared to bottom-up synthesis, they provide more advanced control enabling mass production and compatibility with CMOS fabrication technologies promising for large-scale integration. However, top-down methods are known to be more amenable to producing defects and traps on channel surface, leading to higher level of $1/f$ noise [@Rajan2013-2]. On the other hand, bottom-up methods mostly rely on stochastic processes and do not allow an entire control over fabrication, leading to reproducibility problems, which is the main challenge to produce bioFETs with homogeneous characteristics [@Matsumoto2013].
Immobilization of bioreceptors on the semiconductor channel can be realized through covalent or non-covalent binding, depending on the type of semiconductor and receptor molecules [@Lu2013]. Surface binding of receptors is a random process, and it is not possible with the current state-of-the-art technologies to exactly control the number of immobilized receptors and their orientation on the surface. This implies another reproducibility problem that should be accounted for while implementing a bioFET-based receiver.
The effect of imperfect fabrication and the inevitable randomness in main system parameters, such as the concentration of surface receptors and the net charge of ligands, could be reflected as noise processes into the analytical model.
Scaling and Integration Challenges
----------------------------------
Main functional units of the bioFET-based MC receiver, which are the biorecognition layer and the semiconductor channel, are already made of nanoscale materials; and thus, they do not pose a problem for scaling of the receiver. The major components that set the limitation for the receiver size are the source and drain contacts together with the reference electrode which is generally used for the aim of stabilizing the surface potential for improved detection performance [@Rajan2013-2].
It is an ongoing challenge to miniaturize the reference electrode by keeping its stable operation. For the aim of integrating bioFETs to small scale on-chip devices, pseudo reference electrodes has been introduced with smaller sizes compared to a true reference electrode. However, they compromise voltage stability to some extent, and need to be further scaled down to work on a nanoscale device [@Rajan2013-2]. Recently, a promising approach has been proposed in [@Jain2012] to completely remove the need for a reference electrode. Combining the piezoelectric effect with the bioFET concept, the authors came up with a novel architecture named Flexure-FET which achieves to detect molecular concentration with high sensitivity, even by not requiring for ligands to be charged.
Application Challenges
----------------------
A major challenge arises when the receiver is employed in an *in vivo* application. The physiological conditions imply solutions with high ionic concentrations, abundance of interferers and contaminants, and existence of disruptive flows and fluctuating temperature, which may degrade the receiver’s performance in several aspects. First, high ionic concentration, which is $\sim$100mM for physiological conditions, creates a strong screening effect reducing the Debye length to $\sim$1nm, and thus, impedes the sensitivity of the receiver [@Rajan2013]. Moreover, interferers that have affinities for the receptors could create strong background noise; and contaminants and disruptive flows may alter the binding kinetics, impede the stability of the receptors, even separate them from the dielectric layer.
Even though using highly charged ligands and very small size receptors like aptamers can solve the screening problem in theory, recently proposed frequency domain technique promises for much more realistic solutions. In [@Zheng2010], the authors have shown that the power spectrum of binding events is Lorentzian-shaped, and easy to distinguish from the $1/f$ noise in frequency domain, because of the weakening of $1/f$ noise at high frequencies. They reveal that frequency domain detection outperforms the conventional time domain technique in terms of sensitivity in highly ionic solutions.
Future Research Avenues {#Future}
=======================
Progress in biosensor research has been mostly based on empirical studies, and the proposed designs generally lack the complementary analytical models. This leads to a problem for reproducibility of experimental results, and thus, to a plethora of studies that are not consistent with each other. There are only a few attempts to analytically model a biosensor, most of which are cited in this paper [@Hassibi2007] [@Rajan2013-2] [@Kuscu2015c]. However, these works together with our model assume that the ligands to be detected are in equilibrium with receptors. This is acceptable in the context of biosensors, because they are usually envisioned to be operated in laboratory conditions as being immersed in pre-prepared solutions with long incubation times. However, for the case of communications, considering the continuous and *in situ* operation of the envisioned nanonetwork applications, this assumption could be violated when the bandwidth of the incoming signal is comparably high or the diffusion effects cannot be neglected. Therefore, the first research avenue should be development of comprehensive analytical models both for time and frequency domains that capture the transient dynamics, the effect of environmental conditions, also the mobility of nanomachines.
A limited number of works attempted to develop realistic simulation frameworks for diffusion-based MC and bioFETs. The current MC simulators are designed based on NS-3, and are mainly focused on the communication channel [@Llatser2014]. The models used in the simulators assume ideal MC receivers with perfect sampling of concentration. On the other hand, bioFET simulators are developed mainly for the purpose of proper evaluation and classification of experimental results [@Hediger2012]. These frameworks lack underlying analytical models, thus, are not able to provide the required design and optimization tools. At this point, a simulation framework for bioFET-based MC receivers needs to be developed using the analytical models; and then, the framework should be combined with the MC simulators, which are continuously being improved, to provide a more realistic simulation environment for end-to-end MC.
Analytical models and simulation frameworks should be accompanied by experimental validations. This requires the development of receiver prototypes based on bioFET architectures optimized for communication purposes. Microfluidics provides an invaluable platform to transport molecules in a controlled fashion. It has proven feasible for conducting MC experiments with controllable system parameters [@Bicen2013], which makes microfluidics a promising means of further optimizing the MC receiver prototypes in real communication scenarios.
Conclusion
==========
Physical design of a molecular receiver has been elaborated for the first time in the literature. Detection of messages in a molecular communication system mandates the label-free and *in situ* sensing of molecules. A comprehensive review of the state-of-the-art biosensing methods revealed that the basic requirements of an MC receiver can be met only with a design in nanobioelectronic domain. FET-based biosensing mechanism is particularly favoured as the underlying mechanism of MC receiver due to the fast, reversible and simple operation. We conceptually presented the main operation principles of a FET-based MC receiver, and evaluated the reception performance in terms of sensitivity and SNR. These preliminary results demonstrating the reliable reception of molecular messages are encouraging for the further development of nanobioelectronic MC receiver. Future research avenues include the derivation of an improved analytical model for the receiver operation comprising the transient kinetics of molecular binding and transportation dynamics, optimization of the receiver performance for the envisioned applications through simulations and experiments, development of optimal detection schemes, and the integration of the receiver into a nanomachine.
[1]{} I. F. Akyildiz et al., “Nanonetworks: A new communication paradigm," *Comput. Netw.,* vol. 52, no. 12, pp. 2260-2279, 2008.
M. Kuscu et al., “Fluorescent molecules as transceiver nanoantennas: The first practical and high-rate information transfer over a nanoscale communication channel based on FRET," *Sci. Rep.,* vol. 5, pp. 7831, 2015.
T. Nakano et al., “Molecular communication and networking: opportunities and challenges," *IEEE Trans. Nanobiosci.,* vol. 11, no. 2, pp. 135-148, 2012.
M. S. Kuran et al., “Modulation techniques for communication via diffusion in nanonetworks," in *Proc. IEEE ICC,* Kyoto, Japan, June 2011.
M. Pierobon and I. F. Akyildiz, “Capacity of a diffusion-based molecular communication system with channel memory and molecular noise," *IEEE Trans. Inf. Theory,* vol. 59, no. 2, pp. 942-954, 2013.
T. Nakano et al., “Molecular communication among biological nanomachines: A layered architecture and research issues," *IEEE Trans. Nanobiosci.,* vol. 13, no. 3, pp. 169-197, 2014.
D. Kilinc and O. B. Akan, “Receiver design for molecular communication," *IEEE J. Sel. Areas Commun.,* vol. 31, no. 12, pp. 705-714, 2013.
F. W. Schellera et al., “Research and development in biosensors," *Curr. Opin. Biotechnol.,* vol. 12, no. 1, pp. 35-40, 2001.
A. Poghossian and M. J. Schoning, “Label-free sensing of biomolecules with field-effect devices for clinical applications," *Electroanalysis,* vol. 26, no. 6, pp. 1197-1213, June 2014.
V. Perumal and U. Hashim, “Advances in biosensors: Principle, architecture and applications," *J. Appl. Biomed.,* vol. 12, no. 1, pp. 1-15, 2014.
D. Grieshaber et al., “Electrochemical biosensors - Sensor principles and architectures," *Sensors,* vol. 8, no. 3, pp. 1400-1458, 2008.
S. M. Borisov and O. S. Wolfbeis, “Optical biosensors," *Chem. Rev.,* vol. 108, no. 2, pp. 423-461, 2008.
J. Tamayo et al., “Biosensors based on nanomechanical systems," *Chem. Soc. Rev.,* vol. 42, no. 3, pp. 1287-1311, 2013.
D. Cantrell, “T cell antigen receptor signal transduction pathways," *Annu. Rev. Immunol.,* vol. 14, pp. 259-274, 1996.
J. Wang, “Electrochemical glucose biosensors," *Chem. Rev.,* vol. 108, no. 2, pp. 814–825, 2008.
K. R. Rogers, “Principles of affinity-based biosensors," *Mol. Biotechnol.,* vol. 14, no. 2, pp. 109-129, 2000.
W. Yang et al., “Carbon nanomaterials in biosensors: Should you use nanotubes or graphene?," *Angew. Chem. Int. Ed.,* vol. 49, no. 12, pp. 2114-2138, 2010.
M. Curreli et al., “Real-time, label-free detection of biological entities using nanowire-based FETs," *IEEE Trans. Nanotechnol.,* vol. 7, no. 6, pp. 651-667, 2008.
P. R. Nair and M. A. Alam, “Design considerations of silicon nanowire biosensors," *IEEE Trans. Electron Devices,* vol. 54, no. 12, pp. 3400-3408, 2007.
A. Hassibi et al., “On noise processes and limits of performance in biosensors," *J. Appl. Phys.,* vol. 102, pp. 014909, 2007.
A. Hassibi et al., “Biological shot-noise and quantum-limited signal-to-noise ratio in affinity-based biosensors," *J. Appl. Phys.,* vol. 97, pp. 084701, 2005.
M. Pierobon and I. F. Akyildiz, “Noise analysis in ligand-binding reception for molecular communication in nanonetworks," *IEEE Trans. Signal Process.,* vol. 59, no. 9, pp. 4168-4182, 2011.
M. Pierobon and I. F. Akyildiz, “Intersymbol and co-channel interference in diffusion-based molecular communication," in *Proc. IEEE ICC,* Ottowa, Canada, 2012.
M. Pierobon and I. F. Akyildiz, “A statistical–physical model of interference in diffusion-based molecular nanonetwork," *IEEE Trans. Commun.,* vol. 62, no. 6, pp. 2085-2095, 2014.
N. K. Rajan et al., “Performance limitations for nanowire/nanoribbon biosensors," *Wiley Interdiscip. Rev. Nanomed. Nanobiotechnol.,* vol. 5, no. 6, pp. 629-645, 2013.
M. J. Deen et al., “Noise considerations in field-effect biosensors," *J. Appl. Phys.,* vol. 100, pp. 074703, 2006.
N. K. Rajan, “Limit of detection of silicon bioFETs," Ph.D. dissertation, Yale Univ., New Haven, CT, USA, 2013.
J.-Q. Lu and X.-G. Zhang, “Nucleotide capacitance calculation for DNA sequencing," *Biophys. J.,* vol. 95, no. 9, pp. 60-62, 2008.
F. Patolsky and C. M. Lieber, “Nanowire nanosensors," *Mater. Today,* vol. 8, no. 4, pp. 20–28, 2005.
D. Sarkar et al., “MoS$_2$ field-effect transistor for next-generation label-free biosensors," *ACS Nano,* vol. 8, no. 4, pp. 3992–4003, 2014.
L. Torsi et al., “Organic field-effect transistor sensors: A tutorial review," *Chem. Soc. Rev.,* vol. 42, no. 22, pp. 8612-8628, 2013.
K. Maehashi and K. Matsumoto, “Label-free electrical detection using carbon nanotube-based biosensors," *Sensors,* vol. 9, no. 7, pp. 5368-5378, 2009.
K.-I Chena et al., “Silicon nanowire field-effect transistor-based biosensors for biomedical diagnosis and cellular recording investigation," *Nano Today,* vol. 6, no. 2, pp. 131-154, 2011.
S. J. Park et al., “Ultrasensitive flexible graphene based field-effect transistor (FET)-type bioelectronic nose," *Nano Lett.,* vol. 12, no. 10, pp. 5082–5090, 2012.
P. Lin and F. Yan, “Organic thin-film transistors for chemical and biological sensing," *Adv. Mater.,* vol. 24, no. 1, pp. 34-51, 2012.
H. S. Song et al., “Bioelectronic tongue using heterodimeric human taste receptor for the discrimination of sweeteners with human-like performance," *ACS Nano,* vol. 8, no. 10, pp. 9781–9789, 2014.
L. P. Gine and I. F. Akyildiz, “Molecular communication options for long range nanonetworks," *Comput. Netw.,* vol. 53, pp. 2753–2766, 2009.
E. Luzi et al., “New trends in affinity sensing: aptamers for ligand binding," *TrAC Trends Anal. Chem.,* vol. 22, no. 11, pp. 810-818, 2003.
P. R. Nair and M. A. Alam, “Screening-limited response of nanobiosensors," *Nano Lett.,* vol. 8, no. 5, pp. 1281–1285, 2008.
R. Elnathan et al., “Biorecognition layer engineering: Overcoming screening limitations of nanowire-based FET devices," *Nano Lett.,* vol. 12, no. 10, pp. 5245–5254, 2012.
A. M. Berezhkovskii and A. Szabo, “Effect of ligand diffusion on occupancy fluctuations of cell-surface receptors," *J. Chem. Phys,* vol. 139, pp. 121910, 2013.
M. Kuscu and O. B. Akan, “Modeling and analysis of SiNW bioFET as molecular antenna for bio-cyber interfaces towards the Internet of bio-nanothings," in *Proc. IEEE WF-IoT 2015,* Milan, Italy, Dec. 2015.
W. Bialek and S. Setayeshgar, “Physical limits to biochemical signaling," *P. Natl. Acad. Sci. USA,* vol. 102, no. 29, pp. 10040–10045, 2005.
K. Georgakopoulou et al., “Modeling of fluctuation processes on the biochemically sensorial surface of silicon nanowire field-effect transistors," *J. Appl. Phys.,* vol. 117, pp. 104505, 2015.
G. S. Bang et al., “High-fidelity formation of a molecular-junction device using a thickness-controlled bilayer architecture," *Small,* vol. 4, no. 9, pp. 1399–1405, 2008.
M. Okada et al., “Ionic strength affects diffusive permeability to an inorganic phosphate ion of negatively charged dialysis membranes," *ASAIO Trans.,* vol. 36, no. 3, pp. M324-7, 1990.
X. Duan et al., “Quantification of the affinities and kinetics of protein interactions using silicon nanowire biosensors," *Nat. Nanotechnol.,* vol. 7, pp. 401-407, 2012.
M. R. Hediger et al., “BioFET-SIM web interface: Implementation and two applications," *PLoS ONE,* vol. 7, no. 10, pp. e45379, 2012.
H. Kobayashi et al., “Oxide thickness dependence of energy shifts in the Si 2*p* levels for the SiO$_2$/Si structure, and its elimination by a palladium overlayer," *Appl. Phys. Lett.,* vol. 73, no. 7, pp. 933-935, 1998.
N. K. Rajan et al., “Optimal signal-to-noise ratio for silicon nanowire biochemical sensors," *Appl. Phys. Lett.,* vol. 98, no. 26, pp. 264107, 2011.
C. S. Tsai, *Biomacromolecules: Introduction to Structure, Function and Informatics*, Hoboken, NJ: John Wiley & Sons, Inc, 2006.
M. Horesh et al., “A temperature-differential affinity biosensor: model and $D$-optimal performance limits," *IEEE Sensors J.,* vol. 11, no. 9, pp. 2007-2015, 2011.
L.-S. Meng et al., “MIMO communications based on molecular diffusion," in *Proc. IEEE GLOBECOM,* pp. 5380 - 5385, Anaheim, CA, USA, December 2012.
I. F. Akyildiz et al., “A survey on sensor networks," *IEEE Commun. Mag.,* vol. 40, no. 8, pp. 102-114, 2002.
J. M. Jornet and I. F. Akyildiz, “Graphene-based plasmonic nano-antenna for terahertz band communication in nanonetworks," *IEEE J. Sel. Area. Comm.* vol. 31, no. 12, pp. 685-694, 2013.
I. F. Akyildiz et al., “The internet of Bio-Nano things," *IEEE Commun. Mag.* vol. 53, no. 3, pp. 32-40, 2015.
M. Kuscu and O. B. Akan, “The Internet of molecular things based on FRET," *IEEE Internet Things J.,* to be published.
A. Matsumoto and Y. Miyahara, “Current and emerging challenges of field effect transistor based bio-sensing," *Nanoscale,* vol. 5, no. 22, pp. 10702-10718, 2013.
N. Lu et al., “Ultra-sensitive nucleic acids detection with electrical nanosensors based on CMOS-compatible silicon nanowire field-effect transistors," *Methods,* vol. 63, no. 3, pp. 212-218, 2013.
A. Jain et al., “Flexure-FET biosensor to break the fundamental sensitivity limits of nanobiosensors using nonlinear electromechanical coupling," *PNAS,* vol. 109, no. 24, pp. 9304–9308, 2012.
G. Zheng et al., “Frequency domain detection of biomolecules using silicon nanowire biosensors," *Nano Lett.,* vol. 10, pp. 3179–3183, 2010.
I. Llatser et al., “N3Sim: Simulation framework for diffusion-based molecular communication nanonetworks," *Simul. Model. Pract. Th.,* vol. 42, pp. 210-212, 2014.
A. O. Bicen and I. F. Akyildiz, “System-theoretic analysis and least-squares design of microfluidic channels for flow-induced molecular communication," *IEEE Trans. Signal Process.,* vol. 61, no. 20, pp. 5000-5013, 2013.
[^1]: The authors are with the Next-generation and Wireless Communications Laboratory (NWCL), Department of Electrical and Electronics Engineering, Koc University, Istanbul, 34450, Turkey (e-mail: {mkuscu, akan}@ku.edu.tr).
[^2]: This work was supported in part by the European Research Council (ERC) under grant ERC-2013-CoG \#616922.
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The discovery of superconductivity below 1 K within a limited pressure range in the ferromagnet UGe$_2$ [@Saxena00; @Tateiwa01; @Huxley01; @Bauer01] provides an unanticipated example of coexistence of superconductivity and strong ferromagnetism. The electronic pairing mechanism needed for superconductivity is believed to be magnetic in origin. However, it is amazing that ferromagnetically ordered uranium magnetic moments with so large magnitude ($\sim 1.4 \, \mu_{\rm B}$ at ambient pressure as deduced from magnetization measurements) are directly involved [@Ginsburg57]. Since the pairing must involve the conduction electrons, it is important to characterize their magnetic properties. Because of the restrictions imposed by the magnetic form factor, this can not be done by diffraction techniques. As the muons localize in interstitial sites, they have the potentiality to yield information on the conduction electrons. Here we show, using the muon spin relaxation technique, that UGe$_2$ is actually a dual system where two sub-states of $f$ electrons coexist. We indeed report the existence at ambient pressure of itinerant long-range magnetic correlations with magnetic moments of $\sim 0.02 \, \mu_{\rm B}$ and a spectral weight in the megahertz range. A quantitative understanding of this state is moreover reached assuming that these correlations involve only long wavelength fluctuation modes.
UGe$_2$ is a ferromagnet with a Curie temperature $T_{\rm C} \simeq 52$ K which crystallizes in the orthorhombic ZrGa$_2$ crystal structure (space group Cmmm) [@Oikawa96; @Boulet97]. Magnetic measurements indicate a strong magnetocrystalline anisotropy [@Menovsky83; @Onuki92; @Huxley01] with easy magnetization axis along the ${\bf a}$ axis.
We present results obtained by the muon spin relaxation ($\mu$SR) technique. Fully polarized muons are implanted into the studied sample. Their spin (1/2) evolves in the local magnetic field, ${\bf B}_{\rm loc}$, until they decay into positrons. Since the positron is emitted preferentially in the direction of the muon spin at the decay time, it is possible to follow the evolution of the muon spin polarization [@Dalmas97; @Amato97]. The measured physical parameter is the so-called asymmetry which characterizes the anisotropy of the positron emission. Below $T_{\rm C}$, if ${\bf B}_{\rm loc}$ has a component perpendicular to the initial muon beam polarization, ${\bf S}_\mu$ (taken parallel to $Z$), we expect the asymmetry to display spontaneous oscillations with an amplitude maximum for ${\bf B}_{\rm loc} \perp {\bf S}_\mu$. On the other hand, if ${\bf B}_{\rm loc} \parallel {\bf S}_\mu$, the asymmetry can be written as the product of an initial asymmetry related to sample, $a_{\rm s}$, and the muon spin relaxation function, $P_Z(t)$, which monitors the dynamics of ${\bf B}_{\rm loc}$.
UGe$_2$ crystals were grown from a polycrystalline ingot using a Czochralski tri-arc technique [@Menovsky83]. We present results for two samples. Each consists of pieces cut from the crystals, put together to form a disk and glued on a silver backing plate. They differ by the orientation (either parallel or perpendicular) of the ${\bf a}$ axis relative to the normal to the sample plane. The measurements were performed at the EMU spectrometer of the ISIS facility, from 5 K up to 200 K, mostly in zero-field. Additional $\mu$SR spectra were recorded with a longitudinal field.
We found that the temperature dependence of $a_{\rm s}$ for ${\bf S}_\mu \parallel {\bf a}$ is consistent with ${\bf B}_{\rm loc} \parallel {\bf a}$. In agreement with that conclusion, a spontaneous muon spin precession resulting in wiggles in the asymmetry is observed for ${\bf S}_\mu \perp {\bf a}$. Defining $T_{\rm C}$ as the temperature at which the wiggles disappear, we found $T_{\rm C}$ = 52.49 (2) K. This value coincides with the maximum of the relaxation rate (to be evidenced below) for ${\bf S}_\mu \parallel {\bf a} $ and ${\bf S}_\mu \perp {\bf a} $.
In this letter we focus on the description of data taken around the Curie point.
All the spectra were analyzed as a sum of two components: $a P_Z^{\rm exp}(t) = a_{\rm s} P_Z(t) + a_{\rm bg}$. The first component describes the $\mu$SR signal from the sample and the second accounts for the muons stopped in the background, [*i.e.*]{} the cryostat walls and sample holder. In zero-field, for all relevant temperatures and for the two orientations of ${\bf S}_\mu$ relative to ${\bf a}$, $P_Z(t)$ is well described by an exponential function: $P_Z(t)$ = $\exp(- \lambda_Z t)$ where $\lambda_Z$ measures the spin-lattice relaxation rate at the muon site. An example is shown in Fig. \[fig\_spectra\]. $a_{\rm bg}$, which is basically temperature independent, was measured for ${\bf S}_\mu \perp {\bf a}$ and $T< T_{\rm C}$ as the constant background signal. We got $a_{\rm bg}$ = 0.077 [@notebg]. For ${\bf S}_\mu \parallel {\bf a}$, it could only be estimated from the sample size since the relaxation was never strong enough to measure it directly. We took $a_{\rm bg}$ = 0.064. The uncertainty on this $a_{\rm bg}$ leads to an uncertainty on the absolute value of $\lambda_Z({\bf S}_\mu \parallel {\bf a})$ of $\sim 10 \%$.
=82 mm
In Fig. \[lambda\] we display $\lambda_Z(T)$ measured in zero-field for ${\bf S}_\mu \perp {\bf a}$ and ${\bf S}_\mu \parallel {\bf a}$. For both geometries, $\lambda_Z(T)$ exhibits a maximum at $T_{\rm C}$. It is due to the critical slowing down of the spin dynamics. Surprisingly the anisotropy between the orientations is very weak although UGe$_2$ is known to be extremely anisotropic [@noteani]. Furthermore we show in the following lines that $\lambda_Z(T)$ near $T_C$ is quantitatively understood in the framework of the Heisenberg model with dipolar interactions, whereas UGe$_2$ is considered as an Ising system. The magnetic signal that we observe has therefore a different origin than the well documented uranium magnetic state observed e.g. by macroscopic measurements.
$\lambda_Z(T)$ has been computed several years ago [@Yaouanc93] for the critical regime of dipolar Heisenberg ferromagnets and has been successfully compared to experiments [@Yaouanc93; @Yaouanc96; @Henneberger99]. It is based on the derivation of the static and dynamical scaling laws from mode coupling theory [@Frey94]. The two scaling variables at play depend on two material parameters: $\xi_0$, the magnetic correlation length at $T=2 T_{\rm C}$, and $q_D$, the dipolar wave vector which is a measure of the strength of the exchange interaction relative to the dipolar energy. This model initially derived for the paramagnetic phase applies also below $T_C$ [@Yaouanc96].
Specifically, the model predicts that $\lambda_Z(T)$ = ${\cal W}[a_{\rm L} I^{\rm L}(T) + a_{\rm T} I^{\rm T}(T)]$ where $I^{\rm {L,T}}$ [@Dalmas94] are scaling functions obtained from mode coupling theory and $a_{\rm {L,T}}$ are parameters determining respectively the amount of longitudinal (L) and transverse (T) fluctuations probed by the measurements. The L,T indices denote the orientation relative to the wave vector of the fluctuation mode. $a_{\rm L,T}$ only depend on muon site properties. The result of the fit of $\lambda_Z(T)$ is shown in the inserts of Fig. \[lambda\]. The divergence of $\lambda_Z$ at $T_{\rm C}$ is strongly reduced by the effect of the dipolar interaction [@Frey94]. The temperature scale gives the product $q_{\rm D} \xi_0$ [@Yaouanc93]. For ${\bf S}_\mu \perp {\bf a}$, we get $q_{\rm D} \xi_0$ = 0.021 (2), and for ${\bf S}_\mu \parallel {\bf a}$, $q_{\rm D} \xi_0^+$ = 0.043 (2) and $q_{\rm D} \xi_0^-$ = 0.020 (2). The index $+$ ($-$) on $\xi_0$ specifies that we consider the paramagnetic (ferromagnetic) state. $ \xi_0({\bf S}_\mu \parallel {\bf a}) >
\xi_0({\bf S}_\mu \perp {\bf a})$ in the paramagnetic state, suggesting that the magnetic correlations are somewhat anisotropic. The fact that $\xi_0^+ > \xi_0^-$ is an expected feature [@Dalmas97]. The relaxation rate scale yields ${\cal W}^+ a_{\rm L}$ = 0.140 (4) MHz and ${\cal W}^- a_{\rm L}$ = 0.20 (2) MHz for ${\bf S}_\mu \parallel {\bf a}$. The transverse contribution to $\lambda_Z$ for both $T < T_{\rm C}$ and $T > T_{\rm C}$ is more difficult to estimate since $a_{\rm T}$ is found much lower than $a_{\rm L}$. Reasonable fits are obtained with $a_{\rm T}/a_{\rm L}$ = 0.036 (14). We have computed $a_{\rm L}$ and $a_{\rm T}$ for different possible muon sites and found only one site satisfying $a_{\rm T} < a_{\rm L}/2$. This is site $2b$ (in Wyckoff notation) of coordinates (0, 1/2, 0) for which $a_{\rm L}$ = 1.2486, $a_{\rm T}$ = 0.0386. We then deduce ${\cal W}^+$ = 0.112 (3) MHz and ${\cal W}^-$ = 0.161 (16) MHz. The scale deduced from the measurements with ${\bf S}_\mu \perp a$ is about twice as large, pointing out again to the weak anisotropy of the magnetic correlations.
In order to further characterize the relaxation near $T_{\rm C}$, we performed at a given temperature longitudinal field measurements for the two orientations of ${\bf S}_\mu$ relative to ${\bf a}$. The field responses for the two geometries are similar. An illustration is given in Fig. \[fig\_spectra\]. Surprisingly, the spectra are field dependent at extremely low $B_{\rm ext}$, proving that the probed magnetic fluctuations are quasi-static (fluctuation rate in the MHz range) and since $\lambda_Z$ is small, the associated magnetic moment must be small as well. Quantitatively, the field dependence of $P_Z(t)$ can not be described consistently either by a simple exponential relaxation form (see the lower panel of Fig. \[fig\_spectra\]) nor by a relaxation function computed with the strong collision model assuming an isotropic Gaussian component field distribution [@Hayano79]. On the other hand, the relaxation is well explained if we assume that the distribution of the local field at the muon, $B_{\rm loc}$, is squared-Lorentzian [@Walker80]. We write $P_Z(t)$ = $P_Z(\Delta_{\rm Lor}, \nu_f ,t)$ where $\Delta_{\rm Lor}$ characterizes the width of the field distribution and $\nu_f$ its fluctuation rate [@Uemura85]. A global fit of the spectra ($B_{\rm ext}$ = 0, 0.2, 0.4, 0.6, 0.8, 1.0 and 2.0 mT) taken at a given temperature is possible. For ${\bf S}_\mu \perp {\bf a}$ at $T$ = 52.59 (2) K, the description of the seven spectra is done with $\Delta_{\rm Lor}$ = 70 $\mu$T and $\nu_f$ = 0.10 MHz. For ${\bf S}_\mu \parallel {\bf a}$ at $T$ = 52.47 (2) K the two parameters are $\Delta_{\rm Lor}$ = 40 $\mu$T and $\nu_f$ = 0.50 MHz: the zero-field spectra have therefore been recorded in the motional narrowing limit ($\nu_f/ (\gamma_\mu \Delta_{\rm Lor}) > 1$ where $\gamma_\mu$ is the muon gyromagnetic ratio; $\gamma_\mu$ = 851.6 Mrad s$^{-1}$T$^{-1}$). This justifies the formalism used to treat $\lambda_Z(T)$ close to $T_{\rm C}$.
=82 mm
We now present an interpretation of our results.
We first note that the detected fluctuations can not arise directly from the localized uranium $5f$ electrons since $\nu_f$ would then be in the THz window as estimated from $\nu_f \simeq k_{\rm B} T_{\rm C} /\hbar$, rather than in the MHz range as measured. We also already mentioned that the observed $\mu$SR signal has not the properties expected from the known macroscopic properties. These apparently conflicting results can be understood if the $5f$ electrons are viewed as two electron subsets. This picture has already been argued for UCu$_5$ [@Schenck90] and UPd$_2$Al$_3$ [@Caspary93; @Feyerherm94; @Metoki98; @Bernhoeft98; @Sato01]. However, for UGe$_2$ the signatures of both subsets are found at a single temperature, the Curie temperature, whereas for UCu$_5$ and UPd$_2$Al$_3$ the temperatures at which the two subsets are detected are far apart. So UGe$_2$ presents a novel variant of the two electron subset model. Within this picture, the anisotropy of the magnetization arises from the localized $5f$ spectral density and the magnetic fluctuations probed by $\mu$SR is a signature of the band-like electrons. We do not detect the signature of the localized $5f$ electrons, because of the strong motional narrowing of the related relaxation rate. It is observed for the ferromagnet YbNiSn; see Ref. .
The effect of the dipolar interaction on the quasi-elastic linewidth, $\Gamma (q)$, of the fluctuations has already been observed for the weak itinerant ferromagnet Ni$_3$Al [@Semadeni00]. In particular, at criticality $\Gamma (q) \propto q^{5/2}$, as expected from scaling [@Frey94]. Thus it is not completely surprising to detect its influence on $\lambda_Z(T)$ for the band-like electrons of UGe$_2$. Quantitatively, the data have been described in the established framework of critical dynamics . We shall now prove that the detected magnetization density arises entirely from long wavelength, [*i.e.*]{} small $q$, fluctuations. The magnetic properties of weak itinerant ferromagnets are explained with the latter hypothesis [@Lonzarich85; @Moriya85]. In our model the values of ${\cal W}$ and $\nu_f$ and of the magnitude of the band-like uranium magnetic moment, $m_{\rm U}$, are controlled by two wavevectors: $q_{\rm D}$ already introduced and the cut-off wavevector, $q_c$, which sets the upper bound for the wavevector of the fluctuations involved in the build-up of the magnetization density. For simplicity we consider that the magnetic properties of this electronic subset are isotropic. We shall detail the analysis of the data taken with ${\bf S}_\mu \parallel {\bf a}$. The same approach works equally well for the data recorded with ${\bf S}_\mu \perp {\bf a}$. As explained below, we get an overall consistent picture setting $q_{\rm D} = 1.0 \, \times 10^{-3}$ Å$^{-1}$ and $q_c = 0.1 $ Å$^{-1}$.
The magnetization arising from the conduction electrons can be viewed as a stochastic variable with a variance $\langle (\delta{\cal M})^2 \rangle$. From the fluctuation-dissipation (Nyquist’s) theorem , $\langle (\delta{\cal M})^2 \rangle$ obeys the sum rule $$\begin{aligned}
\langle (\delta{\cal M})^2 \rangle = {3 \, k_{\rm B} T \over 2\pi^2\mu_0}
\int_0^{q_c} \chi (q) q^2 {\rm d}q,
\label{mag}\end{aligned}$$ if the energy of the magnetic fluctuations is smaller than the thermal energy. $\mu_0$ is the permeability of free space. Assuming an Ornstein-Zernike form for the wavevector dependent susceptibility, $\chi (q)$, and since $q_{\rm D}$ is very small, $\langle (\delta{\cal M})^2 \rangle \simeq$ $3\, k_{\rm B} T \, q_{\rm D}^2\, q_c / (2 \pi^2 \mu_0)$. Since $m_{\rm U} = v_0 \sqrt{\langle (\delta{\cal M})^2 \rangle }$ where $v_0$ is the volume per uranium atom ($v_0$ = 61.6 Å$^3 $), we infer $m_{\rm U} = 0.02 \, \mu_{\rm B}$ at $T_{\rm C}$. Interestingly, the analysis of polarized neutron scattering data suggests for the conduction electrons a magnetic moment of $0.04\,(3) \, \mu_{\rm B}$ at low temperature [@neutron].
The scale ${\cal W}$ for $\lambda_Z$ can then be computed within the framework presented above. Numerically, from Eq. 5.10c of Ref. , we get ${\cal W}$ = 0.16 MHz, close to the measured values. With the same theory, $\hbar \Gamma (q) = \Omega q^{5/2}$ with $\Omega$ = 18 meVÅ$^{2.5}$ at criticality and for small $q_{\rm D}$ (see Eq. 4.14b of Ref. ). Since the measured dynamics is mainly driven by the fluctuations at $q_{\rm D}$ [@Dalmas94], we estimate $\nu_f \simeq \Gamma (q_{\rm D})$ = 0.87 MHz, not far from the measured value.
We now discuss the magnitude of $\Delta_{\rm Lor}$. If the distribution of $B_{\rm loc}$ was Gaussian, the zero-field width of the distribution would be $\Delta_{\rm Gauss}$ = 1.7 mT for muon at site $2b$, and $m_{\rm U} = 0.02 \, \mu_{\rm B}$ computed using the Van Vleck type formalism of Ref.[@Hayano79]. However the distribution is squared-Lorentzian rather than Gaussian. Such a distribution is observed in systems with diluted and disordered magnetic moments [@Walker80]. According to Uemura [*et al.*]{} , $\Delta_{\rm Lor} = \sqrt{\pi/2} \, c \, \Delta_{\rm Gauss}$ where $c$ is the concentration of moments at the origin of the distribution. This relation leads to $c$ = 1.9 %, consistent with the usual fact that a tiny fraction of the total number of valence electrons are able to contribute to the magnetic susceptibility.
From the $q_{\rm D}$ value we derive the exchange interaction. We obtain $2 {\cal J} = k_{\rm B} T_{\rm C} /4.2$ (see Eq. 4.4b of Ref. and Ref. ). For comparison, the same method gives $2 {\cal J}/ k_{\rm B} T_{\rm C} = 1/11$ and $1/20$ for metallic Fe and Ni, respectively. Therefore the evaluation of the exchange energy is quite reasonable. From the measured product $q_{\rm D}\xi_0^+$, we get $\xi_0^+ \simeq 43$ Å. This means that the interaction between the itinerant magnetic moments is relatively long-range, even far outside the critical regime. Although about an order of magnitude larger than for conventional ferromagnets, $\xi_0^+ $ compares favorably with the neutron result for Ni$_3$Al: $\xi_0^+$ = 24 (9) Å[@Semadeni00]. For the same compound, we derive from that $q_c$ = 0.2 Å$^{-1} $, a value twice as large as found for UGe$_2$. The moment carried by the itinerant electrons is about four times smaller for UGe$_2$ than for Ni$_3$Al [@Semwal99]. Nearest neighbor U atoms form zigzag chains parallel to ${\bf a}$ [@Huxley01]. This may lead to magnetic frustration and thus explains the disordered nature of the distribution of $B_{\rm loc}$. A proper understanding of the origin of the squared-Lorentzian distribution requires more work.
One may question the uniqueness of our interpretation. We first note that the observed $\lambda_Z$ can not arise from an impurity phase since the measured critical dynamics occurs right at the well known Curie temperature of UGe$_2$. It could be argued that the observed signal is the signature of a weak disorder in the uranium magnetic moments. This has already been seen in UAs [@Asch89] where the $\mu$SR signal below the Néel temperature has been attributed to a diluted source of small magnetic moments. Their quasi-static nature is related to the absence of spin excitations. However, the moments we observe in UGe$_2$ are quasi-static even above $T_{\rm C}$.
In conclusion we have shown that at ambient pressure UGe$_2$ is a dual system where an electronic subset of itinerant states coexist with the subset of localized $5f$ electrons responsible for the well known bulk magnetic properties. Its associated magnetic moment is quite small and characterized by a very slow spin dynamics. A quantitative picture for that subset is achieved by assuming that only fluctuations at long wavelength are at play. It would be of interest to follow the small moment itinerant state as a function of pressure to determine whether the Cooper’s pairs arise from it. However, it seems difficult to perform that task with $\mu$SR, unless the spin-lattice relaxation rate increases appreciably at high pressure.
S.S. Saxena [*et al.*]{}, Nature [**406**]{}, 587 (2000).
N. Tateiwa [*et al.*]{}, J. Phys.: Condens. Matter [**13**]{}, L17 (2001).
A. Huxley [*et al.*]{}, Phys. Rev. B [**63**]{}, 144519 (2001).
E.D. Bauer [*et al.*]{}, J. Phys.: Condens. Matter [**13**]{}, L759 (2001).
V.L. Ginsburg, Sov. Phys. JETP [**4**]{}, 153 (1957).
K. Oikawa [*et al.*]{}, J. Phys. Soc. Jpn. [**65**]{}, 3229 (1996).
P. Boulet [*et al.*]{}, J. Alloys Compounds [**247**]{}, 104 (1997).
A. Menovsky [*et al.*]{}, in [*High Field Magnetism*]{}, page 189, (ed. Date, M.) (North-Holland, Amsterdam, 1983).
Y. Onuki [*et al.*]{}, J. Phys. Soc. Jpn. [**61**]{}, 293 (1992).
P. Dalmas de Réotier and Yaouanc, J. Phys.: Condens.Matter [**9**]{}, 9113 (1997).
A. Amato, Rev. of Mod. Phys. [**69**]{}, 1119 (1997).
In a previous short report $\lambda_Z(T)$ was presented for ${\bf S}_\mu \perp a$. The values were slightly smaller, reflecting the uncertainty on $a_{\rm bg}$ which was estimated and not measured directly.
A. Yaouanc [*et al.*]{}, Physica B [**259-261**]{}, 126 (1999).
When present, the anisotropy of the critical magnetic fluctuations can be observed; see for example P.C.M. Gubbens [*et al.*]{}, Hyp. Int. [**85**]{}, 245 (1994).
A. Yaouanc, P. Dalmas de Réotier, and E. Frey, Phys. Rev. B [**47**]{}, 796 (1993).
A. Yaouanc [*et al.*]{}, Phys. Rev. B [**53**]{}, 350 (1996).
S. Henneberger [*et al.*]{}, Phys. Rev. B [**60**]{}, 9630 (1999).
E. Frey and F. Schwabl, Adv. Phys. [**43**]{}, 577 (1994).
P. Dalmas de Réotier, A. Yaouanc, and E. Frey Phys. Rev. B [**50**]{}, 3033 (1994).
R.S. Hayano [*et al.*]{}, Phys. Rev. B [**20**]{}, 850 (1979).
L.R. Walker and R.E. Walstedt, Phys. Rev. B [**22**]{}, 3816 (1980).
Y.J. Uemura [*et al.*]{}, Phys. Rev. B [**31**]{}, 546 (1985).
A. Schenck [*et al.*]{}, Phys. Rev. Lett. [**65**]{}, 2454 (1990).
R. Caspary [*et al.*]{}, Phys. Rev. Lett. [**71**]{}, 2146 (1993).
R. Feyerherm [*et al.*]{}, Phys. Rev. Lett. [**73**]{}, 1849 (1994).
N. Metoki [*et al.*]{}, Phys. Rev. Lett. [**80**]{}, 5417 (1998).
N. Bernhoeft [*et al.*]{}, Phys. Rev. Lett. [**81**]{}, 4244 (1998).
N.K. Sato [*et al.*]{}, Nature [**410**]{}, 340 (2001).
F. Semadeni [*et al.*]{}, Phys. Rev. B [**62**]{}, 1083 (2000).
G.G. Lonzarich, and L. Taillefer, J. Phys. C [**18**]{}, 4339 (1985).
T. Moriya, [*Spin Fluctuations in Itinerant Electron Magnetism*]{} (Springer Verlag, Berlin, 1985).
see [*e.g.*]{} R.M. White, [*Quantum Theory of Magnetism*]{} (McGraw-Hill, New York, 1970).
N. Kernavanois [*et al.*]{}, Phys. Rev. B [**64**]{}, 174509 (2001); K. Kuwahara [*et al.*]{}, Physica B [**312-313**]{}, 106 (2002).
A. Semwal and S.N. Kaul, Phys. Rev. B [**60**]{}, 12799 (1999).
L. Asch [*et al.*]{}, Europhys. Lett. [**10**]{}, 673 (1989).
|
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abstract: 'In this work we numerically study critical phases in translation-invariant $\mathbb{Z}_N$ parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a $\mathbb{Z}_N$ spin model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation and translational invariance ensures that the spin model is always self-dual. We first study the low-energy spectrum of chains with only nearest-neighbor coupling, which are mapped onto standard self-dual $\mathbb{Z}_N$ clock models. For $3\leq N\leq 6$ we match the numerical results to the known conformal field theory(CFT) identification. We then analyze in detail the phase diagram of a $N=3$ chain with both nearest and next-nearest neighbor hopping and six critical phases with central charges being $4/5$, 1 or 2 are found. We find continuous phase transitions between $c=1$ and $c=2$ phases, while the phase transition between $c=4/5$ and $c=1$ is conjectured to be of Kosterlitz-Thouless type.'
author:
- Wei Li
- Shuo Yang
- 'Hong-Hao Tu'
- Meng Cheng
bibliography:
- './parafermion.bib'
title: 'Criticality in Translation-Invariant Parafermion Chains'
---
Introduction
============
In recent years non-Abelian anyons have been a focus of intense theoretical and experimental investigations [@TQCreview]. Their exotic properties have deepened our understanding of quantum many-body phases and also found potential applications in building topological quantum computers. However, quantum phases that host these exotic quasiparticles, namely non-Abelian topological phases, are usually quite elusive in nature and often require delicate conditions (e.g. complicated forms of many-particle interactions) to occur. Recently, proposals of engineering non-Abelian phases from more conventional materials have greatly stimulated this field of research [@Fu_2008; @Zhang2008; @Sau_PRL2010; @Alicea'12]. For example, Majorana zero modes, being analogues of Ising anyons, have been proposed to exist at ends of semiconductor nanowires in proximity to s-wave superconductors [@1DwiresLutchyn; @1DwiresOreg], as well as at the magnetic/superconducting domain walls on the edge of two-dimensional topological insulators [@MajoranaQSHedge]. The effort has culminated in the experimental observation of possible signatures of Majorana zero modes in semiconductor/superconductor heterostructures [@Mourik2012; @Fink2012; @Deng2012; @Das2012; @Churchill2013].
Following this line of ideas, it has been proposed that certain extrinsic defects in two-dimensional topologically ordered phases can bind exotic zero modes which are natural generalizations of Majorana zero modes [@Barkeshli_arXiv2013a; @Barkeshli_arXiv2013b]. Various physical realizations have been proposed, including magnetic/superconducting domain walls on the edge of two-dimensional fractionalized topological insulators [@Cheng_PRB2012; @Clarke_NatCommun2013; @Lindner_PRX2012; @Klinovaja_arxiv] and dislocations in bilayer quantum Hall systems [@Barkeshli_PRB2013; @Barkeshli_PRX2012] or toric-code type models [@You_PRB2012; @You_PRB2013; @Teo_arxiv2013a]. A common feature among all these seemingly different realizations is that the zero modes can be effectively described by second-quantized operators obeying parafermionic algebra [@Fradkin; @Fendley_JStat2012], which in the simplest case reduce to the well-known Majorana operators. They are therefore referred as parafermion zero modes subsequently. The parafermion zero modes also exhibit non-Abelian braiding statistics [@Clarke_NatCommun2013; @Lindner_PRX2012; @Bonderson_PRB2013; @Burrello_PRA2013; @You_PRB2013; @Teo2], with quantum dimensions squared to an integer.
A recent theoretical development pushes the limit of this engineering approach even further, where it was suggested that even more exotic topological phase, such as the famed Fibonacci phase, can be built by a delicate control of interactions between an array of such defects. The basic fact that underlies this construction is that a chain of interacting $\mathbb{Z}_3$ parafermion zero modes can be tuned to a critical point described by a $\mathbb{Z}_3$ parafermion CFT. Then by assembling many such critical chains together and coupling neighboring chains in an appropriate way [@coupledwire; @Neupert_arxiv2014], a superconducting analogue of the Fibonacci phase can emerge [@Mong_PRX2014; @Stoudenmire_unpub; @Vaezi_arxiv2013].
These interesting developments call for a more systematic investigation of the collective behavior of parafermion zero modes, in particular beyond the realm of exact integrability. Unlike Majorana zero modes, a quadratic Hamiltonian of parafermion zero modes is by no means a “free theory”. They are inherently strongly interacting and even a simple “quadratic” Hamiltonian consist of bilinears of parafermion zero modes can exhibit a rich phase diagram. The study of the physics of one-dimensional non-Abelian anyonic chains was pioneered in \[\], where the phase diagram of a chain of interacting Fibonacci anyons was presented, and is subsequently generalized to other anyon models [@GoldenChain2; @Gils_PRL2009; @Gils_NP2009; @Poiblanc_PRB2011; @Pfeifer_PRB2012; @Gils_PRB2013]. More recently there have been several works on gapped phases of parafermion systems both in one and two dimensions [@Motruk_PRB2013; @Bondesan_arxiv2013; @Milsted_arxiv; @Z3Kitaev]. In this work we focus on the phase diagram of a translation-invariant quadratic Hamiltonian describing hopping parafermions in one dimension. [By mapping the parafermion Hamiltonian to a $\mathbb{Z}_N$ spin model using a Jordan-Wigner-type transformation [@Fradkin; @Fendley_JStat2012], we see that translation invariance ensures that the spin model is always self-dual, which suggests that these models are critical.]{} We first analyze the critical phases when there are only nearest-neighbor hoppings. It is well-known that these Hamiltonians can be mapped to self-dual $\mathbb{Z}_N$ clock models [@Fendley_JStat2012] which have been studied thoroughly in the context of classical statistical mechanics. For $3\leq N\leq 5$ we match the low-energy spectrum with theoretical predictions. In particular, we highlight the subtleties in identifying the CFT spectra due to the non-diagonal CFT partition functions when $N=3$.
We then study the phase diagram of a $\mathbb{Z}_3$ parafermion chain with both nearest-neighbor(NN) and next-nearest-neighbor(NNN) couplings, which leads to a spin model that has not been considered before. Translation invariance still guarantees criticality, but we observe numerically that as the ratio between the NN and NNN couplings are tuned, critical phases with different central charges including $c=4/5, 1, 2$ are realized. We also characterize the phase transitions between these critical phases. We find that the transitions between $c=1$ and $c=2$ phases are continuous.
The paper is organized as follows: in Sec. \[sec:model\] we introduce the parafermion zero modes and define the model Hamiltonian that is the focus of the paper. We also briefly review the generalized Jordan-Wigner transformation. In Sec. \[sec:phases\] we establish the phase diagram of $\mathbb{Z}_N$ parafermion chains with NN couplings. In Sec. \[sec:z3\] we study $\mathbb{Z}_3$ parafermion chains with both NN and NNN couplings. Sec. \[sec:conclusion\] concludes the paper.
Model and Definition {#sec:model}
====================
Let us first define formally what parafermion zero modes are. A $\mathbb{Z}_N$ parafermion mode is defined as a unitary operator $\gamma$ such that $\gamma^N=1$. For many parafermions, if a certain ordering prescription is chosen, we then have the commutation relation: $$\gamma_i\gamma_j=\omega^{\operatorname{sgn}(j-i)}\gamma_j\gamma_i,\: \omega=e^{\frac{2\pi i}{N}}.
\label{eq:commutation}$$ When $N=2$ this is the familiar anti-commutation relation of Majorana zero modes. Notice that $2n$ parafermions can be represented by $N^{n}$-dimensional Hilbert space, therefore in a sense each parafermion zero mode carries “$\sqrt{N}$”-dimensional states.
In one dimension, lattice sites are naturally ordered. A generic hopping Hamiltonian can be written down: $$H=\sum_{ij} (t_{ij}\gamma_i^\dag\gamma_j+\text{h.c.}).
\label{eq:ham}$$ Although the Hamiltonian is quadratic, it is by no means free/non-interacting when $N>2$ due to the parafermionic commutation relation between the operators. If we try to diagonalize the Hamiltonian by Fourier transformation, the commutation relation between the momentum-space modes becomes utterly complicated. Therefore the model is intrinsically a strong-coupling problem. To have a glimpse of the rich physics contained in this model, we consider hoppings up to the NNN bonds, See Fig. \[fig:chain\](a) for an illustration of this parafermion chain.
We notice that the definitions of the operators $\gamma_j$ allow a $\mathbb{Z}_N$ gauge redundancy: $\gamma_j\rightarrow \omega^{n_j}\gamma_j$ where $n_j\in\mathbb{Z}$. The hopping amplitudes also have the same redundancy: $$t_{ij}\rightarrow t_{ij}\omega^{n_i-n_j}.
\label{}$$ For example, in an open chain, $t_{ij}$ and $\omega^{|j-i|} t_{ij}$ are identical up to gauge transformations. However, $t_{ij}$ and $-t_{ij}$ in generally are not related for odd $N$. In fact, the “equivalence classes” of hopping amplitudes are labeled by gauge-invariant quantities such as $t_{i,i+1}t_{i+1,i+2}t_{i+2,i}=t_1^2t_2^*$. For example, we can perform a gauge transformation $\gamma_i\rightarrow \omega^{-i}\gamma_i$, and $t_1\rightarrow \omega t_1$ and $t_2\rightarrow \omega^2t_2$. Such a transformation can also leave the boundary condition of the parafermions twisted if they sit on a ring. However we mainly focus on open boundary condition and we expect twisted periodic boundary conditions do not affect the major low-energy characterizations of the bulk.
![Illustration of (a) the model of parafermion chain and (b) the corresponding $\mathbb{Z}_N$ spin model.[]{data-label="fig:chain"}](fig0.pdf){width="\columnwidth"}
We heavily rely on numerical methods to understand the low-energy physics of this model. In order to carry out numerical simulations, the model is transformed into a $\mathbb{Z}_N$ spin model with a generalized version of Jordan-Wigner transformation [@Fradkin; @Fendley_JStat2012]. To be specific, we assume that open boundary condition is imposed. We then define the following tranformation: $$\begin{gathered}
\gamma_{2i}=\sigma_i\prod_{j<i}\tau_j,
\gamma_{2i+1}=\beta\sigma_i\tau_i\prod_{j<i}\tau_j.
\end{gathered}
\label{}$$ Here $\sigma_i, \tau_i$ are spin operators that act on a $N$-dimensional Hilbert space for each site, satisfying $$\sigma_j^N=\tau_j^N=1, \sigma_j^\dag\sigma_j=\tau_j^\dag\tau_j=1, \sigma_j\tau_j=\omega\tau_j\sigma_j.
\label{}$$ The spin operators on different sites commute. The constant $\beta$ must satisfy $\beta^N=\omega^{\frac{N(N-1)}{2}}=(-1)^{N-1}$ so that $\gamma_{2i+1}^{N}=1$.
We now apply this transformation to the parafermion chain Hamiltonian with NN and NNN couplings: $$H=\sum_i\big(t_1\gamma_i^\dag\gamma_{i+1}+t_2\gamma_{i}^\dag\gamma_{i+2}+\text{h.c.}\big).
\label{}$$
The parafermion bilinears become $$\begin{gathered}
\gamma_{2j}^\dag\gamma_{2j+1}=\beta\tau_j, \gamma_{2j+1}^\dag\gamma_{2j+2}=\beta^*\omega^*\sigma^\dag_{j}\sigma_{j+1}\\
\gamma_{2j}^\dag\gamma_{2j+2}=\sigma^\dag_j\tau_j\sigma_{j+1},\gamma_{2j+1}^\dag\gamma_{2j+3}=\omega^*\sigma_{j}^\dag\sigma_{j+1}\tau_{j+1}
\end{gathered}
\label{}$$ As a result, we obtain the following $\mathbb{Z}_N$ spin model \[see Fig. \[fig:chain\](b)\]: $$\begin{split}
H=&\sum_{j}(t_1\beta^*\omega^*\sigma_j^\dag\sigma_{j+1}+t_1\beta\tau_j+\text{h.c.})\\
&-\sum_j(t_2\omega^*\sigma_j^\dag\sigma_{j+1}\tau_{j+1}+t_2\sigma_j^\dag\tau_j\sigma_{j+1}+\text{h.c.}).
\end{split}$$ It is convenient to set $\beta=\omega^{-\frac{N+1}{2}}$ and redefine $t_1=-\beta^* J_1, t_2=-J_2$, $$\begin{split}
H=&-\sum_{j}(J_1\sigma_j^\dag\sigma_{j+1}+J_1\tau_j+\text{h.c.})\\
&-\sum_j(J_2\omega^*\sigma_j^\dag\sigma_{j+1}\tau_{j+1}+J_2\sigma_j^\dag\tau_j\sigma_{j+1}+\text{h.c.}).
\end{split}
\label{eq-ZN-spin-model}$$ We will be working with this form of the Hamiltonian in the rest of the paper and mainly focus on the case where both $J_1$ and $J_2$ are real for simplicity.
Let us examine the symmetries of the Hamiltonian. The parafermion model, as well as the spin model obtained by applying the Jordan-Wigner transformation, both have a global $\mathbb{Z}_N$ symmetry generated by $$Q=\prod_{j}\tau_j=\prod_{j}\gamma_{2j}^\dag\gamma_{2j+1}.
\label{}$$ $Q$ can be regarded as the global $\mathbb{Z}_N$ charge. Later the $\mathbb{Z}_N$ quantum numbers will be exploited in the numerical simulation.
We now turn to space-time symmetry. We can define space inversion and time reversal transformations as follows: $$\begin{gathered}
\mathcal{I}:\sigma_j\leftrightarrow \sigma_{-j}, \tau_j\leftrightarrow\tau_{-j}\\
\mathcal{T}:\sigma_j\leftrightarrow \sigma_j^\dag, \tau_j \leftrightarrow \tau_j
\end{gathered}
\label{}$$ The $J_2$ term breaks both the inversion and the time-reversal symmetry.
We now show that the spin model obtained in this way is always self-dual. Let us define the dual disorder variables: $$\mu_{j}=\prod_{l\leq j}\tau_l, \nu_{j}=\sigma_{j}^\dag\sigma_{j+1}.
\label{}$$ Under the duality transformation, we have $$\begin{gathered}
\sigma_j^\dag\sigma_j\rightarrow \nu_j,
\tau_j\rightarrow \mu_{j-1}^\dag \mu_{j}\\
\sigma_j^\dag\sigma_{j+1}\tau_{j+1}\rightarrow \nu_j\mu_j^\dag\mu_{j+1}.
\end{gathered}
\label{}$$ So the Hamiltonian is invariant.
We notice that the self-duality of the spin model corresponds exactly to the translation invariance of the parafermion model [@Mong_Potts]. Therefore a translation-invariant parafermion chain always maps to a self-dual spin model. Although there is no rigorous proof that self-duality implies criticality, we are not aware of any counterexamples in one-dimensional systems. We will see in the following sections that our model indeed exhibits criticality.
$\mathbb{Z}_{N}$ model with only NN couplings {#sec:phases}
=============================================
In this section, we start from the $\mathbb{Z}_{N}$ model (\[eq-ZN-spin-model\]) with $J_{2}=0$, where only NN couplings are present. For $J_{1}=1$, the $\mathbb{Z}_{N}$ parafermion chain maps exactly to the self-dual $\mathbb{Z}_{N}$ clock model. The study of the phase diagram of $\mathbb{Z}_{N}$ clock model has a long history. Utilizing a field theoretical approach, it has been argued that [@lecheminant2002] the quantum model in one dimension can be related to the classical planar XY model with $\mathbb{Z}_{N}$ symmetry-breaking fields in two dimensions in certain anisotropic limit. The Euclidean action of the classical model is equivalent to that of a sine-Gordon model given by [@lecheminant2002] $$\mathcal{S}=\frac{1}{2}\int \mathrm{d}^{2}{{{{\mathbf r}}}}\,[(\nabla \varphi)^2+g\cos
(\sqrt{N}\varphi)+\tilde{g}\cos (\sqrt{N}\theta)], \label{eq:Sine-Gordon}$$where $\varphi ({{{{\mathbf r}}}})$ is a bosonic field and $\theta ({{{{\mathbf r}}}})$ is the conjugate field. The two cosine potentials in (\[eq:Sine-Gordon\]) always have the same scaling dimensions, thus competing with each other. The duality transformation of the spin model corresponds to $\varphi\leftrightarrow \theta$ in the field theory. Hence when $g=\tilde{g}$ the field theory is self-dual. It is useful to first consider the weak-coupling limit and perform a renormalization group analysis of the perturbations to the Gaussian fixed point. For $N<4$, the two perturbations $\cos \sqrt{N}\varphi$ and $\cos\sqrt{N}\theta$ are both relevant. So they drive the theory to a new strong-coupling fixed point whose central charge is less than $1$ according to Zamolodchikov’s $c$-theorem [@ctheorem]. Therefore the infra-red fixed-point is necessarily a CFT minimal model. It is known that the new fixed-point is described by an Ising CFT with central charge $c=1/2$ for $N=2$, and a $\mathbb{Z}_{3}$ parafermion CFT with central charge $c=4/5$ for $N=3$. For $N=4$, the perturbation is marginal and the low-energy fixed-point will be identified below. For $N\geq 5$ the cosine perturbations in (\[eq:Sine-Gordon\]) become irrelevant, thus the low-energy fixed point is again Gaussian.
We now come back to lattice models. For $N=3$, the clock model (\[eq-ZN-spin-model\]) coincides with the famous $3$-state Potts model, which has the special property of being integrable [@Sierra_book]. $J_1=1$ will be called the ferromagnetic(FM) coupling, since the $\mathbb{Z}_3$ spins are aligned in the same direction in the ground state of the Hamiltonian with just the term $-J_1\sum_j (\sigma_j^\dag\sigma_{j+1}+\sigma_{j+1}^\dag\sigma_j)$, and correspondingly $J_1=-1$ the antiferromagnetic(AF) coupling. For the FM case, one can deduce from the Bethe ansatz solutions that the chain has gapless excitations [@albertini1992] and the low-energy effective theory is the $\mathbb{Z}_{3}$ parafermion CFT [@dotsenko1984], also known as the minimal model $\mathcal{M}(6,5)$ with a central charge $c=4/5$. However, the field content of the three-state Potts criticality differs from the genuine $\mathcal{M}(6,5)$ CFT. Only $6$ primary fields out of the $10$ with conformal dimensions $h=0,2/5,7/5,3,1/15,2/3$ are responsible for the ferromagnetic three-state Potts model. In fact, the field content is completely specified by the following non-diagonal modular-invariant partition function [@cardy1986]: $$Z_{F} =|\chi _{0}+\chi_{3}|^2
+|\chi _{\frac{2}{5}}+\chi_{\frac{7}{5}}|^2
+2|\chi _{\frac{1}{15}}|^2+2|\chi _{\frac{2}{3}}|^2,
\label{eq:Z3character}$$where $\chi_{h}=\mathrm{tr}_{h}(q^{L_{0}-c/24})$ is the holomorphic CFT character with $\mathrm{tr}_h$ the trace in the conformal block $h$ (labeled by the conformal dimension).
For the $\mathbb{Z}_{3}$ clock model with antiferromagnetic coupling, it was proposed in \[\] based on Bethe ansatz that the critical theory should be the $\mathbb{Z}_{4}$ parafermion CFT with central charge $c=1$. The partition function relevant for the antiferromagnetic three-state Potts model has been shown to be the non-diagonal combination of characters [@gepner1987]$$Z_{A} =|\chi _{0}+\chi _{1}|^2
+4|\chi _{\frac{3}{4}}|^2
+2|\chi _{\frac{1}{3}}|^2+2|\chi _{\frac{1}{12}}|^2,
\label{eq:Z4character}$$where five primary fields with conformal dimensions $h=0,3/4,1,1/3,1/12$ show up. Notice that if we represent the $\mathbb{Z}_4$ parafermion CFT as the coset $\mathbb{SU}(2)_4/\mathbb{U}(1)$, then only fields with integer $\mathbb{SU}(2)$ spins appear in the partition function. This implies that the CFT can be obtained from $\mathbb{SU}(2)_4/\mathbb{U}(1)$ by “condensing” the highest spin primary fields in $\mathbb{SU}(2)_4$, which results in $\mathbb{SU}(3)_1$ theory [@BaisPRB2009]. With this perspective, the CFT of the antiferromagnetic $\mathbb{Z}_3$ Potts chain should better be described as $\mathbb{SU}(3)_1\times\overline{\mathbb{U}(1)}_2\simeq \mathbb{U}(1)_6$ [^1].
![Block entanglement entropy of the $\mathbb{Z}_3$ FM($\theta=0$) and AF($\theta=\pi$) Potts model. (a) $\theta=0$ with $c=\frac{4}{5}$, and (b) $\theta=\pi$ with $c=1$. Open boundary conditions are adopted in both cases.[]{data-label="fig:entropy"}](fig3bc.pdf){width="0.95\columnwidth"}
In order to verify these field theoretical predictions, we perform numerical simulations for $\mathbb{Z}_{N}$ clock models (\[eq-ZN-spin-model\])with only nearest-neighbor interactions, based on density-matrix renormalization group (DMRG) and exact diagonalization (ED) techniques. The numerical methods that we adopt here also form the basis for our further investigations of more complicated models in subsequent sections. In the DMRG method, we approximate the ground states and sometimes also several low-lying excited states of the Hamiltonian with open boundary conditions by matrix-product states. For the ground states of 1D critical quantum chains with length $L$ and open boundaries, it has been shown[@Holzhey; @Vidal; @Calabrese-Cardy] that the von Neumann entanglement entropy of a block of $x$ consecutive spins scales as $$S=\frac{c}{6}\log_2 {\left( \frac{L}{\pi }\sin \frac{\pi x}{L}\right) }+S_{0},
\label{eq:Cardy}$$where $c$ is the central charge of the CFT and $S_{0}$ is a non-universal constant. In the case of periodic boundary condition, the coefficient of the logarithmic scaling of entanglement entropy in (\[eq:Cardy\]) should be modified to $c/3$. In DMRG calculations, we fit the numerically computed von Neumann entropy with this formula, from which one can read off the central charge $c$.
Once $c$ is determined, we can compute the energy spectra of a finite-size chain with periodic boundary condition and further constrain the conformal dimensions of primary fields in the CFT. This is based on the following result: For a 1D critical chain described by a CFT, the energy spectra are given by [@affleck1986a; @blote1986] $$E=\varepsilon _{\infty }L-\frac{\pi vc}{6L}+\frac{2\pi v}{L}(h+{\overline{h}}+n+{\overline{n}}), \label{eq:finite-size}$$where $\varepsilon _{\infty }$ is the ground-state energy per site in the thermodynamic limit, $v$ is the sound velocity, $h$ and ${\overline{h}}$ are conformal dimensions of the CFT primary fields, and $n$ and ${\overline{n}}$ are non-negative integers. In practice, we can find $v$ accurately from the finite-size scaling of the ground state energy. Then comparing the numerically computed energy spectra with (\[eq:finite-size\]) allows to extract conformal dimensions of the CFT primary fields, which are characteristic quantities for identifying the CFT.
Notice that caution should be taken when one tries to extract the holomorphic conformal dimension $h$ from . From the excited energy spectra we can only obtain $h+{\overline{h}}$ directly. If the partition function is diagonal, all CFT states have zero conformal spins meaning $h={\overline{h}}$. However, in the present case the relevant partition functions of both $\mathbb{Z}_3$ and $\mathbb{Z}_4$ parafermion CFTs are non-diagonal, which means that operators with non-zero conformal spins appear in the spectrum. For example, for the $\mathbb{Z}_4$ parafermion CFT one should find two degenerate levels corresponding to $(h,{\overline{h}})=(1,0),(0,1)$, and the other levels should all be diagonal meaning that $h={\overline{h}}$ (modulo the $n$ and ${\overline{n}}$ shifts ).
For the ferromagnetic $\mathbb{Z}_{3}$ Potts model, we confirm that the central charge is well fitted to $c=4/5$ in our DMRG calculations. The numerically computed von Neumann entanglement entropy is shown in Fig. \[fig:entropy\] (a). The (rescaled) finite-size spectrum of a periodic chain with $L=14$ sites is shown in Fig. \[fig:ed\](a). Using Eq. (\[eq:finite-size\]), the CFT primary fields appearing in (\[eq:Z3character\]) are found in the low-lying spectrum, including the non-diagonal combinations $(3,0)$ and $(0,3)$, see Fig. \[fig:ed\](a). Our extrapolated ground-state energy per site $
\varepsilon _{\infty }= -2.43599$ (from DMRG calculations) and sound velocity $v=2.58441$ (from ED results with size $L=14$) both agree very well with the exact values $\varepsilon _{\infty }=-\frac{2
\sqrt{3}}{\pi }-\frac{4}{3}$ and $v=\frac{3\sqrt{3}}{2}$ from the Bethe ansatz solution [@albertini1992].
![Low-energy spectra of the quantum $\mathbb{Z}_{3}$ Potts model of size $L=14$ with periodic boundary conditions for (a) the ferromagnetic coupling $J_{1}=1$ and (b) the antiferromagnetic coupling $J_{1}=-1$. The spectra have been shifted and rescaled by the exact values of the ground-state energy and the sound velocity according to Eq. (\[eq:finite-size\]), so that the comparison to the CFT predictions is more transparent. The open circles (squares) denote the energy levels with $\mathbb{Z}_{3}$ quantum number $Q=0$ ($Q=\pm 1$). The energy levels corresponding to the CFT primary fields are labeled by their conformal dimensions $(h,\bar{h})$.[]{data-label="fig:ed"}](ed1.pdf){width="\columnwidth"}
For the antiferromagnetic $\mathbb{Z}_{3}$ Potts model, we find $c=1$ from the numerical fit of entanglement entropy \[see Fig. \[fig:entropy\](b)\]. We have observed that the ground-state energy has an even-odd dependence on the system size, so we extract conformal dimensions from a chain with even system size $L=14$ \[see Fig. \[fig:ed\](b)\]. This is in agreement with the ED ground state being located at momentum $\pi$. Our numerical results confirm the $\mathbb{U}(1)_6$ CFT prediction for the antiferromagnetic three-state Potts model, as well as the relevant primary fields in the partition function (\[eq:Z4character\]). Moreover, the numerically computed ground-state energy per site $\varepsilon _{\infty }=-1.816071$ (from DMRG calculations) and sound velocity $v=1.2883$ (from ED results with size $L=14$) are also in very good agreement with the exact values $\varepsilon _{\infty }=-\frac{\sqrt{3}}{\pi }-\frac{3\sqrt{3}}{
2}+\frac{4}{3}$ and $v=\frac{3\sqrt{3}}{4}$ [@albertini1992]. All low-lying excited levels can be matched up with CFT predictions, including two non-diagonal combinations $(1,0)$ and $(0,1)$.
$N$ Coupling CFT Remarks
---------- ---------- -------------------------------- ---------------------------------
$2$ AF/FM Ising
$3$ FM $\mathbb{Z}_3$ PF non-diagonal partition function
$3$ AF $\mathbb{U}(1)_6$ non-diagonal partition function
$4$ AF/FM $\mathbb{U}(1)_4/\mathbb{Z}_2$ $R=2$
$\geq 5$ AF/FM $\mathbb{U}(1)_{2N}$ $R=\sqrt{2N}$
: Summary of the low-energy CFT descriptions of the $\mathbb{Z}_N$ clock model. “PF” is short for parafermion CFT. $R$ is the compactification radius of the $\mathbb{U}(1)$ boson CFT. []{data-label="tab:Zn"}
Now we turn to $N>3$. We confirm that, for $N=4,5,6$, the central charges of the $\mathbb{Z}_{N}$ clock models with $J_{1}=\pm 1$ are all equal to $1$, in consistent with the field theoretical prediction based on . In addition, we have extracted the conformal dimensions of corresponding primary fields from the low-lying excited states. For $N=4$, the lowest six primary fields have conformal dimensions $h=1/16,1/16,1/8,1/2,1/2,9/16$, in perfect agreement with the $\mathbb{Z}_{2}$ orbifold of a $\mathbb{U}(1)$ boson compactified on a circle of radius $R=2$ [@orbifold], which is just two copies of Ising CFTs. For $N=5$, regardless of the sign of the coupling the lowest two conformal dimensions read $h=1/20$ and $1/5$, in agreement with the CFT of a compactified boson on a circle of radius $R=\sqrt{10}$ [@BorisenkoPRE2011]. Thus, we expect that for $N\geq 5$ all $\mathbb{Z}_{N}$ clock models with $J_{1}=\pm 1$ are described by $c=1$ free-boson CFT with a compactification radius $R=\sqrt{2N}$. We summarize these results in Table \[tab:Zn\].
$\mathbb{Z}_3$ model with up to NNN couplings {#sec:z3}
=============================================
In this part we present the phase diagram of a $\mathbb{Z}_3$ parafermion chain with NN and NNN hoppings, which is summarized in Fig. \[fig:pdiag\]. We parametrize the two hopping strengths by $J_1=\cos\theta$ and $J_2=\sin\theta$. The Hamiltonian is solved numerically by the DMRG method with open boundary conditions. As expected, the whole phase diagram are filled by critical phases. This is readily seen by calculating the ground state entanglement entropy as a function of the block size $x$ and fitting it with Eq. . From the scaling we also read off the central charge $c$ of the critical phase.
![Phase diagram of the $\mathbb{Z}_3$ Potts model. There are six critical phases labeled by different central charges. There are two exactly solvable points at $\theta=0$ and $\theta=\pi$ [@Sierra_book], which extend to a $c=\frac{4}{5}$ phase and a $c=1$ phase, respectively. In addition, there are two $c=2$ critical phases, roughly centered around the $\theta=\frac{\pi}{2}$ and $\theta=\frac{3\pi}{2}$ points. There also exist two $c=1$ phases between the $c=2$ phases and the $c=\frac{4}{5}$ phase. The positions of transition points separating the $c=1$ and $c=2$ phases are determined by locating the positions of peaks in second-order energy derivatives. The boundaries between $c=\frac{4}{5}$ and $c=1$ phases are more subtle and can be extracted from the entanglement data.[]{data-label="fig:pdiag"}](phase-diag.pdf){width="0.8\columnwidth"}
To accurately pin down the phase boundaries, we first calculate numerically the ground-state energy density (i.e. energy per site) and its first- and second-order derivatives with respect to $\theta$ to locate the phase transition points which at the same time reveal the nature of the phase transitions. In Fig. \[fig:energy\] we show the energy per site $e_0$, its first and second-order derivatives with respect to $\theta$ as a function of $\theta$. One can clearly see that the first-order derivative $\frac{{\mathrm{d}}e_0}{{\mathrm{d}}\theta}$ is continuous and there are discontinuities in the second-order derivative $\frac{{\mathrm{d}}^2 e_0}{{\mathrm{d}}\theta^2}$ at $\theta\approx 0.36\pi, 0.57\pi,1.17\pi, 1.62\pi$. We then calculate the central charge in different regions of the phase diagram in order to identify the phases. We show the block entanglement entropy for selected points in Fig. \[fig:c=1\]. This also provides an alternative check of phase boundaries. We find that the phase transitions between $c=1$ and $c=2$ phases are very likely to be continuous, and in these cases the two ways of obtaining the phase boundaries agree with each other perfectly.
![Ground-state energy per site $e_0$ and its (first- and second-order) derivatives with respect to the parameter $\theta$. $e_o$ and its derivatives are seen to converge with increasing system sizes nearly everywhere, except for $\theta$ in the vicinity of transition points (diverging peaks of ${\mathrm{d}}^2e_0/{\mathrm{d}}\theta^2$). The four peaks (at $0.356\pi$, $0.572\pi$, $1.168 \pi$, and $1.624\pi$) in the ${\mathrm{d}}^2e_0/{\mathrm{d}}\theta^2$ clearly signal continuous transitions.[]{data-label="fig:energy"}](fig2.pdf){width="0.95\columnwidth"}
![Block entanglement entropy: (a) $\theta=\pi/2, 3\pi/2$ with $c=2$; (b) selected points in the two phases with $c=1$ above and below the $c=\frac{4}{5}$ phase ($0.05 \pi < \theta < 0.36 \pi$, and $-0.05 \pi < \theta < -0.4\pi$).[]{data-label="fig:c=1"}](fig5.pdf){width="0.95\columnwidth"}
However, the energy and its derivatives do not show any features near the “transition” between the $c=\frac{4}{5}$ phase and the neighboring $c=1$ phases. This part of the phase diagram near $\theta=0$ is particularly relevant to the recent studies of the Fibonacci phase [@Mong_PRX2014; @Vaezi_arxiv2013; @Vaezi_fib2014], so we carefully perform finite-size scaling of the central charge to map out the phase diagram in this region, see Fig. \[fig:Z3\]. We confirm that the $c=\frac{4}{5}$ $\mathbb{Z}_3$ parafermion CFT region extends roughly from $-0.04 \pi$ to $0.04 \pi$, beyond which it is taken over by $c=1$ phases. In order to confirm there is indeed a phase transition, we calculate the bipartite entanglement entropy around the “transition” point and observe a clear jump as shown in Fig. \[fig:bipartite entanglement\], which implies a dramatic change of the ground state wavefunction between the $c=\frac{4}{5}$ and $c=1$ phases.
![The fittings of the central charge from the scaling of block entanglement entropies around the $\mathbb{Z}_3$ point $\theta=0$. Both OBC and PBC are used. The results are consistent with the existence of a finite region with $c\approx \frac{4}{5}$, roughly ranging from $\theta=-0.03 \pi$ to $\theta=0.03 \pi$. []{data-label="fig:Z3"}](fig4.pdf){width="0.95\columnwidth"}
![Bipartite entanglement entropy ($S_b$) by cutting the chain in the middle. $S_b$ shows a clear jump (at around $\theta \approx \pm 0.05 \pi$) between $c=\frac{4}{5}$ and $c=1$ regions. MPS bond dimension is set to $D=300$ in this calculation.[]{data-label="fig:bipartite entanglement"}](fig6.pdf){width="0.9\columnwidth"}
Regarding the nature of the transition between the $c=\frac{4}{5}$ and $c=1$ phases, there can be three possibilities theoretically, all consistent with the numerical results: (1) The singularity appears in third- or higher-order derivatives. (2) There is a Kosterlitz-Thouless transition where the derivatives of the energy with respect to the tuning parameter are smooth to all orders. (3) It is a crossover instead of a phase transition. Although we are limited by the accuracy of numerical simulations, we believe a third-order phase transition is not very likely. The third option is also unfavored due to the abrupt change in the ground state entanglement. We therefore conjecture that the phase transition is of the Kosterlitz-Thouless type.
Interestingly, near the $\theta=\frac{\pi}{2},\frac{3\pi}{2}$ points where there are only NNN couplings one may naively think that the chain can be decoupled as two copies of the model with only NN hopping. This expectation is however not true. Due to the unusual commutation algebra (\[eq:commutation\]) between the parafermions, the two “copies” do not commute and are still highly entangled. This is quite different from the case of the Majorana hopping model($N=2$ parafermion chain). When there are only NNN coupling the even sites and the odd sites decouple from each other and form two $c=\frac{1}{2}$ CFTs. In the present case, for both AF and FM NNN couplings, we find $c=2$, obtained by the entanglement fitting in Fig. \[fig:c=1\](a).
![(a) The spin-spin correlation function (CF) $\langle \sigma_i \sigma^{\dagger}_j \rangle$ exhibits oscillations with varying period depending on $\theta$. The inset (log-log plot) illustrates the algebraic decay of absolute values of CF with distance. (b) The static structure factor of the CFs, revealing explicitly the long period of oscillations.[]{data-label="fig:cf_sf"}](fig-cf.pdf){width="0.9\columnwidth"}
The nature of these $c\geq 1$ phases remains unclear. Due to strong finite-size effect we are unable to identify the CFTs of these phases except their central charges. In the following we calculate the spin-spin correlation functions $C(x)\equiv\langle\sigma_i\sigma_{i+x}^\dag\rangle$, which may reveal useful information about the conformal dimensions of the scaling fields in the CFT. We notice that in general the CFT field identification of lattice operators is a highly nontrivial problem [@Mong_Potts], so caution should be taken in interpreting the numerical results.
Let us start from the $\theta=0,\pi$ exactly solvable points. It has long been known that at $\theta=0$ the $\sigma_i$ operators actually turn into the twist field in the $\mathbb{Z}_3$ parafermion CFT with scaling dimension $\frac{2}{15}$, so $C(x)\sim x^{-4/15}$ [@Mong_Potts]. Similarly, at $\theta=\pi$ the $\sigma_i$ turns into the twist field in the $\mathbb{Z}_4$ parafermion CFT with scaling dimension $\frac{1}{12}$ and $C(x)\sim x^{-1/3}$, which we have verified numerically. In both cases the identification of the continuum limit of $\sigma_i$ is rather straightforward.
Once we move away from the integrable points, the behavior of the spin correlation function becomes more complicated. We find that in the $c=1$ phase, $C(x)$ exhibits oscillations whose characteristic wavevectors depend on $\theta$ \[see Fig. \[fig:cf\_sf\](a)\]. This can be seen most easily from the peaks of the static structure factor defined as $S(k) = \sum_{x=1}^{L-1} \cos k x\, C(x)$ \[see Fig. \[fig:cf\_sf\](b)\]. This behavior is reminiscent of correlation functions in a Luttinger liquid, which often exhibit oscillations on the scale of Fermi wavelength. We also fit the decay exponent of the envelop function of $C(x)$ for two different values of $\theta$ and in both cases the values are close to $-1/3$. It is tempting to conjecture that the CFT in this phase is closely related to $\mathbb{U}(1)_6$ CFT, but our data is still too preliminary to draw any conclusions. We leave investigations of the CFT for future works.
Conclusions and Discussions {#sec:conclusion}
===========================
In this work we numerically study the criticality of a translation-invariant chain of $\mathbb{Z}_N$ parafermion zero modes by mapping to a $\mathbb{Z}_N$ spin model. We completely characterize the low-energy CFT of the $\mathbb{Z}_N$ parafermion chain with NN couplings by a combination of DMRG and ED methods and the results are in perfect agreement with theoretical predictions. We also determine the phase diagram of the $\mathbb{Z}_3$ parafermion chain with up to NNN couplings. We show that the introduction of a relatively small NNN coupling (compared to the NN coupling) can significantly alter the low-energy properties. Phase transitions between different critical phases are also characterized.
We now briefly discuss the physical implications of the results. Parafermion zero modes can be realized at the edge of some Abelian fractional quantum Hall states. For example, by patterning alternating regions gapped out by electron tunneling or s-wave pairing on the edge of a spin-unpolarized $\nu=2/3$ FQH state, $\mathbb{Z}_3$ parafermion zero modes are localized on the domain walls [@Mong_PRX2014] (a similar setup without superconductivity is considered in Ref. \[\]). Virtual tunneling of quasiparticles across the gapped regions then splits the degeneracy, and the effective low-energy Hamiltonian is given by . The tunneling amplitudes decay exponentially with the separation between domain walls, i.e. $t_{ij}\sim e^{-|x_i-x_j|/\xi}$, where $x_i$ is the position of the parafermion zero modes and $\xi$ is the correlation length. To the leading approximation, if the domain walls are evenly separated, they collectively realize a $\mathbb{Z}_3$ parafermion CFT. Our results show that $\mathbb{Z}_3$ parafermion CFT is destablized if $t_{\text{NNN}}/t_{\text{NN}}$ is larger than a critical value which we estimate to be $\tan 0.04\pi\approx 0.12$, which roughly corresponds to the separation between NN sites being $\sim 2\xi$.
We also emphasize that the criticality is protected by translation invariance. This should be compared to the topological symmetry that protects gapless phases in other one-dimensional models of non-Abelian anyonic chains [@GoldenChains; @GoldenChain2; @Pfeifer_PRB2012]. In fact, one can realize such a $\mathbb{Z}_N$ parafermion chain on the edge of a translation-symmetry enriched topological phase naturally. An exactly solvable model of this type on a square lattice has been recently studied in \[\]. The topological order in the bulk is identical to that of the $\mathbb{Z}_N$ toric code (or equivalently, a $\mathbb{Z}_N$ lattice gauge theory coupled to matter). However, the translation symmetry has a nontrivial interplay with the topological order: the elementary electric charge and the magnetic charge are exchanged under lattice translations. As a result, $\mathbb{Z}_N$ parafermion zero modes appear on the lattice dislocations. This model also has gapless edge modes if the edge preserves the translation invariance of the system. One can show that the edge can be described by exploiting a parafermionic parton representation of the model. The translation symmetry on the edge is inherited from that of the bulk. One might wonder whether the generalized Jordan-Wigner transformation breaks the translation invariance by hand when the parafermion zero modes are grouped to form $\mathbb{Z}_N$ spins [@Teo].
Acknowledgment
==============
M.C. thanks enlightening conversations with Jeffery Teo, David Clarke and Zhenghan Wang and especially Hong-Chen Jiang for very useful discussions on numerical results. H.H.T. thanks Germ[á]{}n Sierra, Michele Burrello and Eddy Ardonne for helpful discussions. M.C. acknowledges the hospitality of the Max-Planck Institute for Quantum Optics in Garching where part of the work was carried out. M.C. acknowledges the hospitality of Perimeter Institute for Theoretical Physics during the finalization of the manuscript. This work has been supported by the DFG through SFB-TR12, the EU project SIQS, and the DFG Cluster of Excellence NIM. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.
[^1]: We thank E. Ardonne for very helpful correspondence on the CFT of antiferromagnetic $\mathbb{Z}_3$ Potts model.
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abstract: 'Due to a continuum of electronic states present in periodic systems, the description of molecular dynamics on surfaces poses a serious computational challenge. One of the most used families of approaches in these settings are friction theories, which are based on the approach. Yet, a mean-field treatment of electronic degrees of freedom in the method makes this approach inaccurate in some cases. Our aim is to clarify when breaks down for molecular dynamics on surfaces. Answering this question provides limits of applicability for more approximate friction theories derived from . We assess the method on one-dimensional, numerically exactly solvable models with a large but finite number of electronic states. Using the Landau-Zener formula and the Massey parameter, an expression that determines when breaks down is deduced.'
author:
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---
Introduction
============
Molecular dynamics in periodic systems in general and on metallic surfaces in particular is complicated by the presence of a continuum of electronic states. This continuum can lead to break-down of the Born-Oppenheimer approximation, which is based on the assumption that the energy separation between electronic states is much larger than the nuclear kinetic energy. This means that a proper treatment of such systems must take into account nonadiabatic effects. The most computationally feasible candidates for simulations in extended systems are methods [@ilya; @metal_fssh; @shenvi; @Gherib:2015ix], such as the approach [@doltsinis], and Tully’s algorithm [@fssh; @doltsinis; @Wang:2016bza]. Yet, due to the continuum of electronic states, and are not straightforwardly applicable to periodic systems. First, the continuum of states must be discretized [@discretization], and even then thousands of electronic states can be involved in dynamics,[@iesh] rendering such calculations computationally challenging.
To avoid explicit treatment of a large number of electronic states, frictional theories were introduced. In these theories, starting from the formalism, the electronic continuum is approximately integrated,[@fric] and nuclear trajectories evolve on the adiabatic ground state with an additional frictional force arising from nonadiabatic transitions. The frictional approaches have been extensively studied over the years [@fric; @tens_fric; @mode_fric; @dou_fric; @many_body_fric; @q_fric]. The problem with frictional approximations appears when multiple electronic surfaces with very different nuclear dependence become energetically available. This often takes place for different adsorbate states, for example, dynamics of $\mathrm{Cl}$/$\mathrm{Cl}^-$ adsorbates on the Cu (110) surface in scanning tunneling microscope induced reactions.[@polanyi; @polanyi2] Here, the underlying may not be as good as in the case when only one adsorbate state is involved.
Some inaccuracies introduced by frictional approximations were corrected by switching to a collective variable approach [@ilya]. This approach does not treat electronic DOF fully implicitly but rather reduces their explicit consideration to only few (two or three) collective electronic coordinates. However, in the current formulation this approach still cannot surpass the EH method which it is based on. Applying the collective variable idea to the framework would be ideal in cases when several adsorbate states are involved in surface dynamics, since this framework allows for proper account of dynamics on individual electronic states [@Loaiza2:inprep].
Still, and -based methods are frequently used in modeling surface dynamics, and it is instructive to develop a numerical criterion indicating when going beyond the EH framework is necessary. This is the main objective of the present work. We begin by reviewing the EH and SH methods with an illustration of EH failure on Tully’s extended coupling model in Sec. \[subsec:MQC\]. Using the formula, we deduce a parameter that can indicate when dynamics breaks down in Sec. \[subsec:EHBI\]. Section \[sec:sim\] illustrates performance of the new indicator on one-dimensional surface dynamics inspired models.
Theory {#sec:theory}
======
Mixed quantum-classical methods {#subsec:MQC}
-------------------------------
To introduce notation and to provide self-sufficient description of methodology we start with a brief reminder of the and methods.[@fssh; @doltsinis; @tullymqc] In both of these methods, nuclear coordinates ${\ensuremath{\mathbf{R}}}$ are treated classically and are substituted by corresponding time-dependent functions ${\ensuremath{\mathbf{R}}}={\ensuremath{\mathbf{R}}}(t)$. Using this substitution one can formulate the total non-stationary electronic wavefunction as $$\label{eq:ansatz}
\ket{\Psi({\ensuremath{\mathbf{R}}}(t),t)}=\sum_{j}c_j(t)\ket{\phi_j[{\ensuremath{\mathbf{R}}}(t)]},$$ where $\ket{\phi_j[{\ensuremath{\mathbf{R}}}(t)]}$ are eigenfunctions of the electronic Hamiltonian $\hat H_e[{\ensuremath{\mathbf{R}}}(t)]$, $\hat H_e[{\ensuremath{\mathbf{R}}}(t)]\ket{\phi_j[{\ensuremath{\mathbf{R}}}(t)]}=E_j[{\ensuremath{\mathbf{R}}}(t)]\ket{\phi_j[{\ensuremath{\mathbf{R}}}(t)]}$, and $E_j[{\ensuremath{\mathbf{R}}}(t)]$ are corresponding adiabatic PESs. To formulate equations of motion for time-dependent coefficients $c_j(t)$ the time-dependent Schrödinger equation $$\label{eq:TDSE}
i{\frac{\partial }{\partial t}}\ket{\Psi({\ensuremath{\mathbf{R}}}(t),t)}=\hat H_e[{\ensuremath{\mathbf{R}}}(t)]\ket{\Psi({\ensuremath{\mathbf{R}}}(t),t)}$$ is projected onto the adiabatic wavefunctions $\ket{\phi_j[{\ensuremath{\mathbf{R}}}(t)]}$ $$\label{eq:elec_dyn}
\dot{{\ensuremath{\mathbf{c}}}}=-(i{\ensuremath{\mathbf{H_e}}} +\dot{{\ensuremath{\mathbf{R}}}} \cdot {\ensuremath{\mathbf{\Gamma}}}){\ensuremath{\mathbf{c}}},$$ where ${\ensuremath{\mathbf{\Gamma}}}$ is a matrix with elements ${\ensuremath{\mathbf{\Gamma}}}_{jk}=\braket{\phi_j[{\ensuremath{\mathbf{R}}}] | \nabla_{{\ensuremath{\mathbf{R}}}} \phi_k[{\ensuremath{\mathbf{R}}}]}$ corresponding to the vectors, and ${\ensuremath{\mathbf{c}}}$ is a vector with entries $c_j(t)$. In the adiabatic representation, ${\ensuremath{\mathbf{H_e}}}$ is a diagonal matrix with elements ${\ensuremath{\mathbf{H_e}}}_{,kj}=\braket{\phi_k({\ensuremath{\mathbf{R}}}) | \hat H_e ({\ensuremath{\mathbf{R}}}) | \phi_j({\ensuremath{\mathbf{R}}})}=E_k ({\ensuremath{\mathbf{R}}})\delta_{kj}$.
In and methods the nuclear dynamics, ${\ensuremath{\mathbf{R}}}(t)$, is governed by Newton equations of motion. For , the force is based on the gradient of the averaged electronic energy $${\ensuremath{\mathbf{F}}}=-\nabla_{{\ensuremath{\mathbf{R}}}}\braket{\Psi | \hat H_e | \Psi}.$$ Using the adiabatic expansion for $\ket{\Psi}$ in [Eq. (\[eq:ansatz\])]{} and \_[Ř]{}=(E\_k-E\_j)v\_[jk]{}+ one obtains the nuclear equation of motion in the Ehrenfest method as $$M\ddot{{\ensuremath{\mathbf{R}}}}=-\sum_k \vert c_k\vert^2\nabla_{{\ensuremath{\mathbf{R}}}} E_k+\sum_{k\neq j}c_k^*c_j(E_k-E_j){\ensuremath{\mathbf{\Gamma}}}_{kj}.$$ This equation can also be derived from the conservation of the total energy $E=\frac{1}{2}M {\left| \dot{{\ensuremath{\mathbf{R}}}} \right|}^2+\braket{\Psi | \hat H_e | \Psi}$.
In , nuclear dynamics is governed by forces from a single at each moment of time $$M\ddot{{\ensuremath{\mathbf{R}}}}=-\nabla_{{\ensuremath{\mathbf{R}}}} E_j({\ensuremath{\mathbf{R}}}).$$ However, to allow for nonadiabatic dynamics, a probability $P_{j\rightarrow k}$ of changing a corresponding to a state $\ket{\phi_j}$ to that of a state $\ket{\phi_k}$ is introduced for every trajectory, a so-called hopping probability, $$\label{eq:hop_prob}
P_{j\rightarrow k}=\frac{2{\operatorname{Re}}(c_j^*c_k\dot{{\ensuremath{\mathbf{R}}}}\cdot {\ensuremath{\mathbf{\Gamma}}}_{jk})\Delta t}{\vert c_j \vert ^2},$$ where $\Delta t$ is a time-step. In both methods an ensemble of trajectories is used to model the quantum nuclear distribution.
Thus, and have the same electronic dynamic equation Eq., but different nuclear dynamics. These differences in treatments of nuclear DOF can result in significant differences in nuclear dynamics in cases where the following two conditions are satisfied: 1) the coupling region is followed by the region where participating PESs have different slopes, 2) probabilities to find the system on either of participating PESs are similar. A good, illustrative example of these differences is dynamics in the Tully extended coupling model [@fssh] (Fig. \[fig:extended\]a). The difference in PES slopes after the coupling region and the chosen initial energy causes the nuclear wave-packet to split, the part on the excited is reflected while that on the ground state is transmitted (Fig. \[fig:extended\]b). gives a nuclear distribution almost indistinguishable from the exact quantum one, while fails to capture the branching of the nuclear distribution.
![a) (red and blue solid) for Tully’s extended coupling model [@fssh], the initial energy (horizontal dashed), and (green dashed). b) Initial (dashed gray, its magnitude is divided by $2$) and final nuclear position distributions (blue for , red for , and black dots for exact quantum). The initial Wigner distribution is centered at $R=-10$ and has average momentum $p=8.5$.[]{data-label="fig:extended"}](e_extended_fig1a.pdf "fig:"){width="7cm"} ![a) (red and blue solid) for Tully’s extended coupling model [@fssh], the initial energy (horizontal dashed), and (green dashed). b) Initial (dashed gray, its magnitude is divided by $2$) and final nuclear position distributions (blue for , red for , and black dots for exact quantum). The initial Wigner distribution is centered at $R=-10$ and has average momentum $p=8.5$.[]{data-label="fig:extended"}](extended_fig1b.pdf "fig:"){width="7cm"}
For the comparison with the exact quantum dynamics, we used the method.[@tannor].
Ehrenfest break-down indicator {#subsec:EHBI}
------------------------------
To provide more quantitative measure for possibility of the Ehrenfest method failure in situations where involved PESs have different slopes, we will use the formula [@zener] to estimate the probabilities of finding the system on different PESs. The equation does not require coefficients for the electronic states entering [Eq. (\[eq:hop\_prob\])]{}, and therefore, is more convenient for estimates. Of course, use of the expression introduces certain constraints, however, such constraints are generally consistent with considered processes of interest: molecular dynamics of an adsorbate that has at least two different molecular electronic states, whose PESs approach each other in localized nuclear configuration regions. Thus, all models considered in this work have nonadiabatic couplings localized near crossings of diabatic potentials.
The potential assumes that the transition region is so small that the energy difference between diabats may be seen as a linear function in time ($2\delta t$), while the off-diagonal coupling $\Delta$ is a constant, and is defined, along with all its auxiliary variables, in Appendix A. The formula for calculating the probability of changing from diabatic surface $a$ to $b$ after passing through a diabatic potentials (Fig. \[fig:LZ\]) crossing point is
$$\label{eq:LZ}
P_{b\leftarrow a}=1-\exp\Big ({\frac{-2\pi \Delta^2}{\dot R {\left| F_b-F_a \right|}}} \Big ),$$
where $\dot R$ is the nuclear velocity evaluated at the crossing point, and $F_b$ and $F_a$ are the diabatic forces at that point. In the adiabatic picture, $P_{b\leftarrow a}$ corresponds to the probability of staying on the adiabatic that was very similar to the diabatic potential of the initial diabatic state ($a$) and ($b$).
![Landau-Zener crossing model with $\delta=1$ and $\Delta=0.5$. Adiabatic (diabatic) are solid (dashed) lines.[]{data-label="fig:LZ"}](LZ_fig2.pdf){width="7cm"}
The argument of the exponential function is called the Massey parameter[@massey] $$\label{eq:massey_d}
\xi_{\rm dia}=\frac{2\pi\Delta^2}{\dot R{\left| F_b-F_a \right|}}.$$ It can be used as an indicator for adiabatic ($\xi \gg 1$) or diabatic ($\xi \ll 1$) behavior. It was originally derived in the diabatic basis [@zener], but it can also be obtained in the adiabatic representation using a perturbative approach [@adiabatic_lz]. Also, using the LZ model, it is possible to transform the Massey parameter $\xi_{\rm dia}$ to the adiabatic representation (see Appendix A for details) $$\label{eq:massey_a}
\xi_{\rm adi}=\frac{\pi(E_2-E_1)}{4{\left| \braket{\phi_1|\partial_t\phi_2} \right|}}=\frac{\pi(E_2-E_1)}{4\dot R {\left| \Gamma_{12} \right|}},$$ where $E_i$’s are adiabatic energies, $\braket{\phi_1|\partial_t\phi_2}$ is a time-derivative coupling, and $\Gamma_{12}$ is the ; all quantities are taken at the diabatic crossing point or at the minimal gap point. The adiabatic formulation is beneficial for analyzing the data from first-principles calculations that do not have underlying diabatic models. Because of this, we will use $\xi\equiv\xi_{\rm adi}$.
From previous consideration we know that will be problematic when probabilities of finding the system on competing pathways are similar, we judiciously consider the Massey parameter $\xi =1$ to be an indicator of this case.
To calculate Massey’s parameter from either [Eq. (\[eq:massey\_d\])]{} or [Eq. (\[eq:massey\_a\])]{}, the value of the nuclear velocity can be estimated from the energy conservation condition. We have two different values for $\dot R$: $\dot R_{\rm adi}$, which assumes a motion on the ground adiabatic state, and $\dot R_{\rm dia}$ for a motion in the first diabatic state. Both velocities would then represent an unperturbed motion in their respective bases. Given an initial total energy $\epsilon$ for a trajectory, the velocities are evaluated as $$\label{eq:Rdot}
\dot R=\Big ( \frac{2}{M}(\epsilon-E)\Big )^{\frac{1}{2}},$$ where for $\dot R = \dot R_{\rm dia} (\dot R_{\rm adi})$ we use $E=E_a(E_1)$ at the diabatic crossing point. Since we are working with the adiabatic values, we will be using the velocity estimate using $E = E_1$ in [Eq. (\[eq:Rdot\])]{}.
Results and Discussion {#sec:sim}
======================
To illustrate correlation between our indicator and performance of mixed quantum-classical (MQC) methods we performed simulations with the EH, SH, and SO methods on two one-dimensional models. All MQC simulations were run using $2000$ trajectories with initial conditions sampled from a Wigner Gaussian distribution with the standard deviation of $1/\sqrt{2}$ for both nuclear positions and momenta. Dynamics were propagated using the fourth-order Runge-Kutta method for $\approx 121$ fs, with a time-step of $\Delta t=1.0 (0.1)$ a.u. for a sinusoidal (metallic surface) model. In all cases, dynamics were checked to yield converged results with respect to the time-step. The particle mass was set to be of $2000$ a.u.
All dynamics were done in the adiabatic basis except in the low coupling case shown in Fig. \[fig:sinus\]c,d. For this case, the motion is highly diabatic, and as discussed in Ref. , yields better results in the representation with fewer hops, which is the diabatic representation in this case.
All coding was done using the Julia language, and the code is available at <https://github.com/iloaiza/MQC>.
Sinusoidal model
----------------
A sinusoidal model has a diabatic potential of the form $$V(R)=\left[ {\begin{array}{cc}
A\sin (kR) & \Delta \\
\Delta & -A\sin (kR) \\
\end{array} } \right].$$ This potential will serve to model periodic systems, while offering a very simple parametric dependence that allows us to easily explore several coupling regimes by varying $\Delta$ with fixed $A=0.02$ and $k=0.5$.
Based on particle’s initial energy we distinguish the following dynamical regimes in this model: : the initial energy is lower than the adiabatic potential energy barrier height on the ground adiabatic state. This case is trivial since there is no appreciable non-adiabatic dynamics taking place, instead the particle is trapped on the ground surface site. (2) Adiabatic regime: there is enough energy to cross the adiabatic barrier but not for appreciable probability of nonadiabatic transition to the excited state. (3) Nonadiabatic regime: the particle has enough energy to be promoted to the excited state but not enough to cross the excited state barrier. (4) High energy regime: the initial energy is higher than the excited barrier. For our investigation only the second, third and fourth regimes are of interest.
#### Adiabatic regime:
Figure \[fig:sinus\_0022\] shows the sinusoidal potential and the nuclear distributions for all three methods. Even when the energy is not enough to get to the excited adiabatic state, by tuning the coupling so that $\xi\approx 1$ we create a population in the excited state, making dynamics slower; whereas dynamics will experience rejected hops, following a trajectory on the ground state that reproduces the results accurately.
{width="7cm"} {width="7cm"}
#### Nonadiabatic regime:
Figure \[fig:sinus\] shows nuclear distributions for the different coupling regimes. All three methods yield very similar results if Massey’s parameter is not near $1$. For large and small couplings, the nuclear wave-packet moves either highly adiabatically or diabatically and both and reproduce quantum simulations well (Figs. \[fig:sinus\]d,f).
Branching of nuclear trajectories will be particularly noticeable for the intermediate coupling case that corresponds to $\xi=1.9$ (Fig. \[fig:sinus\]a). At every diabatic crossing, the coupling splits the ensemble into two parts, one is following the adiabatic route on the ground state and the other is transferred to the upper state and is reflected by its repulsive part. These competing pathways with different behaviors are not captured well with dynamics (Fig. \[fig:sinus\]b).
{width="7cm"} {width="7cm"} {width="7cm"} {width="7cm"} {width="7cm"} {width="7cm"}
#### High energy regime:
Even setting $\xi=1$ does not break down the EH method for the high initial energy case (Fig. \[fig:crit\_bal\]), all three methods produce similar dynamics. The rationale for this behavior is similarity in slopes of ground and excited PESs over a distance spanning several minima. However, one can notice that the exact distribution has two parts (faster and slower) while the distribution is centered right at their separation point. This result hints that at longer times the separation between two parts of the exact distribution may grow while the distribution will be approximating their average.
![a) Sinusoidal model with coupling $c=0.0044$ and $\xi=0.96$: (solid) and initial energy (horizontal dashed). b) Initial nuclear position distribution (dashed gray, its magnitude is divided by $2$, the average momentum $p_0=15$), and final nuclear position distributions: (blue, mirrored for clarity), (red) and SO (black dots).[]{data-label="fig:crit_bal"}](e_crit_ballistic_fig5a.pdf "fig:"){width="7cm"} ![a) Sinusoidal model with coupling $c=0.0044$ and $\xi=0.96$: (solid) and initial energy (horizontal dashed). b) Initial nuclear position distribution (dashed gray, its magnitude is divided by $2$, the average momentum $p_0=15$), and final nuclear position distributions: (blue, mirrored for clarity), (red) and SO (black dots).[]{data-label="fig:crit_bal"}](crit_ballistic_fig5b.pdf "fig:"){width="7cm"}
Metallic surface model
----------------------
To model dynamics on a metallic surface where more electronic states are accessible, we use the model that was originally introduced in Ref. for representing a chemisorbed atom on a 1D metallic chain of atoms. The total diabatic potential is built in three steps. First, an elementary building block is chosen as a harmonic potential describing the chemical bonding in an adatom-metal dimer: $$V_0(R)=\frac{M\Omega^2R^2}{2},$$ where $\Omega$ is a harmonic frequency. Second, this elementary potential is replicated by defining $V_{kD}(R)=V_0(R-kD)$ for $-n\leq k \leq n$, and all replicas are placed in interacting diabatic potential matrix $$\label{eq:single_layer}
V(R)=\left[ {\begin{array}{cccc}
V_{-nD}(R) & \beta_1 & \beta_2 & \dots \\
\beta_1 & V_{(-n+1)D}(R) & \beta_1 & \vdots \\
\vdots & \beta_1 & \ddots & \\
\dots & \dots & & V_{nD}(R)
\end{array} } \right]$$ with $\beta_i$ coupling constants, and $D$ the distance between the atoms of the chain. Third, this set of diabatic potentials is replicated $m+1$ times vertically in the energy direction with addition of a small spatial shift $d$ and diabatic coupling $\alpha$ between nearest neighboring replicas. The scalar potential $V_{kD}$ in Eq. is then replaced by a $(m+1)\times(m+1)$ tridiagonal matrix with diagonal elements $[\mathbb{V}_{kDd\alpha}]_{i,i}(R)=V_{kD}(R+(i-1)d)$, and each coupling constant becomes a $(m+1)\times(m+1)$ diagonal matrix $$\mathbf{V}(R)=\left[ {\begin{array}{ccc}
\mathbb{V}_{-nDd\alpha}(R) & \bbbeta_1 & \dots \\
\bbbeta_1 & \ddots & \vdots \\
\end{array} } \right].$$ All parameters used for this model are given in Table 1.
Parameter Value (a.u.)
----------------------------------- ---------------------
Number of atoms in a chain $2n+1$ 9
Harmonic frequency $\Omega$ $0.0028$
Atom chain distance $D$ $5.0$
Inter-layer couplings:
$\beta_1$ $0.03, 0.011$
$\beta_2$ $0.02$
$\beta_{i>2}$ $0.01$
Intra-layer coupling $\alpha$ 0.004
Intra-layer offset $d$ $1.0\times 10^{-4}$
Number of layers $m+1$ 10
: Parameters of the metallic surface model
\[tab:parameters\]
To summarize, the metallic model has two types of couplings: $\alpha$, between the almost parallel PESs (intra-layer), which are to model electron-hole excitations. $\beta_i$ between the on different equilibrium positions on the surface (inter-layer), and yield adiabatic with very different slopes, as it can be seen in Fig. \[fig:multi\]a.
There are two main energetic regimes which will be of interest. (1) Frictional regime: The initial energy is sufficient to stimulate nonadiabatic transitions between almost parallel PESs constituting the first layer but is insufficient for significant population of an excited adatom state corresponding to the second layer (Fig. \[fig:layered\_low\_energy\]). (2) Two-layer regime: The initial energy allows the system to have similar probabilities for the first and second layers (Fig. \[fig:multi\]). Even higher initial energies are not expected to add new dynamical regimes because excitations to higher layers will only increase possible deviations of dynamics on different layers.
[*a. Frictional regime:*]{} Figure \[fig:layered\_low\_energy\] shows the results for an inter-layer coupling of $\beta_1=0.03$. Both and yield very similar results: the electronic population has spread over the layered states, and both methods capture most of the nuclear branching. Both methods show a frictional behaviour: Fig. \[fig:layered\_low\_energy\]c shows the deviation against a trajectory on the adiabatic ground state (i.e. Born-Oppenheimer dynamics). The adiabatic Massey parameter is $\xi=874$, meaning there are no competing pathways in different layers, which is consistent with the results being almost identical to the ones.
![a) Metallic surface model, $\xi=874$ (between the ground state and the first excited state of the next layer): (solid) and initial energy (horizontal dashed). b) Initial (dashed gray, divided by $4$, zero average momentum) and final nuclear positions for nuclear trajectories (blue for , red for , and black dots for ). results were mirrored for clarity. c) Deviations from a Born-Oppenheimer trajectory with zero initial momentum and -22 a.u. initial position.[]{data-label="fig:layered_low_energy"}](e_low_energy_fig6a.pdf "fig:"){width="7cm"} ![a) Metallic surface model, $\xi=874$ (between the ground state and the first excited state of the next layer): (solid) and initial energy (horizontal dashed). b) Initial (dashed gray, divided by $4$, zero average momentum) and final nuclear positions for nuclear trajectories (blue for , red for , and black dots for ). results were mirrored for clarity. c) Deviations from a Born-Oppenheimer trajectory with zero initial momentum and -22 a.u. initial position.[]{data-label="fig:layered_low_energy"}](low_energy_fig6b.pdf "fig:"){width="7cm"} ![a) Metallic surface model, $\xi=874$ (between the ground state and the first excited state of the next layer): (solid) and initial energy (horizontal dashed). b) Initial (dashed gray, divided by $4$, zero average momentum) and final nuclear positions for nuclear trajectories (blue for , red for , and black dots for ). results were mirrored for clarity. c) Deviations from a Born-Oppenheimer trajectory with zero initial momentum and -22 a.u. initial position.[]{data-label="fig:layered_low_energy"}](fric_fig6c.pdf "fig:"){width="7cm"}
[*b. Two-layer regime:*]{} When we used $\beta_1=0.03$, we obtained a ballistic-like motion, both and yield dynamics that are extremely similar to the ones: we are in an adiabatic regime ($\xi=6.0$), and is able to model this correctly, including the intra-layer effects. Figure \[fig:multi\] shows the results for the $\beta_1=0.011$ regime. As expected, fails to account for the branching of the nuclear trajectories, obtaining results that are very different from the exact quantum ones, while recovers results that are a lot more accurate. The inter-layer coupling is then responsible for the breakdown of dynamics, and our indicator can be applied for such transitions.
![a) Metallic surface model with inter-layer coupling $\beta_1=0.011$ and $\xi=0.90$ (between the ground state and the first excited state of the next layer): (solid), initial energy (horizontal dashed). b) Initial (dashed gray, its magnitude is divided by $4$, zero average momentum) and final nuclear position distributions (blue for , red for , and black dots for ). results were mirrored for clarity.[]{data-label="fig:multi"}](e_layers_fig7a.pdf "fig:"){width="7cm"} ![a) Metallic surface model with inter-layer coupling $\beta_1=0.011$ and $\xi=0.90$ (between the ground state and the first excited state of the next layer): (solid), initial energy (horizontal dashed). b) Initial (dashed gray, its magnitude is divided by $4$, zero average momentum) and final nuclear position distributions (blue for , red for , and black dots for ). results were mirrored for clarity.[]{data-label="fig:multi"}](layerscrit_fig7b.pdf "fig:"){width="7cm"}
Conclusions
===========
We have analyzed break-down of the EH approach in several one-dimensional models containing multiple electronic states and amenable to numerically exact treatment. The main condition for EH failure is accessibility of several electronic PESs with different nuclear forces. The numerical indicator for identifying such cases, the Massey parameter, has been suggested and assessed. When the Massey parameter is much larger or much smaller than 1, and yield very accurate results. On the other hand, when the Massey parameter approaches 1 for several competing pathways, the breaks down failing to simulate nuclear dynamics following one of the competing pathways. can properly treat such cases and outperforms in modeling dynamics for the considered models. The Massey parameter can be calculated using diabatic \[Eq. (\[eq:massey\_d\])\] or adiabatic parameters \[Eq. (\[eq:massey\_a\])\], and although the two versions are numerically somewhat different, they qualitatively agree in most of the cases. Even though, the considered models were inspired by possible PESs of periodic systems, our results applicable not only to dynamics on surfaces: the given criterion could be applied to any system with nonadiabatic transitions and localized .
Since friction theories are based on the approach, the breakdown of will lead to the breakdown of any friction-based method. When the Massey parameter approaches 1, any method that does not incorporate several possible paths will be unable to properly model the dynamics. This result is consistent with that of @shvsfric, where it has been concluded that a friction approach seems to fail in the case where there are non-equivalent pathways. Such behavior can be explained by the breakdown of the underlining EH theory as opposed to an intrinsic failure of the friction approach.
Acknowledgements {#acknowledgements .unnumbered}
================
I.L.G. is grateful to Ilya G. Ryabinkin and Rami Gherib for stimulating discussions. A.F.I. acknowledges the financial support from the Ontario Ministry of Research and Innovation through an Early Researcher Award.
Appendix A: The Landau-Zener model {#sec:appendix .unnumbered}
==================================
Here we reformulate the Massey parameter for the LZ model in the adiabatic quantities. First, let us derive the time-derivative coupling between the adiabatic states of the LZ model. The model potential is a time-dependent matrix in the diabatic representation $$V(t)=\left[ {\begin{array}{cc}
\delta t & \Delta \\
\Delta & -\delta t \\
\end{array} } \right].$$ The eigenfunctions (adiabatic states) of this potential are $$\ket{\phi_1}=\begin{bmatrix}
\sin(\frac{\theta}{2}) \\
-\cos(\frac{\theta}{2})
\end{bmatrix};\ \ket{\phi_2}=\begin{bmatrix}
\cos(\frac{\theta}{2}) \\
\sin(\frac{\theta}{2})
\end{bmatrix} ,$$ where $\tan(\theta)=\Delta/(\delta t)$. The corresponding eigenvalues are $E_1=-\sqrt{\delta^2t^2+\Delta^2}$ and $E_2=\sqrt{\delta^2t^2+\Delta^2}$, and the time-derivative coupling is $$\begin{aligned}
\tau&=\braket{\phi_1|\partial_t \phi_2}=\frac{\braket{\phi_1|\partial_t V|\phi_2}}{\omega} \\
&=\frac{1}{\omega}
\begin{bmatrix}
\sin(\frac{\theta}{2}) \ -\cos(\frac{\theta}{2})
\end{bmatrix}
\begin{bmatrix}
\delta & 0 \\
0 & -\delta
\end{bmatrix}
\begin{bmatrix}
\cos(\frac{\theta}{2}) \\
\sin(\frac{\theta}{2})
\end{bmatrix} \\
&=\frac{\delta \sin(\theta)}{\omega},\end{aligned}$$ where $\omega=E_2-E_1$. Accounting for $\sin(\theta)=\Delta/\sqrt{\delta^2t^2+\Delta^2}$ and $\omega=2\sqrt{\delta^2t^2+\Delta^2}$ we obtain $$\tau=\frac{\delta\Delta}{2(\delta^2t^2+\Delta^2)}.$$ Second, at zero time, when the particle reaches the diabatic intersection point, $\tau$ can be turned into a quantity that will be useful for our purpose by division on $\omega$ $$\label{eq:ad_tau}
\frac{\tau(t=0)}{\omega}=\frac{\delta\Delta}{4(\delta^2t^2+\Delta^2)^{\frac{3}{2}}}\Big |_{t=0}=\frac{\delta}{4\Delta^2}.$$ Third, let us reformulate the diabatic Massey parameter by calculating the diabatic force difference. We can write the force matrix $F=-\nabla V$, and using the chain rule for the derivative we get $$\label{eq:force}
F=\frac{1}{\dot R}{\frac{\partial V}{\partial t}}=\frac{1}{\dot R}\begin{bmatrix}
\delta & 0 \\
0 & -\delta
\end{bmatrix}.$$ Therefore the diabatic force difference is given by ${\left| F_b-F_a \right|}=2\delta\dot R^{-1}$. Note that the model assumes a quick transition through a crossing, meaning the nuclear velocity will not suffer any significant changes: it can be thought of as a constant, a good first order approximation if the force is acting for a short time. Replacing [Eq. (\[eq:force\])]{} into the diabatic Massey parameter: $$\xi=\frac{2\pi\Delta^2}{\dot R{\left| F_b-F_a \right|}}=\frac{\pi\Delta^2}{\delta}.$$ Using Eq. we can rewrite this as $$\xi=\frac{\pi}{4}\frac{4\Delta^2}{\delta}=\frac{\pi\omega}{4\tau(t=0)},$$ obtaining the adiabatic Massey parameter shown in Eq..
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|
---
author:
- 'Xiaojun Huang[^1] and Wanke Yin'
date: 'March, 2007 '
title: '**A Bishop surface with a vanishing Bishop invariant**'
---
**Abstract**
We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant $s<\infty$ is of infinite dimension. We also prove that the equivalence class of the germ of a generic real analytic Bishop surface near a complex tangent with a vanishing Bishop invariant can not be determined by a finite part of the Taylor expansion of its defining equation. This answers, in the negative, a problem raised by J. Moser in 1985 after his joint work with Webster in 1983 and his own work in 1985. Such a phenomenon is strikingly different from the celebrated theory of Moser-Webster for elliptic Bishop surfaces with non-vanishing Bishop invariants. We also show that a formal map between two real analytic Bishop surfaces with the Bishop invariant $\lambda=0$ and with the Moser invariant $s\not = \infty$ is convergent. Hence, two real analytic Bishop surfaces with $\lambda=0$ and $s<\infty$ are holomorphically equivalent if and only if they have the same formal normal form (up to a trivial rotation). Notice that there are many non-convergent formal transformations between Bishop surfaces with $\lambda=0$ and $s=\infty$. Notice also that a generic formal map between two real analytic hyperbolic Bishop surfaces is divergent as shown by Moser-Webster and Gong. Hence, Bishop surfaces with a vanishing Bishop invariant and $s\not = \infty$ behave very differently, in this respect, from hyperbolic Bishop surfaces or elliptic Bishop surfaces with $\lambda=0 $ and $ s=\infty$. We also show that a Bishop surface with $\lambda=0$ and $s<\infty$ generically has a trivial automorphism group and has the largest possible automorphism group if and only if it is biholomorphic to the model surface $M_s=\{(z,w)\in {\mathbb C}^2:\ w=|z|^2+z^s+\-{z}^s\}$. Notice that, by the Moser-Webster theorem, an elliptic Bishop surface with $\lambda\not = 0$, always has automorphic group ${\mathcal Z}_2$. Hence, Bishop surfaces with $\lambda=0$ and $s\not = \infty$ have the similar character as that of strongly pseudoconvex real hypersurfaces in the complex spaces of higher dimensions.
Introduction and statements of main results
===========================================
In this paper, we study the precise holomorphic structure of a real analytic Bishop surface near a complex tangent point with the Bishop invariant vanishing. A Bishop surface is a generically embedded real surface in the complex space of dimension two. Points on a Bishop surface are either totally real or have non-degenerate complex tangents. The holomorphic structure near a totally real point is trivial. Near a point with a complex tangent, namely, a point with a non-trivial complex tangent space of type $(1,0)$, the consideration could be much more subtle. The study of this problem was initiated by the celebrated paper of Bishop in 1965 \[Bis\], where for a point $p$ on a Bishop surface $M$ with a complex tangent, he defined an invariant $\lambda$ now called the Bishop invariant. Bishop showed that there is a holomorphic change of variables, that maps $p$ to $0$, such that $M$, near $p=0$, is defined in the complex coordinates $(z,w)\in {\mathbb C}^2$ by $$w=z\-{z}+\lambda (z^2+\-{z}^2)+o(|z|^2),
\label{eqn:000}$$ where $\lambda\in [0,\infty]$. When $\lambda=\infty$, (\[eqn:000\]) is understood as $w=z^2+\-{z}^2+o(|z|^2).$ It is now a standard terminology to call $p$ an elliptic, hyperbolic or parabolic point of $M$, according to whether $
\lambda\in [0,1/2)$, $\lambda \in (1/2,\infty)$ or $\lambda=1/2,\infty$, respectively.
Bishop discovered an important geometry associated with $M$ near an elliptic complex tangent $p$ by proving the existence of a family of holomorphic disks attached to $M$ shrinking down to $p$. He also proposed several problems concerning the uniqueness and regularity of the geometric object obtained by taking the union of all locally attached holomorphic disks. These problems, including their higher dimensional cases, were completely answered through the combining efforts of many people. (See \[KW1\], \[BG\], \[KW2\], \[MW\], \[Mos\], \[HK\], \[Hu3\]; in particular, see \[KW1\], \[MW\], \[Hu3\]).
Bishop invariant is a quadratic invariant, capturing the basic geometric character of the surface. The celebrated work of Moser-Webster \[MW\] first investigated the more subtle higher order invariants. Different from Bishop’s approach of using the attached holomorphic disks, Moser-Webster’s starting point is the existence of a more dynamically oriented object: an intrinsic pair of involutions on the complexification of the surface near a non-exceptional complex tangent. Here, recall that the Bishop invariant is said to be non-exceptional if $\lambda\not = 0, 1/2, \infty$ or if $\lambda\nu^2-\nu +\lambda=0$ has no roots of unity in the variable $\nu$. Moser-Webster proved that, near a non-exceptional complex tangent, $M$ can always be mapped, at least, by a formal transformation to the normal form defined in the complex coordinates $(z,w=u+iv)\in {\mathbb C}^2$ by: $$u=z\-{z}+(\lambda +\epsilon u^s)(z^2+\-{z}^2)\ , \ v=0\ ,\
\epsilon\in\{0,1,-1\}\ ,\ s\in {\mathbb Z}^+.$$
Moser-Webster also provided a convergence proof of the above mentioned formal transformation in the non-exceptional elliptic case: $0<\l<1/2$. However, the intriguing elliptic case with $\lambda=0$ has to be excluded from their theory. Instead, Moser in \[Mos\] carried out a study for $\lambda=0$ from a more formal power series point of view. Moser derived the following formal pseudo-normal form for $M$ with $\lambda=0$: $$w=z\-{z}+z^s+\-{z}^s+2Re\{\sum_{j\ge s+1}a_{j}z^j\}.$$ Here $s$ is the simplest higher order invariant of $M$ at a complex tangent with a vanishing Bishop invariant, which we call the Moser invariant. Moser showed that when $s=\infty$, $M$ is then holomorphically equivalent to the quadric $M_{\infty}=\{(z,w)\in
{\mathbb C}^2: w=|z|^2\}$.
Moser’s formal pseudo-normal form is still subject to the simplification of a very complicated infinitely dimensional group $aut_{0}(M_{\infty})$, the formal self-transformation group of $M_{\infty}$. And it was left open from the work of Moser \[Mos\] to derive any higher order invariant other than $s$ from the Moser pseudo-normal form. At this point, we mention that $aut_{0}(M_{\infty})$ contains many non-convergent elements. Based on this, Moser asked two basic problems concerning a Bishop surface near a vanishing Bishop invariant in his paper \[Mos\]. The first one is concerning the analyticity of the geometric object formed by the attached disks up to the complex tangent point. This was answered in the affirmative in \[HK\]. Hence, the work of \[HK\], together with that of Moser-Webster \[MW\], shows that, as far as the analyticity of the local hull of holomorphy is concerned, all elliptic Bishop surfaces are of the same character. The second problem that Moser asked is concerning the higher order invariants. Notice that by the Moser-Webster normal form, an analytic elliptic Bishop surface with $\lambda\not =0$ is holomorphically equivalent to an algebraic one and possesses at most two more higher order invariants. Moser asked if $M$ with $\lambda=0$ is of the same character as that for elliptic surfaces with $\lambda\not =0$. Is the equivalence class of a Bishop surface with $\lambda=0$ determined by an algebraic surface obtained by truncating the Taylor expansion of its defining equation at a sufficiently higher order level? Gong showed in \[Gon2\] that under the equivalence relation of a smaller class of transformation group, called the group of holomorphic symplectic transformations, $M$ with $\lambda=0$ does have an infinite set of invariants. However, under this equivalence relation, elliptic surfaces with non-vanishing invariants also have infinitely many invariants. Gong’s work later on (see, for example, \[Gon2-3\] \[AG\]) demonstrates that as far as many dynamical properties are concerned, exceptional or non-exceptional hyperbolic, or even parabolic complex tangents are not much different from each other.
In this paper, we derive a formal normal form for a Bishop surface near a vanishing Bishop invariant, by introducing a quite different weighting system. This new weighting system fits extremely well in our setting and may have applications in many other problems. We will obtain a complete set of invariants under the action of the formal transformation group. We show, in particular, that the modular space for Bishop surfaces with a vanishing Bishop invariant and with a fixed (finite) Moser invariant $s$ is an infinitely dimensional manifold in a Frèchet space. This then immediately provides an answer, in the negative, to Moser’s problem concerning the determination of a Bishop surface with a vanishing Bishop invariant from a finite truncation of its Taylor expansion. Furthermore, it can also be combined with some already known arguments to show that most Bishop surfaces with $\lambda=0,\ s\not =\infty$ are not holomorphically equivalent to algebraic surfaces. Hence, one sees a striking difference of an elliptic Bishop surface with a vanishing Bishop invariant from elliptic Bishop surfaces with non-vanishing Bishop invariants. The general phenomenon that the infinite dimensionality of the modular space has the consequence that any subclass formed by a countable union of finite dimensional spaces is of the first category in the modular space seems already clear even to Poincaré \[Po\]. In the CR geometry category, we refer the reader to a paper of Forstneric \[For\] in which the infinite dimensionality of the modular space of generic CR manifolds is used to show that CR manifolds holomorphically equivalent to algebraic ones form a very thin set among all real analytic CR manifolds. Similar to what Forstneric did in \[For\], our argument to show the generic non-algebraicity from the infinite dimensionality of the modular space also uses the Baire category theorem. It is not clear to us if the new normal form obtained in this paper for a real analytic Bishop surface with $\lambda=0,\ s<\infty$ is always convergent. However, we will show that if the formal normal form is convergent, then the map transforming the surface to its normal form must be convergent in case the Moser invariant $s\not =\infty$. Remark that there are many non-convergent formal maps transforming real analytic Bishop surfaces with a vanishing Bishop invariant and with $s=\infty$ to the model surface $M_{\infty}$ defined before. (See \[MW\] \[Mos\] \[Hu2\]). Hence, our convergence theorem reveals a non-trivial role that the Moser invariant has played in the study of the precise holomorphic structure of a Bishop surface with $\lambda=0$. At this point, we would like to mention that there are many other different problems where one also considers the convergence of formal power series, though very different methods and approaches need to be employed in different settings. To name a few, we here mention the papers of Baouendi-Ebenfelt-Rothschild \[BER\]\[BMR\]\[MMZ\], Webster \[We\], Stolovitch \[St\] and the references therein. In the research described in \[BER\]\[BMR\]\[MMZ\], one tries to understand the convergence of formal CR maps between not too degenerate real analytic CR manifolds. In \[We\] \[Sto\], one encounters other type of convergence problems in the normalization of real submanifolds in ${\Bbb C}^n$. Our convergence argument uses the Moser-Webster \[MW\] polarization, as in the non-vanishing Bishop invariant case treated by Moser-Webster. However, different from the Moser-Webster situation, we do not have a pair of involutions, which were the starting point of the Moser-Wbetser theory. The main idea in the present paper for dealing with our convergence problem is to find a new surface hyperbolic geometry, by making use of the flattening theorem of Huang-Krantz \[HK\].
We next state our main results, in which we will use some terminology to be defined in the next section:
**Theorem 1.1:**
**Theorem 1.2**:
Define ${\mathcal Z}_s$ for the group of transformations consisting of maps of the form $\{\psi_\theta: (z,w) \mapsto (e^{i\theta}z,w), \hskip 5pt
e^{is\theta}=1\}$.
We next give several immediate consequences of Theorems 1.1 and 1.2:
**Corollary 1.3:**
[**Corollary 1.4**]{}: [*Let $M_1$ and $M_2$ be real analytic Bishop surfaces with $\lambda=0$ and $s\not =\infty$ at $0$. Suppose that $M_1$ has a formal normal form: $$w'=z'\bar{z'}+{z'}^s+\bar{z'}^s+2Re\{\sum\limits_{k=1}^{\infty}\sum\limits_{j=2}^{s-1}
a_{ks+j}{z'}^{ks+j}\};$$ and suppose that $M_2$ has a formal normal form: $$w'=z'\bar{z'}+{z'}^s+\bar{z'}^s+2Re\{\sum\limits_{k=1}^{\infty}\sum\limits_{j=2}^{s-1}
b_{ks+j}{z'}^{ks+j}\}.$$ Then $(M_1,0)$ is biholomorphic to $(M_2,0)$ if and only if there is a constant $\theta$, with $e^{s\theta\sqrt{-1}}=1$, such that $a_{ks+j}=e^{\theta j\sqrt{-1}}b_{ks+j}$ for any $k\ge 1$ and $j=2,\cdots,s-1$.*]{}
[**Theorem 1.5**]{}: [*A generic real analytic Bishop surface with a vanishing Bishop invariant and $s\not =\infty$ is not holomorphically equivalent to an algebraic surface in ${\mathbb{C}}^2$.*]{}
[**Acknowledgment**]{}: The key part of this work was completed when the first author was visiting, in January of 2006, the School of Mathematics, Wuhan University, China and when both authors were enjoying the month long visit at the Institute of Mathematical Sciences, The Chinese University of Hong Kong in the Spring of 2006. The first author would like very much to thank his friends Professors Hua Chen and Gengsheng Wang at Wuhan University for their hospitality during the visit. Both authors would also like to express their appreciation to IMS at the Chinese University of Hong Kong for its generous supports and helps provided during the authors’ visit.
Uniqueness of formal maps between approximately normalized surfaces
====================================================================
In what follows, we use $(z,w)$ or $(z',w')$ for the coordinates for ${\mathbb C}^2$. Let $A(z,\-{z})$ be a formal power series in $(z,\-{z})$ without constant term. We say that the order of $A(z,\bar{z})$ is $k$ if $A(z,\-{z})=\sum_{j+l=k}A_{j\-{l}}z^j\-{z}^l+o(|z|^k)$ with at least one of the $A_{j\-{l}}\in {\mathbb C}$ ($j+l=k)$ not equal to $0$. In this case, we write $\ord(A(z,\-{z}))=k$. We say $\ord(A(z,\-{z}))\ge k$ if $A(z,\-{z})=O(|z|^k)$.
Consider a formal real surface M in $\mathbb{C}^2$ near the origin. Suppose that $0$ is a point of complex tangent for $M$. Then, after a linear change of variables, we can assume that $T^{(1,0)}_0M=\{w=0\}$. If there is no change of coordinates such that $M$ is defined by an equation of the form $w=O(|z|^3)$, we then say $0$ is a point of $M$ with a non-degenerate complex tangent. In this case, Bishop showed that there is a change of coordinates in which $M$ is defined by (\[Bis\] \[Hu1\]) $$w=z\-{z}+\lambda (z^2+\-{z}^2)+O(|z|^3).$$ Here $\lambda\in [0,\infty]$ and when $\lambda=\infty$, the equation takes the form: $w=z^2+\-{z}^2+O(|z|^3).$ $\lambda$ is the first absolute invariant of $M$ at $0$, called the Bishop invariant. Bishop invariant is a quadratic invariant, resembling to the Levi eigenvalue in the hypersurface case. When $\lambda\in [0,1/2)$, we say that $M$ has an elliptic complex tangent at $0$. In this paper, we are only interested in the case of an elliptic complex tangent. We need only to study the case of $\lambda=0$; for, in the case with $\lambda\in (0,1/2)$, the surface has been well understood by the work of Moser-Webster \[MW\]. When $\lambda=0$, Moser-Webster and Moser showed in \[MW\] \[Mos\] that there is an integer $s\ge 3$ or $s=\infty$ such that $M$ is defined by $$w=z\bar{z}+z^s+\bar{z}^s+E(z,\bar{z}),
\label{eqn:Jam000}$$ where $E$ is a formal power series in $(z,\-{z})$ with $\ord(E) \geq s+1.$ When $s=\infty$, we understand the defining equation as $w=z\-{z}$, namely, $M$ is formally equivalent to the quadric $M_{\infty}=\{w=z\-{z}\}$. $s$ is the next absolute invariant for $M$, called the Moser invariant. The case for $s=\infty$ is also well-understood through the work of Moser \[Mos\]. Hence, in all that follows, our $M$ will have $\lambda=0$ and a fixed $s<\infty$.
A formal map $z'=F(z,w),\ w'=G(z,w)$ without constant terms is called an invertible formal transformation (or simply, a formal transformation) if $\frac{\partial (F,G)}{\partial
(z,w)}(0,0)$ is invertible. When a formal map has no constant term, we also say that it preserves the origin.
**Lemma 2.1**:
[*Proof of Lemma 2.1*]{}: (i) is the content of Lemma 3.2 of \[Hu1\]. To prove (ii), we write $F=(az+f,cw+g)$, where by (i), we can assume that $$f(z,w)=O(|z|^2+|w|)\ \ ,\ \
g(z,w)=O(|w|^2+|zw|+|z|^3).$$ Notice that $$f(0,E(0,\bar{z}))=O(\bar{z}^s)\ \ ,\ \
\overline{{f}}(\bar{z},\bar{E}(\bar{z},0))=O(\bar{z}^2)\ \
,\ \ g(0,E(0,\bar{z}))=o(\bar{z}^s).$$ Applying the defining equation of $M'$ , we have, on $M$, the following: $$\begin{array}{lll}
cw+g(z,w)&=&|a|^2|z|^2+\bar{a}\bar{z}f(z,w)
+az\overline{f}(\bar{z},\bar{w})
+f(z,w)\overline{f}(\bar{z},\bar{w})\\
&&+\left(az+f(z,w)\right)^s+\left(\bar{a}\bar{z}
+\overline{f}(\bar{z},\bar{w})\right)^s+o(|z|^s).
\end{array}$$ Regarding $z$ and $\bar{z}$ as independent variables in the above equation and then letting $z=0,w=E(0,\bar{z}),\bar{w}=\bar{E}(\bar{z},0)$, we obtain $$c\bar{z}^s+o(\bar{z}^s)=(\bar{a}\bar{z})^s+o(\bar{z}^s).$$ Hence, it follows that $c=\bar{a}^s$. Together with $c=|a|^2$ and $s
\geq 3$, we get $$c=1\ ,\ a=e^{i\theta}\ ,\ \hbox{where $\theta$ is a constant}.$$ Now we turn to the proof of (iii). Notice that $$G(z,w)=|F(z,w)|^2+E^*(F(z,w),\overline{F(z,w)})\hskip 5pt for \hskip
5pt (z,w) \in M.$$ Since $E^*$ is now assumed to be formally real valued, we have $$G(z,w)=\overline{G(z,w)} \ \ \ \hbox{on}\ M.$$ Write $$G(z,w)=\sum\limits_{\a,\b}^{\infty}a_{\a\b}z^\a w^\b.$$ We will prove inductively that $a_{\a\b}=\-{a_{\a\b}}$ for $\a=0$ and $a_{\a\b}=0$ otherwise. First, for each $m>>1$, write $E=E_{(m)}(z,\-{z})+E_m$ with $E_{(m)}(z,\-{z})$ a polynomial of degree at most $m-1$ and $E_m=O(|z|^m)$. Then for any $m>>1$, there are integers $N_1(m)>>m$ and $N_2(m)>>m$ such that $$\sum\limits_{\a,\b=0}^{N_2(m)}a_{\a\b}z^\a
w^\b=\sum\limits_{\a,\b=0}^{N_2(m)}\-{a_{\a\b}z^\a}w^\b +o(|z|^m)\ \
,\ \ w=z\-{z}+E_{(N_1(m))}(z,\-{z}).$$ Next, suppose that $N_0=\a_0+2\beta_0$ is the smallest number such that $a_{\a\b}$ is real-valued for $\a=0$, and zero otherwise whenever $\a+2\b<N_0$. (If such an $N_0$ does not exist, then Lemma 2.1 (iii) holds automatically). Choose $m>>N_0$. For $0<r<<1$, define $\sigma_{N_1}(\xi,r)$ to be the biholomorphic map from the unit disk in $\mathbb C$ to the smoothly bounded simply connected domain: $\{\xi\in {\mathbb C}:\ |\xi|^2+r^{-2}E_{(N_1)}(r\xi,r\-{\xi})<1\}$ with $\sigma_{N_1}(\xi,r)=\xi(1+O(r)).$ Since the disk $(r\sigma_{N_1}(\xi,r),r^2)$ is attached to $M_{N_1}$ defined by $ w=z\-{z}+E_{(N_1)}(z,\-{z})$, it follows that $$\sum\limits_{\a+2\b=N_0}a_{\a\b}r^{N_0}\xi^\a =\sum\limits_{\a+2\b=N_0}\-{a_{\a\b}\xi^\a}r^{N_0}+o(r^{N_0}), \ \ |\xi|=1.$$ Letting $r\ra 0$, we get $$\sum\limits_{\a+2\b=N_0}a_{\a\b}\xi^\a =\sum\limits_{\a+2\b=N_0}\-{a_{\a\b}\xi^\a}, \ \ |\xi|=1,$$ from which we see that when $\a+2\b=N_0$, $a_{\a\b}$ is real for $\a=0$, and zero otherwise. This contradicts the choice of $N_0$ and thus completes the proof of Lemma 2.1 (iii). $\endpf$
The main purpose of this section is to prove the following uniqueness result for mappings between approximately normalized surfaces:
**Theorem 2.2**:
One of the crucial ideas for the proof of Theorem 2.2 is to set the weight of $\-{z}$ differently from that of $z$. More precisely, we set the weight of $z$ to be $1$ and that of $\bar{z}$ to be $s-1$. For a formal power series $A(z,\-{z})$ with no constant term, we say that $wt(A(z,\-{z}))=k$, or $wt(A(z,\-{z}))\ge k$, if $A(tz,t^{s-1}\-{z})=t^kA(z,\-{z})$, or , $A(tz,t^{s-1}\-{z})=O(t^k)$, respectively, as $t\in {\mathbb R}\ra 0$. In all that follows, we use $\Theta_l^j$ to denote a formal power series in $z$ and $\bar{z}$ of order at least $j$ and weight at least $l$. (Namely, $\Theta^j_l(tz,t\-{z})=O(t^j)$ and $\Theta_l^j(tz,t^{s-1}\-{z})=O(t^l)$ as $t\ra 0$). We use $\mathbb{P}_l^j$ to denote a homogeneous polynomial in $z$ and $\bar{z}$ with the exact order $j$ and weight at least $l$. [*We emphasize that $\Theta_l^j$ and $\mathbb{P}_l^j$ may be different in different contexts*]{}.
In what follows, we also define the normal weight of $z, w$ to be $1,2$, respectively. For a formal power series $h(z,w,\-{z},\-{w})$, we use $wt_{nor}(h)\ge k$ to denote the vanishing property: $h(tz,t^2w,t\-{z},t^2\-{w})=O(t^k)$ as $t\ra 0$. Let $h(z,w)$ be a formal power series in $(z,w)$ without constant term. Then we have the formal expansion: $$h(z,w)=\sum\limits_{l=1}^{\infty}h_{nor}^{(l)}(z,w)$$ where $$h_{nor}^{(l)}(tz,t^2w)=t^lh_{nor}^{(l)}(z,w)$$ is a polynomial in $(z,w)$. Notice that $h_{nor}^{(l)}(z,w)$ is homogeneous of degree $l$ in the standard weighting system which assigns the weight of $z$ and $w$ to be $1$ and $2$, respectively. In what follows, we write $$h_l(z,w)=\sum\limits_{j=l}^{\infty}h_{nor}^{(j)}(z,w)\ \hbox{ and
}\ h_{(l)}=\sum_{j=1}^{l-1}h_{nor}^{(j)}(z,w).
\label{eqn:James-00010}$$
: We need to prove that any solution $(f,g)$ of the following equation has the property that $wt_{nor}(f(z,w))\ge 2n+1$, $wt_{nor}(g(w))\ge 2n+2$ under the normalization conditions as in the theorem: $$\begin{array}{lll}
w+g(w)&=&(z+f(z,w))(\bar{z}+\overline{f(z,w)}) +2Re \large\{
(z+f(z,w))^s\\
&&+\sum\limits_{k=1}^{n}\sum\limits_{j=2}^{s-1}b_{ks+j}(z+f(z,w))^{ks+j}
\large\}+E_2(f(z,w),\-{f(z,w)})
\end{array}
\label{eqn:Jam01}$$ where $w=z\bar{z}+z^s+\bar{z}^s+E(z,\bar{z})$ with $$E=2Re\left(\sum\limits_{k=1}^{n}\sum\limits_{j=2}^{s-1}a_{ks+j}z^{ks+j}\right)+E_1(z,\-{z}).$$ With an immediate simplification, (\[eqn:Jam01\]) takes the form: $$\begin{array}{lll}
g(w)&=&\bar{z}f(z,w)+z\overline{f(z,w)}+|f(z,w)|^2
+2Re\big\{(z+f(z,w))^s-z^s\\
&&+\sum\limits_{k=1}^{n}\sum\limits_{j=2}^{s-1}\left(b_{ks+j}(z+f(z,w))^{ks+j}-a_{ks+j}z^{ks+j}\right)\big\}
+o(|z|^{ns+s-1})
\end{array}
\label{eqn:Jam02}$$
In the proof of Theorem 2.2, we set the following convention. For any positive integer $N$, we define $a_N$ and $b_{N}$ to be as in Theorem 2.2 if $N=ks+j$ with $k\le n,\ 2\le j\le s-1$, and to be $0$ otherwise. For the rest of this section, we will define a positive integer $N_0$ as follows:
Suppose that there is a pair of integers $(j_0,k_0)$ such that $s<k_0s+j_0(\le ns+s-1)$ is the smallest number satisfying $a_{k_0s+j_0} \neq b_{k_0s+j_0}.$ We then define $N_0=k_0s+j_0$. Otherwise, we define $N_0=sn+s$.
The proof of Theorem 2.2 is carried out in two steps, according to the vanishing order of $f$ being even or odd.
**Step I of the proof of Theorem 2.2:** In this step, we assume that either $$\ord\left(f(z,w(z,\bar{z})\right)=2t$$ is an even number or $f\equiv 0$, where $w(z,\bar{z})=z\bar{z}+z^s+\bar{z}^s+E(z,\bar{z})$. Write $g(w)=c_lw^l+o(w^l)$. Denote by $\widehat{N_0}=\hbox{min}\{N_0,\ \ord(f), sn+s-1\}$. (If $f\equiv
0$, we define $\ord(f)=\infty$.) Then (\[eqn:Jam02\]) gives the following: $$c_lz^l\bar{z}^l+O(|z|^{2l+1})=2Re[(b_{N_0}-a_{N_0})z^{N_0}]+O(|z|^{\widehat{N_0}+1}).$$ From this, we can easily conclude the following:
(2.I). Suppose that $2t\ge N_0$ and $c_l\not =0$. Then $2l>\min\{N_0,sn+s-1\}$ and $b_{N_0}=a_{N_0}$. By our choice of $N_0$, $N_0$ must be $ns+s$. Hence, the theorem in this case readily follows.
(2.II). When $2t<N_0$, then $2l\ge \min\{2t+2, sn+s\}$ under the assumption that $c_l\not = 0$. Thus $l>t\ge 1$ (if $c_l\not = 0$).
Suppose that $N_0=2t+1$ in Case (2.II). Assuming that $N_0<ns+s$ and collecting terms with degree $2t+1$ in $(\ref{eqn:Jam02})$, we obtain
$$\bar{z}f_{nor}^{(2t)}(z,z\bar{z})+z\overline{f_{nor}^{(2t)}(z,z\bar{z})}+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)=0$$
This clearly forces that $a_{N_0}=b_{N_0}$. Thus, we must have $N_0=ns+s$ and Theorem 2.2 also follows easily in this setting.
*Hence, we will assume, in what follows:*
(2.III). $ns+s>N_0\ge 2t+2,\ l>t\ge 1$.
Collecting terms with (the ordinary) degree 2t+1 in (\[eqn:Jam02\]), we get: $$\bar{z}f_{nor}^{(2t)}(z,z\bar{z})+z\overline{f_{nor}^{(2t)}(z,z\bar{z})}=0
\label{eqn:Jam03}$$ Writing $f_{nor}^{(2t)}(z,w)=\sum\limits_{k+2l=2t}a_{kl}z^kw^l$ and substituting it back to (2.12), we then get:\
$$f_{nor}^{(2t)}(z,w)=aw^t-\bar{a}z^2w^{t-1}$$ for $a \neq 0$. Hence $$f(z,w)=f_{nor}^{(2t)}(z,w)+f_{2t+1}(z,w)=aw^t-\bar{a}z^2w^{t-1}+f_{2t+1}(z,w)
\label{eqn:Jam03-00}$$ Next, a simple computation shows that $wt(w) \geq s,\ \ord(w(z,\-{z})) \geq 2, \ wt(f_{nor}^{(2t)}) \geq
st+2-s,\ wt(\overline{f_{nor}^{(2t)}}) \geq
st,\ g=g_{2t+2},\ f=f_{nor}^{(2t)}+f_{2t+1}(z,w).$ Also if $ \ l_1+l_2 \geq s$ with $\ l_2>
1,$ or $ l_1+l_2 >s$ with $l_2 \ge 1$, then $wt(z^{l_1}f_{nor}^{{(2t)l_2}})=l_1+l_2(ts+2-s) \geq ts+2$. Moreover, $wt(z^{l_1}f_{nor}^{{(2t)}l_2}f_{2t+1}^{l_3}) \geq s$ if $l_1+l_2+l_3 \geq s-1,\ l_2^2+l_3^2 \neq 0$.
We can verify the following $$|f(z,w)|^2=2Re(\-{f_{nor}^{(2t)}}f_{2t+1})+\Theta^{2t+2}_{st+2}+\Theta^2_{st+2}f_{2t+1}.$$
Substituting (\[eqn:Jam03-00\]) into (\[eqn:Jam02\]), we get: $$\begin{array}{rcl}
g_{2t+2}(w)&=&2Re\{(\-{z}+sz^{s-1})f\}+|f(z,w)|^2+
2Re\{\sum_{l=2}^{s}
\mathbb{P}_{s-l}^{s-l}f^{l}\}\\
%\bar{z}(f_{nor}^{(2t)}+f_{2t+1})+z(\overline{f_{nor}^{(2t)}}+\overline{f_{2t+1}})\\
% &+|f_{nor}^{(2t)}|^2+\overline{f_{nor}^{(2t)}}\cdot f_{2t+1}+f_{nor}^{(2t)}\overline{f_{2t+1}}
% +f_{2t+1}\overline{f_{2t+1}}+2Re\{sz^{s-1}f\}
&&+2Re(\sum\limits_{
%\stackrel{\tau=0 \ or}
{ \tau=ks+j<
N_0}}\sum_{l=0}^{\tau-1}
\mathbb{P}_{l}^{l}f^{\tau-l})
+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)+\Theta^{N_0+1}_{N_0+1}\\
&=&2Re\{(\bar{z}+sz^{s-1})f_{nor}^{(2t)}+(\bar{z}+sz^{s-1}
+\-{f_{nor}^{(2t)}})f_{2t+1}(z,w)\} \\ &
&+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)
+\Theta_{s}^{2}f_{2t+1}(z,w)+\Theta_{s}^{2}\-{f_{2t+1}(z,w)}
+\Theta_{N_s}^{2t+2}
%\\ &+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{2t+1}(z,w)} +\Theta_{N_t+1}^{N_0+1}
% +\Theta_{N_t+1}^{N_0+1}
\end{array}$$ Here $N_0$ is defined as before and $N_s:= \hbox{min}\{ts+2, N_0+1\}$.
Notice that $$\begin{array}{l}
\bar{z}f_{nor}^{(2t)}+z\overline{f_{nor}^{(2t)}}+2Re\{sz^{s-1}f_{nor}^{(2t)}\}\\
=2Re\{\bar{z}(aw^t-\bar{a}z^2w^{t-1})+sz^{s-1}(aw^t-\bar{a}z^2w^{t-1})\}\\
=-\bar{a}z^2\bar{z}w^{t-1}+z\bar{a}w^t-sz^{s-1}\bar{a}z^2w^{t-1}+\Theta_{ts+2}^{2t+2}\\
=(1-s)\bar{a}z^{s+1}w^{t-1}+\Theta_{ts+2}^{2t+2}
%\label{eqn:Jam05}
\end{array}$$
Hence, we obtain $$\begin{array}{rll}
g_{2t+2}(w)&=&(1-s)\bar{a}z^{s+1}(z\bar{z}+z^s)^{t-1}+(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{2t+1}(z,w)\\
&&+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)
+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{2t+1}(z,w)}\\
&&+2Re\{\-{f_{nor}^{(2t)}}f_{2t+1}(z,w)\} + \Theta_{N_s}^{2t+2}
\label{eqn:Jam05}
\end{array}$$ If $t=1$, collecting terms of degree $s+1$ in (\[eqn:Jam05\]) and noticing that $N_0>s+1$ by the given condition, we get $$\begin{array}{lll}
\sum_{2j}\delta_{2j}^{s+1}g_{nor}^{(2j)}(z\-{z})&=&
(1-s)\bar{a}z^{s+1}+\bar{z}f_{nor}^{(s)}(z,z\-{z})+z\overline{f_{nor}^{(s)}(z,z\-{z})}\\
&&
+az^2\-{f_{nor}^{(s-1)}(z,z\-{z})}+\-{az^2}f_{nor}^{(s-1)}(z,z\-{z})+\mathbb{P}_{s+2}^{s+1}.
\end{array}$$ Here $\d_{2j}^{s+1}$ takes value $1$, when $2j=s+1$, and $0$ otherwise.
Since $s+2 \geq s+1$, $\mathbb{P}_{s+2}^{s+1}=\bar{z}A$ with $A$ a polynomial. Thus it follows easily that $(1-s)\bar{a}z^{s+1}$ divides $\bar{z}$. This is a contradiction and thus $t
>1$. In particular, (\[eqn:Jam05\]) can be written as
$$\begin{array}{lll}
g_{2t+2}(w)&=&(1-s)\bar{a}z^{s+1}(z\bar{z}+z^s)^{t-1}+(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{2t+1}(z,w)\\
&&+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)
+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{2t+1}(z,w)}+
\Theta_{N_s}^{2t+2}
\label{eqn:Jam06}
\end{array}$$
We next prove the following:
**Lemma 2.3**: Assume that $2t+j(s-2)+2 \leq m \leq
2t+(j+1)(s-2)+1$ with $0 \leq j \leq t-1$ and $m\le N_0$. Then $$\begin{aligned}
\begin{array}{lll}
g_m(w)&=&\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z}+z^s)^{t-j-1}+(\bar{z}+sz^{s-1}
+\Theta_{s}^{2})f_{m-1}(z,w)\\
&&+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{m-1}(z,w)}+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)+\Theta_{N_s}^{m}
\label{eqn:Jam060}
\end{array}\end{aligned}$$
[*Proof of Lemma 2.3*]{}: The argument presented above gives the proof of the lemma with $m=2t+2$. We complete the proof of the lemma in three steps.
[**Step I of the proof of Lemma 2.3**]{}: This step is not needed when $s=3$. Denote $ m_0=2t+j(s-2)+2$, where $j$ is an integer with $0\le j\le t-1$. Suppose that $m_0\le N_0$. We also assume that there is an integer $m$ such that $m\ge m_0$, $m+1\le 2t+(j+1)(s-2)+1$ (such an $m$ certainly does not exist if $s=3$), $m+1\le N_0$ and moreover the formula (\[eqn:Jam060\]) holds for this $m$. Collecting terms of degree $m$ in (\[eqn:Jam060\]), we get $$g^{(m)}(z\bar{z})=\bar{z}f_{nor}^{(m-1)}(z,z\bar{z})+z\overline{f_{nor}^{(m-1)}(z,z\bar{z})}+\hat{\mathbb{P}}_{N_s}^{m}
\label{eqn:Jam01-1}$$ Notice that $\hat{\mathbb{P}}_{N_s}^{m}(=\mathbb{P}_{N_s}^{m})$ must be real valued, and notice that $g^{(m)}(z\-{z})$ is also of weight at least $N_s$. We can write $$g^{(m)}(z\bar{z})-\mathbb{P}_{ts+2}^{m}=\sum\limits_{\stackrel{\alpha+\beta=m}{\alpha+\beta(s-1)\geq
N_s}} a_{\alpha\bar{\beta}}z^{\alpha}\bar{z}^{\beta}
\label{eqn:James01}$$ Write\
$$\begin{array}{ll}
f_{nor}^{(m-1)}(z,z\-{z})=\sum\limits_{\widetilde{\alpha}+2\widetilde{\beta}=m-1}
b_{\widetilde{\alpha}\widetilde{\beta}}z^{\widetilde{\alpha}}(z\bar{z})^{\widetilde{\beta}}
=\sum\limits_{\widetilde{\alpha}+2\widetilde{\beta}=m-1}
b_{\widetilde{\alpha}\widetilde{\beta}}z^{\widetilde{\alpha}+\widetilde{\beta}}\bar{z}^{\widetilde{\beta}}.
\label{eqn:Jam07}
\end{array}$$ Then $$\sum\limits_{\widetilde{\alpha}+2\widetilde{\beta}=m-1}
b_{\widetilde{\alpha}\widetilde{\beta}}z^{\widetilde{\alpha}+\widetilde{\beta}}
\bar{z}^{\widetilde{\beta}+1}+\sum\limits_{\widetilde{\alpha}+2\widetilde{\beta}=m-1}
\overline{b_{\widetilde{\alpha}\widetilde{\beta}}}\bar{z}^{\widetilde{\alpha}+\widetilde{\beta}}
z^{\widetilde{\beta}+1} =
\sum\limits_{\stackrel{\alpha+\beta=m}{\alpha+\beta(s-1)\geq N_s}}
a_{\alpha\bar{\beta}}z^{\alpha}\bar{z}^{\beta}$$ We see that if $m$ is even, then $2b_{\widetilde{\alpha}\widetilde{\beta}}=a_{\alpha\bar{\beta}}+ic$ when $\alpha=\beta={m \over
2},\ \widetilde{\alpha}=1,\ \widetilde{\beta}={m \over 2}-1,\ c \in
\mathbb{R}$. The other relations are as follows: $$b_{\widetilde{\alpha}\widetilde{\beta}}=b_{\alpha\bar{\beta}},\ \ \hbox{if}\
\widetilde{\alpha}+\widetilde{\beta}=\alpha,\
\widetilde{\alpha}+2\widetilde{\beta}=m-1,\ \widetilde{\beta}+1=\beta,\widetilde{\alpha}
> 1,\ \alpha+(s-1)\beta \geq N_s.$$ From this, one can easily see that $$wt(f_{nor}^{(m-1)}(z,\-{z}))\ge \hbox{min}\{\wt{\a}+\wt{\b}+(s-1)\wt{\b}\}=\hbox{min}\{{\a}+(s-1){\b}-s+1\}\ge N_s-s+1.
\label{eqn:Jam08}$$
Substituting (\[eqn:Jam07\]) into (\[eqn:Jam060\]), we get $$\begin{array}{lll}
g_{m+1}(w)&=&(1-s)^{j+1}\bar{a}z^{(j+1)s+1}(z\bar{z}+z^s)^{t-j-1}+(\bar{z}+sz^{s-1}
+\Theta_{s}^{2})f_m(z,w)\\
&&+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_m(z,w)}+\Theta_{N_s}^{m+1}
+(sz^{s-1}+\Theta_{s}^{2})f_{nor}^{(m-1)}\\
&&+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)+(s\bar{z}^{s-1}+\Theta_{s}^{2})\overline{f_{nor}^{(m-1)}}
\end{array}$$ By (\[eqn:Jam08\]), we get $$(sz^{s-1}+\Theta_{s}^{2})f_{nor}^{(m-1)}+(s\bar{z}^{s-1}+\Theta_{s}^{2})\overline{f_{nor}^{(m-1)}}
=\mathbb{P}_{N_s}^{m+1}.$$ Hence $$\begin{array}{lll}
g_{m+1}(w)&=&(1-s)^{(j+1)}\bar{a}z^{(j+1)s+1}(z\bar{z}+z^s)^{t-j-1}+(\bar{z}+sz^{s-1}
+\Theta_{s}^{2})f_m(z,w)\\
&& +(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_m(z,w)}+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)+\Theta_{N_s}^{m+1}.
\end{array}$$ By induction, we showed that if the lemma holds for $m_0$ defined above, then it holds for any $m$ with $m_0 \leq m \leq 2t+(j+1)(s-2)+1$ and $m\le N_0$.
[**Step II of the proof of Lemma 2.3**]{}: In this step, suppose that we know that the lemma holds for $m \in [
2t+j(s-2)+2,2t+(j+1)(s-2)+1]$ with $m\le N_0$, where $j$ is a certain non-negative integer bounded by $t-2$. We then proceed to prove that the lemma holds also for $m
\in [ 2t+(j+1)(s-2)+2,2t+(j+2)(s-2)+1]$, whenever $m\le N_0$.
Suppose that $2t+(j+1)(s-2)+1<N_0$. By the assumption, we have $$\begin{array}{lll}
g_{2t+(j+1)(s-2)+1}(w)&=&\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z}+z^s)^{t-j-1}
\\ && +(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{2t+(j+1)(s-2)}(z,w)\\
&&+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{2t+(j+1)(s-2)}(z,w)}\\
&&+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right) +\Theta_{N_s}^{2t+(j+1)(s-2)+1}.
\label{eqn:Jam09}
\end{array}$$ Collecting terms of degree $2t+(j+1)(s-2)+1$ in (\[eqn:Jam09\]), we get $$\begin{array}{lll}
g^{(2t+(j+1)(s-2)+1)}(z\bar{z})&=&\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z})^{t-j-1}
+\hat{\mathbb{P}}_{N_s}^{2t+(j+1)(s-2)+1}\\
&&+\bar{z}f_{nor}^{(2t+(j+1)(s-2))}(z,z\bar{z})+z\overline{f_{nor}^{(2t+(j+1)(s-2))}(z,z\bar{z})}.
\label{eqn:Jam010}
\end{array}$$ Here we denote by $\hat{\mathbb{P}}_{N_s}^{2t+(j+1)(s-2)+1}$ a certain homogeneous polynomial of degree $2t+(j+1)(s-2)+1$ with weight at least $N_s$.
Now, we solve (\[eqn:Jam010\]) as follows. Write $\Lambda=2t+(j+1)(s-2)$. Notice that $$I:=-\hat{\mathbb{P}}_{N_s}^{\Lambda+1}
+a(1-s)^{j+1}\bar{z}^{(j+1)s+1}(z\bar{z})^{t-j-1}+g^{(\Lambda+1)}(z\bar{z})$$ is real valued and $I= \mathbb{P}_{N_s}^{\Lambda+1}$. Then (\[eqn:Jam010\]) can be rewritten as $$\begin{array}{lll}
I&=&\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z})^{t-j-1}+a(1-s)^{j+1}\-{z}^{(j+1)s+1}(z\bar{z})^{t-j-1}
\\
&&+\bar{z}f_{nor}^{(2t+(j+1)(s-2))}(z,z\bar{z})+z\overline{f_{nor}^{(2t+(j+1)(s-2))}(z,z\bar{z})}.
\label{eqn:Jam010-2}
\end{array}$$
Write $$I=\sum\limits_{\stackrel{ l+k=\Lambda+1}{l+(s-1)k \geq N_s}}
a_{l\bar{k}}z^l\bar{z}^k.$$
Since $a_{l\bar{k}}=\overline{a_{k\bar{l}}}$, we also require that $k+(s-1)l \geq N_s$. We next can get the following general solution of (\[eqn:Jam010-2\]): $$\begin{array}{ll}
&f_{nor}^{(2t+(j+1)(s-2))}(z,w)=f^{(\Lambda)}_1+f^{(\Lambda)}_2\ \ \ \ \ \hbox{with}\\ & f^{(\Lambda)}_1=-\bar{a}(1-s)^{j+1}z^{(j+1)s+2}w^{t-j-2}
% +\sum\limits_{\stackrel{l+k=2t+(j+1)(s-2)+1}{t-j
% \leq k \leq t+{1\over2}((k+1)(s-2)+1)}}h_{lk}z^{l}w^k,
\\ & f^{(\Lambda)}_2=\sum_{\wt{l}+2\wt{k}=\Lambda}h_{\wt{l}\wt{k}}z^{\wt{l}}w^{\wt{k}}
\end{array}$$ where $h_{\wt{l}\wt{k}}'s$ are determined by the following: $$\sum\limits h_{\wt{l}\wt{k}}z^{\wt{l}+\wt{k}}\bar{z}^{\wt{k}+1}+\sum\limits
\overline{h_{\wt{l}\wt{k}}}z^{\wt{l}+\wt{k}+1}\bar{z}^{\wt{k}}=
\sum_{l,k}a_{{l}{k}}z^{{l}}\bar{z}^{{k}}.
\label{eqn:Jam12}$$ Hence, we see that if $h_{\wt{l}\wt{k}}\not =0$, then either $\wt{l}=1, \ 2\wt{k}=\Lambda$ (in case $\Lambda$ is even) or $\wt{l}+\wt{k}=l, \ \wt{k}+1=k.$ Here $l,\ k$ satisfy the properties described above. Based on such an analysis and as argued before, we can conclude the following: $$(sz^{s-1}+\Theta_{s}^{2})f_2^{(\Lambda)}(z,z\bar{z})
+(s\bar{z}^{s-1}+\Theta_s^2)\overline{f_2^{(\Lambda)}(z,z\bar{z})}
= {\Theta}_{N_s}^{\Lambda+2}.
\label{eqn:Jam13}$$
Hence, from (\[eqn:Jam09\])-(\[eqn:Jam13\]), we get $$\begin{array}{lll}
g_{\Lambda+2}(w)+g_{nor}^{(\Lambda+1)}(w)&=&(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{\Lambda+1}(z,w)
+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{\Lambda+1}(z,w)}\\
&&+\Theta_{N_s}^{\Lambda+2}+\hat{\mathbb{P}}_{N_s}^{\Lambda+1}
+\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z}+z^s)^{t-j-1}\\
&&+(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{nor}^{(\Lambda)}(z,w)
+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{nor}^{(\Lambda)}(z,w)}\\
&&+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right).
\end{array}$$ Notice that $$\begin{array}{ll}
&g_{nor}^{(\Lambda+1)}(z\bar{z})=\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z})^{t-j-1}
+\bar{z}f_{nor}^{(\Lambda)}(z,z\bar{z})+z\overline{f_{nor}^{(\Lambda)}(z,z\bar{z})}
+\hat{\mathbb{P}}_{N_s}^{\Lambda+1},\\
&g_{nor}^{(\Lambda+1)}(w)-g_{nor}^{(\Lambda+1)}(z\bar{z}) \in
\Theta_{N_s}^{\Lambda+2}.
\end{array}$$ We get\
$$\begin{array}{ll}
g_{\Lambda+2}(w)=&(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{\Lambda+1}(z,w)+2Re\left((b_{N_0}-a_{N_0})z^{N_0}\right)\\
&+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{\Lambda+1}(z,w)}
+\Theta_{N_s}^{\Lambda+2}+J,
\end{array}$$ where\
$$\begin{array}{ll}
J=&(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{nor}^{(\Lambda)}(z,w)
+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{nor}^{(\Lambda)}(z,w)}\\
&
+\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z}+z^s)^{t-j-1}-
\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z})^{t-j-1}\\
&
-(\bar{z}f_{nor}^{(\Lambda)}(z,z\bar{z})+z\overline{f_{nor}^{(\Lambda)}(z,z\bar{z})}).
\end{array}$$ Here we notice that $$\begin{array}{l}
\bar{z}f_{nor}^{(\Lambda)}(z,w)
+z\overline{f_{nor}^{(\Lambda)}(z,w)}
-(\bar{z}f_{nor}^{(\Lambda)}(z,z\bar{z})+z\overline{f_{nor}^{(\Lambda)}(z,z\bar{z})})\\
\hskip 10pt+\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z}+z^s)^{t-j-1}-
\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z})^{t-j-1}\\
=-\bar{a}(1-s)^{j+1}z^{(j+1)s+1}z\bar{z}(z\bar{z}+z^s)^{t-j-2}+\Theta_{N_s}^{\Lambda+2}\\
\hskip 10pt+\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z})^{t-j-1}+\Theta_{N_s}^{\Lambda+2}\\
\hskip 10pt+\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z}+z^s)^{t-j-1}-
\bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z})^{t-j-1}\\
=\bar{a}(1-s)^{j+1}z^{(j+2)s+1}(z\bar{z}+z^s)^{t-j-2}+\Theta_{N_s}^{\Lambda+2}.
\end{array}$$ Hence we have $$\begin{array}{lll}
J&=&
(sz^{s-1}+\Theta_{s}^{2})f_1^{(\Lambda)}(z,w)
+(s\bar{z}^{s-1}+\Theta_s^2)\overline{f_1^{\Lambda}(z,w)}
\\&&
%+ \-{z}(f_1^{(\Lambda)}(z,w)-f_1^{(\Lambda)}(z,z\-{z}))+ {z}(\-{f_1^{(\Lambda)}(z,w)}-\-{f_1^{(\Lambda)}(z,z\-{z})})
+\bar{a}(1-s)^{j+1}z^{(j+2)s+1}(z\bar{z}+z^s)^{t-j-2}
%- \bar{a}(1-s)^{j+1}z^{(j+1)s+1}(z\bar{z})^{t-j-1}
+\Theta_{N_s}^{\Lambda+2}\\ &=&
\bar{a}(1-s)^{j+2}z^{(j+2)s+1}w^{t-j-2}+\Theta_{N_s}^{\Lambda+2}.
\label{eqn:Jam14}
\end{array}$$ This proves the lemma when $m=2t+(j+2)s+2$. Now, the result obtained in the previous step completes the proof of the claim in this step.
: We now can complete the proof of the lemma by inductively using results obtained in Steps I-II. Indeed, since we know that the Lemma holds for $m=2t+2$, we see, by Step I, that the lemma holds for any $m\le N_0$ with $m\in
[2t+2, 2t+(s-2)+1]$. Then, applying first Step II and then applying Step I again, we see the lemma holds for any $m\le N_0$ with $m\in [2t+j(s-2)+2, 2t+(j+1)(s-2)+1]$ and $j=1$. Now, by an induction argument on $j$, we see the proof of the lemma. $\endpf$
We next complete the proof of Theorem 2.2 in case $\ord(f)=2t$. First, if $m=ts+1<N_0$, we then have, by Lemma 2.3: $$\begin{aligned}
\begin{array}{lll}
g_{ts+1}(w)&=&\bar{a}(1-s)^tz^{ts+1}+\Theta_{ts+2}^{ts+1}
+(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{ts}(z,w)\\
&&+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{ts}(z,w)}.
\end{array}\end{aligned}$$ Collecting terms of degree $ts+1$ in the above equation, we obtain:
$$g_{nor}^{(ts+1)}(z\bar{z})=\bar{a}(1-s)^tz^{ts+1}
+\mathbb{P}_{ts+2}^{ts+1}+\bar{z}f_{nor}^{(ts)}(z,z\bar{z})+z\overline{f_{nor}^{(ts)}(z,z\bar{z})}.
\label{eqn:Jam15}$$
Since $ts+2>ts+1$, we can write $\mathbb{P}_{ts+2}^{ts+1}=\bar{z}A(z,\bar{z})$ for some polynomial function $A$. Hence, the equation above is solvable only if a=0, which is a contradiction.
Second, suppose $2t+1 < N_0\le ts+1 $. By the normalization assumption in the theorem, we notice that $N_0\not = ts+1$. Hence, we must have $2t+1 < N_0< ts+1 $
Assume that $j$ is the integer such that $ 2t+j(s-2)+2 \leq k_0s+j_0 \leq 2t+(j+1)(s-2)+1 $. Then by Lemma 2.3 and collecting terms of degree $N_0$ in $(\ref{eqn:Jam060})$, we have $$\begin{aligned}
\begin{array}{lll}
g_{nor}^{(N_0)}(z\bar{z})&=&2Re\{(b_{N_0}-a_{N_0})z^{N_0}\}
+\delta(1-s)^{j+1}\bar{a}z^{(j+1)s+1}(z\bar{z})^{t-j-1}
\\ &&+\bar{z}f_{nor}^{(N_0)}(z,z\bar{z})+z\overline{f_{nor}^{(N_0)}(z,z\bar{z})}
+\Theta_{N_0+1}^{N_0}.
\end{array}\end{aligned}$$ Here $\delta=0$ if $N_0 < 2t+(j+1)(s-2)+1$ and $\delta=1$ if $N_0 = 2t+(j+1)(s-2)+1$.\
With the same argument above, we can see a contradiction too. Hence, to reach no contradiction, we must have $b_{N}=a_{N}$ for any $N\le ns+s-1$. We thus conclude that $ts+1\ge ns+s$ and $t\ge n+1$. This finally completes the proof. : In this step, we show that we can also have the result stated in Theorem 2.2 when $\ord(f)$ is a finite odd number by applying the same argument as in Step I.
Suppose that $\ord(f)=2t+1$, then we can still assume that $2t+2 \leq N_0$ as argued in Step I, where $N_0$ is defined in a similar way. Assume that $ts+s+1 < N_0$. Collecting terms of degree 2t+2 in (\[eqn:Jam02\]), we get $$g_{nor}^{(2t+2)}(z\bar{z})=\bar{z}f_{nor}^{(2t+1)}(z,z\bar{z})+z\overline{f_{nor}^{(2t+1)}(z,z\bar{z})}.
\label{eqn:Jam20}$$ Its solution is given by $$f_{nor}^{(2t+1)}(z,w)=bzw^t \hskip 0.5cm , \hskip 0.5cm
g_{nor}^{(2t+2)}(w)=(b+\bar{b})w^{t+1}. \label{eqn:Jam21}$$ Substituting the solution in (\[eqn:Jam21\]) to (\[eqn:Jam20\]) and letting $A=(s-1)b-\bar{b}$, we get $$\begin{array}{lll}
g_{2t+3}(w)&=&Az^s(z\bar{z}+z^s)^t+(\bar{z}+sz^{s-1}+\Theta_{s}^{2})f_{2t+2}(z,w)\\
&&+(z+s\bar{z}^{s-1}+\Theta_s^2)\overline{f_{2t+2}(z,w)}
+\Theta_{ts+s+1}^{2t+3}.
\label{eqn:Jam22}
\end{array}$$ Repeating the same induction argument as in the proof of Lemma 2.3, we get $$\begin{array}{lll}
g_{ts+s}(w)&=&A(1-s)^tz^{ts+s}+\Theta_{ts+s+1}^{ts+s}+(\bar{z}+sz^{s-1}
+\Theta_{s}^{2})f_{ts+s-1}(z,w)\\
&&+(z+s\bar{z}^{s-1}+\Theta_{s}^{2})\overline{f_{ts+s-1}(z,w)}.
\label{eqn:Jam22-1}
\end{array}$$ Collecting terms of degree ts+s in (\[eqn:Jam22-1\]), we obtain $$g_{nor}^{(ts+s)}(z\bar{z})=A(1-s)^tz^{ts+s}+\mathbb{P}_{ts+s+1}^{ts+s}
+\bar{z}f_{nor}^{(ts+s-1)}(z,z\bar{z})+z\overline{f_{nor}^{(ts+s-1)}(z,z\bar{z})}.$$ As before, it is solvable only when A=0 thus b=0, which gives a contradiction. The case for $ts+s\ge N_0$ can be similarly studied to conclude that $ts+s\ge ns+s$ and thus $2t+1\ge 2n+1$. This completes the proof of Theorem 2.2. $\endpf$
**A complete set of formal invariants, proofs of Theorem 1.1, Corollary 1.3 and Theorem 1.5**
==============================================================================================
In this section, we will establish a formal normal form for the formal surface defined in (\[eqn:Jam000\]), by applying a formal transformation preserving the origin. This will give a complete classification of germs of formal surfaces $(M,0)$ with $\l=0,\ s<\infty$ in the formal setting, which, in particular, can be used to answer an open question raised by J. Moser in 1985 (\[pp 399, Mos\]).
As another application of our complete set of formal invariants, we show that a generic Bishop surface with the Bishop invariant vanishing is not equivalent to an algebraic surface, by applying a Baire category argument similar to the study in the CR setting (see the paper of Forstneric \[For\]). Notice that this phenomenon is strikingly different from the theory for elliptic Bishop surfaces with non-vanishing Bishop invariants, where Moser-Webster proved their celebrated theorem, that states that any elliptic Bishop surface with a non-vanishing Bishop invariant has an algebraic normal form. Let $M$ be a formal Bishop surface in $\mathbb{C}^2$ defined by $$w=H(z,\-{z})=z\bar{z}+2Re\{\sum_{j=s}^{N}a_jz^j\}+E_{N+1}(z,\bar{z}),
\label{eqn:Annie001}$$ where $s\ge 3$ is a positive integer and $E_{N+1}$ is a formal power series in $(z,\-{z})$ with $\ord(E_{N+1}) \geq N+1.$ Moreover, $a_s=1$ and for $ m>s,\ m\le N$, $$a_m=0\ \ \ \hbox{if}\ \ m=0,1\ \hbox{mod s}.$$ Our first result of this section is the following normalization theorem:
[**Theorem 3.1**]{}: [*With the above notation, there is a polynomial map $$\left\{
\begin{array}{ll}
z'=z+f(z,w),\hskip 10pt&f(z,w)=O(|w|+|z|^2) \\
w'=w+g(z,w), &g(z,w)=O(|w|^2+|z|^3+|zw|)
\end{array}
\right.$$ that transforms the formal Bishop surface $M$ defined in (\[eqn:Annie001\]) to the formal Bishop surface defined by $$w'=H^*(z',\-{z'})=z'\bar{z'}+2Re\{\sum_{j=s}^{N+1}b_jz'^j\}+E^*_{N+2}(z',\bar{z'}).
\label {eqn:Annie003}$$ Here $E^*_{N+2}=O(|z|^{N+2})$, $a_j=b_j$ for $s\le j\le N$ and $$b_{N+1}=0\ \ \hbox{if}\ \ N+1=0,1\ \hbox{mod s}.$$ Moreover, when $N+1\not =0,1\ \hbox{ mod}(s)$, $wt_{nor}(f)\ge N$ and $wt_{nor}(g)\ge N+1$; when $N=ts$, $wt_{nor}(f)\ge 2t,\ \ wt_{nor}(g)\ge 2t+1.$ and when $N=ts-1$, $wt_{nor}(f)\ge 2t-1,\ \ wt_{nor}(g)\ge 2t.$*]{}
Before proceeding to the proof, we recall a result of Moser, which will be used for our consideration here. For any $m\ge 4$ and holomorphic polynomials $$f_{nor}^{(m-1)}(z,w),\ g_{nor}^{(m)}(z,w),\ \phi^{(m)}(z),$$ we define an operator, which we call the Moser operator ${\mathcal L}$, as follows: $${\mathcal L}(f_{nor}^{(m-1)}(z,w), g_{nor}^{(m)}(z,w), \phi^{(m)}(z)):=g_{nor}^{(m)}(z,z\-{z})-2Re\{\-{z}f_{nor}^{(m-1)}(z,z\-{z})+\phi^{(m)}(z)\}.$$ The following lemma is essentially the content of Proposition 2.1 of \[Mos\]:
: Let $G(z,\-{z})$ be a homogeneous polynomial of degree $m$. Then $${\mathcal L}(f_{nor}^{(m-1)}(z,w), g_{nor}^{(m)}(z,w), \phi^{(m)}(z))=G(z,\-{z})$$ has a unique solution: $\{f_{nor}^{(m-1)}(z,w), g_{nor}^{(m)}(z,w),
\phi^{(m)}\}$ under the normalization condition: $f_{nor}^{(m-1)}=z^2f^*$ with $f^*$ a holomorphic polynomial. Moreover, when $G$ has no harmonic terms, then ${\mathcal
L}\left(f_{nor}^{(m-1)}(z,w), g_{nor}^{(m)}(z,w),
0\right)=G(z,\-{z})$ also has a unique solution $\{f_{nor}^{(m-1)}(z,w), g_{nor}^{(m)}(z,w)\}$ under the same normalization condition just mentioned.
The proof of the Theorem 3.1 follows from a similar induction argument that we used in the previous section.
: We complete the proof in three steps.
[**Step 1**]{}: We first show that there is a polynomial map: $z'=z+f_{nor}^{(N)}(z,w),\ w'=w+g_{nor}^{(N+1)}(z,w)$, which maps $M$ to a surface defined by the following equation: $$w=z\bar{z}+2Re\{\sum_{j=s}^{N+1}b_jz^j\}+E^*_{N+2}(z,\bar{z})
\label{eqn:Annie004}$$ with $b_j=a_j$ for $s\le j\le N$. Substituting the map into (\[eqn:Annie004\]) and collecting terms of degree $N+1$, we see that the existence of the map is equivalent to the existence of solutions of the following functional equation:
$${\mathcal L}(f_{nor}^{(N)}(z,w), g_{nor}^{(N+1)}(z,w), b_{N+1}z^{N+1})=-E^{(N+1)}_{N+1}(z,\-{z}).
\label{eqn:Annie005}$$
By Lemma 3.2, we know that (\[eqn:Annie005\]) is indeed solvable and is uniquely solvable under the normalization condition as in Lemma 3.2.
For the rest of the proof of the theorem, we can assume that $E_{N+1}=2Re\{b_{N+1}z^{N+1}\}+o(|z|^{N+1}).$
[**Step 2**]{}: In this step, we assume that $N+1=1\ \hbox{mod s}$. Write $N=ts$. We then show that there is a polynomial map of the form: $$\begin{array}{ll}
&z'=z+\sum_{l=0}^{N-2t}\{f^{(2t+l)}(z,w)\},\ \\ &w'=w+\sum_{\tau=0}^{N+1-2t-2}\{g_{nor}^{(2t+2+\tau)}(w)\}
\label {eqn:Annie006-02}
\end{array}$$ such that under this transformation, $M$ is mapped to a formal surface $M'$ defined by (\[eqn:Annie003\]) with $b_{N+1}=0$. The map is also uniquely determined by imposing the normalization condition as in Lemma 3.2 for $f^{(j)}$ with $2t<j\le N+1$.
As in Step I, this amounts to studying a series of normally weighted homogeneous functional equations with the normally weighted degree running from $2t$ to $N+1$. Substituting (\[eqn:Annie006-02\]) into (\[eqn:Annie003\]) and then collecting terms of degree $2t+1$, we obtain the equation (\[eqn:Jam03\]), which can be solved as: $$f_{nor}^{(2t)}(z,w)=aw^t-\bar{a}z^2w^{t-1}$$ with $a$ to be (uniquely) determined later.
Now, suppose we are able to solve $f_{nor}^{(2t+l)},\
g_{nor}^{(2t+1+l)}$ for $2t+l=2t,\cdots, m-1\le st-2.$ Substituting (\[eqn:Annie006-02\]) into (\[eqn:Annie003\]) and then collecting terms of degree $m+1$, we obtain an equation similar to (\[eqn:Jam01-1\]), as argued in the proof of Lemma 2.3: $$g^{(m+1)}(z\bar{z})=\bar{z}f_{nor}^{(m)}(z,z\bar{z})+z\overline{f_{nor}^{(m)}(z,z\bar{z})}+\hat{\mathbb{P}}_{ts+2}^{m+1}
\label{eqn:Annie006-01}$$ Notice that $\hat{\mathbb{P}}_{ts+2}^{m+1}(=\mathbb{P}_{ts+2}^{m+1})$ must be real valued and is uniquely determined by the known data. This equation, in terms of the Moser operator, can be rewritten as: $${\mathcal L}\left(f_{nor}^{(m)}(z,z\bar{z}),g^{(m+1)}(z\bar{z}),0\right)=\hat{\mathbb{P}}_{ts+2}^{m+1}.
\label{eqn:Annie006}$$ Since $\hat{\mathbb{P}}_{ts+2}^{m+1}$ is real-valued and divisible by $\-{z}$, it does not contain any harmonic terms. By Lemma 3.2, it can be solved, and can be uniquely solved under the normalization condition in Lemma 3.2. By induction, we can uniquely obtain $f_{nor}^{(m)},\ g_{nor}^{(m+1)}$ for $m\le ts-1$. Substituting (\[eqn:Annie006-02\]) into (\[eqn:Annie003\]) and then collecting terms of degree $m=ts+1$, we obtain an equation similar to (\[eqn:Jam15\]), which can be rewritten as: $$\begin{array}{ll}
&{\mathcal L}(g_{nor}^{(ts+1)}(z\bar{z}),f_{nor}^{(ts)}(z,z\-{z}),0) =2Re\{\bar{a}(1-s)^tz^{ts+1}\}
\\ &\hskip
1cm
+\hat{\mathbb{P}}_{ts+2}^{ts+1}-a(1-s)^t\-{z}^{ts+1}-2Re(b_{ts+1}z^{ts+1}).
\label{eqn:Annie007}
\end{array}$$
As in the proof of Theorem 2.2, the real-valued homogeneous polynomial $\hat{\mathbb{P}_{ts+2}^{ts+1}}-a(1-s)^t\-{z}^{ts+1}$ has a $\-{z}$ factor and thus has no harmonic terms. Hence, if we choose $a=\-{b_{ts+1}}/(1-s)^t$, then (\[eqn:Annie007\]) is uniquely solvable, under the normalization condition in Lemma 3.2. This completes the proof of the claim in this step.
[**Step 3**]{}: In this step, we assume that $N+1=0\ \hbox{mod s}$. Write $N=(t+1)s-1$. We then show that there is a unique polynomial map of the form: $$\begin{array}{ll}
&z'=z+\sum_{l=0}^{N-1-2t}\{f_{nor}^{(2t+l+1)}(z,w)\},\ \\ &w'=w+\sum_{\tau=0}^{N+1-2t-2}\{g_{nor}^{(2t+2+\tau)}(w)\}
\label {eqn:Annie006}
\end{array}$$ such that under this transformation, $M$ is mapped to a formal surface $M'$ defined by (\[eqn:Annie003\]) with $b_{N+1}=0$. Here $f^{(m)}_{nor}$ satisfies the normalization condition in Lemma 3.2 for $m\not = 2t+1$.
The argument for this step is the same as that for Step 2. We first have to choose $$f_{nor}^{(2t+1)}(z,w)=bzw^t, \hskip 1cm
g^{(2t+2)}_{nor}(w)=(b+\bar{b})w^{t+1}$$ with $b$ to be uniquely determined later. Arguing exactly in the same way as in Step 2, we can inductively find the unique solution (under the normalization condition) for $f_{nor}^{(2t+l)},\ g_{nor}^{(2t+1+l)}$ with $2t+l=2t+2,\cdots, <st+s-1.$ At the level with degree $ts+s$, we have the following equation:
$$\begin{array}{ll}
&2Re(b_{N+1}z^{N+1})+g_{nor}^{(ts+s)}(z\bar{z})=((s-1)b-\bar{b})(1-s)^tz^{ts+s}
\\ &\hskip
1cm+\hat{\mathbb{P}}_{ts+s+1}^{ts+s}+\bar{z}f^{(ts+s-1)}(z,z\bar{z})
+z\overline{f^{(ts+s-1)}(z,z\bar{z})}. \label{eqn:Annie-002}
\end{array}$$
Now, arguing the same way as in Step 2, the equation (\[eqn:Annie-002\]) is uniquely solvable by taking $b$ such that $(s-1)b-\-{b}=b_{N+1}$ and by imposing the normalization condition as in Lemma 3.2 to $f^{(ts+s-1)}_{nor}$.
Now, the map in Theorem 3.1 can be chosen as the map in Step 1 if $N+1\not = 0,1\ \h{mod}(s)$. When $N+1=0,\ \h{or}\ 1\ \h{mod}(s)$, the map in Theorem 3.1 can be defined by composing the map in Step 2 or that in Step 3, respectively, with the map in Step 1. We see the proof of Theorem 3.1. Moreover, with such fixed procedures and normalizations described in the above steps, there are a set of universal polynomials $\{P_{kl}(a_{\a\b})\}_{1\le \a+\b\le k+l}$ (depending only on $s$ and $N$) such that the coefficients of the map $(z',w')=(z,w)+(f,g)=(z,w)+\sum_{k,l}b_{kl}z^kw^l$ in Theorem 3.1 are determined by
$$b_{kl}=P_{kl}(a_{\a\b}),\ \ 1\le \a+\b\le k+l
\label{eqn:Annie-002-03}$$
where $H=\sum_{\a,\b\ge 0}a_{\a\b}z^\a\-{z}^\b.$
The last sentence in Theorem 3.1 follows from the procedures that we used to prove the existence part. $\endpf$
We next choose the map $z'=z+f,\ w'=w+g$ in Theorem 3.1 such that its coefficients are determined by (\[eqn:Annie-002-03\]). Let $z=z'+f^*(z',w')$ and $w=w'+g^*(z',w')$ be its inverse transformation. Notice that the coefficients of $(f^*,g^*)$ in its Taylor expansion up to degree, say $m$, are universal polynomial functions of the coefficients of $(f,g)$ up to degree $m$ for any $m$. Hence we have the defining equation of $M^*$, the image of $M$, as follows: $$w'+g^*(z',w')=H(z'+f^*(z',w'), \-{z'+f^*(z',w')}).$$ Applying an implicit function theorem to solve for $w'$ and making use of the uniqueness of the graph function, we see that the coefficients in the Taylor expansion of $H^*$ up to degree $m$ must also be polynomial functions of the coefficients of $H$ of degree not exceeding $m$ in its Taylor expansion. Repeating such a normalization procedure that we did for $M$ to $M^*$ and by an induction argument, we get the following theorem: (The uniqueness part follows from Theorem 2.2.)
[**Theorem 3.3**]{}: [*Let $M$ be a formal Bishop surface defined by $$w=H(z,\-{z})=z\bar{z}+z^s+\bar{z}^s+E(z,\bar{z}),
\label{eqn:Jam00}$$ where $s\ge 3$ is a positive integer and $E(z,\-{z})=\sum_{\a+\b\ge s+1}^{\infty}a_{\a{\b}}z^\a\-{z^\b}$. Then there is a unique formal transformation of the form: $$\left\{
\begin{array}{ll}
z'=z+f(z,w),\hskip 10pt&f(z,w)=O(|w|+|z|^2) \\
w'=w+g(z,w), &g(z,w)=O(|w|^2+|z|^3+|zw|)
\end{array}
\right.$$ that transforms $M$ to the formal Bishop surface defined by $$w'=H^*(z',\-{z'})=z'\bar{z'}+z'^s+\-{z'}^s+ 2Re\{\sum_{ j=2,\cdots,s-1;\ k\ge 1}^{\infty}\lambda_{ks+j}z'^{ks+j}\}.
\label {eqn:Annie100}$$ The normal form in (\[eqn:Annie100\]), up to a transformation of the form $z''=e^{i\theta}z',\ w''=w$ with $e^{is\theta}=1$, uniquely determines the formal equivalence class of $M$. Moreover, there are a set of universal polynomial functions $$\{\Lambda_{ks+j}(Z_{\a\b})\}_{s+1\le \a+\b\le ks+j;\ j =2,\cdots,s-1;\ k\ge 1}$$ depending only on $s$, such that: $$\begin{array}{ll}
%&\{\Lambda_{ks+j}(Z_{\a\b})\}_{s+1\le \a+\b\le ks+j;\ j =2,\cdots,s-1;\ k\ge 1}\\ &
%\hbox{with}\
\lambda_{ks+j}=\Lambda_{ks+j}(a_{\a\b})_{s+1\le \a+\b\le ks+j;\ j =2,\cdots,s-1;\ k\ge 1}.
\label{eqn:Annie101}
\end{array}$$*]{}
[*Proofs of Theorem 1.1 and Corollary 1.3*]{}: Theorem 1.1 follows immediately from Theorem 3.3 and Lemma 2.1 (ii). The proof of Corollary 1.3 (a), (b), (d) also follows easily from Theorem 3.1. To see Corollary 1.3 (c), we let $\mathcal G$ be a proper subgroup of ${\mathcal Z}_s$. Define $J_G:=\{j:\ 2\le j\le s-1,\ e^{i\theta j}=1,\
\h{for any}\ (e^{i\theta}z,w)\in {\mathcal G}\}$. Let $M_G$ be defined by $$w=z\-{z}+z^s+\-{z}^s+2Re\{\sum_{j\in J_G}a_{s+j}z^{s+j}\},$$ with $a_{s+j}\not =0$. Then we will verify that $aut_0(M_G)={\mathcal G}$. To this aim, write ${\mathcal G}^*$ to be the collection of $\xi's$ with $(z,w)\ra (\xi z,w)$ belonging to ${\mathcal G}$. By Corollary 1.3 (a), we need only to show that if $\xi^{*s}=1$ and $\xi^{*j}=1$ for any $j\in J_G$, then $\xi^*\in {\mathcal G}^*$. Write $k=|{\mathcal G}^*|$. Then $s=km$ with $m(\in {\mathbf N})>1$. For any $\xi(\in {\mathcal G}^*)\not = 1$, since the order of $\xi$ must be divisible by $k$, we see that $\xi^k=1$. Therefore, ${\mathcal G}^*$ forms a complete set of the solutions of $z^k=1$. Now, it is clear that $J_G=\{k,\cdots, (m-1)k\}$. Hence, we see that $\xi^{*k}=1$. Thus, $\xi^*\in {\mathcal G}^*$. This completes the proof of Corollary 1.3 (c).
Now, by Corollary 1.3 (a), we see that for $M$ as in Corollary 1.3 (e), $M$ must be formally equivalent to $M_s$. Assuming Theorem 1.2, which we will prove in the next section, we also conclude that $M$ is biholomorphically equivalent to $M_s$. Corollary 1.3 (f) is a simple consequence of the results in (a) and (e). $\endpf$
[**Corollary 3.4**]{}: [*Let $M$ be a real analytic Bishop surface defined by an equation of the form: $$w=H(z,\-{z})=z\-{z}+2Re\{z^s+\sum_{k\ge 1,\ j=2,\cdots, s-1}a_{ks+j}z^{ks+j}\}\ \ \h{with infinitely many\ }\ a_{ks+j}\not = 0.$$ Then for any $N>s$, $M$ is not equivalent to the Bishop surface $M_N$ defined by $$w=H_{(N+1)}(z,\-{z})=z\-{z}+2Re\{z^s+\sum^{ks+j\le N}_{k\ge 1,\ j=2,\cdots, s-1}a_{ks+j}z^{ks+j}\}.$$ Here $H_{(N+1)}$ is the $N^{th}$-truncation from the Taylor expansion of $H$ at $0$. In fact, $M_{(N+1)}$ is equivalent to $M_{(N'+1)}$ with $N'>N$ if and only if $a_{ks+j}= 0$ for any $N<ks+j\le N'$.* ]{}
Corollary 3.4 answers, in the negative, the second problem that J. Moser asked in his paper (\[pp 399, Mos\]).
As a less obvious application of Theorem 3.3, we next show that a generic Bishop surface with the Bishop invariant vanishing at $0$ and with $s<\infty$ is not even formally equivalent to any algebraic surface in ${\mathbb C}^2$. For this purpose, we borrow the idea used in the CR setting based on the Baire category argument. For the consideration in the CR setting by using the Baire category theorem, the reader is referred to the paper of Forstneric \[For\].
Write ${\mathcal M}_s$ for the collection of all formal Bishop surfaces defined as in (\[eqn:Jam00\]): $$w=H(z,\-{z})=z\bar{z}+2Re(z^s)+\sum_{\a+\b\ge s+1}a_{\a\b}z^\a\-{z^\b}.
\label{eqn:Annie001-1}$$
Write ${\mathcal F}:=\{\vec{a}=(a_1,\cdots,a_n,\cdots): \ a_j\in {\mathbb C}\}$, equipped with the usual distance function: $$dist(\vec{a},\vec{b})=\sum_{j=1}^{\infty}\frac{|a_j-b_j|}{2^j(1+|a_j-b_j|)}.$$ We know that $\mathcal F$ is a Frèchet space. There is a one-to-one correspondence between ${\mathcal M}_s$ and $\mathcal F$, which assigns each $M\in {\mathcal M}_s$ to an element: $\vec{M}=(a_{\a\b})\in {\mathcal F}$ labeled in the lexicographical order. Therefore, we can, in what follows, identify ${\mathcal M}_s$ as a Frèchet space. We define the operator $\mathcal J$ such that it sends any $M\in {\mathcal M}_s$ to $(\lambda_{ks+j})_{j\not = 0,1;k\ge 1}$, where $(\l_{sk+j})$ is described as in Theorem 3.3. By (\[eqn:Annie101\]), we easily see that $\mathcal J$ is a continuous map from ${\mathcal M}_s$ to $\mathcal F$.
$(M,p)$ in ${\mathbb C}^2$ is called the germ of an algebraic surface if $M$ near $p$ possesses a real polynomial defining equation. If $p\in M$ is a point with an elliptic complex tangent, whose Bishop invariant is $0$ and whose Moser invariant is $s<\infty$, then there is a change of coordinates (see \[Hu1\], for instance) such that $p=0$ and $M$ near $0$ is defined by an equation of the form: $$w=z\-{z}+B(z,\-{z},w,\-{w}),\ \ B(z,\-{z},w,\-{w})=
\sum_{ 3\le \a+\b+2\gamma+2\tau}c_{\a\b\gamma\tau}z^\a \-{z}^\b w^{\gamma} \-{w}^\tau,
\label{eqn:Annie-00-0}$$ where $B$ is a polynomial in its variables. By using the implicit function theorem and using the argument in the step 1 of the proof of Theorem 3.1, it is not hard to see that there is a fixed procedure to transform (\[eqn:Annie-00-0\]) into a surface defined by an equation as in (\[eqn:Annie001-1\]), in which $a_{\a\b}$ are presented by polynomials of $c_{\a\b\gamma\tau}$ and $H(z,\-{z})$ becomes what we call a Nash algebraic function to be defined as follows:
We call a real analytic function $h(z,\-{z})$ near $0$ a Nash algebraic function if either $h\equiv 0$ or there is an irreducible polynomial $P(z,\-{z};X)$ in $X$ with polynomial coefficients in $(z,\-{z})$ such that $P(z,\-{z};h(z,\-{z}))\equiv 0.$ Certainly, we can always assume that the coefficients of $(z,\xi,X)$ (in $P(z,\xi,X)$) of terms with highest power in $X$ have maximum value $1$. The degree of $h$ is defined as the total degree of $P$ in $(z,\-{z},X)$.
For $d,\ n,\ m\ge 1$, we define ${\mathcal A}^d_{B}(n,m)\subset {\mathcal M}_s$ to be the subset of Bishop surfaces defined in (\[eqn:Annie001-1\]), where $H(z,\-{z})'s$ are Nash algebraic functions derived from the $B's$ in (\[eqn:Annie-00-0\]) in the procedure described above with the degree of $B's$ bounded by $d$, that further satisfy the following properties:
[**Cond (1)**]{}: $H(z,\xi)'s$ are holomorphic over $|z|^2+|\xi|^2<1/m^2$;
[**Cond(2)**]{}: $\max_{(|z|^2+|\xi|^2)<1/m^2}|H(z,\xi)|\le n$ and $|c_{\a\b\gamma\tau}|\le n$.
Write ${\mathcal A}^d_{B}=\cup_{n,m=1}^{\infty}{\mathcal
A}^d_{B}(n,m)$ and ${\mathcal A}_B=\cup_{d=1}^{\infty}{\mathcal
A}^d_{B}$. It is a consequence of Theorem 3.3 that $M$, defined in (\[eqn:Jam00\]), is formally equivalent to an algebraic surface if and only if ${\mathcal J}(M)\in {\mathcal J}({\mathcal A}_B)$. (Therefore, $M$ defined in (\[eqn:Jam00\]) is not formally equivalent to an algebraic surface if and only if ${\mathcal J}(M)\not \in {\mathcal J}({\mathcal A}_B)$.)
Now, for any sequence $\{M_j\}\subset {\mathcal A}_B^d(n,m)$ with $M_j: w=H_j(z,\-{z})=z\-{z}+z^s+\-{z}^s+o(|z|^s),$ by a normal family argument and by passing to a subsequence, we can assume that $H_j(z,\xi)\ra H_0(z,\-{z})$ over any compact subset of $\{|z|^2+
|\xi|^2<1/m^2\}$. If follows easily that $M_0$ defined by $w=H_0$ is also in ${\mathcal A}_B^d(n,m)$. Moreover, $D^\a_zD^\b_\xi H_j(0)\ra
D^\a_zD^\b_\xi H_0(0)$ for any $(\a, \b)$. By (\[eqn:Annie101\]), ${\mathcal J}(M_j)\ra {\mathcal J}(M_0)$ in the topology of $\mathcal F$. Therefore, we easily see that ${\mathcal J}({\mathcal
A}_B)$ is a subset of $\mathcal F$ of the first category.
Next, for any $R>0$, we let $${\mathcal S}_R:=\{\vec{\lambda}=(\l_{sk+j})_{k\ge 1;j = 2,\cdots,s-1}\}: \ \|\vec{\l}\|_R:=\sum_{ks+j}|\l_{ks+j}|R^{ks+j}<\infty\}.$$
It can be verified that ${\mathcal S}_R$ is a Banach space under the above defined $\|\cdot\|_R$-norm. (In fact, it reduces to the standard $l^1$-space when $R=1$.) We now claim that ${\mathcal K}^d_{B}$, defined as the closure of ${\mathcal J}\left({\mathcal A}^d_B(n,m)\right)\cap {\mathcal S}_R$ in ${\mathcal S}_R$ in its Banach norm, has no interior point.
Suppose, to the contrary, that a certain $\epsilon$-ball $\mathcal B$ of $\vec{a_0}=(\l^0_{sk+j})_{k\ge 1;j = 2,\cdots, s-1}$ in ${\mathcal S}_R$ is contained in ${\mathcal K}^d_{B}$. We must then have ${\mathcal B}\subset {\mathcal J}\left({\mathcal A}^d_B(n,m)\right)\cap {\mathcal S}_R$. Indeed, for any $\vec{a}\in {\mathcal B}$, let ${\mathcal J}(M_j)\ra \vec{a}$ with $M_j\in {\mathcal A}^d_{B}(n,m)$. By the argument in the above paragraph, we can assume, without loss of generality, that $M_j\ra M_0\in {\mathcal A}^d_{B}(n,m)$ in the $\mathcal F$-norm. By (\[eqn:Annie101\]), we see that ${\mathcal J}(M_0)=\vec{a}$. Choose $\vec{a}=\{\l_{ks+j}\}$ such that $|\l_{ks+j}-\l^0_{ks+j}|\cdot(2R)^{ks+j}<\epsilon$ for any $ks+j$. For any $N\ge 1$, then we see that there is a certain $H=z\-{z}+z^s+\-{z^s}+\sum_{s+1\le \a+\b}a_{\a\b}z^\a\-{z^\b}$ Nash algebraic near $0$ such that $$\lambda_{ks+j}=\Lambda_{ks+j}(a_{\a\b}), \ \ N\ge ks+j\ge s+1, \ \a+\b\le ks+j,\ \ \Lambda=(\Lambda_{ks+j})_{s+1\le ks+j\le N}.
\label{eqn:Annie-002-02}$$ Here $H$ is obtained from $B$ in (\[eqn:Annie-00-0\]) with degree of $B$ bounded by $d$. Since $a_{\a\b}$ are polynomial functions of $c_{\a\b\gamma\tau}$, we can conclude a contradiction from (\[eqn:Annie-002-02\]). Indeed, since the variables on the right hand side of (\[eqn:Annie-002-02\]) are polynomially parametrized by less than $d^4$ free variables ($c_{\a\b\gamma\tau}$), the image of (\[eqn:Annie-002-02\]) can not fill in an open subset of ${\mathbb R}^{N-s}$ as $N>>1.$
Therefore, we proved that ${\mathcal
A}_{B}=\cup_{d,n,m=1}^{\infty}{\mathcal A}^d_{B}(n,m)$ is a set of the first category in ${\mathcal S}_R$. By the Baire category theorem, we conclude that most elements in ${\mathcal S}_R$ are not from ${\mathcal J}\left({\mathcal A}_B\cap {\mathcal
S}_R\right)$. For any $\vec{a}=(\l_{sk+j})\not \in {\mathcal
J}\left({\mathcal A}_B\cap {\mathcal S}_R\right)$, the Bishop surface defined by: $w=z\-{z}+z^s+\-{z}^s+2\hbox{Re}(\sum_{k\ge1;j\not =
0,1}\l_{ks+j}z^{ks+j})$ is not equivalent to any algebraic surface in ${\mathbb C}^2$. When $R$ varies, we complete a proof of Theorem 1.5. $\endpf$
A real analytic surface in ${\mathbb C}^2$ is called a Nash algebraic surface if it can be defined by a Nash algebraic function. By the same token, we can similarly prove the following:
: Most real analytic elliptic Bishop surfaces with the Bishop invariant $\l=0$ and the Moser invariant $s<\infty$ at $0$ are not equivalent to Nash algebraic surfaces in $\mathbb{C}^2$.
[*Proof of Theorem 3.5*]{}: To prove Theorem 3.5, we define ${\mathcal A}^d_{B}(n,m)$ in the same way as before except that we now only require that $H(z,\-{z})=z\-{z}+z^s+\-{z}^s+\sum_{\a+\b\ge s+1}a_{\a\b}z^{\a}\-{z}^\b$ is a general Nash algebraic function with total degree bounded by $d$ and with the same conditions described as in Cond (1) and the first part of Cond (2). The last part of Cond (2) is replaced by the condition that $|b_{\a\b\gamma}|\le n$, where $P(z,\-{z},X)=\sum b_{j}(z,\-{z})X^j=\sum_{\a\b\gamma}b_{\a\b\gamma}z^\a\-{z}^\b X^\gamma$ is a minimal polynomial of $H$ with the same coefficient restriction as imposed before.
We fix an $H_0$ and its minimal polynomial $P_0(z,\-{z};X)$. (We will fix certain coefficient of $P$ in the top degree terms of $X$ to be $1$ to make the minimal polynomial $P_0$ unique). Let ${\mathcal A}^d_{B}(n,m; H_0,\delta)$ be a subset of ${\mathcal A}^d_{B}(n,m)$, where $M=\{ w=H(z,\-{z})\}\in {\mathcal A}^d_{B}(n,m; H_0,\delta)$ if and only if $|b_{\a\b\gamma}-b^0_{\a\b\gamma}|\le \delta$. Here $P=\sum b_{\a\b\gamma}z^\a\-{z}^\b X^\gamma$ and $P_0=\sum b^0_{\a\b\gamma}z^\a\-{z}^\b X^\gamma$ are the minimal polynomials of $H$ and $H_0$, respectively. We assume that $P$ is normalized in the same manner as for $P_0$. (Certainly, we can always do this if $\delta<<1$.)
Consider an $H$ and its minimal polynomial $P$ associated with an element from ${\mathcal A}^d_{B}(n,m; H_0,\delta)$. Let $R$ be the resultant of $P$ and $P'_X$ with respect to $X$. We know that $R$ is a non-zero polynomial of $(z,\-{z})$ of degree bounded by $C_1(d)$, a constant depending only on $d$. Write $H=H^*_{(N)}+H^{**}_N$ with $H_{(N)}^{*}$ the Taylor polynomial of $H$ up to order $N-1$ and $H^{**}_N$ the remainder. Then from $P(z,\-{z},H^*_{(N)}+H^{**}_N)=0$, we obtain
$$P^{**}(z,\-{z}, X^{**})=0\ \ \hbox{ with}\ \ X^{**}=H_N^{**}.
\label{eqn:Annie-add}$$
Here $P^{**}$ is a polynomial of total degree bounded by $C_2(d,
N)$, a constant depending only on $d$ and $N$, and its coefficients are determined polynomially by the coefficients of $P$ and $H_{(N)}^{*}$. Notice that $D_{X^{**}}\left(
P^{**}(z,\-{z},X^{**})\right)|_{X^{**}=0}=D_{X}\left (P(z,\-{z},
X)\right)_{X=H_{(N)}^{*}}$. Since there are polynomials $G_1$ and $G_2$ such that $G_1P+G_2P'_X=R$ and since $P(z,\-{z},H^{*}_{(N)})=o(|z|^N)$, we conclude that the degree $k_0$ of the lowest non-vanishing order term of $P'_X(z,\-{z},
H_{(N)}^{*})$ is bounded by $C_1(d)$, depending only on $d$. Choose an $N\ge C_1(d)$ and a sufficiently small positive number $\delta$. We can apply a comparing coefficient method to (\[eqn:Annie-add\]) to conclude that each $a_{\a_0\b_0}$ for $\a_0+\b_0\ge N$ is determined by $b_{\a\b\gamma}$ and $a_{\a\b}$ with $\a+\b\le N-1$ through at most $C(k_0,N)$ rational functions in $b_{\a\b\gamma}$ and $a_{\a\b}$ ($\a+\b\le N-1$) with $C(k_0,N)$ depending only on $k_0, N$. Now, (\[eqn:Annie-002-02\]) can be used in the same manner to show that the interior of the closure of ${\mathcal J}({\mathcal A}^d_{B}(n,m;H_0,\delta))\cap {\mathcal S}_R$ in ${\mathcal S}_R$ is empty. It is easy to see that ${\mathcal J}({\mathcal A}_{B})$ can be written as a countable union of these sets. We see that ${\mathcal J}({\mathcal A}_{B})$ is a set of the first category in ${\mathcal S}_R$. This completes the proof of Theorem 3.5. $\endpf$
[**Remark 3.6 (A)**]{}: The crucial point for Theorem 3.5 to hold is that the modular space of surfaces with a vanishing Bishop invariant and $s<\infty$ is parameterized by an infinitely dimensional space. Hence, any subclass of ${\mathcal M}_s$, that is represented by a countable union of finite dimensional subspaces of ${\mathcal M}_s$, is a thin set of ${\mathcal M}_s$ under the equivalence relation. This idea, that the infinite dimensionality of the modular space would generally have the consequence of the generic non-algebraicity for its elements, dates back to the early work of Poincaré \[Po\]. In the CR setting, Forstneric in \[Fo\] has used the infinitely dimensional modular space of CR manifolds and the Baire category argument to give a short and quick proof that a generic CR submanifold in a complex space is not holomorphically equivalent to any algebraic manifold. Some earlier studies related to non-algebraicity for CR manifolds can be found, for instance, in \[BER\] \[Hu2\] \[Ji\]. However, by a result of the first author with Krantz \[HK\] and a result of the first author in \[Hu3\], a Bishop surface with an elliptic complex tangent can always be holomoirphically transformed into the algebraic Levi-flat hypersurface ${\mathbb C}\times {\mathbb R}$ and also into the Heisenberg hypersurface in ${\mathbb C}^2$. [**(B)**]{}. In the normal form (\[eqn:Annie100\]), the condition that $\lambda_{ks+j}=0$ for $j=0,1,\ k=1,2,\cdots$ can be compared with the Cartan-Chern-Moser chain condition in the case of strongly pseudoconvex hypersurfaces (see \[CM\]). In the hypersurface case, the chain condition is also described by a finite system of differential equations. It would be very interesting to know if, in our setting here, there also exist a finite set of differential equations describing our chain condition.
**Surface hyperbolic geometry and a convergence argument**
==========================================================
In this section, we study the convergence problem for the formal consideration in the previous section. Our starting point is the flattening theorem of Huang-Krantz \[HK\], which says that an elliptic Bishop surface with a vanishing Bishop invariant can be holomorphically mapped to ${\mathbb C}\times {\mathbb R}$.
Hence, to study the convergence problem, we can restrict ourselves to a real analytic Bishop surface $M$ defined by $$w=z\bar{z}+z^s+\bar{z}^s+E(z,\bar{z}),\ E(z,\bar{z})=\overline{E(z,\bar{z})}=o(|z|^s),\ 3\le s<\infty.
\label{eqn:Jenny-001}$$ Recall that the Moser-Webster complexification $\mathfrak{M}$ of $M$ is the complex surface near $0 \in \mathbb{C}^4$ defined by:\
$$\left\{
\begin{array}{l}
w=z\zeta+z^s+\zeta^s+E(z,\zeta)\\
\eta=z\zeta+z^s+\zeta^s+E(z,\zeta).
\end{array}
\right.
\label{eqn:Jenny-002}$$
We define the projection $\pi:\mathfrak{M}\longrightarrow
\mathbb{C}^2$ by sending $(z,\zeta,w,\eta) \in \mathfrak{M}$ to $(z,w)$. Then $\pi$ is generically $s$ to 1. Write $B$ for the branching locus of $\pi$. Namely, $(z,w) \in B$ if and only if $\exists(\zeta_0,\eta_0)$ such that $(z,\zeta_0,w,\eta_0) \in
\mathfrak{M}$ and $\pi$ is not biholomorphic near $(z,\zeta_0,w,\eta_0)$. Write ${\mathfrak B}=\pi^{-1}(B)$.
Then\
$$\begin{array}{l}
(z,w) \in B \\
\Longleftrightarrow \exists\zeta \ \hbox{ such that }\
w=z\zeta+z^s+\zeta^s+E(z,\zeta)\ \hbox{ and }\
z+s\zeta^{s-1}+E_{\zeta}(z,\zeta)=0 \\
\Longleftrightarrow \sharp\{\pi^{-1}(z,w)\}<s.
\end{array}$$ It is easy to see that near $0,\ B$ is a holomorphic curve passing through the origin.\
Now, suppose $M'$ is defined by $w'=z'\bar{z'}+z'^s+\bar{z'}^s+E^{\ast}(z',\bar{z'})$ with $E^{\ast}(z',\bar{z'})=\overline{E^{\ast}(z',\bar{z'})}$ near 0. Write $\mathfrak{M}'$ for the complexification of $M'$. Suppose that $F:(M,0) \longrightarrow (M',0)$ is a biholomorphic map. Then $F$ induces a biholomorphic map $\mathcal {F}$ from $(\mathfrak{M},0)$ to $(\mathfrak{M}',0)$ such that $\pi' \circ \mathcal {F}=F \circ
\pi$. From this, it follows that $F(B)=B'$, where $B'$ is the branching locus of $\pi'$ near the origin.
We next give the precise defining equation of $B$ near $0$. From the equation $z+s\zeta^{s-1}+E_{\zeta}(z,\zeta)=0 $, we can solve, by the implicit function theorem, that $$z=h_1(\zeta)=-s\zeta^{s-1}+o(\zeta^{s-1}),
\label{eqn:Jenny-003}$$ where $h_1(\zeta)$ is holomorphic near 0. Substituting (\[eqn:Jenny-003\]) into (\[eqn:Jenny-002\]), we get $$w=h_2(\zeta)=(1-s)\zeta^s+o(\zeta^s).
\label{eqn:Jenny-004}$$ From ( \[eqn:Jenny-004\] ), we get $$-\frac{w}{s-1}=(h_3(\zeta))^s\ \hbox{ with }\ h_3(\zeta)=\zeta+o(\zeta).$$ Hence, we get $$\begin{array}{ll}
&\zeta=h_3^{-1}((-\frac{w}{s-1})^{\frac{1}{s}})
=(-1)^{\frac{1}{s}}(\frac{1}{s-1})^{\frac{1}{s}}w^{\frac{1}{s}}+o(w^{\frac{1}{s}})\\
&z=h_1((-1)^{\frac{1}{s}}(\frac{w}{s-1})^{\frac{1}{s}}+o(w^{\frac{1}{s}}))=s(-1)^{-\frac{1}{s}}w^{\frac{s-1}{s}}\cdot(s-1)^{\frac{1-s}{s}}
+o(w^{\frac{s-1}{s}}).
\end{array}$$ Here, $h_j's$ are holomorphic functions near 0. Next, let $w=u \geq 0$ and write $$\begin{array}{ll}
&A_j(u)=h_1\circ h_3^{-1}\left (e^{-\frac{(2j+1)\pi
\sqrt{-1}}{s}}(\frac{u}{s-1})^{1/s}\right )\\
& =se^{\frac{(1+2j)\pi
\sqrt{-1}}{s}}u^{\frac{s-1}{s}}\cdot(s-1)^{\frac{1-s}{s}}
+o(u^{\frac{s-1}{s}}),\ \ j=0,1,\cdots,s-1.
\end{array}
\label{eqn:Jenny-005}$$
**Lemma 4.1**: For $0 < u \ll 1,\ A_j(u) \in
D(u)$. Here $$D(u)=\{z \in \mathbb{C}^1: w=z\bar{z}+z^s+\bar{z}^s+E(z,\bar{z}) <
u\}.$$
[*Proof of Lemma 4.1*]{}: The proof follows clearly from the following estimate: $$|A_j(u)|^2+Re\{ 2A_j^s(u)+E(A_j(u),\overline{A_j(u)}) \}
=O(u^{\frac{2(s-1)}{s}}) \ll u$$ as far as $0<u \ll 1$ and $s \geq 3$. $\endpf$
The following fact will be crucial for our later discussions:
$\{(A_j(u),u)\}_{j=0}^{s-1}=B \cap \{w=u\}$ and $A_j(u)$ is real analytic in $u^{1/s}$ for each fixed $j$.
Consider a surface $(M, p)$ in ${\mathbb C}^2$. We say that $M$ near $p$ is defined by a complex-valued function $\rho$, if $M$ near $p$ is precisely the zero set of $\rho$ and $\{Re(\rho),Im(\rho)\}$ has constant rank two near $p$ as functions in $(x,y,u,v)$. For a surface $(M,p)$ defined by $\rho$ and a biholomorphic map $F$ from a neigborhood of $p$ to a neighborhood of $p'$, we say that $F(M)$ approximates $(M^*,p')$ defined by $\rho^*=0$ to the order $m$ at $p'$ if there are smooth functions $h_1$ and $h_2$ with $|h_1|^2-|h_2|^2\not = 0$ at $p'$ such that $\rho\circ F^{-1}(Z)=h_1\cdot \rho^*+h_2\cdot\-{\rho^*}+o(|Z-p'|^m).$
**Lemma 4.2**: Let $M,\ M'$ be Bishop surfaces near $0$ as defined above. Suppose that $F(M)$ approximates $M'$ to the order $\wt{N}=Ns+s-1$ at $0$ with $N>1$. Then $$|F(A_j(u),u)-(A_j'(u'),u')| \ale |u|^{N}, \ \hbox{for} \
j=0,\cdots,s-1,\ u>0.$$ Here $F=(z+f,w+g)$ is a holomorphic map with $f=O(|w|+|z|^2),\
g(z,w)=g(w)=O(w^2)$ and $u'=u+g(u)$.
[*Proof of Lemma 4.2*]{}: Let $\Phi_1$ be a biholomorphic map, which maps M into $M_{nor}^{N}$ defined by $$w=z\bar{z}+2Re\{z^s+\sum\limits_{k=1}^{N}\sum\limits_{j=2}^{s-1}a_{ks+j}z^{ks+j}\}
+o(|z|^{sN+s-1}),$$ and let $\Phi_2$ be a biholomorphic map from $M'$ to ${M'}_{nor}^{N}$ with ${M'}_{nor}^{N}$ defined by\
$$w'=z'\bar{z'}+2Re\{{z'}^s+\sum\limits_{k=1}^{N}\sum\limits_{j=2}^{s-1}{a'}_{ks+j}{z'}^{ks+j}\}
+o(|z'|^{sN+s-1}).$$ Define $\Psi=\Phi_2 \circ F \circ \Phi_1^{-1}$. Here we assume $\Phi_1,\ \Phi_2$ satisfy the normalization as in Theorem 3.1 at the origin. Then $\Psi(M_{nor}^{N})$ approximates ${M'}_{nor}^{N}$ up to order $\wt{N}$.
By Theorem 2.2, we conclude that $$a_{ks+j}=a'_{ks+j} \ for \ ks+j \leq {\wt{N}} \ \ \hbox{and}\ \
\Psi=Id+O(|(z,w)|^{ N}), \ \hbox{with}\ \ \wt{N}=Ns+s-1.$$
In what follows, we write $A_j(u),\ A_j^{\ast}(u),\ A_j^{nor}(u),\ A_j^{\ast nor}(u)$ for those quantities, defined as in (\[eqn:Jenny-005\]), corresponding to $M,\ M',\ M_{nor}^{N},\ {M'}_{nor}^{N}$, respectively. Write $h_j^{nor}$ and $h_j^{\ast nor}$ for those holomorphic functions, defined before, corresponding to $M_{nor}^{N}$ and ${M'}_{nor}^{N}$, respectively. Then from the way these functions were constructed, we have $$h_j^{nor}(\zeta)=h_{j}^{\ast nor}(\zeta)+O(|\zeta|^{N}) \
for \ j=1,2,3.$$ Hence, $$A_j^{nor}(u)=A_j^{\ast
nor}(u+\widetilde{g}(u))+O(u^{N}),$$ where $\Psi=(z+\widetilde{f}(z),w+\widetilde{g}(w))$. This immediately gives the following: $$F(A_j(u),u)=(A_j^{\ast}(u'),u')+O(u^{{N}}),$$ where $u'=u+g(u),F=(z+f(z,w),w+g(w))$. $\endpf$
Summarizing the above, we have the following:
**Proposition 4.3:**
Let $z=r\sigma(\tau,r)$ with $u=r^2$ be the conformal map from the disk $r\Delta:=\{\tau\in{\mathbb C}: \ |\tau|<r\}$ to $D(u)$ with $\sigma(0,r)=0,\
\sigma_{\tau}'(0,r) > 0$. Here, as defined before, $$D(u)=\{z \in
\mathbb{C}^1: z\bar{z}+z^s+\bar{z}^s+E(z,\bar{z}) < u=r^2\}.$$ Similarly, let $z=r\sigma^*(\tau^*,r)$ with $u=r^2$ be the conformal map from the disk $r\Delta$ to $D^*(u)$ with $\sigma^*(0,r)=0,\ {\sigma^{*}}'_{\tau}(0,r) > 0$. Here, $$D^*(u)=\{z \in \mathbb{C}^1: z\bar{z}+z^s+\bar{z}^s+E^*(z,\bar{z})
< u=r^2\}.$$ Then we know that $\sigma(\tau,r)=\tau(1+O(r))$ and $\sigma$ is real analytic in $(\tau,r)$ over $\Delta_{1+\varepsilon}
\times (-\varepsilon,\varepsilon)$ with $0 < \varepsilon \ll 1$. (See \[Hu3\]). Similar property also holds for $\sigma^*$.
Let $\tau_j(u) \in \Delta$ be such that $r\sigma(\tau_j(u),r)=A_j(u)$. Then $$\tau_j(u)=\sigma^{-1}(\frac{A_j(u)}{u^{\frac{1}{2}}},\sqrt{u})
=\frac{A_j(u)}{u^{\frac{1}{2}}}(1+O(\sqrt{u}))$$ Notice that $\frac{A_j(u)}{u^{\frac{1}{2}}}=\sum\limits_{l=s-2}^{\infty}C_{l,j}u^{\frac{l}{2s}}$. Namely, $\frac{A_j(u)}{u^{\frac{1}{2}}}$ is analytic in $u^{\frac{1}{2s}}$. Here $$C_{s-2,j}=s(s-1)^{\frac{1-s}{s}}e^{\frac{\pi
\sqrt{-1}(1+2j)}{s}}. \label{eqn:Jenny-006}$$
Now, the hyperbolic distance between $A_1(u)$ and $A_2(u)$ as points in $D(u)$ is the same as the one between $\tau_1$ and $\tau_2$ as points in $\Delta$. Let $L_{1(j+1)}(u)=e^{d_{hyp}(\tau_0,\tau_j)}-1$. In particular, $L_{12}(u)=e^{d_{hyp}(\tau_0,\tau_1)}-1$ Then since $$d_{hyp}(\tau_0,\tau_1)={1 \over
2}\ln\left(\frac{1+|\frac{\tau_0-\tau_1}{1-\bar{\tau_0}\tau_1}|}
{1-|\frac{\tau_0-\tau_1}{1-\bar{\tau_0}\tau_1}|}\right),\ \hbox{and}$$ $$L_{12}(u)=s(s-1)^{\frac{1-s}{s}}
|e^{\frac{\sqrt{-1}\pi}{s}}-e^{\frac{3\sqrt{-1}\pi}{s}}|
u^{\frac{s-2}{2s}}+o(u^{\frac{s-2}{2s}}),$$ we see that $L_{12}(u)$ is analytic in $u^{{1 \over 2s}}$.
Next, suppose $F:M \longrightarrow M'$ is a biholomorphic map with $F=(\wt{f},\wt{g})=(z,w)+(O(|w|+|z|^2),O(w^2))$. Then $\widetilde{f}=z+f$ is a conformal map from $D(u)$ to $D^*(u')$ with $u'=u+g(u)$. Hence the hyperbolic distance between $A_1(u)$ to $A_2(u)$ is the same as that of the hyperbolic distance from $A_1^{\ast}(u')$ to $A_2^{\ast}(u')$, for $F(A_j(u),u)=(A_j^{\ast}(u'),u')$.
Now, suppose that $F$ is a biholomorphic map with $F=(\wt{f},\wt{g})=(z,w)+(O(|w|+|z|^2),O(w^2))$ such that $F(M)$ approximates $M'$ at 0 up to order $\wt{N}=Ns+s-1>s$. As before, we can assume that $M,\ M'$ are already normalized up to order $\wt{N}$. Then $F=Id+O(|z,w|^{N}),\
M=\{w=z\bar{z}+2Re\{\varphi_0(z)\}+o(|z|^{\wt{N}})\}$, $M'=\{w=z\bar{z}+2Re\{\varphi_0(z)\}+o(|z|^{\wt{N}})\}$, where $\varphi_0
(z)=z^s+o(z^s),\ u'=u+g(u)=u+o(|u|^{N})$ and $\varphi_0^{(sk+j)}(0)=0$ for $j = 0,1\ \hbox{mod }(s).$
From the way $\sigma$ and $\sigma^*$ were constructed, we can show that (see \[Lemma 2.1, Hu3\]): $$\sigma^{\ast}(\tau,u')-\sigma(\tau,u)=\tau O(u^{N }).$$ Indeed, this follows from the following more general result:
[**Lemma 4.4**]{}: Let $\sigma(\xi,r)=\xi\cdot (1+O(r))$ and $\sigma^*(\xi,r)=\xi\cdot (1+O(r))$ be the biholomorphic map from the unit disk $\Delta$ to $$\begin{array}{ll}
&D(r):=\{\xi\in {\mathbb C}(\approx \-{\Delta}): \
|\xi|^2+rF_1(r,\xi,\-{\xi})<1\},\\ &D^*(r):=\{\xi\in {\mathbb C}
(\approx \-{\Delta}): \ |\xi|^2+rF_1(r,\xi,\-
{\xi})+r^mF_2(r,\xi,\-{\xi})<1\},
\end{array}$$ respectively. Here $F_j(r,\xi,\-{\xi})$ are real-valued real analytic functions in a neighborhood of $\{0\}\times \-{\D}
\times{\-\D}$. Then there is a constant $C$, depending only on $F_j$, such that $$|\sigma^{\ast}(\xi,r)-\sigma(\xi,r)|\le C|\xi| r^{m},\ \ \xi\in\-{\D}.$$ [*Proof of Lemma 4.4*]{}: From the way $\sigma$ and $\sigma^*$ were constructed (see \[Lemma 2.1, Hu3\]), there are $U, U^*\in C^{\omega}(\p\D\times (-\epsilon_0,\epsilon_0))$ with $0<\epsilon_0<<1$ such that $$\sigma(\xi,r)=\xi\left(1+U(\xi,r)+{\mathcal H}(U(\cdot,r))\right),\ \sigma^*(\xi,r)=\xi\left(1+U^*(\xi,r)+{\mathcal H}(U^*(\cdot,r))\right), \ \xi\in \p\D.$$ Here ${\mathcal H}$ is the standard Hilbert transform and $U,\ U^*$ satisfy the following equations: $$U=G_1(r,\xi,U,{\mathcal H} (U)),\ \ U^*=G_1(r,\xi,U^*,{\mathcal H}(U^*))+r^mG_2(r,\xi,U^*,{\mathcal H}(U^*)),$$ where $G_j(r,\xi,x,y)$ are real analytic in $(r,\xi,x,y)$ with $G_j \ale |r|+|x|^2+|y|^2$. Notice by the implicit function (see \[Lemma 2.1, Hu3\]), $\|U\|_{1/2},\ \|U^*\|_{1/2}\le C_1|r|$ with $\|\cdot\|_{1/2}$ the Hölder-$\frac{1}{2}$ norm. Next, we have $$\begin{array}{ll}
&U^*-U=\int_{0}^{1}\frac{\p G_1}{\p x}(r,\xi,\tau U^*+(1-\tau)U, \tau {\mathcal H} (U^*)+(1-\tau){\mathcal H}(U))(U^*-U)\hbox{d}\tau
\\ & +\int_{0}^{1}\frac{\p G_1}{\p y}(r,\xi,\tau U^*+(1-\tau)U,\tau {\mathcal H}(U^*)+(1-\tau){\mathcal H}( U))({\mathcal H}(U^*)-{\mathcal H}(U))\hbox{d}\tau
\\ & +r^mG_2(r,\xi,U^*,{\mathcal H}(U^*)).
\end{array}$$ By noticing that the Hilbert transform is bounded acting on the Hölder space, we easily conclude the result in the lemma by letting $|r|<<1$. $\endpf$
Now, by Lemma 4.4, we see that $\tau_j^{\ast}(u')=\tau_j(u)+O(u^{N})$. Therefore $$L_{12}(u')-L_{12}(u)=O(u^{N }).$$ In particular, if $F:M \longrightarrow M'$ is a formal equivalence map with $F=(\wt{f},\wt{g})=(z,w)+(O(|w|+|z|^2),O(w^2))$. Then $$L_{12}^{\ast}(u')=L_{12}(u)\ \hbox{ in the formal sense}.
\label{eqn:Jenny-008}$$
We next prove the following:
**Lemma 4.5:** Let $F:M \longrightarrow M'$ be a formal equivalence map with $F=(\wt{f},\wt{g})=(z,w)+(O(|w|+|z|^2),O(w^2))$. Write $F=(\widetilde{f},\widetilde{g})=(z+f,w+g)$ as before. Then $\widetilde{g}$ is convergent.
[*Proof of Lemma 4.5*]{}: By (\[eqn:Jenny-008\]), we have $$L_{12}^{\ast}(\widetilde{g}(u))=L_{12}(u)\ \ \hbox{in the formal
sense}.$$ Write $u=V^{2s}$ and $\widetilde{g}(u)=U^{2s}$. Then $$L_{12}^{\ast}(U^{2s})=L_{12}(V^{2s}).$$ Notice that $L_{12}^{\ast}(U^{2s})$ and $L_{12}(V^{2s})$ now are analytic in $U$ and $V$, respectively. Moreover, $$L_{12}^{\ast}(U^{2s})=(\psi^{\ast}(U))^{s-2},\ L_{12}(V^{2s})=(\psi(V))^{s-2}$$ with $\psi,\ \psi^{\ast}$ invertible holomorphic map of $(\mathbb{C},0)$ to itself, and with $\psi'(0)={\psi^{\ast}}'(0)$. Hence, we get $$\widetilde{g}(u)=((\psi^{\ast -1} \circ \psi)(u^{{1 \over
2s}}))^{2s}.$$ On the other hand, $(\psi^{\ast -1} \circ
\psi(z)^{\frac{1}{2s}})^{2s}$ defines a multiple valued holomorphic function near the origin. By the Puiseux expansion, we get $$(\psi^{\ast -1} \circ \psi(u^{{1 \over
2s}}))^{2s}=\sum\limits_{j=2s}^{\infty}c_ju^{{j \over 2s}}$$ However, $(\psi^{\ast -1} \circ \psi(u^{{1 \over 2s}}))^{2s}$ also admits a formal power series expansion. We conclude that $c_j=0$ if $2s$ does not divide $j$. This proves the convergence of $\widetilde{g}(u)$. $\endpf$
We next prove the following theorem:
**Theorem 4.6:**
[*Proof of Theorem 4.6*]{}: We can assume that $\wt{f}=z+f$ with $wt_{nor}(f)\ge 2$ and $\wt{g}=w+g$ with $wt_{nor}(g)\ge 4$. By Lemma 4.4 and by considering $F_0\circ F$ instead of $F$, where $F_0(z,w)=(z,g^{-1}(w))$, we can assume, without loss of generality, that $\widetilde{g}=w$. We will prove the convergence of $\widetilde{f}$ by the hyperbolic geometry associated to the surface discussed above. By Proposition 4.3 (2), we first notice that $$\widetilde{f}(A_j(u),u)=A_j^{\ast}(u)\ \hbox{ in the formal sense.}$$ Namely, $\widetilde{f_{(N)}}(A_j(u),u)=A_j^{\ast}(u)+o(u^{N'})$ for any $N$. Here, $\widetilde{f_{(N)}}$ is the $N^{th}$-truncation in the Taylor expansion of $f$ at $0$; and $N'$ depends only on N with $N' \rightarrow \infty$ as $N
\rightarrow \infty$.
Write $\widetilde{M}$ and $\widetilde{M'}$ for the holomorphic hull of $M$ and $M'$, respectively. We next construct a holomorphic map from $\widetilde{M} \setminus M$ to $\widetilde{M'} \setminus M'$ as follows:
For $(z,u) \in D(u) \times \{u\}$, let $\tau(u) \in \Delta$ be such that $r\sigma(\tau(u),r)=z,\ u=r^2$. Let $\Psi(\cdot,r)$ be a biholomorphic map from $\Delta$ to itself such that $\Psi(\tau_j(u),r)=\tau_j^{\ast}(u)$ for $j=0,1$. Here, to see the existence of $\Psi(\cdot,r)$ , it suffices for us to explain that $d_{hyp}(\tau_0(u),\tau_1(u))=d_{hyp}(\tau_0^{\ast}(u),\tau_1^{\ast}(u))$. But, this readily follows from (\[eqn:Jenny-008\]) and Lemma 4.4. Now, let $$\Psi_1=\frac{\tau-\tau_0(u)}{1-\bar{\tau_0}(u)\tau},
\Psi_1^{\ast}=\frac{\tau-\tau_0^{\ast}(u)}{1-\bar{\tau_0^{\ast}}(u)\tau},
\ \Theta(\tau,r)=e^{-i\theta(r)+i\theta^{\ast}(r)}\tau,$$ where\
$$\begin{array}{l}
\theta(r)=
arg\{\frac{\tau_1(u)-\tau_0(u)}{1-\overline{\tau_0}(u)\tau_1(u)}
\frac{1}{u^{\frac{s-2}{2s}}}\} \\
\theta^{\ast}(r)=
arg\{\frac{\tau_1^{\ast}(u)-\tau_0^{\ast}(u)}{1-\overline{\tau_0^{\ast}}
(u)\tau_1^{\ast}(u)}\frac{1}{u^{\frac{s-2}{2s}}} \}.
\end{array}
\label{eqn:Jenny-010}$$ Then
$$\Psi(\tau,r)=\Psi_1^{\ast -1}(\tau,r) \circ \Theta(\tau,r) \circ
\Psi_1(\tau,r).
\label{eqn:Jenny-011}$$
$\Psi(\tau,r)$ is analytic in $(\tau,u^{{1 \over 2s}})
\in \Delta_{1+\varepsilon_0} \times
(-\varepsilon_0,\varepsilon_0)$. (See \[Lemma 2.1, Hu3\]). We notice that when $f$ is a priori known to be convergent, we then have, by the uniqueness property of the Möbius transformation, that
$$\wt{f}(r\sigma(\xi,r),r^2)=r\sigma^{*}(\Psi(\xi,r),r^2).
\label{eqn:Jenny-012}$$
Consider the angle $\Theta_j$ ($j=2,\cdots,s-1$) from the geodesic connecting $\tau_j$ to $\tau_0$ to the geodesic connecting $\tau_j$ to $\tau_1$. As a function of $u$ (or $r$), we see, as in the definition of $\Psi(\xi,r)$, that $$\Theta_j(u)=arg\{\frac{\tau_1(u)-\tau_j(u)}{\tau_0(u)-\tau_j(u)}\cdot \frac{1-\-{\tau_j(u)}\tau_0(u)}{1-\-{\tau_j(u)}\tau_1(u)}\}=arg\{\frac{C_{s-2,2}-C_{s-2,j}}{C_{s-2,1}-C_{s-2,j}}\}+O(u^{1/(2s)}).$$ We can similarly define $\Theta_j^*$ for $M'$. Then the same argument which we used to show that $L_{12}(u)=L_{12}^*(u)$ can be used to prove that $$\Theta_j(u)\equiv \Theta^*_j(u),\ \ \h{and} \ \ L_{1(j+1)}(u)=L^*_{1(j+1)}(u).$$
Now, we can use a Möbius transformation to map $\tau_j$ to the origin and $\tau_2$ to a point in the positive real line. Then we easily see that $\Theta_j$ and $L_{1(j+1)}$ uniquely determine $\tau_j(u)$. As an immediate consequence of such a consideration, we conclude that
**Lemma 4.7**: $\Psi(\tau_j(u),r)=\tau_j^{\ast}(u) \ \ \hbox{for } j=0,\cdots,s-1$.
Now, for $(z,u) \in \widetilde{M}
\setminus M$ close to the origin, we define $$f^{\ast}(z,u)=\sqrt{u}\sigma^{\ast}(\Psi(\sigma^{-1}(\frac{z}{\sqrt{u}},\sqrt{u}),\sqrt{u}),\sqrt{u})$$ Then $f^{\ast}(z,u)$ is analytic in $\widetilde{M} \setminus M$. We next prove the following: **Lemma 4.8:**$\forall \alpha \geq
0,\frac{\partial^{\alpha} f^{\ast}}{\partial
z^{\alpha}}(0,u)=\frac{\partial \widetilde{f}}{\partial
z^{\alpha}}(0,u)$ in the formal sense. Namely, letting $\widetilde{f_{(N)}}$ be the polynomial consisting of terms of degree $\leq N$ in the Taylor expansion of $\widetilde{f}$ at $0$, then $\exists N'(N)\rightarrow \infty$ as $N \rightarrow \infty$ such that\
$$\frac{\partial^{\alpha} f^{\ast}}{\partial
z^{\alpha}}(0,u)=\frac{\partial^{\alpha} \widetilde{f_{(N)}}}{\partial
z^{\alpha}}(0,u)+o(u^{N'}).$$
[*Proof of Lemma 4.8*]{}: Let $S(u)$ be the hyperbolic polygon in $D(u)$ with vertices $A_j(u)(j=0,1,\cdots,s-1)$, whose sides consist of the geodesic segments connecting the vertices. Let $S^{\ast}(u)$ be the one corresponding to $M'$. We notice that for any points $P,Q\in \D$, then the geodesic segment connecting P to Q is $$\gamma_{P,Q}(t)=\frac{t\frac{Q-P}{1-Q\bar{P}}+P}{1+t\bar{P} \cdot
\frac{Q-P}{1-Q\bar{P}}}, \hskip 1cm 0\leq t \leq 1.$$ Hence, by the same argument used in the proof of Lemma 4.2 and by making use of the property that $\wt{f}$ formally maps vertices to the corresponding ones, we see that for any point $P\in \p S(u)$, we have $$f^{\ast}(P)
% \mid_{\partial S(u)}
= \widetilde{f_{(N)}}(P)
% \mid_{\partial S(u)}
+Error(P).$$
Here $$|Error(P)|\le C u^{N'} \ \ \hbox{with}\ \ N'(N)\rightarrow
\infty \ \hbox{as} \ N\rightarrow \infty$$ and $C$ is a constant independent of $P$.
Now, by the Cauchy formula, $$\frac{\partial^{\alpha} f^{\ast}}{\partial z^{\alpha}}(0,u)=
\frac{\alpha!}{2\pi\sqrt{-1}}\int_{\partial
S(u)}\frac{f^{\ast}(\zeta,u)}{\zeta^{\alpha+1}}d\zeta$$ and $$\frac{\partial^{\alpha} \widetilde{f_{(N)}}}{\partial z^{\alpha}}(0,u)=
\frac{\alpha!}{2\pi\sqrt{-1}}\int_{\partial
S(u)}\frac{\widetilde{f_{(N)}}(\zeta,u)}{\zeta^{\alpha+1}}d\zeta$$ Notice that for $z \in \partial S(u),|z| \gtrsim u^{{s-1 \over s}}$ , it thus follows that $$|\frac{\partial^{\alpha} f^{\ast}}{\partial z^{\alpha}}(0,u)-
\frac{\partial^{\alpha} \widetilde{f_{(N)}}}{\partial z^{\alpha}}(0,u) |
\ale O(u^{N'-{s-1 \over s}\alpha}).$$ This completes the proof of Lemma 4.8. $\endpf$
We continue our proof of Theorem 4.6. We notice that\
$(i): \sigma^{\ast}(\zeta,\sqrt{u})$ is analytic in $(\zeta,\sqrt{u})$ near $(0,0)$,\
$(ii): \Psi(\tau,\sqrt{u})$ is analytic in $\tau$ and $u^{{1 \over 2s}}$ near $(0,0)$ and,\
$(iii): \sigma^{-1}(\frac{z}{\sqrt{u}},\sqrt{u})$ is analytic in $(\frac{z}{\sqrt{u}},\sqrt{u})$ near $(0,0)$, too.\
Write $$\Psi(\tau,\sqrt{u})=\sum\limits_{\alpha,\beta =
0}^{\infty}a_{\alpha\beta}\tau^{\alpha}u^{\frac{\beta}{2s}}$$ and $$\Psi(\tau;Y_1)=\sum\limits_{\alpha,\beta =
0}^{\infty}a_{\alpha\beta}\tau^{\alpha}Y_1^{\beta}$$ Then $$H(X,Y_1,Y_2)=Y_2\sigma^{\ast}(\Psi(\sigma^{-1}(X,Y_2);Y_1),Y_2)$$ is analytic in $X,Y_1,Y_2$ near 0. Write $$H(X,Y_1,Y_2)=\sum\limits_{\alpha,\beta,\gamma=
0}^{\infty}b_{\alpha\beta\gamma}X^{\alpha}Y_1^{\beta}Y_2^{\gamma}
\label{eqn:Jenny-add-02}$$ Then $$f^{\ast}(z,u)=H(\frac{z}{\sqrt{u}},u^{\frac{1}{2s}},\sqrt{u})
=\sum\limits_{\alpha,\beta,\gamma=
0}^{\infty}b_{\alpha\beta\gamma}z^{\alpha}u^{\frac{\gamma-\alpha}{2}
+\frac{\beta}{2s}}$$ Hence,\
$$\frac{\partial^{\alpha} \widetilde{f}}{\partial z^{\alpha}}(0,u)=
\sum\limits_{\alpha,\beta,\gamma=
0}^{\infty}b_{\alpha\beta\gamma}\alpha! u^{\frac{\gamma-\alpha}{2}
+\frac{\beta}{2s}}$$ in the formal sense. It thus follows that if $b_{\alpha\beta\gamma} \neq
0,\frac{\gamma-\alpha}{2} +\frac{\beta}{2s}=\beta'$ is a non-negative integer. Hence $f^{\ast}(z,u)=\sum\limits_{\alpha,\beta,\gamma=
0}^{\infty}b_{\alpha\beta\gamma}z^{\alpha}u^{\beta'}$.\
Now, $|b_{\alpha\beta\gamma}| \ale R^{|\alpha|+|\beta|+|\gamma|}$ with $R \gg 1$ by ( \[eqn:Jenny-add-02\] ). We see that $|b_{\alpha\beta\gamma}| \ale R^{2s|\alpha|+2s\beta'} \ale
(R^{2s})^{\alpha+\beta'}.$ This shows that $f^{\ast}(z,u)$ is holomorphic in $(z,u)$ near 0; and thus $\widetilde{f}$ is convergent, too. The proof of Theorem 4.6 is finally completed. $\endpf$
[*Proofs of Theorem 1.2 and Corollary 1.4*]{}: Theorem 1.2 and Theorem 4.6 have the same content. The proof of Corollary 1.4 follows from Theorem 1.1 and Theorem 1.2. $\endpf$
As another application of Theorem 1.2, we have the following
[**Corollary 4.9**]{}: [*Let $(M,0)$ be a real analytic elliptic Bishop surface with the Bishop invariant vanishing and the Moser invariant $s<\infty$ at $0$. Then any element in $aut_0(M)$ is a holomorphic automorphism of $(M,0)$.* ]{}
[widest-label]{}
P. Ahern and X. Gong, Real analytic submanifolds in ${\mathbf C}^n$ with parabolic complex tangents along a submanifold of codimension one, preprint, 2006.
S. Baouendi, P. Ebenfelt and L. Rothschild, Local geometric properties of real submanifolds in complex space, [*Bull. Amer. Math. Soc.*]{} (N.S.) 37 (2000), no. 3, 309–33.
S. Baouendi, N. Mir and L. Rothschild, Reflection ideals and mappings between generic submanifolds in complex space, [*J. Geom. Anal. 12*]{} (2002), 543-580.
E. Bedford and B. Gaveau, Envelopes of holomorphy of certain 2-spheres in ${\mathbf C}^2$, [*Amer. J. Math.*]{} (105), 975-1009, 1983.
E. Bishop, Differentiable manifolds in complex Euclidean space, [*Duke Math. J.*]{} (32), 1-21, 1965.
É. Cartan, Sur les variétés pseudo-conformal des hypersurfaces de l’espace de deux variables complexes, [*Ann. Mat. Pura Appl. (4) 11*]{}, 17-90(1932).
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, [*Acta Math. 133*]{}, 219-271(1974).
Y. Eliashberg, Filling by holomorphic discs and its applications, Geometry of Low-Dimensional Manifolds, [*London Math. Soc. Lecture Notes Vol. 151*]{}, 1997.
F. Forstneric, Most real analytic Cauchy-Riemann manifolds are nonalgebraizable, [*Manuscripta Math. 115*]{} (2004), 489–494.
X. Gong, On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents, [*Comment. Math. Helv*]{}. 69 (1994), no. 4, 549–574.
X. Gong, Normal forms of real surfaces under unimodular transformations near elliptic complex tangents, [*Duke Math. J. 74*]{} (1994), no. 1, 145–157.
X. Gong, Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics, [*Indiana Univ. Math. J. 53*]{} (2004), no. 1, 83–95.
M. Gromov, Pseudo holomorphic curves in symplectic geometry, [*Invent Math. Vol. 82*]{}, 1985, 307-347.
X. Huang, Local Equivalence Problems for Real Submanifolds in Complex Spaces, [*Lecture Notes in Mathematics 1848 (C.I.M.E. series)*]{}, Springer-Verlag, pp 109-161, Berlin-Heidelberg-New York, 2004.
X. Huang, On some problems in several complex variables and Cauchy-Riemann Geometry, Proceedings of ICCM (edited by L. Yang and S. T. Yau), [*AMS/IP Stud. Adv. Math. 20*]{}, 383-396, 2001.
\[Hu3\] [Hu3 1998]{} X. Huang, On an n-manifold in ${\bf C}^n$ near an elliptic complex tangent, [*J. Amer. Math. Soc.*]{} (11), 669–692, 1998.
X. Huang and S. Krantz, On a problem of Moser, [*Duke Math. J.*]{} (78), 213-228, 1995.
S. Ji, Algebraicity of real analytic hypersurfaces with maxium rank, [*Amer. Jour. Math. Vol 124*]{}, 255-264, 2002.
C. Kenig and S. Webster, The local hull of holomorphy of a surface in the space of two complex variables, [*Invent. Math. 67*]{}, 1-21, 1982.
C. Kenig and S. Webster, On the hull of holomorphy of an n-manifold in ${\mathbb C}^n$, [*Annali Scoula Norm. Sup. de Pisa IV Vol. 11 (No. 2)*]{}, 261-280, 1984
F. Meylan, N. Mir and D. Zaitsev, Approximation and convergence of formal CR-mappings, International Mathematics Research Notices 2003, no. 4, 211-242.
J. Moser, Analytic surfaces in ${\CC}^2$ and their local hull of holomorphy, [*Annales Aca -demiæFennicae Series A.I. Mathematica*]{} (10), 397-410, 1985.
J. Moser and S. Webster, Normal forms for real surfaces in ${\CC}^2$ near complex tangents and hyperbolic surface transformations, [*Acta Math.*]{} (150), 255-296, 1983.
H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, [*Ren. Cire. Mat. Palermo, II. Ser. 23*]{}, 185-220, 1907.
L. Stolovitch, Family of intersecting totally real manifolds of $(\Bbb C^n,0)$ and CR-singularities, preprint, 2006.
S. Webster, Pairs of Intersecting Real Manifolds in Complex Space, [*Asian Jour. Math*]{} Vol. 7 (No. 4), 449-462, 2003.
X. Huang (huangx@math.rutgers.edu), School of Mathematics, Wuhan University, Wuhan 430072, China; and Department of Mathematics, Hill Center-Busch Campus, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA;
Wanke Yin, School of Mathematical Sciences, Wuhan University, Wuhan 430072, China.
[^1]: Supported in part by NSF-0500626
|
---
abstract: 'Convolutional neural networks (CNN) pre-trained on ImageNet are the backbone of most state-of-the-art approaches. In this paper, we present a new set of pre-trained models with popular state-of-the-art architectures for the Caffe framework. The first release includes Residual Networks (ResNets) with generation script as well as the batch-normalization-variants of AlexNet and VGG19. All models outperform previous models with the same architecture. The models and training code are available at <http://www.inf-cv.uni-jena.de/Research/CNN+Models.html> and <https://github.com/cvjena/cnn-models>.'
author:
- |
Marcel Simon, Erik Rodner, Joachim Denzler\
Computer Vision Group\
Friedrich-Schiller-Universität Jena, Germany\
[{marcel.simon, erik.rodner, joachim.denzler}@uni-jena.de]{}
bibliography:
- 'paper.bib'
title: 'ImageNet pre-trained models with batch normalization'
---
Introduction
============
The rediscovery of convolutional neural networks (CNN) [@krizhevsky12alexnet] in the past years is a result of both the dramatically increased computational speed and the advent of large scale datasets as part of the big data trend. The computational speed was mainly boosted by the efficient use of GPUs for common computer vision functions like convolution and matrix multiplication. Large scale datasets [@russakovsky15ilsvrc; @coco; @visualgenome; @cocoCaptions; @plummer2015flickr30k; @cityscapes], on the other hand, provide the amount of data required for training large scale models with more than a hundred million parameters.
This combination allowed for huge advances in all fields of computer vision research ranging from traditional tasks like classification [@he15resnet; @Simon15:NAC; @Simon14:PDD; @azizpour14CNNanalysis; @Freytag16_CFW; @le15deeplysupervised], object detection [@ren15fasterrcnn; @sermanet13overfeat; @Gidaris_2015_ICCV], and segmentation [@long15fcn; @Brust15:CPN; @crf-nn], to new ones like image captioning [@johnsonKL15; @Rohrbach_2013_ICCV; @maoHTCYM15; @yuPYBB16; @xu15cpation], visual question answering [@Antol_2015_ICCV; @NIPS2015_5641; @XiongMS16] and 3D information prediction [@Eigen_2015_ICCV; @Wang_2015_CVPR]. Most of these works are based on models, which are pre-trained on the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) dataset [@russakovsky15ilsvrc]. The classification task of the last year’s ILSVRC contains 1.2 million training images categorized into one thousand categories, which represent a wide variety of everyday objects. Pre-training on this dataset proved to be a crucial step for obtaining highly accurate models in most of the tasks mentioned above.
While computational speed was dramatically increased by the use of GPUs, training a large model like VGG19 [@simonyan14vgg] still takes several months on a high-end GPU. We hence release a continuously growing set of pre-trained models with popular architectures for the Caffe framework [@jia2014caffe]. In contrast to most publicly available models for this framework, our release includes the batch normalization [@ioffe15batchnorm] variants of popular networks like AlexNet [@krizhevsky12alexnet] and VGG19 [@simonyan14vgg]. In addition, we provide training code for reproducing the results of residual networks [@he15resnet] in Caffe, which was not provided by the authors of the paper [@resnetModels]. The release includes all files required for reproducing the model training as well as the log file of the training of the provided model.
Batch normalization for CNNs
============================
Especially for larger models like VGG19, batch normalization [@ioffe15batchnorm] is crucial for successful training and convergence. In addition, architectures with batch normalization allow for using much higher learning rates and hence yield in models with better generalization ability. In our experiments, we found that higher learning rates show a slower initial convergence speed, but end up at a lower final error rate. This was the case for both AlexNet and VGG19.
The advantage of batch normalization is present even for fine-tuning in certain applications. For example, Amthor [@Amthor16_IDD] report that their multi-loss architectures only converged reliably if batch normalization was added to the networks. However, adding batch normalization afterwards to models trained without batch normalization yields in a severe increase in error rates due to mismatching output statistics. Instead, fine-tuning with our batch normalization models is directly possible, which allows for easy adaption to new tasks.
Implementation details
======================
We modified AlexNet and VGG19 by adding a batch normalization layer [@ioffe15batchnorm] between each convolutional and activation unit layer as well as between each inner product and activation unit layer. We followed the suggestions of [@ioffe15batchnorm] and removed the local response normalization and dropout layers. In addition, we also omitted the mean subtraction during training and replaced it by an batch normalization layer on the input data. This results in an adaptively calculated mean in training and relieves users from manually subtracting the mean during feature computation. In addition, this approach has the advantage that the mean adapts automatically during fine-tuning and no manual mean calculation and storage is required.
We train for 64 epochs on the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) 2012 – 2016 dataset [@russakovsky15ilsvrc], which contains roughly 1.2 million images and one thousand object categories. A batch size of 256 and initial learning rate of 0.05 (AlexNet), 0.01 (VGG19) and 0.1 (ResNet) was used. The learning rate follows a linear decay over time. Due to batch normalization, it is important that the batch size is greater than sixteen to obtain robust statistics estimations in the batch normalization layers. In the Caffe framework, this means the batch size in the network definition needs to be sixteen or larger, the solver parameter `iter_size` does not compensate a too small batch size in the network definition. If you want to fine-tune a model but do not have enough GPU memory, you can enable the use of global statistics in training in order to lift this batch size requirement. This will disable the statistics estimation in each forward pass and global statistics will be used instead.
All images are resized such that the smaller side has length 256 pixel and the aspect ratio is preserved. During training, we randomly crop a $224\times 224$ (ResNet, VGG19) or $227\times 227$ (AlexNet) pixel square patch and feed it into the network. During validation, a single centered crop is used. We did not use any kind of color, scale or aspect ratio augmentation.
During training of residual networks, we also observe a sudden divergence at random time points in training as explained by Szegedy [@szegedy16inception4]. In this case, we restarted the training using the last snapshot. Due to a different random seed, the order of the images is different and hence the training does not diverge at this time point anymore.
Please note, that the final models are not cherry picked based on the validation error. We provide the final model after the full training is completed. We did not intervene with training and especially did not manually changed the learning rate, as usually done if the step policy is used for the learning rate.
Results
=======
The top-1 and top-5-error of the trained models are shown in Table \[tab:model-acc\].
----------- ----------- ---------- ----------- ----------
Model
Ours Original Ours Original
AlexNet **39.9%** 42.6% **18.1%** 19.6%
VGG19 **26.9%** 28.7% **8.8%** 9.9%
ResNet-10 **36.1%** – **14.8%** –
ResNet-50 **24.6%** 24.7% **7.6%** 7.8%
----------- ----------- ---------- ----------- ----------
: **Single-crop** top-1 and top-5 error of our models on the validation set of ILSVRC 2012. \[tab:model-acc\]
As observed in previous works [@ioffe15batchnorm], the error rates benefit from the added batch normalization layers. All provided models slightly improve the error rate achieved by previously trained models [@matconvnetPretrained]. In case of AlexNet, for example, we even observe an error decrease of over 2.6%.
In addition to the final results, we also visualize the single-crop top-1 error on the validation set during the training of AlexNet in Fig. \[fig:alexnet-training\].
![**Single crop** top-1 error of AlexNet on the validation set of ILSVRC 2012 with respect to the training time. We used a linear learning rate decay as shown by the gray curve, which explains the steep decrease in error towards the end of the training. \[fig:alexnet-training\]](images/alexnet_err_lr_plot.pdf){width="\linewidth"}
As shown in the figure, the error decreases consistently and fairly quickly during training. Since we use linear learning rate decay, there is a steep error decrease towards the end of the training. While it might look like the error could decrease even further, this is not true. The reason is that the learning rate approaches 0 during the end of the training. Even if the learning rate is kept constant, no improvement can be observed. This is supported by several experiments we performed.
Conclusions
===========
This paper presents a new set of pre-trained models for the ImageNet dataset using the Caffe framework. We focus on the batch-normalization-variants of AlexNet and VGG19 as well as residual networks. All models outperform previous pre-trained models. In particular, we were able to reproduce the ImageNet results of residual networks. All models, log files and training code are available at <http://www.inf-cv.uni-jena.de/Research/CNN+Models.html> and <https://github.com/cvjena/cnn-models>.
Acknowledgments
===============
The authors thank Nvidia for GPU hardware donations. Part of this research was supported by grant RO 5093/1-1 of the German Research Foundation (DFG)
|
---
abstract: |
We present a novel determination of the astrophysical uncertainties associated to the secondary antiproton flux originating from cosmic-ray spallation on the interstellar gas. We select a set of propagation models compatible with the recent B/C data from PAMELA, and find those providing minimal and maximal antiproton fluxes in different energy ranges. We use this result to determine the most conservative bounds on relevant Dark Matter (DM) annihilation channels: We find that the recent claim of a DM interpretation of a gamma-ray excess in the Galactic Center region cannot be ruled out by current antiproton data.
Finally, we discuss the impact of the recently released preliminary data from AMS-02. In particular, we provide a reference model compatible with proton, helium and B/C spectra from this experiment. Remarkably, the main propagation parameters of this model are in agreement with the best fit presented in our earlier statistical analyses. We also show that the antiproton-to-proton ratio does not exhibit any significant anomaly at high energy with respect to our predictions.
author:
- Carmelo Evoli
- Daniele Gaggero
- Dario Grasso
bibliography:
- 'exoticap.bib'
title: Secondary antiprotons as a Galactic Dark Matter probe
---
Introduction
============
The quest for some anomaly in the cosmic-ray (CR) antiproton spectrum has captured the interest of the astro-particle community for more than two decades and is presently one of the main targets of the AMS-02 observatory [@Incagli:2010].
The rationale of this effort is the following: If the astrophysical sources, mainly supernova remnants (SNRs), do not inject primary ${\bar p}$ in a relevant amount – and provided that the secondary antiproton flux due to CR spallation on the diffuse interstellar gas can be accurately computed – then the measured CR antiproton spectrum becomes a valuable probe of new physics (see, e.g., [@Lavalle:2012ef] and references therein). In particular, a fascinating scenario is that $\bar{p}$ are copiously yielded by WIMP dark matter (DM) particles annihilating in the dark halo of the Galaxy [@Silk:1984].
In this context, an important role is played by secondary-over-primary ratios of CR nuclear species: Most noticeably the boron-to-carbon (B/C) ratio is one of the most useful tracers of Galactic CR propagation since boron is expected to be not produced in stars and the cross sections for B production from its main primaries (C and O) are better known. This channel offers a rather robust constraint on CR models and allows a precise determination of the secondary ${\bar p}$ flux produced in the Inter-Stellar Medium (ISM) (see, e.g., [@Maurin:2001; @Moskalenko02; @DiBernardo10; @Putze:2010; @Trotta:2011]) Moreover, it is crucial to constrain the amount of antiprotons produced and then accelerated in SNRs [@Cholis2013; @Mertsch2014].
In this paper we make use of the recent measurements of the B/C ratio, together with the proton, helium and light nuclei absolute spectra provided by the PAMELA collaboration [@Adriani:2011cu; @Adriani:2014xoa] to constrain the propagation parameters, and to accurately compute the secondary antiproton spectrum in the $0.1 - 10^3$ GeV energy range.
Our main goal is to achieve a comprehensive evaluation of the relevant uncertainties affecting this computation, extending the work presented in [@Evoli:2011id]. We focus our attention primarily on: 1) the systematic errors on the antiproton production and inelastic scattering cross sections; 2) the large uncertainty on the propagation parameters in the Galaxy and in the Heliosphere.
Since several papers have been already published on this subject, we briefly mention here the main novelties of our approach.
- We use, for the first time, all the relevant CR measurements taken during the same period and from the same experiment (PAMELA).
- We use a recent computation [@Kappl:2014hha] of the antiproton production and inelastic cross sections based on the newly available NA49 experimental data [@NA49].
- We account for charge dependent solar modulation, including charge sign dependent drifts, by means of the recently developed [Helioprop]{} code [@Maccione:2012cu].
Our approach goes beyond providing an updated range of allowed propagation setups to be used to propagate dark matter annihilation products. The widely used MIN-MED-MAX propagation models are defined as those models giving the maximal, median, and minimal supersymmetric (primary) antiproton flux and are compatible with B/C analysis [@Donato:2003]. Such set of models, however, cannot be confused, as often done in the literature (see however [@Donato:2008jk; @Bringmann:2006im]), with those providing the allowed range of secondary antiprotons. Indeed, while the primary antiproton flux is strongly dependent on the diffusive halo scale height and on the physical conditions in the Galactic center region, once the propagation parameters are set to fit the B/C, the secondary flux is almost independent on those quantities. Since the models providing the maximal/minimal secondary antiproton flux are dependent on the antiproton energy, we obtain for each energy bin the allowed flux compatible with CR nuclear measurements and the cross section uncertainties. We then constrain DM models using the minimum secondary antiproton flux evaluated in that way as background.
Our results allow us to determine up-to-date conservative constraints on a representative sample of DM models and to compare these limits with the recent claim of a gamma-ray excess in the GC associated with DM annihilations in that region.
In the last section, the preliminary results recently released by AMS-02 up to $450$ GeV/n [@AMS_preliminary] are considered as well. In particular, we investigate the presence of an excess in the $\bar{p}/p$ spectrum at energies higher than previously measured by PAMELA. Given that the AMS-02 B/C data are still preliminary, we do not aim at providing a dedicated statistical analysis. On the other hand, we select a single model providing the best fit of AMS-02 B/C data, and we tune the injection spectra to reproduce the proton and Helium data from the same experiment. Finally, we compare the predicted antiproton fluxes for this model with the new measurements. We discuss these results in the last Section.
CRs in the Galaxy
=================
CR propagation in the ISM {#sec:prop}
-------------------------
We model CR propagation in the Galaxy using the publicly available numerical code [DRAGON]{} [@Evoli:2008dv; @Dragonweb]. This code solves the diffusion-convection transport equation for all CR species in the Galaxy taking into account all the relevant processes, including energy losses and spallation due to the interaction with the ISM. We adopt the nuclear cross-section database taken from the public version of [GALPROP]{} [^1] [@Moskalenko1998].
We assume cylindrical symmetry in the Galaxy. The propagation region is a classical cylindrical box with radial extension $R = 20$ kpc and height ($L$) varying in the range ($2-16$) kpc. The lower limit of the halo size is determined by the observed diffuse synchrotron emission and by comparison between the computed low-energy secondary positron spectrum and PAMELA data [@DiBernardo:2012zu; @Lavalle:2014kca]. The upper limit is chosen in agreement with the available measurements of the $^{10}$Be/$^9$Be ratio. In the next section we will show that secondary antiprotons from CRs are almost unchanged by varying $L$ in the given range, making the choice of the allowed domain not crucial in this context. We optimize the spatial resolution, as well as the time steps involved in the numerical algorithm, in order to evaluate the $\chi^2$ against PAMELA data with a precision $< 1$%.
We adopt a two-dimensional ($r$ and $z$) propagation configuration since it is commonly used to study secondary CR antiproton production and also to keep the computational time short enough to scan over a large number of models. A more realistic three-dimensional approach could give a more accurate prediction of the CR fluxes, by taking the spiral arm distribution of CR sources and of the ISM gas into account [@Gaggero2013; @Kissmann:2015]. However we confirmed that these differences are much smaller than the other uncertainties under investigation in this work.
We adopt the following standard expression for the diffusion coefficient $D$:
$$\label{eq:diff_coeff}
D(\rho) = D_0 ~\beta^\eta \left(\frac{\rho}{\rho_0}\right)^\delta \;,$$
where $\rho \equiv p/Z$ is the rigidity of the nucleus characterized by charge $Z$ and momentum $p$, and $\eta$ parametrerizes the low-rigidity behavior of $D$. While kinetic quasilinear theory predicts $\eta=1$ for the dependence of the diffusion coefficient on the particle speed, several effects may give rise to a different effective behavior (see e.g. [@Ptuskin06; @EvoliYan2014]). Bearing these considerations in mind, we leave $\eta$ as a free parameter to be fitted against low-energy CR data.
Diffusive reacceleration is parameterized in terms of the Alfvén velocity $v_A$ and we assume that this process takes place in the entire propagation region. In fact, this circumstance is expected if CRs are responsible for generating their own turbulence, e.g., by CR-induced streaming instability [@Cesarsky; @Blasi:2012yr]. Our approach is then different from semi-analytical methods where reacceleration is active only in the Galactic disk with height $\sim 100$ pc (see, e.g., [@Donato01]).
We also account for a convective wind with velocity $V_C(z)$ vanishing at $z = 0$ and growing linearly with $z$ with the gradient $dV_c/dz$. Although $dV_c/dz$ can be as large as $\sim 100$ km s$^{-1}$ kpc$^{-1}$ in the inner few kpcs of the Galactic disk [@Everett; @Breitschwerdt], the observables we are considering in the present analysis do not probe that region. On the other hand, the local convective velocity is severely constrained by $^{10}$Be/$^9$Be measurements [@Strong:2007nh] and, for this reason, we require $dV_c/dz$ to be less then $10$ km s$^{-1}$ kpc$^{-1}$. By making this choice we introduce another difference with respect to semi-analytical approaches where a constant convection velocity is adopted (see e.g. [@Donato01]).
For the CR astrophysical source distribution we consider the following form: $$Q_{i}(E_{k},r,z) = f_{\rm SNR}(r,z)\ q_{0,i}\ \left(\frac{\rho(E_{k})}{\rho_0}\right)^{- \gamma_{i}} \;,
\label{eq:source}$$
Our spatial source distribution ($f_{\rm SNR}$) follows the Galactic supernova remnants profile inferred from pulsars and stellar catalogues as given in [@Ferriere:2001rg]. We allow different source spectral indexes for the different nuclear species ($\gamma_{i}$), and we fix their values – together with the species relative abundances ($q_{0,i}$) – against recent experimental data.
Proton and helium spectra exhibit a change of slope at a rigidity between $\sim 200$ and $300~{\rm GV}$ [@Adriani:2011cu]: We model it by assuming a spectral break in the injection slopes of these nuclei. These features have relevant implications for the secondary antiproton flux. An alternative possibility for the origin of this break is a change in the ISM turbulence power spectrum and hence in the diffusion coefficient. In this context, high-energy CR antiprotons could be successfully used to discriminate among the two interpretations [@Evoli:2011id], but in the energy range where antiprotons are currently measured the two scenarios are quantitatively equivalent.
Secondary antiproton production in the ISM
------------------------------------------
The source term of secondary antiprotons produced from the interaction of CR nuclei with the ISM is given by the convolution of the antiproton production differential cross-section, $d\sigma/dE_k$, and the interstellar CR nuclei differential energy flux, $d\Phi/dE_k$.
The differential $\bar{p}$ production rate per volume and energy takes the form: $$Q_{\bar{p}} (E_k) = \sum\limits_{i={\rm H},{\rm He}} \sum\limits_{j={\rm H},{\rm He}} 4 \pi \int_{E_k^{\rm th}}^\infty d E'_k \left( \frac{d\sigma}{dE_k} \right)_{ij} n_i \phi_j (E'_k)$$
where $E'_k$ and $E_k$ are the kinetic energies per nucleon of the incoming nucleus (with threshold energy $E_{\rm th} = 6 m_p$) and the outgoing antiproton, respectively. $n_i$ denotes the interstellar gas density.
Existing parameterizations of the antiproton production cross-sections [@1982PhRvD..26.1179T] are mainly based on experimental data earlier than 1980. Recently, two works ([@Kappl:2014hha; @diMauro2014]) have improved these old computations making use of new precision data from the NA49 experiment at CERN [@NA49]. In particular, in [@Kappl:2014hha] the authors extended previous calculations by including the new laboratory measurements and a careful treatment of antiprotons arising from antineutron and hyperon decay. By using a revised approach to proton-helium scattering, the authors of [@Kappl:2014hha] are also able to compute the production cross-sections for the processes: $p + p \rightarrow \bar{p} X$, $p + He \rightarrow \bar{p} X$, $He + p \rightarrow \bar{p} X$ e $He + He \rightarrow \bar{p} X$, where the first particle is the impinging primary CR while the second one is the target interstellar material.
We make use of their results for the antiproton cross sections, as well as their estimate of the related errors, in order to evaluate the nuclear uncertainties on our prediction of the secondary antiproton fluxes in the Galaxy. Finally, we also checked that our conclusion are unchanged by adopting the parameterization for the $p+p$ scattering proposed by [@diMauro2014] and based on the full set of available measurements.
Antiprotons from dark matter annihilations {#sec:dm}
------------------------------------------
Other than spallation of CRs on the ISM nuclei, antiprotons can be produced in the Galaxy by DM in pair annihilation or decay events. Antiproton measurements provide an invaluable tool to constrain such primary contribution, since the ratio between DM signal and background from standard astrophysical sources is usually much larger in the antiproton channel with respect to all other indirect detection methods.
We compute the DM contribution to the antiproton flux as described in several previous works, e.g., [@Evoli:2011id; @Cirelli2014]. In brief, we assume that the source function ($Q_{\rm DM}$) for the WIMP DM component scales with the DM particle number density times the probability of annihilation and the antiproton yield per annihilation: $$Q_{\rm DM} (E_k,r,z) \,=\, \frac{1}{2} \, \frac{\rho^2_{\rm DM}(x)}{m^2_{\rm DM}} \, \langle \sigma v \rangle \, \frac{d N_{\bar{p}}}{d E_k} (E_k)$$ where $\langle \sigma v \rangle$ is the thermally averaged annihilation cross section and $\rho_{\rm DM}(x)$ is the DM density profile as function of the galactocentric distance $x = \sqrt{r^2+z^2}$.
The DM profile is only poorly constrained by direct observations and its shape is usually inferred from N-body simulations of gravitational clustering. In our analysis, we adopt two spherically symmetric profiles: a standard Navarro-Frenk-White (NFW) [@NFW1996], and a generalized NFW (gNFW) as defined, e.g., in [@Cirelli2014]. The free parameters for the NFW profile are chosen following the analysis in [@Catena2012], while for the gNFW profile we adopt a contracted profile ($\gamma = 1.2$) as suggested by the recent claim of a DM associated excess in the GC region [@Calore2014].
For the choice of the annihilation channels, we focus here on two sample cases which have been recently investigated in connection to hints of DM signals (both direct and indirect), but potentially giving a sizable antiproton flux as well [@Evoli:2011id]. In particular, we consider as primary annihilation channels: ${\rm DM \, \, DM} \rightarrow b \bar{b}$ and ${\rm DM \, \, DM} \rightarrow W^+W^-$, and we take the corresponding antiproton yields from the [PPPC4DMID]{} [@Cirelli2010; @Ciafaloni2010].
Propagation in the Heliosphere {#solarmod}
------------------------------
Low-energy ($\lesssim$ 10 GeV) charged CRs are influenced by the solar magnetic field in the final stage of their propagation process. The main effect of the interaction with the heliosphere is a momentum decrease and the consequent alteration of the interstellar spectrum.
This process is usually described in the context of the “force field approximation” [@Gleeson68]. According to this model, the energy spectrum $d\Phi_{\rm TOA}/dE_{k, \rm TOA}$ of particles reaching the top of the atmosphere (TOA) with kinetic energy $E_{k, \rm TOA}$ and momentum $p_{\rm TOA}$ is related to the local interstellar spectrum (LIS) $d\Phi_{\rm LIS}/{dE_{k, \rm LIS}}$ as it follows $$\label{eq:Fisk}
\frac{d\Phi_{\rm TOA}}{dE_{k, \rm TOA}} \,=\, \frac{p_{\rm TOA}^2}{p_{\rm LIS}^2} \, \frac{d\Phi_{\rm LIS}}{dE_{k, \rm LIS}},\qquad
E_{k, \rm LIS} = E_{k, \rm TOA} + |Ze| \phi,$$ where $\phi$ is the Fisk potential and parameterizes the kinetic energy losses. Within this approach the effects of modulation for protons and antiprotons are identical.
More realistic drift models, however, predict a clear charge sign dependence for the heliospheric modulation of positively and negatively charged particles. Moreover, the modulation effect depends on the polarity of the solar magnetic field (SMF), which changes periodically every $\sim 11$ years. The SMF is observed with opposite polarities in the northern and southern hemispheres. At the interface between opposite polarity regions a heliospheric current sheet (HCS) is formed. The phenomenologically parameter $\alpha$, known as the “tilt angle”, sizes the angular extension of the HCS oscillations. The magnitude of $\alpha$ depends on the solar activity and has a large influence on the intensity of the modulation.
A more accurate description then may be achieved by means of dedicated numerical simulations of the CR transport in realistic models for the interplanetary magnetic field.
In order to assess the impact of realistic model for solar modulation, we make use of the recently developed [HelioProp]{} code [@Maccione:2012cu]. This code solves the equation describing CR transport in the heliosphere by means of the stochastic differential equation method [@Strauss:2011; @Strauss:2012]. To characterize the model for solar propagation we have to specify the solar magnetic field geometry, the properties of diffusion, and those of winds and drifts.
For the interplanetary diffusion tensor we assume $H \propto {\rm diag}({\lambda}_{\|}, {\lambda}_{\perp, r}, {\lambda}_{\perp, \theta})$, where the parallel and perpendicular components are set with respect to the direction of the local magnetic field. The CR mean free path parallel to the magnetic field as function of the particle rigidity $\rho$ is parameterized as $\lambda_{\|} = \lambda_0 (\rho / 1 {\rm GeV})^\delta (B/B_{\rm sun})^{-1}$, where $B$ is the magnetic field and $B_{\rm sun}$ is its normalization value at the Earth position, for which we adopt $B_{\rm sun} = 5$ nT. For the perpendicular diffusion we assume ${\lambda}_{\perp, r} = {\lambda}_{\perp, \theta} = 0.02 \lambda_{\|}$, as results from numerical simulations [@Giacalone1999].
Finally, for modeling the magnetic field geometry, winds and drifts associated to the antisymmetric components of the diffusion tensor we follow [@Maccione:2012cu].
Method and results
==================
![The good models as discussed in section \[sec:selection\] in the $\delta - D_0$ (left) and $v_A - \eta$ (right) planes. Blue points correspond to models compatible with the PAMELA B/C ratio. Green points show the good models obtained using the PAMELA primary proton in addition.[]{data-label="fig:dvsdelta"}](plots/DvsDelta.pdf "fig:"){width="49.00000%"} ![The good models as discussed in section \[sec:selection\] in the $\delta - D_0$ (left) and $v_A - \eta$ (right) planes. Blue points correspond to models compatible with the PAMELA B/C ratio. Green points show the good models obtained using the PAMELA primary proton in addition.[]{data-label="fig:dvsdelta"}](plots/ETAvsVA.pdf "fig:"){width="49.00000%"}
![The envelope of the secondary antiproton spectra computed with the different propagation models found to reproduce the B/C and primary spectra.[]{data-label="fig:minmax"}](plots/apMinMaxPropOnly.pdf){width="50.00000%"}
![[*Left plot:*]{} The secondary antiproton flux computed for different values of the halo height $L$. [*Right plot:*]{} The normalization of the diffusion coefficient required, for each value of $L$, to reproduce the B/C ratio (the green line is used to guide the eye).[]{data-label="fig:vsL"}](plots/apVsL.pdf "fig:"){width="45.00000%"} ![[*Left plot:*]{} The secondary antiproton flux computed for different values of the halo height $L$. [*Right plot:*]{} The normalization of the diffusion coefficient required, for each value of $L$, to reproduce the B/C ratio (the green line is used to guide the eye).[]{data-label="fig:vsL"}](plots/DvsL.pdf "fig:"){width="45.00000%"}
Selection of CR propagation models {#sec:selection}
----------------------------------
We focus our attention on the free parameters of our propagation model (see section \[sec:prop\]) within the range given in table [\[tab:parameters\]]{}. We notice that the observables we are considering in our analysis are not sensitive to $L$, we then use a fixed value $L=2$ kpc and we evaluate the impact of different values for this parameter afterwards.
[|c|c|c|c|]{} Parameter & Min value & Max value & Units\
$\eta$ & $-1$ & $1$ &\
$D_0$ & $0.1$ & $10$ & $10^{28}$ cm$^2$/s\
$\delta$ & $0.2$ & $0.8$ &\
$dV_c/dz$ & $0$ & $10$ & km/s/kpc\
$V_A$ & $0$ & $100$ & km/s\
For each model selected randomly in the parameter space described above, we fix the injection slopes and the source abundances for primary nuclei heavier than carbon by fitting CREAM data above 10 GeV/n [@Cream2014]. For the carbon, helium and proton parameters, including the Fisk potential $\phi$, we fit the recent PAMELA Carbon [@Adriani:2014xoa] and proton [@Adriani:2011cu] measurements at energy below break at $\sim 200$ GV. We assume Boron is entirely secondary. It was shown by [@Tomassetti:2012] that, neglecting the production and acceleration of secondary nuclei inside SNRs, the $\delta$ may be underestimated by a factor of $\sim 5 - 15$% (see also [@Genolini:2015]). We checked that the Fisk potential gives an accurate description of modulated spectra compared against the more realistic predictions provided by the [Helioprop]{} simulations. Using a charge-dependent formalism for the modulation is relevant only when we compare differently charged particle spectra. We discuss this in detail in section \[sec:charged\].
With the given set of diffusion and source parameters we are now able to calculate the B/C ratio.
We identify a model as a *good* one, if it reproduces the $B/C$ data as well as proton and carbon data within the 3$\sigma$ limits.
In particular, we compute the $\chi^2$’s against $B/C$ ($\chi^2_{BC}$), proton ($\chi^2_p$) and carbon ($\chi^2_C$) measurements for each propagation model and we accept the model if each $\chi^2$ is smaller than the corresponding threshold ($\chi^2_{3\sigma}$) reported in table [\[tab:chi2\]]{}. The $\chi^2_{3\sigma}$ values reported in the table are calculated according to the $\chi^2$ distribution function with a number of degrees of freedom $\mathcal{F}= \mathcal{D} - \mathcal{P}$ with $\mathcal{D}$ equals to the number of data points and $\mathcal{P}$ to the number of parameters. The $3\sigma$ threshold implies a $99.73$% probability to get a smaller $\chi^2$ value or, equivalently, a p-value of $0.27$%.
A similar strategy (using only $\chi^2_{BC}$) has been adopted in [@Donato01] to select the models in order to determine the well-known MIN, MED and MAX setups.
[|c|c|c|c|]{} Observable & Data points $\mathcal{D}$ & \# of parameters $\mathcal{P}$ & $\chi^2_{3 \sigma}$\
B/C & 18 & 5 & 29.79\
Proton flux & 71 & 7 & 98.21\
Carbon flux & 16 & 7 & 21.58\
We repeat the procedure until $N=10^4$ good models are selected.
In figure \[fig:dvsdelta\] we show where the selected models are located in the $\delta$-$D_0$ and $v_A$-$\eta$ planes. We find that PAMELA data allow $\delta$ to vary between $0.2$ and $0.8$ and this parameter is strongly anti-correlated with $D_0$. [ This anti-correlation can be explained as follows: An increase of $\delta$ corresponds to a lower secondary-to-primary ratio at high energies, therefore $D_0$ must be smaller in order to enhance the high-energy secondary production.]{} The low-energy parameter $\eta$ is strongly degenerate with the solar modulation potential and, as a consequence, the available data do not constrain $\eta$ within the chosen range. By contrast, the Alfvén speed parameter ($v_A$) is found to be bounded by $\sim 30$ km$/$s if primary spectra are included in the analysis and no low-energy breaks in the injection slopes are admitted.
The propagation model giving the best fit of the PAMELA B/C and proton data is characterized by the following parameters: $D_0 = 1.6$, $\delta = 0.41$, $v_A = 8.5$, $dV_C/dz = 1.6$, $\gamma_C = 2.56$, $\gamma_H = 2.47$.
Extreme models determination
----------------------------
In order to evaluate the propagation uncertainty in the determination of the secondary antiproton flux, we show in figure \[fig:minmax\] the envelope of the secondary antiproton spectra computed with the propagation models selected beforehand. At lower energies the uncertainty band widens since more parameters are necessary to model CR propagation, while at larger energies the main uncertainty comes from the poor determination of the $\delta$ parameter. Our result is in agreement with [@Bringmann:2006im].
We notice that the propagation model giving the best minimum (maximum) flux of secondary antiprotons is not univocally determined over the entire energy range $0.1 - 100$ GeV. In fact, at the different energies the minimum (maximum) flux is associated with a different propagation setup.
To give an example of the different models selected by varying the energy at which we evaluate the extreme fluxes, we provide in table $3$ the model parameters associated with the minimum and maximum antiproton flux at $1$, $10$ and $100$ GeV.
[ |cccccccccc| ]{} E$_k$ & $\eta$ & D$_0$ & $\delta$ & v$_A$ & dV$_C/$dz & $\gamma_p$ & q$_C$ & $\gamma_C$ & $\phi$\
$[$GeV$]$ & & units & & $[$km/s$]$ & $[$km/s/kpc$]$ & & $[\times 10^3]$ & & $[$GV$]$\
\
1 & 0.30 & 3.32 & 0.30 & 32.2 & 0.04 & 2.58 & 2.74 & 2.53 & 0.77\
10 & 0.68 & 2.85 & 0.38 & 28.6 & 0.03 & 2.54 & 2.83 & 2.48 & 0.86\
100 & -0.16 & 1.17 & 0.75 & 9.31 & 6.78 & 2.38 & 3.40 & 2.17 & 0.57\
\
1 & 0.84 & 0.85 & 0.74 & 0.52 & 5.65 & 2.40 & 3.80 & 2.18 & 0.53\
10 & -0.92 & 0.83 & 0.68 & 7.71 & 4.05 & 2.44 & 4.03 & 2.22 & 0.54\
100 & 0.60 & 2.85 & 0.23 & 27.4 & 6.88 & 2.62 & 2.95 & 2.59 & 0.75\
\[tab:minmax\]
Some trends emerge from this comparison. The minimal models selected at higher energies feature a larger $\delta$, while high reacceleration reduces the antiproton flux at low energies. As expected the maximal models show the opposite behavior, indeed the hardest value allowed for $\delta$ gives the maximum contribution to the antiproton flux at higher energies.
It is important to remark here that our poor knowledge of the halo size does not affect these conclusions. To show this, we select the propagation model giving the B/C best fit and we test different values for $L$ up to $16$ kpc. In order not to lose the perfect agreement with the secondary over primary data, we increase the $D_0$ value accordingly (see the right plot in figure \[fig:vsL\]). As shown in figure \[fig:vsL\], different choices for $L$ in this range do not affect our predictions for the secondary antiproton flux.
Although in this paper we assumed a uniform value of $\delta$ in the whole Galaxy, it was recently shown that diffuse $\gamma$-ray data favor a scenario characterized by radially-dependent CR transport properties [@Gaggero:2014; @Gaggero:2015]. In order to investigate the possible impact of that scenario on our results, we computed the local secondary antiproton spectrum for the KRA$_\gamma$ model considered in those papers finding a negligible correction.
Antiproton production cross-section uncertainties
-------------------------------------------------
We compare here the propagation uncertainties derived in the previous sections with those associated with the antiproton production processes.
In figure \[fig:cs\], we show the [relative difference]{} between the minimum (maximum) secondary antiproton flux and that obtained using the best-fit propagation model. The corresponding region represents the uncertainty on the secondary flux associated with galactic propagation.
We compare this uncertainty band with the relative differences associated with production cross sections. To this end, we compute secondary antiprotons with the new prescriptions recently proposed by [@Kappl:2014hha] and we evaluate them against the traditional fitting relations given in [@1982PhRvD..26.1179T; @Moskalenko1998].
We find that nuclear uncertainties can be as large as $50$% even at $\sim 100$ GeV, and are much larger below few GeVs. However, with the available CR data, the propagation uncertainties dominate over the entire energy range as shown in figure \[fig:cs\].
Upcoming measurements (in particular, from AMS-02 [@Incagli:2010], CALET [@CALET], and ISS-CREAM [@Cream2014]) are expected to significantly improve our knowledge of propagation parameters and then to reduce the associated uncertainties. In that situation, antiproton production cross sections will prevent us to provide predictions for the astrophysical backgrounds as accurate as the forecasted sensitivities.
![Comparison between propagation and nuclear uncertainties. [*Yellow band:*]{} Error on the $\bar{p}$ flux due to the uncertainty in the propagation parameters. [*Blue lines:*]{} The [relative difference between the $\bar{p}$ flux computed using the fiducial cross section from [@Kappl:2014hha] and: (dot-dashed) the maximal model from [@Kappl:2014hha], (solid) the minimal model from [@Kappl:2014hha], (dashed) the parameterization from [@1982PhRvD..26.1179T; @Moskalenko1998].]{} []{data-label="fig:cs"}](plots/csComparison.pdf){width="50.00000%"}
The role of charge-dependent solar modulation {#sec:charged}
---------------------------------------------
![The envelope of the secondary antiproton spectra computed with the charge-dependent modulation ([*black lines*]{}) and compared with that one obtained with the force-field approximation ([*yellow band*]{}).[]{data-label="fig:minmax_CD"}](plots/apComparisonHelioprop.pdf){width="49.00000%"}
As pointed out in section \[solarmod\], charge-dependent solar modulation can be relevant when the TOA antiproton flux is evaluated. Therefore, we compare here our predictions of the extreme fluxes based on the force-field approximation with those obtained with a charge-dependent modulation model.
To modulate the antiproton flux in the charge-dependent scenario, we develop the following strategy:
- For each propagation model, we consider as free parameters: 1) solar magnetic field polarity; 2) $\alpha$ (HCS tilt angle); 3) $\lambda_0$ (normalization of the parallel mean free path); 4) $\delta$ (power-law slope of the heliosphere diffusion coefficient as function of rigidity). Solar polarity and $\alpha$ are fixed by the data-taking period, since they can be obtained by direct measurements [@TiltAngle], while we determine $\lambda_0$ and $\delta$ by fitting the predicted TOA proton flux against the low-energy PAMELA measurements.
- We use the same set of parameters obtained from protons to modulate the LIS antiproton flux.
In figure \[fig:minmax\_CD\] we show the extreme antiproton fluxes as obtained with our charge-dependent modulation model. We immediately notice that the more detailed treatment of solar modulation does not impact significantly on the determination of the minimum flux, while differences up to $\sim 20 - 30$% are shown for the maximum one (see also [@Cirelli2014] for a more detailed discussion on this issue).
To understand this result, we point out that the models giving the minimal antiproton TOA fluxes are the ones with the largest LIS proton spectrum and, therefore, they need a stronger modulation to reproduce the data. Stronger modulation is associated with a larger Fisk potential or a smaller heliospheric parallel mean free path. In this situation, diffusion dominates over charge-dependent drifts and the modulation of particles with different charges becomes equivalent.
Conservative limit on DM models from antiproton data
====================================================
![Uncertainties on the flux of primary antiprotons originating from DM annihilation due to the propagation parameters (yellow band) and to the halo size (red line). For the latter, the ratio between the two extreme cases $L = 16$ kpc and $L = 2$ kpc is considered[]{data-label="fig:dmPropOnly"}](plots/dmPropOnly.pdf){width="55.00000%"}
In this section we derive the most conservative – with respect to all the background uncertainties discussed before – constraints on the DM annihilation cross section for the DM WIMP scenarios introduced in section \[sec:dm\].
To propagate DM antiprotons, we choose $L = 2$ kpc since it is the minimum value compatible with synchrotron diffuse emission observations [@DiBernardo:2012zu; @Bringmann:2011py]. We note here that, while the actual value of $L$ is irrelevant for the secondary antiprotons (see Fig. \[fig:vsL\]), DM antiprotons can change significantly and, in fact, this parameter is the most important one to evaluate this contribution. In particular, thinner halos underproduce the DM $\bar{p}$ flux, and therefore $L=2$ kpc corresponds to the minimum flux expected from a given DM model (see [@Evoli:2011id] for a more detailed discussion) and, for that reason, to the less stringent bound.
In order to quantify these statements, in Fig. \[fig:dmPropOnly\] we show the ratio between the antiproton flux corresponding to $L=16$ kpc and $L=2$ kpc, compared to the uncertainty band due to the poor knowledge of the propagation parameters. The plot clearly shows how, as far as primary antiprotons from DM annihilations are concerned, the uncertainty on the halo size is strongly dominant.
We also remark that we can safely neglect the charge-dependent effects in the determination of the minimum background (see section \[sec:charged\]).
![Antiproton bounds on DM annihilation rate. [*Red lines:*]{} $b \bar{b}$ channel for NFW profile for different assumption for the secondary $\bar{p}$ production. [*Blue lines:*]{} the same for the $WW$ channel. The results obtained with a gNFW profile with $\gamma = 1.2$ are indistinguishable from the NFW ones.[]{data-label="fig:bound"}](plots/boundnew.pdf){width="55.00000%"}
In order to get the most conservative bound with respect to the propagation setup, a naive strategy would be the following: Starting from a minimal background, the DM component is added until a best fit to the data is reached, and then the DM cross section is increased until the quality of the fit is worsened up to the $2\sigma$ level. However, although the models providing the minimal background are compatible with the B/C within $3\sigma$, they do not necessarily provide a satisfactory fit of the antiproton flux. The addition of a DM component could still leave unexplained the antiproton flux at energies above the DM mass.
Hence, we determine the 2$\sigma$ exclusion contour in the plane ($m_{\rm DM}$, $\langle \sigma v \rangle$) as follows.
For each good propagation model (based on B/C and nuclear data), we first compute the secondary background and the DM flux for a given DM mass, channel and profile. We then find the best-fit value for the annihilation cross section.
If the combination of background and DM (computed assuming the best-fit cross section) satisfies the following condition:
$$\chi^2 \, (\sigma_{\rm best fit} v) \, \leq \chi^2_{0.05} \, (\mathcal{F} = 23 - 1 = 22) \simeq 33.92$$
(where the number of degrees of freedom $\mathcal{F}$ is computed taking into account $23$ data points and $1$ parameter), then we retain the model. In other words, we reject propagation models with a p-value lower than $5$%.
If the model passes this test, we compute the value of $\langle \sigma v \rangle$ above which the fit worsens beyond the $2\sigma$ level with respect to the best fit. More precisely, we set the following threshold, corresponding again to a p-value equal to $5$%:
$$\chi^2 \, (\sigma_{\rm max} v) - \chi^2 \, (\sigma_{\rm best fit} v) \, \leq \chi^2_{0.05} \, (\mathcal{F} = 1) \simeq 3.84 \,$$
where now the only degree of freedom is the annihilation cross section.
Finally, a scan over the models is performed: The limit value shown in Fig. $7$ is therefore obtained by taking the maximum cross section (with respect to the different propagation models) satisfying the bound described above for given DM mass.
We remark that this strategy allows to obtain the bound by propagating secondary and DM antiprotons consistently with the same propagation model.
In the same plot, we compare the limits we obtain when the background is computed with the fiducial model for the $\bar{p}$ production cross-section and with their minimal and maximal values. Because of the strategy adopted to derive them, the case in which the most conservative limits are obtained with the minimal production cross sections is not always prevalent: Indeed, it may happen that larger production cross sections select a different subset of propagation models to reproduce the $\bar{p}$ measurements in combination with DM. Some of the propagation models allowed with fiducial or maximal cross-section, but not with the minimal case, may produce a lower DM $\bar{p}$ flux leading to a larger $2\sigma$ annihilation cross section.
In figure \[fig:bound\] the reader can see our results for the maximum allowed annihilation cross section for the $b \bar{b}$ and $W^+W^-$ annihilation channels. The maximum allowed cross section we find for $b \bar{b}$ is around one order of magnitude larger than what the authors of [@Cirelli2014] found for the charge-symmetric modulation case. The main difference in our approach is to make use of a broader analysis of the propagation and nuclear uncertainties in order to determine the background.
Our results can be now compared with the DM interpretation of the recently claimed signal in the gamma-ray channel located in the inner few degrees around the GC [@Hooperon2014].
In [@Hooperon2014] the authors show that a DM particle with mass $\sim 43$ GeV annihilating into $b \bar{b}$ with a cross section $\langle\sigma v\rangle \, \simeq \, 2.2 \cdot 10^{-26} \, {\rm cm^3 s^{-1}}$ (for the Inner Galaxy analysis) and distributed according to a gNFW profile with $\gamma = 1.2$ can accomodate the anomalous excess.
The detailed analysis reported in [@Calore2014] provided a better quantification of the systematic uncertainties affecting the proposed signal; more recently, the Fermi-LAT collaboration released preliminary estimates of the energy spectrum of this excess, based on four qualitatively different background models [@Fermi_GC_analysis]. A wide set of DM masses and annihilation channels are compatible with these new analyses (see, e.g., [@Agrawal2014]). It has been also shown that, in the context of the Minimal Supersymmetric Model framework, these candidates are not in tension with LHC or direct detection constraints. In figure \[fig:bound\] we compare our findings with the favored regions of annihilation cross sections connected to the GC excess as reported in [@Calore2014].
The bottom line of this analysis can be summarized as it follows: Although some fiducial models of CR propagation would produce strong tension with the DM interpretation of the GC excess (see, e.g., [@Bringmann2014; @Cirelli2014]), given the large uncertainties on the propagation parameters (for the secondary $\bar{p}$) and on the halo height (for the DM $\bar{p}$), the antiproton channel cannot be invoked to conclusively exclude this hypothesis.
Discussion on AMS-02 data
=========================
![Our reference model is compared to AMS-02 proton [@Aguilar:2015] ([*left plot*]{}) and helium ([*right plot*]{}) data. With the dotted and dashed lines we show the minimal and maximal breaks compatible with the hardening measured by AMS-02.[]{data-label="fig:p_ams"}](plots/protonsAMS.pdf "fig:"){width="49.00000%"} ![Our reference model is compared to AMS-02 proton [@Aguilar:2015] ([*left plot*]{}) and helium ([*right plot*]{}) data. With the dotted and dashed lines we show the minimal and maximal breaks compatible with the hardening measured by AMS-02.[]{data-label="fig:p_ams"}](plots/heAMS.pdf "fig:"){width="49.00000%"}
In this section we focus on the recently released AMS-02 data, including protons [@Aguilar:2015], and preliminary helium, B/C and $\bar{p}/p$ ratio [@amstalk], with energy range extending to 450 GeV.
In particular, we take a closer look at the new impressively accurate data on the $\bar{p}/p$ ratio and we attempt to evaluate their compatibility with the other hadronic observables. Given the preliminary nature of the released data we do not attempt a statistical analysis of the uncertainties associated with propagation. In this perspective, the final release of the secondary/primary measurements, when systematic and statistical errors are fully accounted for, will be crucial.
![Our reference model compared to AMS-02 preliminary B/C data. [*Solid line:*]{} the TOA spectrum modulated with $\phi = 0.6$ GV; [*dotted line:*]{} the LIS spectrum.[]{data-label="fig:bc_ams"}](plots/bcAMS.pdf){width="55.00000%"}
A propagation model chosen among those considered in section \[sec:selection\], and compatible with preliminary B/C measurements, is shown in figure \[fig:bc\_ams\]. The propagation parameters are: $D_0 = 1.5$, $\delta = 0.42$, $v_A = 27$, $dV_C/dz = 14$, $\gamma_C = 2.46$, $\gamma_H = 2.44/2.31$. For comparison, the same value for $\delta$ was found by [@Genolini:2015] using the same datasets.
Remarkably, the predicted B/C ratio reproduces the AMS-02 data over more than three orders of magnitude in energy. It is worth noting here that the $\delta$ required by the new high-energy measurements is in perfect agreement with the best-fit value obtained in our earlier statistical analysis [@DiBernardo10], based on the available high-energy measurements preceding PAMELA and AMS-02 releases.
We also tune the proton and helium injection slopes to accomodate the AMS-02 data. For the protons, we also consider the minimal and maximal injection slopes at high energy compatible with the data. The reader can see the comparison with the new datasets in figure \[fig:p\_ams\].
Armed with a model fully consistent with all the preliminary nuclear observables, we can finally compare our prediction for the $\bar{p}/p$ ratio with the data.
In figure \[fig:ap\_ams\] we show this comparison. The computation of the secondary flux is performed using the fiducial value of the cross sections provided by [@Kappl:2014hha], and the associated uncertainty is shown as a blue band.
We conclude that, even without considering all the relevant uncertainties associated with propagation or injection slopes, our predictions for the $\bar{p}/p$ are in good agreement with the preliminary data in the entire energy range. Our findings are then in agreement with the conclusions of [@Giesen:2015], although our analysis relies on the B/C data from the same experiment for the assessment of the propagation model.
![Our reference model compared to AMS-02 preliminary $\bar{p}/p$ data. Blue solid line: the $\bar{p}/p$ spectrum computed with the fiducial cross sections from [@Kappl:2014hha], with the optimal hardening in the proton and helium injection spectra. Dotted and dashed lines: the $\bar{p}/p$ spectrum computed with the minimal and maximal hardening in the proton spectrum as in Fig. \[fig:p\_ams\]. The blue band reports the uncertainty associated to the production cross sections.[]{data-label="fig:ap_ams"}](plots/apAMS.pdf){width="55.00000%"}
Conclusions
===========
We presented a revisited study of the dominant uncertainties in the determination of the CR secondary antiproton spectrum.
By performing a scan over the parameter space relevant for CR propagation, we identified a set of models compatible with B/C, proton, helium and carbon data provided by the PAMELA experiment. We were then able to bracket the minimum and maximum secondary antiproton fluxes constrained by local observables and we compared the associated uncertainty band with the errors related to the production cross sections. It is the first time that such analysis has been performed by using comprehensive numerical simulations of CR propagation in the Galaxy and the Heliosphere. More importantly, we used for the first time a complete set of measurements from the same experiment: Using consistent data from the same data-taking period allowed us to reduce the uncertainties due to solar modulation.
[Similarly to previous results, we found that the determination of the (almost unknown) diffusion halo height is irrelevant for the computation of the secondary antiproton flux since this parameter is degenerate with the diffusion coefficient normalization $D_0$.]{} In addition, we found that using the recent PAMELA data, the uncertainty on the propagation model dominates over the nuclear ones.
Our result has important implications for the indirect search of primary $\bar{p}$ from DM annihilations in the galactic halo. Therefore, we provided the most conservative – with respect to the mentioned effects – constraints on the annihilation rate for some popular DM models recently investigated in connection to hints of DM signals in other detection channels.
Our method may be taken as a reference procedure to be exploited when the final measurements for all the relevant channels are published by the AMS-02 collaboration.
At the moment, the preliminary release by the AMS-02 collaboration of nuclear data does not permit to perform a statistical analysis. Nevertheless, we found that the model in agreement with AMS-02 proton, helium, and B/C data is compatible with the $\bar{p}/p$ spectrum. Therefore, we do not report any significant anomaly in this observable. Our result is then consistent with the conclusions presented in [@Giesen:2015].
Acknowledgements
================
We thank Torsten Bringmann, Marco Cirelli, Nicola Mori, Pasquale D. Serpico, Piero Ullio, Alfredo Urbano and Christoph Weniger for useful discussions. We are indebted to Mattia Di Mauro and Martin Winkler for providing us their cross section models and to Francesca Calore for the confidence regions related to the GC analysis. Carmelo Evoli acknowledges support from the “Helmholtz Alliance for Astroparticle Physics HAP” funded by the Initiative and Networking Fund of the Helmholtz Association. Daniele Gaggero acknowledges the SFB 676 research fellowship from the University of Hamburg as well as the hospitality of DESY.
[^1]: <http://galprop.stanford.edu>
|
---
abstract: 'In this paper, we establish a boundary observability estimate for stochastic Schrödinger equations by means of the global Carleman estimate. Our Carleman estimate is based on a new fundamental identity for a stochastic Schrödinger-like operator. Applications to the state observation problem for semilinear stochastic Schrödinger equations and the unique continuation problem for stochastic Schrödinger equations are also addressed.'
author:
- 'Qi Lü[^1]'
title: 'OBSERVABILITY ESTIMATE FOR STOCHASTIC SCHRÖDINGER EQUATIONS AND ITS APPLICATIONS[^2]'
---
stochastic Schrödinger equation, global Carleman estimate, observability estimate, state observation problem, unique continuation property
93B07, 35B45
Introduction and Main Results
=============================
Let $T > 0$, $G \subset \mathbb{R}^{n}$ ($n
\in \mathbb{N}$) be a given bounded domain with a $C^{2}$ boundary $\G$. Let $\G_0$ be a suitable chosen nonempty subset (to be given later) of $\G$. Put $Q \= (0,T) \t G$, $\Si \=
(0,T) \t \G$, and $\Si_0 \= (0,T) \t \G_0$.
Let $(\O, {\cal F}, \{{\cal F}_t\}_{t \geq 0},
P)$ be a complete filtered probability space on which a one dimensional standard Brownian motion $\{ B(t) \}_{t\geq 0}$ is defined. Let $H$ be a Banach space. Denote by $L^{2}_{\cal
F}(0,T;H)$ the Banach space consisting of all $H$-valued $\{ {\cal F}_t \}_{t\geq 0}$-adapted processes $X(\cdot)$ such that $\mathbb{E}(|X(\cdot)|^2_{L^2(0,T;H)}) <
\infty$; by $L^{\infty}_{\cal F}(0,T;H)$ the Banach space consisting of all $H$-valued $\{
{\cal F}_t \}_{t\geq 0}$-adapted bounded processes; and by $L^{2}_{\cal
F}(\O;C([0,T];H))$ the Banach space consisting of all $H$-valued $\{ {\cal F}_t \}_{t\geq
0}$-adapted processes $X(\cdot)$ such that $\mathbb{E}(|X(\cdot)|^2_{C([0,T];H)}) <
\infty$. All of these spaces are endowed with the canonical norm. Put $$H_{T} \= L_{\cal F}^2 (\O; C([0,T];H_{0}^1(G))).$$
Let us consider the following stochastic Schrödinger equation: $$\begin{aligned}
\label{system1}
\left\{
\begin{array}{lll}
\ds idy + \D ydt = (a_1 \cdot \nabla y + a_2 y + f)dt + (a_3 y + g)dB(t) &\mbox { in } Q, \\
\ns\ds y = 0 &\mbox{ on } \Si, \\
y(0) = y_0 &\mbox{ in } G,
\end{array}
\right.\end{aligned}$$ with initial datum $y_0 \in
L^2(\O,\cF_0,P;H_0^1(G))$, suitable coefficients $a_i$ ($i=1,2,3$), and source terms $f$ and $g$. The solution to is understood in the following sense.
\[def1\] We call $y\in H_T$ a solution to the equation if\
1. $y(0) = y_0$ in $G$, P-a.s.;\
2. For any $t \in [0,T]$ and $\eta \in
H_0^1(G)$, it holds that $$\begin{aligned}
\nonumber
&\,& \q\int_{G} iy(t,x)\eta(x)dx - \int_{G}
iy(0,x)\eta(x)dx\nonumber\\
&\,& = \int_0^t \int_G \[ \nabla
y(s,x)\cdot\nabla\eta(x) + \big(a_1
\cdot \nabla y + a_2 y + f\big)\eta(x) \]dxds \nonumber \\
&\,&\q + \int_0^t \int_G (a_3 y + g)\eta(x)
dxdB(s), \,\, \mbox { P-a.s. } \nonumber\end{aligned}$$
We refer to [@Prato Chapter 6] for the well-posedness of the equation in $H_T$, under suitable assumptions (the assumptions in this paper are enough).
Similar to its deterministic counterpart, the stochastic Schrödinger equation plays an important role in quantum mechanics. We refer the readers to [@Bar; @Kol] and the rich references therein for the details of its physical background.
The main purpose of this paper is to establish a boundary observability estimate for the equation in the following setting.
Denote by $\nu(x)$ the unit outward normal vector of $G$ at $x\in \G$. Let $x_0\in\big(\mathbb{R}^n\setminus \overline
G\big)$. In what follows, we choose $$\label{G0}
\G_0=\big\{ x\in \G :\, (x-x_0)\cdot \nu(x)>0
\big\}.$$ We assume that $$\begin{aligned}
\label{coai}
\left\{\begin{array} {ll} \ds i a_1 \in L_{
\mathcal{F}}^{\infty}(0,T;W_0^{1,\infty}(G;\mathbb{R}^{n})),
\\
\ns\ds a_2
\in L_{ \mathcal{F}}^{\infty}(0,T;W^{1,\infty}(G)), \\
\ns \ds a_3 \in L_{\mathcal{
F}}^{\infty}(0,T;W^{1,\infty}(G)),
\end{array}
\right.\end{aligned}$$ and that $$\begin{aligned}
\label{fg}
\left\{
\begin{array}{ll}\ds
f \in L^2_{\mathcal{F}}(0,T;H_0^1(G)), \\
\ns\ds g \in L^2_{\mathcal{F}}(0,T;H^1(G)).
\end{array}
\right.\end{aligned}$$
In the sequel, we put $$\label{cA}
r_1\=|a_1|^2_{L_{
\mathcal{F}}^{\infty}(0,T;W_0^{1,\infty}(G;\mathbb{R}^{n}))}
+ |a_2|^2_{L_{
\mathcal{F}}^{\infty}(0,T;W^{1,\infty}(G))} +
|a_3|^2_{L_{
\mathcal{F}}^{\infty}(0,T;W^{1,\infty}(G))} +
1,$$ and denote by $C$ a generic positive constant depending only on $T$, $G$ and $x_0$, which may change from line to line.
Now we state the main result of this paper as follows.
\[observability\] If the conditions – hold, then any solution of the equation satisfies that $$\label{obser esti2}
\begin{array}{ll}\ds
\q |y_0|_{L^2(\Omega,{ \mathcal{F}}_0, P; H_0^1(G))} \\
\ns\ds \leq e^{C r_1}\Big(\Big|\frac{\partial
y}{\partial \nu}\Big |_{L^2_{ \mathcal{
F}}(0,T;L^2(\Gamma_0))} + |f|_{L^2_{ \mathcal{
F}}(0,T;H_0^1(G))} + |g|_{L^2_{ \mathcal{
F}}(0,T;H^1(G))}\Big).
\end{array}$$
Since $y$ belongs only to $H_T$, its normal derivative $\frac{\pa y}{\pa\nu}$ may not make sense. Fortunately, due to the hidden regularity of the solution to the equation , one can show that $\frac{\pa
y}{\pa\nu}$ exists and belongs to $L^2_{\cF}(0,T;L^2(\G))$(see Proposition \[hregularity\] for more details).
It is well-known that observability estimates (in the spirit of ) for partial differential equations play fundamental role in proving the controllability of the dual control systems. There exist many approaches and results addressing the observability estimate for determinisitc Schrödinger equations. For example, similar results in the spirit of Theorem \[observability\] are obtained by Carleman estimate (e.g. [@Baudouin-Puel; @Lasiecka-Triggiani-Zhang; @Mercado-Osses-Rosier]), by the classical Rellich-type multiplier approach ([@Machtyngier]), by the microlocal analysis approach ([@Lebeau; @Phung]), and so on. We refer to [@Zuazua] for a nice survey in this respect. However, people know very little about the stochastic counterpart. To our best knowledge, [@Luqi4] is the only published result for this problem, where partial results in this paper have been announced without detailed proofs.
Besides its important application to the controllability problem, the observability estimate not only have its own interest (a kind of energy estimate and quantitative uniqueness for the solution) but also has some other important applications. For instance, a typical application of this sort of estimates is to study the state observation problem, that is, to determine the state of a system by a suitable observation. Once the observability is obtained, we may conclude that the state can be uniquely determined from the observed data and continuously depends on it. For instance, once the inequality is established, it follows that $y\in H_T$ is determined by $\ds\frac{\pa y}{\pa
\nu}\Big|_{(0,T)\times \G_0}$ continuously. In Section \[Sec app\], we shall consider a state observation problem for semilinear stochastic Schrödinger equations.
In this paper, we will prove Theorem \[observability\] by applying the global Carleman estimate (See Theorem \[thcarleman est\] below).
We now introduce the weight functions to be used in our Carleman estimate. Let $$\label{psi}
\psi(x) = |x-x_0|^2 + \tau,$$ where $\tau$ is a positive constant such that $\psi \geq \frac{5}{6}|\psi|_{L^{\infty}(G)}$. Let $s>0$ and $\l>0$. Put $$\label{lvarphi}
\ell = s\frac{e^{4\l \psi} - e^{5\l
|\psi|_{L^{\infty}(G)}}}{t^2(T-t)^2}, \qq
\varphi = \frac{e^{4\l \psi} }{t^2(T-t)^2},\qq
\theta=e^\ell.$$
We have the following global Carleman inequality.
\[thcarleman est\] According to – and , there is an $s_1>0$ (depending on $r_1$) and a $\l_1>0$ such that for each $s\geq s_1$, $\l\geq \l_1$ and for any solution of the equation , it holds that $$\begin{aligned}
\label{carleman est}
\begin{array}{ll}
\ds \q\mathbb{E}\int_Q
\theta^2\Big(s^3\l^4\varphi^3 |y|^2 +
s\l\varphi
|\nabla y|^2\Big) dxdt \\
\ns \ds \leq C \Big\{\mathbb{E}\int_Q \theta^2
\Big(|f|^2 +
s^2\l^2\varphi^2 |g|^2 + |\nabla g|^2 \Big)dxdt +
\mathbb{E}\int_0^T\int_{\G_0}\theta^2
s\l\varphi\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G
dt \Big\}.
\end{array}\end{aligned}$$ Further, if $g\in
L^2_\cF(0,T;H^1(G;\mathbb{R}))$, then can be strengthened as the following: $$\begin{aligned}
\label{carleman est1}
\begin{array}
{ll} \ds \q\mathbb{E}\int_Q
\theta^2\Big(s^3\l^4\varphi^3 |y|^2 +
s\l\varphi
|\nabla y|^2\Big) dxdt \\
\ns \ds\leq C \Big\{\mathbb{E}\int_Q
\theta^2 \Big(|f|^2 +
s^2\l^2\varphi^2 |g|^2 \Big)dxdt +
\mathbb{E}\int_0^T\int_{\G_0}\theta^2
s\l\varphi\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G
dt \Big\}.
\end{array}\end{aligned}$$
Carleman estimate is an important tool for the study of unique continuation property, stabilization, controllability and inverse problems for deterministic partial differential equations (e.g. [@Baudouin-Puel; @Lasiecka-Triggiani-Zhang; @Mercado-Osses-Rosier; @Yamamoto; @Zhangxu1; @Zuazua]). Although there are numerous results for the Carleman estimate for deterministic partial differential equations, people know very little about the corresponding stochastic situation. In fact, as far as we know, [@barbu1; @Luqi4; @Luqi5; @Tang-Zhang1; @Zhangxu3] are the only five published papers addressing the Carleman estimate for stochastic partial differential equations. The references [@barbu1; @Luqi5; @Tang-Zhang1] are devoted to stochastic heat equations, while [@Zhangxu3] is concerned with stochastic wave equations. In [@Luqi4], Theorem \[thcarleman est\] was announced without proof.
At first glance, the proof of Theorem \[thcarleman est\] looks very similar to that of the global Carleman estimate for (stochastic) parabolic equations (See [@Fursikov-Imanuvilov1; @Tang-Zhang1]). Furthermore, one can find that the idea behind the proofs in this paper and [@Fursikov-Imanuvilov1; @Tang-Zhang1] are analogous. Nevertheless, the specific proofs have big differences. First, we have to choose different weight functions. Second, we deal with different equations. Such kind of differences lead to considerably different difficulties in the proof of Theorem \[thcarleman est\]. One cannot simply mimic the proofs in [@Fursikov-Imanuvilov1; @Tang-Zhang1] to obtain Theorem \[thcarleman est\]. Indeed, even in the deterministic setting, the proof of the global Carleman estimate for Schrödinger equations are much more complicated than that for the parabolic and hyperbolic equations (see [@Zhangxu5; @Lasiecka-Triggiani-Zhang]).
The rest of this paper is organized as follows. In Section 2, we give some preliminary results, including an energy estimate and the hidden regularity for solutions of the equation . Section 3 is addressed to establish a crucial identity for a stochastic Schrödinger-like operator. Then, in Section 4, we derive the desired Carleman estimate. Section 5 is devoted to prove Theorem \[observability\]. In Section \[Sec app\], as applications of the observability/Carleman estimates developed in this work, we study a state observation problem for semilinear stochastic Schrödinger equations and establish a unique continuation property for the solution to the equation . Finally, we present some further comments and open problems concerned with this paper in Section 7.
Some preliminaries
==================
In this section, we give some preliminary results which will be used later.
To begin with, for the sake of completeness, we give an energy estimate for the equation .
\[Oprop1\] According to –, for all $y$ which solve the equation , it holds that $$\label{energyesti1}
\mathbb{E}|y(t)|^2_{H_0^1(G)} \leq e^{C r_1}
\Big(\mathbb{E}|y(s)|^2_{H^1_0(G)} +
|f|^2_{L^2_{\mathcal{F}}(0,T;H^1_0(G))} +
|g|^2_{L^2_{\mathcal{F}}(0,T;H^1_0(G))}\Big),$$ for any $ s, t\in [0, T]$.
[*Proof* ]{}: Without loss of generality, we assume that $t <s$. To begin with, we compute $
\mathbb{E}| y(t)|^2_{ L^2(G)} - \mathbb{E}|
y(s)|^2_{ L^2(G)}$ and $ \mathbb{E}|\nabla
y(t)|^2_{ L^2(G)} - \mathbb{E}|\nabla y(s)|^2_{
L^2(G)}$. The first one reads $$\label{Eyt}
\begin{array}{ll} \ds
\mathbb{E}| y(t)|^2_{ L^2(G)} - \mathbb{E}| y(s)|^2_{
L^2(G)}\\
\ns \ds = -\mathbb{E}\int_t^s\int_G \big(y
d\bar{y}+\bar{y}dy + dy
d\bar{y}\big)dx\\
\ns\ds = \mathbb{E}\int_t^s\int_G \Big\{ i
y\big(\D \bar{y} - a_1\cdot \nabla\bar{y} - a_2
\bar{y} -\bar{f}\big) - i\bar{y}\big(\D y -
a_1\cdot\nabla y - a_2 y - f\big) \\
\ns \ds\q - \big(a_3 y + g\big)\big(a_3 \bar{y} + \bar{g}\big) \Big\}dxd\si \\
\ns\ds = \mathbb{E}\int_t^s\int_G \Big\{ i
\big[\div(y\nabla\bar{y})-|\nabla y|^2 - \div(|
y|^2 a_1) +
\div(a_1)|y|^2 - a_2|y|^2 - y\bar{f}\, \big] \\
\ns \ds \q - i \big[\div(\bar{y}\nabla
y)-|\nabla y|^2 - \div(| y|^2 a_1) +
\div(a_1)|y|^2 - a_2|y|^2 - f\bar{y} \big] \\
\ns \ds\q - (a_3 y + g)(a_3 \bar{y} + \bar{g}) \Big\}dxd\si\\
\ns\ds \leq \mathbb{E}\int_t^s
2\Big[\big(|a_3|_{L^{\infty}(G)}+1\big)|y|^2_{L^2(G)}+
|f|_{L^2(G)}^2 + |g|^2_{L^2(G)}\Big]dxd\si.
\end{array}$$ The second one is $$\label{Etyt}
\begin{array}{ll}
\q\mathbb{E}|\nabla y(t)|^2_{ L^2(G)} -
\mathbb{E}|\nabla
y(s)|^2_{L^2(G)}\\
\ns\ds = -\mathbb{E}\int_t^s\int_G \big(\nabla
y d\n\bar{y} + \nabla
\bar{y} d\n y + d\nabla y d\nabla \bar{y}\big)dx\\
\ns \ds = -\mathbb{E}\int_t^s\int_G \Big\{
\div(\nabla y d\bar{y}) - \D y d\bar{y} +
\div(\nabla\bar{y} dy) - \D \bar{y}dy +
d\nabla y d\nabla \bar{y} \Big\}dx\\
\ns \ds = -\mathbb{E}\int_t^s\int_G \Big\{\D
y \Big[i\big(\D\bar{y} - a_1\cdot\nabla \bar{y}
-a_2\bar{y} -f \big)\Big]-\D\bar{y}\Big[
i\big(\D y -
a_1\cdot\nabla y - a_2 y - f\big)\Big]\\
\ns \ds \q +\nabla(a_3 y +
g)\nabla(a_3\bar{y}+\bar{g})
\Big\}dxd\si \\
\ns \ds \leq 2\mathbb{E}\int_t^s
\Big\{\big(|a_1|^2_{W^{1,\infty}(G;\mathbb{R}^m)}+|a_3|^2_{W^{1,\infty}(G)}+1\big)|\nabla
y|^2_{L^2(G)}\\
\ns\ds \q +
\big(|a_2|^2_{W^{1,\infty}(G)}+|a_3|^2_{W^{1,\infty}(G)}+1\big)|y|^2_{L^2(G)}
+|f|^2_{H_0^1(G)} +
|g|^2_{H^1_0(G)}\Big\}dxd\si.
\end{array}$$ From and , we have that $$\label{energyest2}
\begin{array}{ll}\ds
\q\mathbb{E}| y(t)|^2_{H_0^1(G)} -
\mathbb{E}|y(s)|^2_{H_0^1(G)} \\
\ns\ds \leq 2(r_1+1)\mathbb{E}\int_t^s\int_G
|y(\si)|^2_{H_0^1(G)}dxd\si +
\mathbb{E}\int_t^s\int_G
\big(|f(\si)|^2_{H_0^1(G)}+|g(\si)|^2_{H^1_0(G)}\big)dxd\si.
\end{array}$$ From this, and thanks to Gronwall’s inequality, we arrive at $$\label{energyest3}
\mathbb{E}| y(t)|^2_{H_0^1(G)}\leq
e^{2(r_1+1)}\Big\{\mathbb{E}|y(s)|^2_{H_0^1(G)}
+ \mathbb{E}\int_0^T\int_G
\big(|f|^2_{H_0^1(G)}+|g|^2_{H^1_0(G)}\big)dxd\si\Big\},$$ which implies the inequality immediately.
The proof of this proposition is almost standard. However, people may doubt the correctness of the inequality for $t<s$ because of the very fact that the equation is time irreversible. Fortunately, the inequality is true for $t<s$. In fact, in the stochastic setting one should divide the time irreversible systems into two classes. The first class of time irreversibility is caused by the energy dissipation. Thus, one cannot estimate the energy of the system at time $t$ by that at time $s$ uniformly when $t<s$. A typical example of such kind of systems is the heat equation. The second class of time irreversibility comes from the stochastic noise. Such kind of system cannot be solved backward, that is, if we give the final data rather than the initial data, then the system is not well-posed (Recall that, this is the very starting point of backward stochastic differential equations). Stochastic Schrödinger equations and stochastic wave equations are typical systems of the second class. For these systems, we can still estimate the energy at time $t$ by that at time $s$ for $t<s$.
Next, we give a result concerning the hidden regularity for solutions of the equation . It shows that, solutions of this equation enjoy a higher regularity on the boundary than the one provided by the classical trace theorem for Sobolev spaces.
\[hregularity\] According to –, for any solution of the equation , it holds that $$\label{hregularity1}
\begin{array}{ll}\ds
\q\Big|\frac{\partial y}{\partial \nu}\Big
|^2_{L^2_{ \mathcal{ F}}(0,T;L^2(\Gamma_0))}\\
\ns\ds \leq e^{C r_1 }
\Big(|y_0|^2_{L^2(\Omega,{ \mathcal{F}}_0, P;
H_0^1(G))} +|f|^2_{L^2_{ \mathcal{
F}}(0,T;H_0^1(G))} + |g|^2_{L^2_{ \mathcal{
F}}(0,T;H^1(G))}\Big).
\end{array}$$
\[rm2\] By means of Proposition \[hregularity\], we know that $\ds\Big|\frac{\partial y}{\partial
\nu}\Big |^2_{L^2_{ \mathcal{
F}}(0,T;L^2(\Gamma_0))}$ makes sense. Compared with Theorem \[observability\], Proposition \[hregularity\] tells us the fact that $\ds\Big|\frac{\partial y}{\partial \nu}\Big
|^2_{L^2_{ \mathcal{ F}}(0,T;L^2(\Gamma_0))}$ can be bounded by the initial datum and non-homogenous terms. This result is the converse of Theorem \[observability\] in some sense.
To prove Proposition \[hregularity\], we first establish a pointwise identity. For simplicity, here and in the sequel, we adopt the notation $\ds y_i \equiv y_{i}(x) \=
\frac{\partial y(x)}{\partial x_i}$, where $x_i$ is the $i$-th coordinate of a generic point $x=(x_1,\cdots, x_n)$ in $\mathbb{R}^{n}$. In a similar manner, we use the notation $z_i$, $v_i$, etc., for the partial derivatives of $z$ and $v$ with respect to $x_i$.
\[prop2\] Let $\mu = \mu(x) =
(\mu^1,\cdots,\mu^n):\mathbb{R}^n \to
\mathbb{R}^n$ be a vector field of class $C^1$ and $z$ an $H^2_{loc}(\mathbb{R}^n)$-valued $\{\mathcal{F}_t\}_{t\geq 0}$-adapted process. Then for a.e. $x \in \mathbb{R}^n$ and P-a.s. $\omega \in \Omega$, it holds that $$\begin{aligned}
\label{identity2}
\begin{array}
{ll} & \ds \mu\cdot\nabla\bar{z}(i dz + \Delta
z dt) +
\mu\cdot\nabla z(-i d\bar{z} + \Delta \bar{z} dt)\\
\ns =& \ds \nabla\cd \Big[ (\mu\cdot\nabla
\bar{z})\nabla z+ (\mu\cdot\nabla z)\nabla
\bar{z} - i (z d\bar{z}) \mu - |\nabla
z|^2\mu \Big]dt + d(i\mu\cd\nabla \bar{z} z)\\
\ns & \ds - 2\sum_{j,k=1}^n \mu^k_j
z_{j}\bar{z}_{k}dt + (\nabla\cdot \mu) |\nabla
z|^2 dt + i(\nabla\cdot \mu) z d\bar{z} -
i(\mu\cd\nabla d\bar z) dz.
\end{array}\end{aligned}$$
[*Proof of Proposition \[prop2\]*]{} : The proof is a direct computation. We have that $$\begin{aligned}
\label{h1}
\begin{array}
{ll} & \ds \sum_{k=1}^n\sum_{j=1}^n
\mu^k\bar{z}_k z_{jj}+
\sum_{k=1}^n\sum_{j=1}^n \mu^k z_k \bar{z}_{jj}\\
\ns = & \ds
\sum_{k=1}^n\sum_{j=1}^n\Big[(\mu^k\bar{z}_k
z_j)_j + (\mu^k z_k\bar{z}_j)_j +
\mu^k_k|z_j|^2 - (\mu^k|z_j|^2)_k - 2\mu^k_j
\bar{z}_k z_j \Big]
\end{array}\end{aligned}$$ and that $$\label{h2}
\begin{array}{ll}\ds
\q i\sum_{k=1}^n(\mu^k\bar{z}_k dz-\mu^k z_k d\bar{z})\\
\ns\ds = i\sum_{k=1}^n\Big[\,d(\mu^k\bar{z}_k
z) - \mu^k z d\bar z_k - \mu^k d\bar{z}_k
dz -(\mu^k zd\bar{z})_k + \mu^k z d\bar z_k + \mu_k^k z d\bar{z}\, \Big]\\
\ns\ds = i\sum_{k=1}^n\Big[\,d(\mu^k\bar{z}_k
z) - \mu^k d\bar{z}_k dz -(\mu^k zd\bar{z})_k
+ \mu_k^k z d\bar{z} \,\Big].
\end{array}$$ Combining and , we get the equality .
By virtue of Proposition \[prop2\], the proof of Proposition \[hregularity\] is standard. We only give a sketch here.
[*Sketch of the Proof of Proposition \[hregularity\]*]{} : Since $\Gamma $ is $ C^2$, one can find a vector field $\mu_0 = (\mu_0^1,
\cdots, \mu_0^n) \in
C^1(\overline{G};\mathbb{R}^n)$ such that $\mu_0 = \nu$ on $\Gamma$(see [@Komornik page 18] for the construction of $\mu_0$). Letting $\mu = \mu_0$ and $z = y$ in Proposition \[prop2\], integrating it in $Q$ and taking the expectation, by means of Proposition \[prop2\], with similar computation in [@Zhangxu1], Proposition \[hregularity\] can be obtained immediately.
An Identity for a Stochastic Schrödinger-like Operator
======================================================
In this section, we obtain an identity for a stochastic schrödinger-like operator, which is similar to the formula in the spirit but it takes a more complex form and play a key role in the proof of Theorem \[thcarleman est\].
Let $\b(t,x)\in
C^{2}(\mathbb{R}^{1+n};\mathbb{R})$, and let $b^{jk}(t,x)\in
C^{1,2}(\mathbb{R}^{1+n};\;\mathbb{R})$ satisfy $$\label{bjk}
b^{jk}=b^{kj},\qq j,k=1,2,\cdots,n.$$ Let us define a (formal) second order stochastic partial differential operator $\cP$ as $$\label{cp}
\ds
\cP z \= i\b(t,x)dz+\sum_{j,k=1}^n(b^{jk}(t,x)z_j)_k dt,
\q i=\sqrt{-1}.$$ We have the following equality concerning $\cP$:
\[identity1\] Let $\ell,\;\Psi\in
C^2(\mathbb{R}^{1+n};\;\mathbb{R})$. Assume that $z$ is an $H^2_{loc}(\mathbb{R}^n,\mathbb{C})$-valued $\{\cF_t\}_{t\geq 0}$-adapted process. Put $ v=\th
z$(recall for the definition of $\th$). Then for a.e. $x \in \mathbb{R}^n$ and P-a.s. $\o\in \O$, it holds that
$$\begin{aligned}
\label{c2a1}
\begin{array}{ll}
&\th(\cP z\overline {I_1}+\overline{\cP z} I_1)+dM+\div V \\
\ns = & \ds 2|I_1|^2 dt +\sum_{j,k=1}^n
c^{jk}(v_k\ov_j+\ov_k v_j)
dt +D|v|^2 dt \\
\ns & \ds +i\sum_{j,k=1}^n\[(\b
b^{jk}\ell_j)_t+
b^{jk}(\b\ell_t)_j\](\ov_kv-v_k\ov) dt
\\
\ns & \ds +i\[\b\Psi+\sum_{j,k=1}^n(\b
b^{jk}\ell_j)_k\](\ov
dv-vd\ov)\\
& \ds + (\b^2\ell_t)dvd\ov + i\sum_{j,k=1}^n
\b b^{jk}\ell_j (dv d\ov_k - dv_kd\ov),
\end{array}\end{aligned}$$
where $$\label{c2a2}
\left\{
\begin{array}{ll}\ds I_1\= - i\b \ell_t
v - 2\sum_{j,k=1}^n b^{jk}\ell_j v_k +
\Psi v, \\
\ns\ds A\=\sum_{j,k=1}^n
b^{jk}\ell_j\ell_k-\sum_{j,k=1}^n(b^{jk}\ell_j)_k
-\Psi,
\end{array}
\right.$$
$$\label{c2a3} \left\{
\begin{array}{ll}\ds
M\=\b^2\ell_t |v|^2 + i\b\sum_{j,k=1}^n b^{jk}\ell_j(\ov_kv-v_k\ov),\\
\ns\ds V\=[V^1,\cdots,V^k,\cdots,V^n],\\
\ns\ds V^k\=-i \b\sum_{j=1}^n\[b^{jk}\ell_j(vd\ov -\ov
dv ) + b^{jk}\ell_t(v_j\ov-\ov_jv) dt\]\\
\ns\ds\qq\,\,\,\, - \Psi\sum_{j=1}^n b^{jk}(v_j\ov+\ov_jv) dt + \sum_{j=1}^n b^{jk}(2A\ell_j+\Psi_j)|v|^2 dt \\
\ns\ds\qq\q+\sum_{j,j',k'=1}^n\(2b^{jk'}b^{j'k}-b^{jk}b^{j'k'}\)\ell_j(v_{j'}\ov_{k'}+\ov_{j'}v_{k'})
dt,
\end{array}
\right.$$
and $$\label{cc2a3}
\left\{
\begin{array}{ll}\ds
c^{jk}\= \sum_{j',k'=1}^n\[2(b^{j'k}\ell_{j'})_{k'}b^{jk'}-(b^{jk}b^{j'k'}\ell_{j'})_{k'}\] - b^{jk}\Psi,\\
\ns\ds D\=(\b^2\ell_t)_t
+\sum_{j,k=1}^n(b^{jk}\Psi_k)_j+2\[\sum_{j,k=1}^n(b^{jk}\ell_jA)_k+A\Psi\].
\end{array}
\right.$$
Since we only assume that $(b^{jk})_{1\leq
j,k\leq n}$ is symmetric and do not assume that it is positively definite, then similar to [@Fu] and based on the identity in Theorem \[identity1\], we can deduce observability estimate not only for the stochastic Schrödinger equation, but also for deterministic hyperbolic, Schrödinger and plate equations, which had been derived via Carleman estimate (see [@FYZ], [@Lasiecka-Triggiani-Zhang] and [@Zhangxu5], respectively).
[*Proof of Theorem \[identity1\]*]{}: The proof is divided into three steps.
[**Step 1.**]{} By the definition of $v$ and $w$, a straightforward computation shows that: $$\begin{aligned}
\label{th1eq1}
\begin{array}
{ll} \ds \theta \cP z &= \ds i\b dv - i\b
\ell_t v dt + \sum_{j,k=1}^n
(b^{jk}v_j)_k dt \\
\ns & \ds \q + \sum_{j,k=1}^n b^{jk}\ell_j
\ell_k v dt - 2\sum_{j,k=1}^n b^{jk}\ell_j v_k
dt - \sum_{j,k=1}^n (b^{jk}\ell_j)_k v dt
\\
\ns &= \ds I_1dt + I_2,
\end{array}\end{aligned}$$ where $$\label{I2}
I_2 = i\b dv + \sum_{j,k=1}^n (b^{jk}v_j)_kdt +
Avdt.$$ Hence we obtain that $$\label{th1eq2}
\theta (Pz\overline{I_1} + \overline{Pz}I_1) =
2|I_1|^2dt + (I_1\overline{I_2} +
I_2\overline{I_1}).$$
[**Step 2.**]{} In this step, we compute $I_1\overline{I_2} + I_2\overline{I_1}$. Denote the three terms in $I_1$ and $I_2$ by $I_1^j$ and $I_2^j$, $j = 1,2,3$, respectively. Then we have that $$\label{th1eq3}
\begin{array}{ll}\ds
\q I_1^1\overline{I_2^1} + I_2^1 \overline{I_1^1} \\
\ns\ds = -i\b \ell_t v \overline{ (i\b dv)} + i\b dv \overline{ (-i\b \ell_t v)} \\
\ns\ds = -d(\b^2 \ell_t |v|^2) + (\b^2
\ell_t)_t|v|^2dt + \b^2 \ell_t dvd\ov.
\end{array}$$ Noting that $$\begin{aligned}
\label{th1eq4}
\left\{
\begin{array}{lll}
\ds 2vd\ov = d(|v|^2) - (\ov dv - vd\ov) - dv d\ov,\\
\ns\ds 2v \ov_k = (|v|^2)_k - (\ov v_k -
v\ov_k),
\end{array}
\right.\end{aligned}$$ we find first $$\label{th1eq4.1}
\begin{array}{ll}
\ds & \ds 2i\sum_{j,k=1}^n (\b b^{jk}\ell_j vd\ov)_k \\
\ns=& \ds i \sum_{j,k=1}^n \Big\{\b b^{jk}\ell_j \[ d(|v|^2) - (\ov dv - vd\ov) - dv d\ov \] \Big\}_k \\
\ns = & \ds i \sum_{j,k=1}^n\Big\{ \big(\b
b^{jk}\ell_j\big)_k d(|v|^2)
+ \b b^{jk}\ell_j\big[d(|v|^2)\big]_k - \big[\b b^{jk}\ell_j (\ov dv - vd\ov)\big]_k\\
\ns& \ds \qq\q - \big( \b b^{jk}\ell_j\big)_k
dv d\ov - \b b^{jk}\ell_jdv_k d\ov - \b
b^{jk}\ell_jdvd\ov_k \Big\},
\end{array}$$ next $$\label{th1eq4.2}
\begin{array}{ll}
\ds \q -2i\sum_{j,k=1}^n \big(\b b^{jk}\ell_j \big)_k vd\ov \\
\ns\ds= -i \sum_{j,k=1}^n \big(\b b^{jk}\ell_j \big)_k \[ d(|v|^2) - (\ov dv - vd\ov) - dv d\ov \] \\
\ns\ds = -i \sum_{j,k=1}^n\[ \big(\b
b^{jk}\ell_j\big)_k d(|v|^2) - \big(\b
b^{jk}\ell_j \big)_k(\ov dv - vd\ov) \ - \big(
\b b^{jk}\ell_j \big)_k dv d\bar v \],
\end{array}$$ then $$\label{th1eq4.3}
\begin{array}{ll}\ds
\q -2i \sum_{j,k=1}^n d\big(\b b^{jk}\ell_j v\ov_k \big) \\
\ns\ds = - i \sum_{j,k=1}^n d\Big\{ \b b^{jk}\ell_j \big[(|v|^2)_k - (\ov v_k - v\ov_k) \big]\Big\}\\
\ns\ds = - i \sum_{j,k=1}^n \Big\{\big(\b
b^{jk}\ell_j \big)_t (|v|^2)_kdt + \b
b^{jk}\ell_j d\big[(|v|^2)_k\big] - d\big[ \b
b^{jk}\ell_j (\ov v_k - v\ov_k) \big] \Big\},
\end{array}$$ and that $$\label{th1eq4.4}
\begin{array}{ll}\ds
\q 2i \sum_{j,k=1}^n \big(\b b^{jk}\ell_j \big)_t v\ov_k dt \\
\ns\ds = i \sum_{j,k=1}^n d \big(\b b^{jk}\ell_j \big)_t \big[(|v|^2)_k - (\ov v_k - v\ov_k) \big]dt \\
\ns\ds = i \[\sum_{j,k=1}^n \big(\b
b^{jk}\ell_j \big)_t (|v|^2)_kdt - \big(\b
b^{jk}\ell_j \big)_t (\ov v_k - v\ov_k)dt
\].
\end{array}$$ From –, we get that $$\label{th1eq5}
\begin{array}{ll}\ds
\q (I_1^2 + I_1^3)\overline{I_2^1} +
I_2^1(\overline{I_1^2} + \overline{I_1^3})
\\
\ns\ds = \(- 2\sum_{j,k=1}^n b^{jk}\ell_j v_k +
\Psi v\) \overline{ (i\b dv) } + i\b dv
\overline{ \( - 2\sum_{j,k=1}^n b^{jk}\ell_j
v_k +
\Psi v \) }\\
\ns\ds = 2i\sum_{j,k=1}^n \b b^{jk}\ell_j (v_k
d\bar v - \bar v_k dv) + i\b\Psi (\bar v dv -
vd\bar v)
\\
\ns\ds
= 2i\sum_{j,k=1}^n \[ \big(\b b^{jk}\ell_j vd\ov\big)_k - \big(\b
b^{jk}\ell_j\big)_k vd\ov - \b b^{jk}\ell_j
vd\ov_k
\]
\\ \ns\ds \q -2i \sum_{j,k=1}^n
\[ d\big(\b b^{jk}\ell_j v\ov_k\big) - \big(\b
b^{jk}\ell_j\big)_t v\ov_k dt - \b b^{jk}\ell_j
vd\ov_k \]
\end{array}$$ $$\begin{array}{ll}
\ds\q + 2i\sum_{j,k=1}^n \b b^{jk}\ell_j dv
d\ov_k + i\b\Psi (\bar v dv - vd\bar v)\nonumber\\
\ns\ds = -i \sum_{j,k=1}^n \[ \b b^{jk}\ell_j(\ov dv
- vd\ov ) \]_k dt -i \sum_{j,k=1}^n d\[ \b
b^{jk}\ell_j(v\ov_k - \ov v_k) \] \\
\ns\ds \q - i\sum_{j,k=1}^n (\b b^{jk}\ell_j)_t
(\ov v_k - v \ov_k)dt + i\[ \b\Psi +
\sum_{j,k=1}^n
(\b b^{jk}\ell_j)_k \](\ov dv - vd\ov) \\
\ns\ds \q + i\sum_{j,k=1}^n \b b^{jk}\ell_j (dv
d\ov_k - dv_kd\ov).
\end{array}$$
Noting that $b^{jk} = b^{kj}$, we have that $$\label{th1eq7}
\begin{array}{ll}\ds
\q I_1^1\overline{I_2^2} + I_2^2 \overline{I_1^1}
\\
\ns\ds = -i\b \ell_t v \overline{\sum_{j,k=1}^n(b^{jk}v_j)_k}dt + \sum_{j,k=1}^n(b^{jk}v_j)_k \overline{(-i\b \ell_t v)} \\
\ns\ds = \sum_{j,k=1}^n \[ i\b b^{jk}\ell_t
(v_j \ov - \ov_j v)\]_k dt + i \sum_{j,k=1}^n
b^{jk}(\b \ell_t)_k (\ov_j v - v_j\ov)dt.
\end{array}$$ Utilizing $b^{jk} = b^{kj}$ once more, we find $$\sum_{j,k,j',k'=1}^n b^{jk}b^{j'k'}\ell_j
(v_{j'}\ov_{kk'} +
\ov_{j'}v_{kk'})=\sum_{j,k,j',k'=1}^n
b^{jk}b^{j'k'}\ell_j (v_{j'k}\ov_{k'} +
\ov_{j'k}v_{k'}).$$ Hence, we obtain that $$\label{th1eq8}
\begin{array}{ll}\ds
\q 2\sum_{j,k,j',k'=1}^n b^{jk}b^{j'k'}\ell_j (v_{j'}\ov_{kk'} +
\ov_{j'}v_{kk'})dt \\
\ns\ds = \sum_{j,k,j',k'=1}^n
b^{jk}b^{j'k'}\ell_j (v_{j'}\ov_{kk'} +
\ov_{j'}v_{kk'})dt + \sum_{j,k,j',k'=1}^n
b^{jk}b^{j'k'}\ell_j (v_{j'k}\ov_{k'} +
\ov_{j'k}v_{k'})dt\\
\ns\ds =
\sum_{j,k,j',k'=1}^n b^{jk}b^{j'k'}\ell_j
(v_{j'}\ov_{k'} +
\ov_{j'}v_{k'})_k dt \\
\ns\ds = \!\sum_{j,k,j',k'=1}^n \!\[
b^{jk}b^{j'k'}\ell_j (v_{j'}\ov_{k'} +
\ov_{j'}v_{k'}) \]_kdt -
\!\!\sum_{j,k,j',k'=1}^n
(b^{jk}b^{j'k'}\ell_j)_k (v_{j'}\ov_{k'} + \ov_{j'}v_{k'})dt.\\
\end{array}$$
By the equality , we get that
$$\label{th1eq9}
\begin{array}{ll}\ds
\q I_1^2\overline{I_2^2} + I_2^2
\overline{I_1^2}\\
\ns\ds = - 2\!\sum_{j,k=1}^n b^{jk}\ell_j v_k\!
\overline{ \sum_{j,k=1}^n(b^{jk}v_j)_k }dt - 2\!\sum_{j,k=1}^n(b^{jk}v_j)_k \overline{\sum_{j,k=1}^n b^{jk}\ell_j v_k}dt\\
\ns\ds = - 2\!\! \sum_{j,k,j',k'=1}^n \!\!\[
b^{jk}b^{j'k'}\ell_j (v_{j'}\ov_{k}\! +\!
\ov_{j'}v_{k}) \]_{k'} dt \!+\!
2\!\!\sum_{j,k,j',k'=1}^n\!\!
b^{j'k'}(b^{jk}\ell_j)_{k'} (v_{j'}\ov_{k}\! +\! \ov_{j'}v_{k})dt \\
\ns\ds \q + 2\sum_{j,k,j',k'=1}^n
b^{jk}b^{j'k'}\ell_j (v_{j'}\ov_{kk'} +
\ov_{j'}v_{kk'})dt \\
\ns\ds = - 2\!\! \sum_{j,k,j',k'=1}^n \!\[
b^{jk}b^{j'k'}\ell_j (v_{j'}\ov_{k} \!+\!
\ov_{j'}v_{k}) \]_{k'} dt \!+\!
2\!\!\!\sum_{j,k,j',k'=1}^n\!\!
b^{j'k'}(b^{jk}\ell_j)_{k'} (v_{j'}\ov_{k}\! + \!\ov_{j'}v_{k})dt \\
\ns\ds \q + \!\!\!\sum_{j,k,j',k'=1}^n \!\[
b^{jk}b^{j'k'}\ell_j (v_{j'}\ov_{k'} \!+\!
\ov_{j'}v_{k'}) \]_k dt -
\!\!\sum_{j,k,j',k'=1}^n
(b^{jk}b^{j'k'}\ell_j)_k
(v_{j'}\ov_{k'} + \ov_{j'}v_{k'})dt \\
\ns\ds = - 2\!\! \sum_{j,k,j',k'=1}^n \!\[
b^{jk'}b^{j'k}\ell_j (v_{j'}\ov_{k'}\! +\!
\ov_{j'}v_{k'}) \]_{k} dt \!+\!
2\!\!\sum_{j,k,j',k'=1}^n\!\!
b^{jk'}(b^{j'k}\ell_{j'})_{k'} (v_{j}\ov_{k} \!+ \!\ov_{j}v_{k})dt \\
\ns\ds \q + \!\!\sum_{j,k,j',k'=1}^n \[
b^{jk}b^{j'k'}\ell_j (v_{j'}\ov_{k'}\! +
\ov_{j'}v_{k'}) \]_k dt -
\!\!\sum_{j,k,j',k'=1}^n
(b^{jk}b^{j'k'}\ell_{j'})_{k'} (v_{j}\ov_{k} +
\ov_{j}v_{k})dt.
\end{array}$$
Further, it holds that $$\label{th1eq10}
\begin{array}{ll}\ds
\q I_1^3\overline{I_2^2} + I_2^2 \overline{I_1^3} \\
\ns\ds = \Psi v \overline{ \sum_{j,k=1}^n(b^{jk}v_j)_k }dt + \sum_{j,k=1}^n(b^{jk}v_j)_k \overline{ \Psi v }dt \\
\ns\ds =
\sum_{j,k=1}^n
\[ \Psi b^{jk}(v_j \ov + \ov_j v) \]_k dt -
\sum_{j,k=1}^n
\Psi b^{jk}(v_j \ov_k + \ov_j v_k)dt \\
\ns\ds\q -\sum_{j,k=1}^n \Psi_k b^{jk} ( v_j
\bar v + \bar v_j v) dt
\\
\ns\ds =
\sum_{j,k=1}^n
\[ \Psi b^{jk}(v_j \ov + \ov_j v) \]_k dt -
\sum_{j,k=1}^n
\Psi b^{jk}(v_j \ov_k + \ov_j v_k)dt \\
\ns\ds\q -\sum_{j,k=1}^n \[ b^{jk}\Psi_k |v|^2
\]_j dt + \sum_{j,k=1}^n (b^{jk}\Psi_k)_j
|v|^2dt.
\end{array}$$
Finally, we have that $$\label{th1eq11}
\begin{array}{ll}\ds
\q I_1\overline{I_2^3} + I_2^3 \overline{I_1} \\
\ns\ds = -i\b \ell_t v \overline{Av}dt +
Av\overline{ (-i\b \ell_t v) }dt
\\
\ns\ds = - 2\sum_{j,k=1}^n (b^{jk}\ell_j
A|v|^2)_k dt + 2\[ \sum_{j,k=1}^n (b^{jk}\ell_j
A)_k + A\Psi \]|v|^2dt .
\end{array}$$
[**Step 3.**]{} Combining (\[th1eq2\])–(\[th1eq11\]), we conclude the desired formula (\[c2a1\]).
Carleman Estimate for Stochastic Schrödinger Equations
======================================================
This section is devoted to the proof of Theorem \[thcarleman est\].
: The proof is divided into three steps.
**Step 1.** We choose $\b = 1$ and $(b^{jk})_{1\leq j,k\leq n}$ to be the identity matrix. Put $$\d^{jk} =
\left\{\begin{array}{ll}\ds 1,&\mbox{ if }
j=k,\\ \ns\ds 0,&\mbox{ if } j\neq
k.\end{array}\right.$$ Applying Theorem \[identity1\] to the equation with $\theta$ given by , $z$ replaced by $y$ and $v = \theta z$. We obtain that $$\label{identity2.1}
\begin{array}{ll}
\ds\q\theta\cP y {\( i\b \ell_t \bar{v} -
2\sum_{j,k=1}^n b^{jk}\ell_j \bar{v}_k + \Psi
\bar{v}\)} + \theta\overline{\cP y} {\(- i\b
\ell_t v - 2\sum_{j,k=1}^n b^{jk}\ell_j v_k +
\Psi v\)}\\
\ns \ds \q + \;dM + \div V \\
\ns\ds = 2\Big|- i\b \ell_t v -
2\sum_{j,k=1}^n b^{jk}\ell_j v_k + \Psi
v\Big|^2dt +
\sum_{j,k=1}^nc^{jk}(v_k\ov_j+\ov_k
v_j) dt + D|v|^2dt \\
\ns \ds \q + 2i\sum_{j=1}^n (\ell_{jt} +
\ell_{tj})(\ov_j v - v_j\ov)dt +
i(\Psi + \D \ell)(\ov dv - v d\ov) \\
\ns \ds \q + \ell_t dv d\ov + i\sum_{j=1}^n
\ell_j (d\ov_j dv - dv_j d\ov).
\end{array}$$ Here $$\label{Id2eq1.1}
\begin{array}{ll}
M \3n& \ds = \b^2\ell_t |v|^2 +
i\b\sum_{j,k=1}^nb^{jk}\ell_j(\ov_kv-v_k\ov)\\
\ns & \ds = \ell_t |v|^2 + i\sum_{j=1}^n \ell_j
(\ov_j v - v_j \ov);
\end{array}$$ $$\label{Id2eq1.2}
\begin{array}{ll}
A \3n& \ds =\sum_{j,k=1}^nb^{jk}\ell_j\ell_k -
\sum_{j,k=1}^n(b^{jk}\ell_j)_k -\Psi \\
\ns & \ds = \sum_{j=1}^n (\ell_j^2 -
\ell_{jj}) -\Psi;
\end{array}$$ $$\label{Id2eq1.3}
\begin{array}{ll}
D \3n & \ds =(\b^2\ell_t)_t
+\sum_{j,k=1}^n(b^{jk}\Psi_k)_j
+ 2\[\sum_{j,k=1}^n(b^{jk}\ell_j A)_k + A\Psi\]\\
\ns & \ds = \ell_{tt} + \sum_{j=1}^n \Psi_{jj}
+ 2\sum_{j=1}^n (\ell_j A)_j + 2 A\Psi;
\end{array}$$ $$\label{Id2eq1.4}
\begin{array}{ll}
c^{jk} \3n &\ds =
\sum_{j',k'=1}^n\[2(b^{j'k}\ell_{j'})_{k'}b^{jk'}
- (b^{jk}b^{j'k'}\ell_{j'})_{k'}\Psi\] - b^{jk}\\
\ns & \ds =
\[2(b^{kk}\ell_{k})_{j}b^{jj} -
\sum_{j'=1}^n (b^{jk}b^{j'j'}\ell_{j'})_{j'} -
b^{jk}\Psi\]\\
\ns & \ds = 2\ell_{jk} - \d^{jk}\D \ell -
\d^{jk}\Psi;
\end{array}$$ and $$\label{Id2eq1.5}
\begin{array}{ll}
V_k \3n &\ds = -i
\b\sum_{j=1}^n\[b^{jk}\ell_j(vd\ov -\ov
dv ) + b^{jk}\ell_t(v_j\ov-\ov_jv) dt\]\\
\ns & \ds \q - \Psi\sum_{j=1}^n b^{jk}(v_j\ov+\ov_jv) dt + \sum_{j=1}^n b^{jk}(2A\ell_j+\Psi_j)|v|^2 dt \\
\ns & \ds \q
+\sum_{j,j',k'=1}^n\(2b^{jk'}b^{j'k}-b^{jk}b^{j'k'}\)\ell_j(v_{j'}\ov_{k'}+\ov_{j'}v_{k'})
dt\\
\ns & \ds = -i\big[ \ell_k(vd\ov - \ov
dv) + \ell_t(v_j\ov -\ov_j v)dt \big] - \Psi(v_k\ov + \ov_k v)dt + (2A\ell_k + \Psi_k)|v|^2dt\\
\ns & \ds \q + 2\sum_{j=1}^n \ell_j (\ov_j v_k
+ v_j \ov_k)dt - 2\sum_{j'=1}^n
\ell_k(v_j\ov_j)dt.
\end{array}$$
**Step 2.** In this step, we estimate the terms in the right-hand side of the equality one by one.
First, from the definition of $\ell$, $\f$(see ) and the choice of $\psi$(see ), we have that $$\label{lt1}
\begin{array}{ll}\ds
|\ell_t| & \ds = \Big| s\frac{2(2t-T)}{t^3(T-t)^3}\big( e^{4\l\psi} - e^{5\l |\psi|_{L^\infty(G)}} \big) \Big| \\
\ns& \ds \leq \Big| s\frac{2(2t-T)}{t^3(T-t)^3} e^{5\l |\psi|_{L^\infty(G)}} \Big| \\
\ns &\ds \leq \Big| s\frac{C}{t^3(T-t)^3} e^{5\l \psi} \Big|\\
\ns & \ds \leq Cs\varphi^{1+\frac{1}{2}},
\end{array}$$ and that $$\label{ltt1}
\begin{array}{ll}
\ds |\ell_{tt}| & \ds = \Big| s\frac{20t^2 - 20tT + 6T^2}{t^4(T-t)^4} \big( e^{4\l\psi} - e^{5\l |\psi|_{L^\infty(G)}} \big) \Big| \\
\ns& \ds \leq \Big| s\frac{C}{t^4(T-t)^4} e^{5\l |\psi|_{L^\infty(G)}} \Big| \\
\ns& \ds \leq \Big| s\frac{C}{t^4(T-t)^4} e^{8\l \psi } \Big|\\
\ns &\ds \leq Cs\f^2\leq Cs\f^3.
\end{array}$$
We choose below $\Psi = -\D \ell$, then we have that $$\begin{aligned}
\label{Id2eq2}
A = \sum_{j=1}^m \ell_j^2 = \sum_{j=1}^m \big(4s\l\f \psi )^2 =16s^2\l^2\varphi^2 |\nabla\psi|^2.\end{aligned}$$ Hence, we find $$\label{B}
\begin{array}{ll}\ds
D \3n & \ds = \ell_{tt} + \sum_{j=1}^n
\Psi_{jj} + 2\sum_{j=1}^n (\ell_j
A)_j + 2 A\Psi \\
\ns & \ds = \ell_{tt} + \D(\D\ell) + 2\sum_{j=1}^n\big(4s\l\f\psi_j 16s^2\l^2\f^2|\nabla\psi|^2\big)_j - 32s^2\l^2\f^2|\nabla\psi|^2\D \ell \\
\ns & \ds = 384s^3\l^4\varphi^3|\nabla\psi|^4
- \l^4\varphi O(s) - s^3\varphi^3 O(\l^3) +
\ell_{tt}.
\end{array}$$ Recalling that $x_0\in (\mathbb{R}^n\setminus
\overline G)$, we know that $$|\nabla\psi|>0\;\;\mbox{ in }\overline G.$$ From and , we know that there exists a $\l_0>0$ such that for all $\l>\l_0$, one can find a constant $s_0 =
s_0(\l_0)$ so that for any $s>s_0$, it holds that $$\label{B1}
D|v|^2 \geq
s^3\l^4\varphi^3|\nabla\psi|^4|v|^2.$$ Since $$\begin{array}{ll}\ds
c^{jk} = 2\ell_{jk} - \d^{jk}\D \ell - \d^{jk}\Psi \\
\ns\ds\q\,\,\, = 32s\l^2\varphi\psi_j \psi_k +
16s\l\varphi\psi_{jk},
\end{array}$$ we see that $$\label{cjk}
\begin{array}{ll}\ds
\q \ds\sum_{j,k=1}^n c^{jk}(v_j\ov_k + v_k\ov_j)\\
\ns \ds = 32s\l^2\varphi\sum_{j,k=1}^n\psi_j
\psi_k(v_j\ov_k + v_k\ov_j) + 16s\l\varphi
\sum_{j,k=1}^n \psi_{jk}(v_j\ov_k + v_k\ov_j)\\
\ns\ds =
32s\l^2\varphi\[\sum_{j=1}^n(\psi_jv_j)\sum_{k=1}^n
(\psi_k \ov_k) +
\sum_{k=1}^n(\psi_kv_k)\sum_{j=1}^n (\psi_j
\ov_j) \] + 32s\l\varphi \sum_{j=1}^n(v_j\ov_j
+ \ov_j v_j)\\
\ns \ds = 64s\l^2\varphi |\nabla\psi\cd\nabla v|^2 + 64 s\l\f |\nabla v|^2\\
\ns \ds \geq 64 s\l\f |\nabla v|^2.
\end{array}$$
Now we estimate the other terms in the right-hand side of the equality . The first one satisfies that $$\begin{aligned}
\label{ltj}
\begin{array}
{ll} \ds 2i\sum_{j=1}^n (\ell_{jt} +
\ell_{tj})(\ov_j v - v_j\ov) & \ds =
4i\sum_{j=1}^n s\l\psi_j \ell_t(\ov_j v - \ov v_j)\\
\ns & \ds \leq 2 s\varphi |\nabla v|^2 + 2
s\l^2\varphi^3 |\nabla\psi|^2|v^2|.
\end{array}\end{aligned}$$
The second one reads $$\begin{aligned}
\label{liiPsi}
i(\Psi + \D \ell)(\ov dv - v d\ov) = 0.\end{aligned}$$
For the estimate of the third and the fourth one, we need to take mean value and get that $$\label{dvdov}
\begin{array}{ll}\ds
\mathbb{E}\big(\ell_t dv d\ov\big) \3n& \ds= \mathbb{E}\big[\ell_t(\theta \ell_t ydt + \theta
dy)(\overline{\theta \ell_t ydt + \theta
dy)}\big] = \mathbb{E}(\ell_t \theta^2 dy
d\bar{y})
\\
\ns & \ds \leq 2s\theta^2
\varphi^{\frac{3}{2}}\mathbb{E}( a_3^2|y|^2 +
g^2)dt.
\end{array}$$ Here we utilize inequality .
Since $$\begin{array}{ll}\ds
\mathbb{E}(d\ov_j dv) & = \mathbb{E}\big[\overline{\big( \theta \ell_t v dt + \theta dy \big)}_j \big( \theta \ell_t v dt + \theta dy \big)\big] \\
\ns& \ds = \mathbb{E} \big[\, \overline{(\theta dy)}_j (\theta dy) \big]\\
\ns& \ds = \mathbb{E} \big[\, \overline{\big( s\l\f\psi_j\theta dy + \theta dy_j \big)}\theta dy \big]\\
\ns & \ds = s\l\f\psi_j\theta^2 \mathbb{E}d\bar y dy + \theta^2 \mathbb{E}d\bar y_j dy \\
\ns & \ds = s\l\f\psi_j\theta^2
\mathbb{E}|a_3y + g|^2dt + \theta^2
\mathbb{E}\big[\,\overline{ (a_3 y + g) }_j
(a_3 y + g) \big]dt
\end{array}$$ and $$\begin{array}{ll}\ds
\q\theta^2 \mathbb{E}\big[\,\overline{ (a_3 y +
g) }_j
(a_3 y + g) \big]dt\\
\ns\ds = \theta^2 \mathbb{E}\big[(\overline{a_3
y})_j (a_3 y) + (\overline{a_3 y})_j g + (a_3 y
)\bar g_j + g\bar g_j \big] dt\\
\ns\ds =\theta^2 \mathbb{E}\big[(\overline{a_3
y})_j (a_3 y) + (\overline{a_3 y})_j g + g\bar
g_j \big] dt + [\mE\theta^2(a_3 y )\bar g]_j \\
\ns\ds \q - s\l\f\psi_j\theta^2\mE(a_3 y \bar
g)-\th^2\mE[(a_3y)_j]\bar g,
\end{array}$$ we get that $$\begin{array}{ll}\ds
\mathbb{E}(d\ov_j dv) \3n&\ds=
s\l\f\psi_j\theta^2 \mathbb{E}|a_3y + g|^2dt +
\theta^2 \mathbb{E}\big[(\overline{a_3 y})_j
(a_3 y) + (\overline{a_3 y})_j g + g\bar g_j
\big]
dt\\
\ns&\ds \q + \mE(\theta^2 a_3 y \bar g)_j -
s\l\f\psi_j\theta^2\mE(a_3 y \bar
g)-\th^2\mE[(a_3y)_j\bar g].
\end{array}$$ Similarly, we can get that $$\begin{array}{ll}\ds
\mathbb{E}(dv_j d\ov)\3n& \ds=
s\l\f\psi_j\theta^2 \mathbb{E}|a_3y + g|^2dt +
\theta^2 \mathbb{E}\big[(\overline{a_3 y}) (a_3
y)_j + (a_3 y )_j\bar g + g_j\bar g \big]
dt\\
\ns&\ds \q + \mE(\theta^2 \overline{a_3 y} g)_j
- s\l\f\psi_j\theta^2\mE(\overline{a_3 y}
g)-\th^2\mE[(\overline{a_3 y})_j g].
\end{array}$$ Therefore, fourth one enjoys that $$\label{dvjdv}
\begin{array}{ll}
\ds \q i\mathbb{E}\sum_{j=1}^n \ell_j (d\ov_j
dv -
dv_j d\ov)\\
\ns\ds = s\l\varphi\sum_{j=1}^n \psi_j
\[\mathbb{E}\big(d\ov_j dv\big) - \mathbb{E}\big(dv_j d\ov\big) \] \\
\ns \ds = s\l\varphi \psi \sum_{j=1}^n \psi_j
\theta^2 \mathbb{E}\Big\{\big[(\overline{a_3
y})_j (a_3 y) + (\overline{a_3 y})_j g + g\bar
g_j - s\l\f\psi_j a_3 y \bar
g - (a_3y)_j \bar g\big]\\
\ns \ds \q - \big[(\overline{a_3 y}) (a_3 y)_j
+ (a_3 y )_j\bar g + g_j\bar g - s\l\f\psi_j
(\overline{a_3 y} g)- [(\overline{a_3 y})_j
g]\big]\Big\} dt \\
\ns\ds \q + s\l\varphi \psi \sum_{j=1}^n \psi_j
\mathbb{E}\big(\th^2 a_3 y \bar g - \theta^2
\overline{a_3 y} g \big)_j.
\end{array}$$
**Step 3.** Integrating the equality in $Q$, taking mean value in both sides, and noting –, we obtain that $$\label{inep1}
\begin{array}{ll}
\ds \q\mathbb{E}\int_Q \Big(s^3\l^4\varphi^3
|v|^2 \!+\! s\l^2\varphi |\nabla v|^2\Big) dxdt
+ 2\mathbb{E}\int_Q \Big|\!- i\b
\ell_t v - 2\!\!\sum_{j,k=1}^n b^{jk}\ell_j v_k + \!\Psi v\Big|^2dxdt\\
\ns \ds \leq \mathbb{E}\int_Q \Big\{
\theta\cP y {\Big( i\b \ell_t \bar{v}\! -\!
2\!\! \sum_{j,k=1}^n\! b^{jk}\ell_j \bar{v}_k
+ \!\Psi \bar{v}\Big)} + \theta\overline{\cP y}
{\Big(\!-\! i\b \ell_t v\! - \!2\!
\sum_{j,k=1}^n\!
b^{jk}\ell_j v_k + \!\Psi v\Big)} \Big\}dx\\
\ns \ds \q +\; C\mathbb{E}\int_Q
\theta^2\Big[s^2\l^2 \varphi^2(a_3^2|y|^2 +
g^2) + a_3^2|\nabla y|^2 + |\nabla a_3|^2 y^2
+ |\nabla g|^2\Big] dxdt\\
\ns\ds \q + \;\mathbb{E}\int_Q dM dx +
\mathbb{E}\int_Q \div V dx.
\end{array}$$
Now we analyze the terms in the right-hand side of the inequality one by one.
The first term satisfies that $$\label{intprin}
\begin{array}{ll} \ds \mathbb{E}\int_Q \Big\{ \theta\cP y
{\Big( i\b \ell_t \bar{v} - 2\sum_{j,k=1}^n
b^{jk}\ell_j \bar{v}_k + \Psi
\bar{v}\Big)}\\
\ns \ds \q +\; \theta\overline{\cP y} {\Big(-
i\b \ell_t v -
2\sum_{j,k=1}^n b^{jk}\ell_j v_k + \Psi v\Big)} \Big\}dx \\
\ns \ds = \ds \mathbb{E}\int_Q \Big\{ \theta
(a_1 \cdot \nabla y + a_2 y + f) {\( i\b \ell_t
\bar{v} - 2\sum_{j,k=1}^n b^{jk}\ell_j
\bar{v}_k + \Psi \bar{v}\)}\\
\ns\ds \q +\; \theta {(a_1 \cdot \nabla \bar{y}
+ \overline{a_2 y} + \bar{f})} {\Big(- i\b
\ell_t v - 2\sum_{j,k=1}^n b^{jk}\ell_j v_k +
\Psi
v\Big)} \Big\}dxdt\nonumber\\
\ns \ds \leq 2\mathbb{E}\int_Q
\Big\{\theta^2\big|a_1 \cdot \nabla y + a_2 y +
f\big|^2 + \Big|- i\b \ell_t v -
2\sum_{j,k=1}^n b^{jk}\ell_j v_k + \Psi
v\Big|^2 \Big\}dxdt.
\end{array}$$
From the definition of $\theta$, we know that $v(0)=v(T)=0$. Hence, it holds that $$\label{idm}
\int_Q dM dx = 0.$$ By means of Stokes’ Theorem, we have that $$\begin{aligned}
\label{intV}
\begin{array}
{ll} \ds \mathbb{E}\int_Q \div V dx \3n&\ds =
\ds \mathbb{E}\int_{\Si}
2\sum_{k=1}^n\sum_{j=1}^n\Big[
\ell_j\big(\ov_j v_k + v_j \ov_k\big)\nu^k - \ell_k \nu_k v_j \ov_j
\Big]d\Si\\
\ns &\ds = \ds \mathbb{E}\int_{\Si}
\Big(4\sum_{j=1}^n \ell_j \nu_j \Big| \frac{\pa
v}{\pa \nu} \Big|^2 - 2\sum_{k=1}^n \ell_k
\nu_k \Big|
\frac{\pa v}{\pa \nu} \Big|^2\Big) d\Si\\
\ns &= \ds \mathbb{E}\int_{\Si} 2\sum_{k=1}^n
\ell_k \nu_k \Big|
\frac{\pa v}{\pa \nu} \Big|^2 d\Si \\
\ns &\ds \leq C\mathbb{E}\int_0^T \int_{\G_0}
\theta^2 s\l\varphi \Big| \frac{\pa y}{\pa \nu}
\Big|^2 d\G dt.
\end{array}\end{aligned}$$
By (\[inep1\])–(\[intV\]), we have that $$\begin{aligned}
\label{car1}
\begin{array}{ll}
\q \ds \mathbb{E}\int_Q \Big(s^3\l^4\varphi^3
|v|^2 + s\l\varphi
|\nabla v|^2\Big) dxdt \\
\ns \ds \leq C\,\mathbb{E}\int_Q \theta^2
|a_1 \cdot \nabla y + a_2 y + f|^2 dxdt +
C\,\mathbb{E}\int_0^T\int_{\G_0}\theta^2
s\l\varphi\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G dt\\
\ns \ds \q +\, C\mathbb{E}\int_Q
\theta^2\Big[s^2\l^2 \varphi^2\big(a_3^2|y|^2 +
g^2\big) + a_3^2|\nabla y|^2 + |\nabla a_3|^2
y^2 + |\nabla g|^2\Big] dxd t.
\end{array}\end{aligned}$$
Noting that $y_i = \theta^{-1}(v_i - \ell_i v)
= \theta^{-1}(v_i - s\l\varphi\psi_i v)$, we get $$\label{vtoy}
\theta^2\big(|\nabla y|^2 + s^2\l^2\varphi^2
|y|^2\big)\leq C\big(|\nabla v|^2 +
s^2\l^2\varphi^2 |v|^2\big).$$
Therefore, it follows from (\[car1\]) that $$\label{car2}
\begin{array}{ll}
\ds \q\mathbb{E}\int_Q \Big(s^3\l^4\varphi^3
|y|^2 + s\l\varphi
|\nabla y|^2\Big) dxdt \\
\ns \ds \leq C\mathbb{E}\int_Q \theta^2\Big(
|a_1|^2 || \nabla y|^2 + a_2^2 |y|^2 +
|f|^2\Big) dxdt +
C\mathbb{E}\int_0^T\int_{\G_0}\theta^2
s\l\varphi\Big| \frac{\pa
y}{\pa \nu}\Big|^2d\G dt \\
\ns \ds \q + C\mathbb{E}\int_Q
\theta^2\Big[s^2\l^2 \varphi^2\big(a_3^2|y|^2 +
g^2\big) + a_3^2|\nabla y|^2 + |\nabla a_3|^2
y^2 + |\nabla g|^2\Big] dxdt.
\end{array}$$
Taking $\l_1 =\l_0$ and $s_1 = \max(s_0,
Cr_1)$, and utilizing the inequality , we conclude the desired inequality .
On the other hand, if $g\in
L^2_\cF(0,T;H^1(G;\mathbb{R}))$, then $g\bar
g_j - g_j\bar g=0$ for $j=1,\cds,n$. Thus, from –, we get $$\label{inep1z}
\begin{array}{ll}
\ds \q\mathbb{E}\int_Q \Big(s^3\l^4\varphi^3
|v|^2 + s\l^2\varphi |\nabla v|^2\Big) dxdt +
2\mathbb{E}\int_Q \Big|\!- i\b
\ell_t v \!- \!2\!\sum_{j,k=1}^n b^{jk}\ell_j v_k \! +\! \Psi v\Big|^2dxdt\\
\ns \ds \leq \mathbb{E}\int_Q \Big\{
\theta\cP y {\Big( i\b \ell_t \bar{v} \!-\!
2\!\!\sum_{j,k=1}^n b^{jk}\ell_j \bar{v}_k +
\Psi \bar{v}\Big)} + \theta\overline{\cP y}
{\Big(\!\!-\! i\b \ell_t v\! -\!
2\!\!\sum_{j,k=1}^n
b^{jk}\ell_j v_k\! + \!\Psi v\Big)} \Big\}dx\\
\ns \ds \q +\; C\mathbb{E}\int_Q
\theta^2\Big[s^2\l^2 \varphi^2\big(a_3^2|y|^2 +
g^2\big) + a_3^2|\nabla y|^2 + |\nabla a_3|^2
y^2
\Big] dxdt + \mathbb{E}\int_Q dM dx \\
\ns\ds \qq + \mathbb{E}\int_Q \div V dx.
\end{array}$$ Then, by a similar argument, we find that $$\label{car2z}
\begin{array}{ll}
\ds \q\mathbb{E}\int_Q \Big(s^3\l^4\varphi^3
|y|^2 + s\l\varphi
|\nabla y|^2\Big) dxdt \\
\ns \ds \leq C\mathbb{E}\int_Q \theta^2\Big(
|a_1|^2 || \nabla y|^2 + a_2^2 |y|^2 +
|f|^2\Big) dxdt +
C\mathbb{E}\int_0^T\int_{\G_0}\theta^2
s\l\varphi\Big| \frac{\pa
y}{\pa \nu}\Big|^2d\G dt \\
\ns \ds \q + C\mathbb{E}\int_Q
\theta^2\Big[s^2\l^2 \varphi^2\big(a_3^2|y|^2 +
g^2\big) + a_3^2|\nabla y|^2 + |\nabla a_3|^2
y^2 \Big] dxdt.
\end{array}$$ Now taking $\l_1 =\l_0$ and $s_1 = \max(s_0,
Cr_1)$, and using the inequality , we obtain the desired inequality .
Proof of Theorem \[observability\]
==================================
In this section, we prove Theorems \[observability\], by means of Theorem \[thcarleman est\].
[*Proof of Theorem \[observability\]*]{}: By means of the definition of $\ell$ and $\theta$(see ), it holds that $$\begin{aligned}
\label{final1}
\begin{array}{ll}
\ds \q\mathbb{E}\int_Q \theta^2\Big(\varphi^3
|y|^2 + \varphi
|\nabla y|^2\Big) dxdt\\
\ns \ds \geq
\min_{x\in\overline{G}}\Big(\varphi\Big(\frac{T}{2},x\Big)
\theta^2\Big(\frac{T}{4},x\Big)\Big)\mathbb{E}\int_{\frac{T}{4}}^{\frac{3T}{4}}\int_G\big(|y|^2+|\nabla
y|^2\big)dxdt,
\end{array}\end{aligned}$$
$$\label{final2}
\begin{array}{ll} \ds
\q\mathbb{E}\int_Q \theta^2\big(|f|^2 +
\varphi^2|g|^2 + |\nabla
g|^2\big)dxdt \\
\ns \ds\leq \max_{(x,t)\in
\overline{Q}}\big(\varphi^2(t,x)\theta^2(t,x)\big)\mathbb{E}\int_Q\big(|f|^2
+ |g|^2 + |\nabla g|^2\big)dxdt
\end{array}$$
and that $$\label{final3}
\mathbb{E}\int_0^T\int_{\G_0}\theta^2
\varphi\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G dt
\leq \max_{(x,t)\in
\overline{Q}}\big(\varphi(t,x)\theta^2(t,x)\big)\mathbb{E}\int_0^T\int_{\G_0}
\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G dt.$$
From and –(\[final3\]), we deduce that $$\label{final4}
\begin{array}{ll} \ds
\q\mathbb{E}\int_{\frac{T}{4}}^{\frac{3T}{4}}\int_G\big(|y|^2+|\nabla
y|^2\big)dxdt\\
\ns \ds\leq C r_1 \frac{\max_{(x,t)\in
\overline{Q}}\Big(\varphi^2(t,x)\theta^2(t,x)\Big)}{\min_{x\in\overline{G}}\Big(\varphi(\frac{T}{2},x)\theta^2(\frac{T}{4},x)\Big)}\\
\ns \ds \q\times\left\{
\mathbb{E}\int_Q\big(|f|^2 + |g|^2 + |\nabla
g|^2\big)dxdt + \mathbb{E}\int_0^T\int_{\G_0}
\Big| \frac{\pa y}{\pa
\nu}\Big|^2d\G dt\right\}\\
\ns \ds \leq e^{ Cr_1 }\left\{
\mathbb{E}\int_Q\big(|f|^2 + |g|^2 + |\nabla
g|^2\big)dxdt + \mathbb{E}\int_0^T\int_{\G_0}
\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G
dt\right\}.
\end{array}$$
Utilizing (\[final4\]) and (\[energyesti1\]), we obtain that $$\label{final5}
\begin{array}{ll} \ds
\q\mathbb{E}\int_G\big(|y_0|^2 + |\nabla y_0|^2\big)dx \\
\ns \ds \leq e^{C r_1 }\left\{
\mathbb{E}\int_Q\big(|f|^2 + |\nabla f|^2 +
|g|^2 + |\nabla g|^2\big)dxdt +
\mathbb{E}\int_0^T\int_{\G_0} \Big| \frac{\pa
y}{\pa \nu}\Big|^2d\G dt\right\},
\end{array}$$ which concludes Theorem \[observability\] immediately.
Two applications {#Sec app}
================
This section is addressed to applications of the observability/Carleman estimates shown in Theorems \[observability\]–\[thcarleman est\].
We first study a state observation problem for semilinear stochastic Schrödinger equations. Let us consider the following equation: $$\label{system2}
\!\!\left\{
\begin{array}{ll}
\ds idz + \D zdt=\big[\,a_1 \cdot \nabla z +
a_2 z +F_1(|z|)\big]dt + \big[a_3 z
+F_2(|z|)\big]
dB(t)&\mbox{ in } Q,\\
\ns\ds
z=0&\mbox{ on }\Si,\\
\ns\ds z(0)=z_0, &\mbox{ in }G.
\end{array}
\right.$$ Here $a_i$ ($i=1,2,3$) are given as in , $F_1(\cd)\in C^1(\mathbb{R};
\mathbb{C})$ with $F(0)=0$ and $F_2(\cd)\in
C^1(\mathbb{R}; \mathbb{R})$ are two known nonlinear global Lipschitz continuous functions with Lipschitzian constant $L$, while the initial datum $z_0\in L^2(\O,\cF_0,P;
H_0^1(G))$ is unknown. The solution to the equation is understood similar to Definition \[def1\].
From the choice of $F_1$ and $F_2$, one can easily show that the equation admits a unique solution $z\in H_T$ by the standard fixed point argument. We omit the proof here.
The state observation problem associated to the equation is as follows.
- [**Identifiability**]{}. Is the solution $z\in H_T$ (to ) determined uniquely by the observation $\ds\frac{\pa
z}{\pa\nu}\Big|_{(0,T)\times \G_0}$?
- [**Stability**]{}. Assume that two solutions $z$ and $\hat z$ (to ) are given, and let $\ds\frac{\pa
z}{\pa\nu}\Big|_{(0,T)\times \G_0}$ and $\ds\frac{\pa \hat z}{\pa\nu}\Big|_{(0,T)\times
\G_0}$ be the corresponding observations. Can we find a positive constant $C$ such that $$|\!| z-\hat z |\!| \leq C\|\!\| \frac{\pa
z}{\pa\nu}-\frac{\pa \hat z}{\pa\nu} \|\!\|,$$ with appropriate norms in both sides?
- [**Reconstruction**]{}. Is it possible to reconstruct $z\in H_T$ to , in some sense, from the observation $\ds\frac{\pa
z}{\pa\nu}\Big|_{(0,T)\times \G_0}$?
The state observation problem for systems governed by deterministic partial differential equations is studied extensively (See [@Kli; @Li1; @Yamamoto] and the rich references therein). However, the stochastic case attracts very little attention. To our best knowledge, [@Zhangxu3] is the only published paper addressing this topic. In that paper, the author studied the state observation problem for semilinear stochastic wave equations. By means of Theorem \[observability\], we can give positive answers to the above first and second questions.
We claim that $\frac{\pa
z}{\pa\nu}|_{(0,T)\times \G_0}\in
L^2_\cF(0,T;L^2(\G_0))$ (and therefore, the observation makes sense). Indeed, from the choice of $F_1$, it follows that $$\begin{array}{ll}\ds
\mathbb{E}\int_0^T\!\int_G \big|\n
\big(F_1(|z|)\big)\big|^2dxdt \3n& \ds=
\mathbb{E}\int_0^T\!\int_G \| F_1' (|z|)\n |z|
\|^2dxdt \leq L\mathbb{E}\int_0^T\!\int_G
\big|\n |z|\big|^2dxdt\\
\ns&\ds \leq L\mathbb{E}\int_0^T\int_G \big|\n
z\big|^2dxdt,
\end{array}$$ and $$F(|z(t,\cd)|)=0 \q\mbox{ on } \G \mbox{ for
a.e. }\ t\in [0,T].$$ Hence, $$F_1(|z|)\in L^2_{\cF}(0,T;H_0^1(G)) \mbox{ for
any } z\in H_{T}.$$ Similarly, $$F_2(|z|)\in L^2_{\cF}(0,T;H^1(G)) \mbox{ for
any } z\in H_{T}.$$ Consequently, by Proposition \[hregularity\], we find that $\frac{\pa
z}{\pa\nu}|_{(0,T)\times\G_0}\in
L^2_\cF(0,T;L^2(\G_0))$.
Now, we define a nonlinear map as follows: $$\left\{
\begin{array}{ll}\ds \cM:\
L^2(\O,\cF_0,P;H_0^1(G))\to
L^2_{\cF}(0,T;L^2(\G_0)),\\
\ns\ds \cM(z_0)= {\frac{\pa z}{\pa
\nu}}\Big|_{(0,T)\times \G_0},
\end{array}
\right.$$ where $z$ solves the equation . We have the following result.
\[th2\] There exists a constant $\wt C=\wt C(L,T,G)>0$ such that for any $z_0, \hat z_0\in
L^2(\O,\cF_0,P;H_0^1(G))$, it holds that $$\label{th2eq1}
|z_0-\hat z_0|_{L^2(\O,\cF_0,P;L^2(G))} \le \wt
C|\cM(z_0) -\cM(\hat
z_0)|_{L^2_{\cF}(0,T;L^2(\G_0))},$$ where $\hat z=\hat z(\cd\, ;\hat z_0)\in H_{T}$ is the solution to with $z_0$ replaced by $\hat z_0$.
From the well-posedness of the equation , Theorem \[th2\] indicates that the state $z(t)$ of (for $t\in [0,T]$) can be uniquely determined from the observed boundary data $\ds{\frac{\pa z}{\pa \nu}}
\Big|_{(0,T)\t\G_0}$, $P$- a.s., and continuously depends on it. Therefore, we answer the first and second questions for state observation problem for the system positively.
[*Proof of Theorem \[th2\]*]{}: Set $$y=z-\hat z.$$ Then, it is easy to see that $y$ satisfies $$\left\{
\begin{array}{ll}\ds
idy + \D y dt = \big[ a_1 \cdot \nabla y +
a_2 y +F_1(|z|)-F_1(|\hat z|) \big]dt \\
\ns\ds \hspace{2.2cm} + \big[ a_3 y +
F_2(|z|)-F_2(|z|) \big]dB(t) &\mbox{ in } Q,\\
\ns\ds y=0 &\mbox{ on }\Si,\\
\ns\ds y(0)=z_0-\hat z_0 &\mbox{ in } G.
\end{array}
\right.$$ Also, it is clear that $$F_1(|z|)-F_1(|\hat z|)\in
L^2_{\cF}(0,T;H_0^1(G))$$ and $$F_2(|z|)-F_2(|\hat z|)\in
L^2_{\cF}(0,T;H^1(G)).$$ Hence, we know that $y$ solves the equation with $$\left\{
\begin{array}{ll}\ds f=F_1(|z|)-F_1(|\hat z|),\\
\ns\ds g=F_2(|z|)-F_2(|\hat z|).
\end{array}
\right.$$
By means of the inequality in Theorem \[thcarleman est\], there exist an $s_1>0$ and a $\l_1>0$ so that for all $s\geq s_1$ and $\l\geq \l_1$, it holds that $$\begin{array}{ll}\ds
\q\mathbb{E}\int_Q
\theta^2\Big(s^3\l^4\varphi^3 |y|^2 +
s\l\varphi
|\nabla y|^2\Big) dxdt \\
\ns \ds \leq C \Big\{\mathbb{E}\int_Q \theta^2
\Big(|f|^2 +
s^2\l^2\varphi^2 |g|^2 \Big)dxdt +
\mathbb{E}\int_0^T\int_{\G_0}\theta^2
s\l\varphi\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G
dt \Big\}.
\end{array}$$ By the choice of $f$, we see that $$\begin{array}{ll}\ds
\mathbb{E}\int_Q \theta^2 |f|^2dxdt \3n&\ds\leq
\mathbb{E}\int_Q \theta^2 |F_1(|z|)-F_1(|\hat
z|)|^2dxdt \leq L\mathbb{E}\int_Q \theta^2
(|z|-|\hat z|)^2dxdt\\
\ns&\ds \leq L\mathbb{E}\int_Q \theta^2 |z -
\hat z|^2dxdt \leq L\mathbb{E}\int_Q \theta^2
|y|^2dxdt.
\end{array}$$ Similarly, $$s^2\l^2\mathbb{E}\int_Q \theta^2 \f^2 |g|^2dxdt
\leq L\mathbb{E}\int_Q \theta^2\f^2 |y|^2dxdt.$$ Hence, we obtain that $$\begin{array}{ll}\ds
\q\mathbb{E}\int_Q
\theta^2\Big(s^3\l^4\varphi^3 |y|^2 +
s\l\varphi
|\nabla y|^2\Big) dxdt \\
\ns \ds \leq C\Big\{L \mathbb{E}\int_Q
\theta^2 \Big(|y|^2 +
s^2\l^2\varphi^2 |y|^2 \Big)dxdt +
\mathbb{E}\int_0^T\int_{\G_0}\theta^2
s\l\varphi\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G
dt \Big\}.
\end{array}$$ Thus, there is a $\l_2\geq \max\{\l_1, CL\}$ such that for all $s\geq s_1$ and $\l\geq
\l_2$, it holds that $$\label{10.30eq1}
\mathbb{E}\int_Q \theta^2\Big(s^3\l^4\varphi^3
|y|^2 + s\l\varphi |\nabla y|^2\Big) dxdt \leq
C \mathbb{E}\int_0^T\int_{\G_0}\theta^2
s\l\varphi\Big| \frac{\pa y}{\pa \nu}\Big|^2d\G
dt.$$
Further, similar to the proof of the inequality , we can obtain that for any $0\leq
t \le s\leq T$, it holds $$\label{Eyt1}
\begin{array}{ll}\ds
\mathbb{E}| y(t)|^2_{ L^2(G)} - \mathbb{E}|
y(s)|^2_{ L^2(G)}\3n &\ds \leq
2\mathbb{E}\int_t^s\int_G \Big[ |f|^2 + |g|^2
\Big]dxd\si\\
\ns&\ds \leq CL\mathbb{E}\int_t^s\int_G |y|^2
dxd\si.
\end{array}$$ Then, by Gronwall’s inequality, we find that $$\label{Eyt2}
\mathbb{E}| y(t)|^2_{ L^2(G)} \leq
e^{CL}\mathbb{E} | y(s)|^2_{ L^2(G)}, \mbox{
for any } 0\leq t\leq s \leq T.$$ Combining and , similar as the derivation of the inequality , we obtain the inequality .
Now we consider the unique continuation property for the equation . There are numerous works on the unique continuation property for deterministic partial differential equations. The study in this respect began at the very beginning of the 20th century; while a climax appeared in the last 1950-70’s. The most powerful tools at that period is the local Carleman estimate (See [@Hor1] for example). Nevertheless, most of the existing works are devoted to the local unique continuation property at that time. In the recent 20 years, motivated by Control/Inverse Problems of partial differential equations, the study of the global unique continuation is very active (See [@Castro-Zuazua; @Zhangxu1; @Zhang-Zuazua] and the rich references therein). Compared with the fruitful works on the unique continuation property in the deterministic settings, there exist few results for stochastic partial differential equations. As far as we know, [@Zhangxu4; @Zhangxu2] are the only two published articles addressed to this topic, and there is no result on the global unique continuation property for stochastic Schrödinger equations in the previous literature.
We remark that the powerful approach based on local Carleman estimate in the deterministic settings is very hard to apply to the stochastic counterpart. Indeed, the usual approach to employ local Carleman estimate for the unique continuation needs to localize the problem. Unfortunately, one cannot simply localize the problem as usual in the stochastic situation, since the usual localization technique may change the adaptedness of solutions, which is a key feature in the stochastic setting. In this paper, as a consequence of Theorem \[observability\] (which is based on the global Carleman estimate established in Theorem \[thcarleman est\]), we obtain the following unique continuation property for solutions to the equation .
\[ucp\] For any $\e>0$, let $$O_\e([0,T]\t\G_0)\=\Big\{(x,t)\in Q
:\,\dist\big((x,t),[0,T]\t\G_0\big)\leq \e
\Big\}.$$ Let $f=g=0$, $P$-a.s. For any $y$ which solves the equation , if y = 0 O\_(\[0,T\]\_0), P, then $y=0$ in $Q$, $P$-a.s.
[*Proof*]{}: By , we see that $\ds\frac{\pa y}{\pa\nu}=0$ on $(0,T)\t\G_0$, $P$-a.s. Hence, by means of Theorem \[observability\], we find that $y(0)=0$ in $L^2(\O,\cF_0,P;H_0^1(G))$. Consequently, we conclude that $y=0$ in $Q$, $P$-a.s.
Further comments and open problems
==================================
The subject of this paper is full of open problems. Some of them seem to be particularly relevant and could need important new ideas and further developments:
- [**Observability estimate for backward stochastic Schrödinger equations**]{}
Compared with Theorem \[observability\], it is more interesting and difficult to establish the boundary observability estimate for backward stochastic Schrödinger equations. More precisely, let us consider the following backward stochastic Schrödinger equation: $$\label{bsystem1}
\!\!\left\{
\begin{array}{lll}
\ds idu + \D u dt = (a_1\cd \nabla u + a_2 u + f)dt + (a_3 u + U+ g)dB(t) &\mbox{ in } Q,\\
\ns\ds u = 0 &\mbox{ on } \Si,\\
\ns\ds u(T) = u_T &\mbox{ in } G.
\end{array}
\right.$$ Here the final state $u_T\in
L^2(\O,\cF_T,P;H_0^1(G))$ and $\{\cF_t\}_{t\geq 0}$ is the natural filtration generated by $\{B(t)\}_{t\geq 0}$. We expect the following result:\
[*Under the assumptions –, any solution of the equation satisfies that $$\label{bobser esti2}
\begin{array}{ll}\ds
\q |u_T|_{L^2(\Omega,{ \mathcal{F}}_T, P; H_0^1(G))} \\
\ns\ds \leq e^{C r_1} \Big(\Big|\frac{\partial
u}{\partial \nu}\Big |_{L^2_{ \mathcal{
F}}(0,T;L^2(\Gamma_0))} + |f|_{L^2_{ \mathcal{
F}}(0,T;H_0^1(G))} + |g|_{L^2_{ \mathcal{
F}}(0,T;H^1(G))}\Big),
\end{array}$$ or at least, $$\label{bobser esti3}
\begin{array}{ll}\ds
\q |u(0)|_{L^2(\Omega,{ \mathcal{F}}_0, P; H_0^1(G))} \\
\ns\ds \leq e^{C r_1} \Big(\Big|\frac{\partial
u}{\partial \nu}\Big |_{L^2_{ \mathcal{
F}}(0,T;L^2(\Gamma_0))} + |f|_{L^2_{ \mathcal{
F}}(0,T;H_0^1(G))} + |g|_{L^2_{ \mathcal{
F}}(0,T;H^1(G))}\Big).
\end{array}$$*]{}
Unfortunately, following the method in this paper, one could obtain only an inequality as follows: $$\label{bobser esti4}
\begin{array}{ll}\ds
\q |u_T|_{L^2(\Omega,{ \mathcal{F}}_T, P; H_0^1(G))} \\
\ns\ds \leq e^{C r_1} \Big(\Big|\frac{\partial
u}{\partial \nu}\Big |_{L^2_{ \mathcal{
F}}(0,T;L^2(\Gamma_0))} +
|U|_{L^2_\cF(0,T;H^1(G))} + |f|_{L^2_{\mathcal{
F}}(0,T;H_0^1(G))}\\
\ns\ds \qq\qq + |g|_{L^2_{ \mathcal{
F}}(0,T;H^1(G))}\Big).
\end{array}$$ It seems to us that getting rid of the undesired term $|U|_{L^2_\cF(0,T;H^1(G))}$ in the inequality is a very challenging task.
- [**Construction of the solution $z$ from the observation**]{}
In this paper, we only answer the first and the second questions in the state observation problem. The third one is still open. Since the equation is time irreversible, some efficient approaches (See [@Li1] for example), which work well for time reversible systems, become invalid. On the other hand, we may consider the following minimization problem:
[*Find a $\bar z_0\in
L^2(\O,\cF_0,P;H_0^1(G))$ such that $$\| \frac{\pa \bar z}{\pa\nu} - h
\|_{L^2_\cF(0,T;L^2(\G_0))}=\min_{z_0\in
L^2(\O,\cF_0,P;H_0^1(G))}\| \frac{\pa
z}{\pa\nu} - h \|_{L^2_\cF(0,T;L^2(\G_0))},$$ where $h\in L^2_\cF(0,T;L^2(\G_0))$ is the observation and $z$ ($\bar z$) is the solution to the equation with initial datum $z_0$ ($\bar z_0$).*]{}
It seems that one may utilize the method from optimization theory to study the construction of $z_0$. Because of the stochastic nature, this is an interesting but difficult problem and the detailed analysis is beyond the scope of this paper.
- [**Unique continuation property with less restrictive conditions**]{}
In this paper, we show that, under the condition , $y=0$ in $Q$, $P$-a.s. Compared to the classical unique continuation result for deterministic Schrödinger equations with time independent coefficients (see [@Es1; @Lebeau] for example), the condition is too restrictive. It would be quite interesting but maybe challenging to prove whether the result in [@Es1] is true or not for stochastic Schrödinger equations. In fact, as far as we know, people even do not know whether the results in [@Es1; @Lebeau] are true or not for deterministic Schrödinger equations with time-dependent lower order term coefficients, which is a particular case of the equation .
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper is an improved version of one chapter of the author’s Ph D thesis ([@Luqi2]) accomplished at Sichuan University under the guidance of Professor Xu Zhang. The author would like to take this opportunity to thank him deeply for his help. The author also highly appreciates the anonymous referees for their constructive comments.
[1]{}
, [*Carleman estimate and cotrollability of linear stochastic heat equatons*]{}, Appl. Math. Optim., [**47**]{}(2003), pp. 97–120.
, [*Quantum trajectories and measurements in continuous time. The diffusive case*]{}, Lecture Notes in Physics, 782, Springer, Heidelberg, 2009.
, [*Uniqueness and stability in an inverse problem for the Schrödinger equation*]{}, Inverse Problems, [**18**]{}(2002), pp. 1537–1554.
, [*Unique continuation and control for the heat equation from an oscillating lower dimensional manifold*]{}, SIAM J. Control Optim., [**43**]{}(2004), pp. 1400–1434.
, [*Stochastic Equations in Infinite Dimensions*]{}, Cambridge University Press, Cambridge, 1992.
, [*Uniqueness properties of solutions to Schrödinger equations*]{}, Bull. Amer. Math. Soc., [**49**]{}(2012), pp. 415–442.
, [*A weighted identity for partial differential operators of second order and its applications*]{}, C. R. Acad. Sci. Paris Série I, [**342**]{}(2006), pp. 579–584.
, [*Null controllability for the parabolic equation with a complex principle part*]{}, J. Func. Anal., [**257**]{}(2009), pp. 1333–1354.
, [*Exact controllability for multidimensional semilinear hyperbolic equations*]{}, SIAM J. Control Optim., [**46**]{} (2007), pp. 1578–1614.
, [*Controllability of Evolution Equations*]{}, Lecture Notes Series, 34, Seoul National University, Seoul, 1996.
, [*The Analysis of Linear Partial Differential Operators (III–IV)*]{}, Springer-Verlag, Berlin, (1985).
, [*Estimates of initial conditions of parabolic equations ans inequalities via lateral Cauchy data*]{}, Inverse Probmes, [**22**]{}(2006), pp. 495–514.
, [*Semiclassical analysis for diffusions and stochastic processes*]{}, Lecture Notes in Mathematics, 1724, Springer-Verlag, Berlin, 2000.
, [*Exact Controllability and Stabilization. The Multiplier Method*]{}, Wiley, Chichester; Masson, Paris, 1994.
, [*Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates: Part I. $H^1$-estimates*]{}, J. Inv. Ill-posed Problems, [**11**]{}(2004), pp. 43–123.
, [*Contrôle de léquation de Schrödinger*]{}, J. Math. Pures Appl., [**71**]{}(1992), pp. 267–291.
, [*State observation problem for general time reversible system and applications*]{}, Appl. Math. Comput., [**217**]{}(2010), pp. 2843–2856.
, [*Control and Observation of Stochastic Partial Differential Equations*]{}, Ph D Thesis, Sichuan University, 2010.
, [*Observability estimate for stochastic Schrödinger equations*]{}, C. R. Acad. Sci. Paris, Ser I, [**348**]{}(2010), pp. 1159–1162.
, [*Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems*]{}, Inverse Problems, [**28**]{}(2012), 045008.
, [*Exact controllability for the Schrödinger equation*]{}, SIAM J. Control Optim., [**32**]{}(1994), pp. 24–34.
, [*Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights*]{}, Inverse Problems, [**24**]{}(2008), 015017.
, [*Observability and control of Schrödinger equations*]{}, SIAM J. Control Optim., [**40**]{} (2001), pp. 211–230.
, [*Null controllability for forward and backward stochastic parabolic equations*]{}, SIAM J. Control Optim., [**48**]{}(2009), pp. 2191–2216.
, [*Carleman estimates for parabolic equations and applications*]{}, Inverse Problems, [**25**]{}(2009), 123013.
, [*Explicit observability estimate for the wave equation with potential and its application*]{}, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., [**456**]{}(2000), pp. 1101–1115.
, [*Exact controllability of the semilinear plate equations*]{}, Asymptot. Anal., [**27**]{}(2001), pp. 95–125.
, [*Unique continuation and observability for stochastic parabolic equations and beyond*]{}, in Control Theory and Related Topics (In Memory of Xunjing Li), S. Tang and J. Yong, eds., World Sci. Publ., Hackensack, NJ, 2007, pp. 147–160.
, [*Unique continuation for stochastic parabolic equations*]{}, Diff. Int. Eqs., [**21**]{}(2008), pp. 81–93.
, [*Carleman and observability estimates for stochastic wave equations*]{}, Math. Ann., [**325**]{}(2003), pp. 543–582.
, [*Unique continuation for the linearized Benjamin-Boma-Mahony equation with space dependent potential*]{}, Asymptot. Anal., [**27**]{}(2001), pp. 95–125.
, [*Remarks on the controllability of the Schrödinger equation*]{}, Quantum control: mathematical and numerical challenges, CRM Proc. Lecture Notes, 33, Amer. Math. Soc., Providence, RI, 2003, pp. 193–211.
[^1]: School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China; and Basque Center for Applied Mathematics (BCAM), Mazarredo, 14. 48009 Bilbao Basque Country - Spain, ([luqi59@163.com]{}).
[^2]: This work is partially supported by the NSF of China under grant 11101070, and the ERC Advanced Grant FP7-246775 NUMERIWAVES, the Grant PI2010-04 of the Basque Government, the ESF Research Networking Programme OPTPDE and Grant MTM2008-03541 of the MICINN, Spain.
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abstract: 'We propose an ultradiscrete analogue of the vertex operator in the case of the ultradiscrete KP equation–several other ultradiscrete equations–which maps $N$-soliton solutions to $N+1$-soliton ones.'
address: 'Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, 153-8914 Tokyo, Japan'
author:
- Yoichi Nakata
title: 'Vertex operator for the non-autonomous ultradiscrete KP equation'
---
[*Keywords*]{}: Integrable Systems; Solitons; Discrete Systems; Cellular automaton; KP equation
Introduction
============
The KP equation is widely considered as the paradigm of soliton equations. The main results of soliton theory, including first and foremost the celebrated Sato theory, were discovered in the study of this equation. The discovery of the vertex operator, which maps $N$-soliton solutions to $N+1$-soliton ones, figures prominently among these results.
The discrete KP equation (or Hirota-Miwa equation), which is a discretized version of the KP equation, is also regarded as a fundamental discrete soliton equation. By restricting its solutions it reduces to many well-known discrete soliton equations as, for example, the discrete KdV equation or the discrete Toda equation.
Soliton Cellular Automata form a class of cellular automata that exhibit soliton-like behaviour and possess a rich structure including the existence of explicit $N$-soliton solutions and an infinite amount of conserved quantities [@NTS], like most ordinary soliton equations. The “Box and Ball System" (BBS) [@TS] is the main representative of this class. It is related to the discrete soliton equations, through a limiting procedure called “ultradiscretization" [@TTMS].
The non-autonomous ultradiscrete KP equation is obtained by ultradiscretizing the non-autonomous discrete KP equation [@WTS]. Tokohiro presented its $N$-soliton solution and described the dynamics of the BBS with several kinds of balls in [@TTM]. Shinzawa and Hirota discussed the consistency conditions of the Bäcklund transformation for the autonomous ultradiscrete KP equation in [@SH].
Recently, these systems draw increasing interest due to the establishment of relationships to other mathematical topics, for example, to algebraic geometry and representation theory. It is therefore fruitful to clarify the symmetries and the algebraic structure of ultradiscrete soliton equations, as was done for the continuous ones. Takahashi and Hirota presented an approach based on so-called “permanent type solutions" [@TH] (which are expressed as signature-free Casorati determinants) to discuss particular solutions of ultradiscrete systems. Nagai presented identities for permanent type solutions, which can be considered as ultradiscrete analogues of Plücker relations for determinants in [@Nh]. Another approach for obtaining solutions is the vertex operator for the ultradiscrete KdV equation, which is proposed by the author in [@N]. This approach is believed to be closely related to certain types of symmetries for this system.
In this paper, we propose a vertex operator for the non-autonomous ultradiscrete KP equation and various ultradiscrete soliton equations obtained by reduction. In section 2 we first propose a recursive representation of the soliton solutions of the non-autonomous ultradiscrete KP equation. In section 3, we propose the vertex operator as an operator representation of the recursive one. In section 4, we present various reductions of this equation and discuss their vertex operators and solutions. Finally, in section 5, we give some concluding remarks.
Recursive expression for the solution of the ultradiscrete KP equation
======================================================================
The non-autonomous ultradiscrete KP equation is written as $$\label{bilinear}
\fl\quad T_{l, m+1, n} + T_{l+1, m, n+1} = \max \big(\ T_{l+1, m, n} + T_{l, m+1, n+1} - 2R_{n}, T_{l, m, n+1} + T_{l+1, m+1, n}\ \big),$$ where $R_{n} \ge 0$ depends only on $n$.
\[Thm1\] The function $T^{(N)}_{l, m, n}$ expressed as $$\label{exformula}
T^{(N)}_{l, m, n} = \cases{\max \big( T_{l, m, n}^{(N-1)}, 2 \eta_{N} + T^{(N-1)}_{l-1, m+1, n} \big) &($N \ge 1$)\\0&($N = 0$)\\}$$ solves equation (\[bilinear\]) for $\eta_{N}$ given by $$\label{defeta}
\eta_{N} = C_{N} + l P_{N} - m Q_{N} - \sum_{0}^{n} \Omega_{N, d}.$$ Here, $\sum_{i}^{j} \Omega_{N,d}$ stands for $$\sum_{i}^{j} \Omega_{N, d} = \cases{\sum_{d=i+1}^{j} \Omega_{N, d}&($i<j$)\\0&($i=j$)\\-\sum_{d=j+1}^{i} \Omega_{N, d} &($i>j$)\\},$$ and the parameters $P_{i}, Q_{i}$ and $ \Omega_{i, n} (i=1,\ldots, N)$ satisfy the relations: $$\begin{aligned}
P_{N} &\ge P_{N-1} \ge \ldots \ge P_{1} \ge 0 \label{condp} \\
Q_{N} &\ge Q_{N-1} \ge \ldots \ge Q_{1} \ge 0 \label{condq} \\
\Omega_{i, n} &= \min ( Q_{i}, R_{n-1} ). \label{defomega}\end{aligned}$$
\[Lem1\] Let $$H^{(N)}_{l, m, n} = T^{(N)}_{l, m+j+1, n+k} + T^{(N)}_{l+i+1, m, n} - T^{(N)}_{l+1, m+j, n+k} - T^{(N)}_{l+i, m+1, n}$$ for $i, j, k$ such that $$\label{condvanish}
i P_{N} + j Q_{N} + \sum_{n}^{n+k} \Omega_{N, d} \ge \ldots \ge i P_{1} + j Q_{1} + \sum_{n}^{n+k} \Omega_{1, d} \ge 0.$$ Then it holds that $$H^{(N)}_{l,m,n} \le 2(i P_{N} + j Q_{N} + \sum_{n}^{n+k} \Omega_{N, d} ).$$
By employing the inequality $$\label{maxtrineq}
\max ( a, b ) - \max ( c, d ) \le \max ( a - c, b - d ),$$ we obtain $$\begin{aligned}
\fl\quad T^{(N)}_{l, m+j+1, n+k} - T^{(N)}_{l+i, m+1, n} \le \max \big( T^{(N-1)}_{l, m+j+1, n+k} - T^{(N-1)}_{l+i, m+1, n}, \nonumber\\
\qquad\quad -2(i P_N - j Q_N - \sum_{n}^{n+k} \Omega_{N, d}) + T^{(N-1)}_{l-1, m+j+2, n+k} - T^{(N-1)}_{l+i-1, m+2, n} \big)\end{aligned}$$ $$\begin{aligned}
\fl\quad T^{(N)}_{l+i+1, m, n} - T^{(N)}_{l+1, m+j, n+k} \le \max \big( T^{(N-1)}_{l+i+1, m, n} - T^{(N-1)}_{l+1, m+j, n+k}, \nonumber\\
\qquad\quad 2(i P_N + j Q_N + \sum_{n}^{n+k} \Omega_{N, d}) + T^{(N-1)}_{l+i, m+1, n} - T^{(N-1)}_{l, m+j+1, n+k} \big)\end{aligned}$$ Adding the inequalities yields $$\begin{aligned}
\fl\quad H^{(N)}_{l, m, n} \le \max \big( H^{(N-1)}_{l, m, n}, \ 2(i P_N + j Q_N + \sum_{n}^{n+k} \Omega_{N, d} ), \nonumber\\
-2(i P_N + j Q_N + \sum_{n}^{n+k} \Omega_{N, d} ) + H^{(N-1)}_{l, m, n} + H^{(N-1)}_{l-1, m+1, n}, \ H^{(N-1)}_{l-1, m+1, n} \big)\end{aligned}$$ Taking into account the relations $i P_{N} + j Q_{N} + \sum_{n}^{n+k} \Omega_{N, d} \ge i P_{N-1} + j Q_{N-1} + \sum_{n}^{n+k} \Omega_{N-1, d}$, it can be shown inductively that the four arguments in this maximum are all less than $2(i P_N + j Q_N + \sum_{n}^{n+k} \Omega_{N, d} )$.
\[Lem2\] Let $$H'^{(N)}_{l, m, n} = T^{(N)}_{l, m, n+1} + T^{(N)}_{l, m+2, n} - T^{(N)}_{l, m+1, n} - T^{(N)}_{l, m+1, n+1}.$$ One then has $$\label{lem2ineq}
H'^{(N)}_{l, m, n} \le 2(Q_N - \Omega_{N, n+1})$$ when one requires that the $T^{(i)}_{l, m, n} (i=1, \ldots, N)$ are solutions of (\[bilinear\]). Especially for $\Omega_{N,n} = Q_{N}$, the inequality (\[lem2ineq\]) becomes an equality, i.e.: $H'^{(N)}_{l,m,n} = 0$ .
When $\Omega_{N,n} = R_{n+1}$, we obtain by virtue of the inequality (\[maxtrineq\]): $$\begin{aligned}
H'^{(N)}_{l, m, n} \le \max \big( H'^{(N-1)}_{l, m, n}, H'^{(N-1)}_{l-1, m+1, n}, \nonumber\\
2( Q_N - \Omega_{N, n+1} ) + T^{(N-1)}_{l-1, m+1, n+1} + T^{(N-1)}_{l, m+2, n} - T^{(N-1)}_{l-1, m+2, n} - T^{(N-1)}_{l, m+1, n+1}, \nonumber\\
-2( Q_N - \Omega_{N, n+1} ) + H^{(N-1)}_{l-1, m, n+1}\big|_{(i,j,k)=(0,1,-1)} + H'^{(N-1)}_{l-1,m+1,n}\big).\end{aligned}$$ However, $T^{(N-1)}_{l-1, m+1, n+1} + T^{(N-1)}_{l, m+2, n} - T^{(N-1)}_{l-1, m+2, n} - T^{(N-1)}_{l, m+1, n+1} \le 0$ because $T^{(N-1)}_{l,m,n}$ satisfies (\[bilinear\]). It can then be shown inductively that all arguments in the maximum are less than $2(Q_N - \Omega_{N, n+1})$.
On the other hand, when $\Omega_{N,n} = Q_{N}$, by virtue of (\[condq\]), $T^{(N-1)}_{l, m+1, n}$ is equal to $T^{(N-1)}_{l, m, n+1}$ for all $l, m$ because $$\fl \qquad C_{i} + l P_{i} - (m+1) Q_{N} - \sum_{0}^{n} \Omega_{N, d} = C_{N} + l P_{N} - m Q_{N} - \sum_{0}^{n+1} \Omega_{N, d}$$ for all $i=1, \ldots, N$. We thus obtain that $$H'^{(N)}_{l, m, n} = (T^{(N)}_{l, m, n+1} - T^{(N)}_{l, m+1, n}) + (T^{(N)}_{l, m+2, n} - T^{(N)}_{l, m+1, n+1}) = 0.$$
\[Lem3\] Let $$H''^{(N)}_{l, m, n} = T^{(N)}_{l, m, n+1} + T^{(N)}_{l+2, m, n} - T^{(N)}_{l+1, m, n} - T^{(N)}_{l+1, m, n+1}.$$ One then has $$H''^{(N)}_{l, m, n} \le 2 P_N$$ when all of $T^{(i)}_{l, m, n} (i=1, \ldots, N)$ are solutions of (\[bilinear\]).
The proof is essentially the same as that of Lemma \[Lem2\] when $\Omega_{N, n} = R_{n+1}$.
We now have all the necessary lemmas at our disposal and proceed to the proof of theorem \[Thm1\].
We shall prove the theorem inductively. It is clear that $T^{(0)}_{l, m, n}$ solves equation (\[bilinear\]) because of the non-negativity of $R_n$. Now, let us assume that the theorem holds at $1, \ldots, N-1$. By substituting (\[exformula\]) in equation (\[bilinear\]), each contribution can be written as $$\begin{aligned}
\label{ff1}
\fl \quad T^{(N)}_{l, m+1, n} + T^{(N)}_{l+1, m, n+1} = &\max \big(\ T^{(N-1)}_{l, m+1, n} + T^{(N-1)}_{l+1, m, n+1}, \nonumber\\
&\quad 2(P_{N} - \Omega_{N, n+1}) + 2 \eta_{N} + T^{(N-1)}_{l, m+1, n} + T^{(N-1)}_{l, m+1, n+1}, \nonumber\\
&\quad -2 Q_{N} + 2 \eta_{N} + T^{(N-1)}_{l-1, m+2, n} + T^{(N-1)}_{l+1, m, n+1}, \nonumber\\
&\quad 4 \eta_{N} + 2(P_{N} - Q_{N} - \Omega_{N, n+1}) + T^{(N-1)}_{l-1, m+2, n} + T^{(N-1)}_{l, m+1, n+1} \big),\end{aligned}$$ for the left hand side of (\[bilinear\]), and $$\begin{aligned}
\label{ff2}
\fl \quad T^{(N)}_{l+1, m, n} + T^{(N)}_{l, m+1, n+1} = &\max \big( T^{(N-1)}_{l, m, n} + T^{(N-1)}_{l, m+1, n+1}, \nonumber \\
&\quad 2 P_{N} + T^{(N-1)}_{l, m+1, n} + T^{(N-1)}_{l-1, m+1, n+1}, \nonumber\\
&\quad -2 ( Q_{N} + \Omega_{N, n+1} ) + T^{(N-1)}_{l+1, m, n} + T^{(N-1)}_{l, m+2, n+1}, \nonumber\\
&\quad 4 \eta_{N} + 2(P_{N} - Q_{N} - \Omega_{N, n+1}) + T^{(N-1)}_{l, m+1, n} + T^{(N-1)}_{l-1, m+2, n+1} \big) \\
\label{ff3}
\fl \quad T^{(N)}_{l, m, n+1} + T^{(N)}_{l+1, m+1, n} = &\max \big( T^{(N-1)}_{l, m, n+1} + T^{(N-1)}_{l+1, m, n}, \nonumber\\
&\quad 2 (P_{N} - Q_{N}) + T^{(N-1)}_{l, m, n+1} + T^{(N-1)}_{l, m+2, n}, \nonumber\\
&\quad -2 \Omega_{N, n+1} + T^{(N-1)}_{l-1, m+1, n+1} + T^{(N-1)}_{l+1, m+1, n}, \nonumber\\
&\quad 4 \eta_{N} + 2(P_{N} - Q_{N} - \Omega_{N, n+1}) + T^{(N-1)}_{l-1, m+1, n+1} + T^{(N-1)}_{l, m+2, n} \big)\end{aligned}$$ for the right hand side. In these expressions it looks as if each of the maximum operations in (\[ff1\])–(\[ff3\]) has four arguments. However, by virtue of Lemma \[Lem1\], the third argument in (\[ff1\]) and (\[ff2\]) cannot yield the maximum because it is always less than the second argument.
Then, the relevant arguments of the maximum in (\[ff1\]) are in fact $$\begin{aligned}
&T^{(N-1)}_{l, m+1, n} + T^{(N-1)}_{l+1, m, n+1} \label{lhs1} \\
&2 \eta_{N} + 2(P_{N} - \Omega_{N, n+1}) + T^{(N-1)}_{l, m+1, n} + T^{(N-1)}_{l, m+1, n+1} \label{lhs2} \\
&4 \eta_N + 2(P_{N} - Q_{N} - \Omega_{N, n+1}) + T^{(N-1)}_{l-1, m+2, n} + T^{(N-1)}_{l, m+1, n+1} \label{lhs3}\end{aligned}$$ and those in the maximum of the contributions in (\[ff2\]), (\[ff3\]), as they appear in the right hand side of equation (\[bilinear\]): $$\begin{aligned}
\fl \max ( T^{(N-1)}_{l+1, m, n} + T^{(N-1)}_{l, m+1, n+1} - 2R, T^{(N-1)}_{l, m, n+1} + T^{(N-1)}_{l+1, m, n} ) \label{rhs1} \\
\fl 2 \eta_{N} + \max ( 2 P_{N} - 2 R_{n} + T^{(N-1)}_{l, m+1, n} + T^{(N-1)}_{l, m+1, n+1}, \nonumber\\
\fl \quad 2(P_N - Q_N) + T^{(N-1)}_{l, m, n+1} + T^{(N-1)}_{l, m+2, n}, -2 \Omega_{N, n+1} + T^{(N-1)}_{l-1, m+1, n+1} + T^{(N-1)}_{l+1, m+1, n} ) \label{rhs2} \\
\fl 4 \eta_N + 2(P_N - Q_N - \Omega_{N, n+1}) \nonumber\\
\fl \quad + \max ( T^{(N-1)}_{l, m+1, n} + T^{(N-1)}_{l-1, m+2, n+1} - 2R, T^{(N-1)}_{l-1, m+1, n+1} + T^{(N-1)}_{l, m+2, n} ). \label{rhs4}\end{aligned}$$ Here, (\[lhs1\]) and (\[lhs3\]) are identical to (\[rhs1\]) and (\[rhs4\]) because by assumption, $T^{(N-1)}_{l, m, n}$ solves the equation (\[bilinear\]).
By subtracting (\[lhs2\]) from (\[rhs2\]), we obtain $$\label{mid1}
\fl \quad \quad \quad \max \big( 2(\Omega_{N, n+1} - R_{n}), 2(\Omega_{N, n+1} - Q_{N}) + H'^{(N-1)}_{l,m,n}, -2 P_{N} + H''^{(N-1)}_{l,m,n} \big).$$ The third argument of this maximum is non-positive by virtue of Lemma \[Lem3\].
In the case $\Omega_{N,n+1} = Q_{N} \le R_{n}$, $\Omega_{N-1}$ has to be equal to $Q_{N-1}$ due to condition (\[condq\]). Then, the first argument in (\[mid1\]) is also non-positive and the second argument is $0$, due to Lemma \[Lem2\].
In the case $\Omega_{N,n+1}= R_{n}$, the first argument in the maximum in (\[mid1\]) is $0$ and the second argument is non-positive by virtue of Lemma \[Lem2\]. Thus, (\[mid1\]) is equal to $0$, in all possible cases (i.e., $\Omega_{N,n+1} = Q_{N}$ or $\Omega_{N,n+1} = R_{n}$).
We have therefore shown that all arguments of the maximum in (\[ff1\]) which constitutes the left hand side of (\[bilinear\]), have an equivalent counterpart among (\[rhs1\]), (\[rhs2\]), (\[rhs4\]), i.e. among the three arguments that contribute to the right hand side of (\[bilinear\]). Hence, (\[bilinear\]) is satisfied.
Please note that the proof allows for the possibility that, at different values of $n$, $\Omega_{N,n+1}$ satisfies different equalities ($\Omega_{N,n+1}=R_{n}$ or $\Omega_{N,n+1}=Q_{N}$ for different $n$), because the shift of the independent variables induced by (\[exformula\]) affects only $l$ and $m$, not $n$.
Vertex operator for the KP equation
===================================
In this section we propose an alternative representation of the $N$-soliton solutions, generated by a vertex operator $X$.
The $0$-soliton solution $T(;;)$ is written as: $$T(;;) := 0$$ whereas the $N+1$-soliton solution is generated from the $N$-soliton solution $T(P_1, \ldots, P_{N};Q_{1}, \ldots, Q_{N}; C_{1}, \ldots, C_{N})$ (written as $T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C})$ for brevity) by $$\begin{aligned}
\label{symexformula}
\fl\qquad X(P_{N+1}, Q_{N+1}, C_{N+1}) T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C}) \nonumber\\
:= \max ( T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C}), 2\eta_{N+1} + T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C}-\boldsymbol{A}_{N+1}) ) \\
=: T(P_1, \ldots, P_{N}, P_{N+1};Q_{1}, \ldots, Q_{N}, Q_{N+1}; C_{1}, \ldots, C_{N}, C_{N+1}), \end{aligned}$$ where the parameters $P_{N+1}, Q_{N+1}$ in the vertex operator $X$ must satisfy $$(P_{i}-P_{N+1})(Q_{i}-Q_{N+1}) \ge 0.$$ The phase factor $\eta_{N+1}$ is the same as in (\[defeta\]), and the interaction terms $\boldsymbol{A}_{N+1} = {}^t (A_{N+1,1}, \ldots, A_{N+1, N})$ are $$A_{i, j} = \min (P_i, P_j) + \min (Q_i, Q_j).$$
\[sym\] The action of the operator $X$ is commutative.
By calculating $X(\Omega_b, \eta_b) X(\Omega_a, \eta_a) F({\boldsymbol{\Omega}}; {\boldsymbol{\eta}})$ directly, we obtain $$\begin{aligned}
\fl\qquad X(P_b, Q_b, C_b) X(P_a, Q_a, C_a) T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C}) \nonumber\\
\!\!\!\!\!\!\!\! = \max \big( T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C}), 2 \eta_{b} + T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C} - \boldsymbol{A}_{b}), \nonumber\\
2 \eta_{a} + T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C} - \boldsymbol{A}_{a}), 2 \eta_{a} + 2 \eta_{b} - 2 A_{b, a} + T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C} - \boldsymbol{A}_{a} - \boldsymbol{A}_{b}) \big).\end{aligned}$$ From this relation it is clear that interchanging the subscripts $a$ and $b$ does not change the overall value of the maximum. $\square$
Rewriting this proposition yields the following corollary:
\[cor1\] The $N$-soliton solution $T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C})$ is invariant under the permutation of its parameters, i.e.: $$\begin{aligned}
\!\!\!\!\!\! T(P_1, \ldots, P_{N};Q_{1}, \ldots, Q_{N}; C_{1}, \ldots, C_{N}) \nonumber\\
= T(P_{\sigma(1)}, \ldots, P_{\sigma(N)};Q_{\sigma(1)}, \ldots, Q_{\sigma(N)}; C_{\sigma(1)}, \ldots, C_{\sigma(N)}) \quad ( \sigma \in S_{N})\end{aligned}$$
By virtue of corollary \[cor1\], we can fix the labels of the parameters as in (\[condp\]), (\[condq\]) without loss of generality. By virtue of this ordering, the phase shifts in $A_{i, j}$ in the definition (\[symexformula\]) simplify to $$\min(P_{i}, P_{N}) = P_{i},\ \min(Q_{i}, Q_{N}) = Q_{i} \quad (i=1,\ldots,N-1).$$ It should be noted that the phase shifts $\boldsymbol{\eta} \to \boldsymbol{\eta} + \boldsymbol{P}$ and $\boldsymbol{\eta} \to \boldsymbol{\eta} + \boldsymbol{Q}$ are equivalent to shifts on the independent variables $l \to l + 1$ and $m \to m - 1$, which shows that $T(\boldsymbol{P}; \boldsymbol{Q}; \boldsymbol{C})$ is equivalent to $T^{(N)}_{l, m, n}$.
Reduction to various ultradiscrete soliton equations
====================================================
In this section we present some examples of reductions of the ultradiscrete KP equation to $1+1$ dimensional ultradiscrete equations and we give the vertex operators for these equations.
The Box and Ball System and its varieties
-----------------------------------------
By restricting $T_{l,m,n}$ to $$\label{redhKdV}
T_{l, m, n} = F^{l-Mm}_{n}$$ and denoting $s=l-Mm$ and $n=j$, the non-autonomous ultradiscrete KP equation (\[bilinear\]) is reduced to the so-called non-autonomous ultradiscrete hungry KdV equation: $$\label{uhKdVbilinear}
F^{s+M+1}_{j+1} + F^{s}_{j} = \max ( F^{s+M+1}_{j} + F^{s}_{j+1} - 2R_{j}, F^{s+1}_{j} + F^{s+M}_{j+1} ).$$ By means of the dependent variable transformation $$B^{t}_{i,j} = \frac{1}{2} ( F^{s+1}_{j} + F^{s}_{j+1} - F^{s+1}_{j+1} - F^{s}_{j} ),$$ and denoting $s=Mt+i$, (\[uhKdVbilinear\]) is transformed into $$B^{t+1}_{i,j} = \min \Big( R_{j} - \sum_{k=1}^{i-1} B^{t+1}_{k,j} - \sum_{k=i}^{M} B^{t}_{k,j}, \sum_{n=-\infty}^{j-1} ( B^{t}_{i,n} - B^{t+1}_{i,n} ) \Big),$$ which describes the dynamics of a Box and Ball System with $M$ kinds of balls, as presented in [@TTM]. This system is required to satisfy the following boundary conditions: $$\label{bchKdV}
B^{t}_{i,j} = 0 \quad \mbox{for} \quad j\ll 0$$ In particular, in the case of $M=1$ it reduces to an extension of the standard BBS [@TS], with variable size of boxes at each site.
In our representation (\[symexformula\]), the reduction (\[redhKdV\]) is equivalent to the parameter restriction: $$M P_{N} = Q_{N}.$$ It should be noted that our representation satisfies the boundary condition (\[bchKdV\]) because the first argument of $\max$ in (\[symexformula\]) is never chosen for sufficiently small $j$.
Then, the vertex operator for (\[uhKdVbilinear\]) can be written as $$\begin{aligned}
\label{symexformulahKdV}
\fl\qquad X(P_{N+1}, C_{N+1}) T(\boldsymbol{P}; \boldsymbol{C}) \nonumber\\
:= \max ( T(\boldsymbol{P}; \boldsymbol{C}), 2\eta_{N+1} + T(\boldsymbol{P}; \boldsymbol{C}-\boldsymbol{A}_{N+1}) ),\end{aligned}$$ where the phase factor $\eta_{N+1}$ is $$\eta_{N} = C_{N} + s P_{N} - \sum_{0}^{j} \Omega_{N, d},$$ and $\Omega_{N,j}$ and the interaction terms $A_{i, j}$ are expressed as $$\Omega_{N,j} = \min ( R_{j-1}, M P_{N}), \quad A_{i, j} = (M+1) \min (P_i, P_j).$$
The ultradiscrete Toda equation
-------------------------------
By restricting $T_{l,m,n}$ to $$\label{redToda}
T_{l, m, n} = F^{l+n}_{m+n},$$ $R_n = const.$ and denoting $t=l+n$ and $s=m+n$, (\[bilinear\]) is reduced to the ultradiscrete Toda equation: $$\label{uTodabilinear}
F^{t}_{s+1} + F^{t+2}_{s+1} = \max ( F^{t+1}_{s+2} + F^{t+1}_{s} - 2R, 2 F^{t+1}_{s+1} )$$ By means of the dependent variable transformation $$U^{t}_{s} = \frac{1}{2} ( F^{t}_{s+2} - 2 F^{t}_{s+1} + F^{t}_{s} ),$$ (\[uTodabilinear\]) is transformed into $$\fl\ U^{t+2}_{s+1} - 2 U^{t+1}_{s+1} + U^{t}_{s+1} = \max ( U^{t+1}_{s+2} - R, 0 ) - 2 \max ( U^{t+1}_{s+1} -R, 0 ) + \max ( U^{t+1}_{s} - R, 0 ),$$ which describes the dynamics of the Toda type cellular automaton presented in [@MSTTT].
In our representation (\[symexformula\]), the reduction (\[redToda\]) is equivalent to the parameter restriction: $$\Omega_{N} = Q_{N} - P_{N} \qquad \mbox{i.e.} \qquad P_{N} = Q_{N} - \Omega_{N} = \max ( Q_{N} - R, 0 )$$
The vertex operator of (\[uTodabilinear\]) can be expressed as $$\begin{aligned}
\label{symexformulaToda}
\fl\qquad X(P_{N+1}, C_{N+1}) T(\boldsymbol{P}; \boldsymbol{C}) \nonumber\\
:= \max ( T(\boldsymbol{P}; \boldsymbol{C}), 2\eta_{N+1} + T(\boldsymbol{P}; \boldsymbol{C}-\boldsymbol{A}_{N+1}) ),\end{aligned}$$ where the phase factor $\eta_{N+1}$ is $$\eta_{N} = C_{N} + t \max ( Q_{N} - R, 0 )- s Q_{N},$$ and the interaction term $A_{i, j}$ is written as $$A_{i, j} = \min ( Q_{i}, Q_{j} ) + \max ( \min ( Q_{i}, Q_{j} ) - R, 0 )$$
Concluding Remarks
==================
In this paper, we proposed a recursive representation of the $N$-soliton solutions and vertex operators for the ultradiscrete KP equation. We also proposed expressions for various ultradiscrete equations, obtained by reduction from the KP equation.
In fact, the vertex operator approach is closely related to the existence of certain symmetry algebras for integrable systems and the exact relation of our ultradiscrete operator to the symmetries of ultradiscrete systems is an especially interesting problem we want to address in the future.
Because it uses simple shift and $\max$ operators and not the usual algebraic or combinatorial methods, our representation also has the potential to describe solutions different from the solitonic ones. It is an interesting problem to describe the full class of solutions these equations admit.
References {#references .unnumbered}
==========
[10]{}
A. Nagai, T. Tokihiro, and J. Satsuma. Conserved quantities of box and ball system. , 43A:91–97, 2001.
D. Takahashi and J. Satsuma. A soliton cellular automaton. , 59:3514–3519, 1990.
T. Tokihiro, D. Takahashi, J. Matsukidaira, and J. Satsuma. . , 76:3247–3250, 1996.
R. Willox, T. Tokihiro, and J. Satsuma. Nonautonomous discrete integrable systems. , 11:121–135, 2000.
T. Tokihiro, D. Takahashi, and J. Matsukidaira. . , 33:607–619, 2000.
N. Shinzawa and R. Hirota. . , 36:4667–4675, 2003.
D. Takahashi and R. Hirota. Ultradiscrete soliton solution of permanent type. , 76:104007, 2007.
H. Nagai. . , 41:235204 (12pp), 2008.
Y. Nakata. . , 42:412001 (6pp), 2009.
J. Matsukidaira, J. Satsuma, D. Takahashi, T. Tokihiro, and M. Torii. . , 225:287–295, 1997.
|
---
author:
- 'M. Longhetti'
- 'P. Saracco'
- 'S. Cristiani'
- 'A. Fontana'
- 'E. Giallongo'
- 'M. Nonino'
- 'E. Vanzella'
title: 'Tracing the evolution of massive galaxies up to z $\sim$ 3'
---
J-K $>$ 3 galaxies: unveiling M$>10^{11}$M$_{\odot}$ early-type galaxies at z $\sim 3$
======================================================================================
Objects with unusual red near-IR color J-K$>$3 are extremely rare at magnitude brighter than K=20, while their surface density increases at fainter magnitudes. The nature of these sources has not yet been firmly established even if all the analysis performed so far conclude that they are galaxies at z $>$ 2-3 (\[1\], \[2\], \[3\]). A detailed analysis based on a multi-band data set (from 0.3$\mu$m to 2.15$\mu$m) of three J-K$>$3 sources selected at Ks $<$ 22 on the HDF-S (\[4\]) has shown that: i) the three galaxies are at redshift 2.5$<$z$<$3; ii) they have a stellar mass content of M$_{star}\sim 10^{11}$M$_{\odot}$; iii) they would populate the bright end (L$_{z=0} \sim$ L$^{*}$) of the local luminosity function of galaxies even assuming they evolve passively; iv) they cannot follow an evolution significantly different from passive aging.
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Depicting the evolution of massive galaxies from z=3 to z=0
===========================================================
Thus, these J-K$>$3 objects selected are the z$\sim$3 counterpart of local massive early-type galaxies. Between z=3 and z=0 we expect to find the same population of galaxies at a different evolutionary status, namely with older stellar content. For istance, at z=1-1.5 we expect to find early-type galaxies (t$_{z=3}$ - t$_{z=1.5}$) $\approx$ 2 Gyr older. Such old early-type galaxies have been found among the EROs selected with R-K>5 (e.g.\[5\]). In order to find their counterpart at the intermediate redshift z$\sim$2, we have identified a suitable color selection criterion by exploring the the expected colors evolution as derived by synthetic models (\[6\]). The results are displayed in Fig. 1: for short SF time scale ($\tau <$ 1Gyr galaxies at z$\sim$2 have colors J-K$<$3 and I-H$>$4. In fact, adopting these selection criteria on the HDF-S data, we identified 4 massive evolved galaxies at 1.5$<z<$2.5 which seem to be the link between the J-K$>$3 galaxies (z$\sim3$) and the EROs (z$\sim$1). An example of this passive evolution connection is proposed in the right panel of Fig. 1: 822 (upper panel) a J-K$>$3 galaxy selected in the HDF-S well described by a young SSP at z$\sim$2.9; 1203 (middle panel) one of the four massive (M$_{star} \ge 10^{11}$ M$_{\odot}$) galaxies selected in the HDF-S on the basis of the selection criteria above described, well described by a 1.4 Gyr old SSP at z$\sim$1.8; S7F5\_254 (lower panel) the spectrum of a massive early-type galaxy at z=1.2 (\[5\]) superimposed on the template of a 3 Gyr old SSP.
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-0.5 truecm
The photometry, the stellar masses and the mean ages of the stellar populations of these three massive galaxies at different redshift are apparently those we expect from a massive galaxy fully assembled at z$\sim$3 which evolves passively in time. This result is based, in fact, on a couple of objects and may be simply a coincidence. However, it is worthy to note that the number density of the three classes of objects represented in the figure and that of the local massive galaxies are well compatible with the depicted scenario. Actually, the number density of J-K$>$3 and K$<$22 galaxies (2.5$<$z$<$3, M$_{star} \ge 10^{11}$ M$_{\odot}$ galaxies) is $\rho=1.2 \times 10^{-4}$ Mpc$^{-3}$ (\[4\]) and that of J-K$<$3 and I-H$>$4 (1.5$<$z$<$2.5) calculated from the HDF-S data results to be the same within the errors. The number density of the early-type galaxies at 1.0$<$z$<$1.5 (R-K$>$5) is consistent with the previous value once the same stellar mass threshold (M$_{star} \ge 10^{11}$ M$_{\odot}$) is taken into account (\[4\]). The local density of M$_{star} \ge 10^{11}$ M$_{\odot}$ elliptical galaxies is $\rho \simeq 3 \times 10^{-4}$ Mpc$^{-3}$ (\[7\]): at least 40% of the local population of bright massive early-type galaxies can be explained as the results of a simple passive evolution started from galaxies already assembled at z$\sim$3.
-0.3 truecm
[1.]{}
-0.3 truecm M. Dickinson, C. Hanley, R. Elston, et al. 2000, ApJ, 531, 624 P.B. Hall, M. Sawicki, P. Martini, et al. 2001, ApJ, 121, 1840 T. Totani, Y. Yoshii, F. Iwamuro, et al. 2001, ApJ, 558, L87 P. Saracco, M. Longhetti, E. Giallongo, et al. 2003, astro-ph/0310131 P. Saracco, M. Longhetti, P. Severgnini, et al. 2003, A&A, 398, 127 M. Longhetti, P. Saracco, E. Giallongo, et al. 2004, in preparation R.O. Marzke, L.N. Da Costa, P.S. Pellegrini, et al. 1998, ApJ, 503,617
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---
abstract: 'Pixel sensors based on commercial high-voltage CMOS processes are an exciting technology that is considered as an option for the outer layer of the ATLAS inner tracker upgrade at the High Luminosity LHC. Here, charged particles are detected using deep n-wells as sensor diodes with the depleted region extending into the silicon bulk. Both analog and digital readout electronics can be added to achieve different levels of integration up to a fully monolithic sensor. Small scale prototypes using the ams CMOS technology have previously demonstrated that it can achieve the required radiation tolerance of and detection efficiencies above . Recently, large area prototypes, comparable in size to a full sensor, have been produced that include most features required towards a final design: the H35demo prototype produced in ams H35 technology that supports both external and integrated readout and the monolithic ATLASPix1 pre-production design produced in ams aH18 technology. Both chips are based on large fill-factor pixel designs, but differ in readout structure. Performance results for H35DEMO with capacitively-coupled external readout and first results for the monolithic ATLASPix1 are shown.'
address:
- ' Département de Physique Nucléaire et Corpusculaire, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland'
- ' Argonne National Laboratory, Argonne, IL 60439, USA'
- ' Brookhaven National Laboratory, P.O. Box 5000, Upton, NY 11973-5000, USA'
- ' Institut de Física d’Altes Energies, The Barcelona Institute of Science and Technology, Edifici CN, UAB campus, 08193 Bellaterra (Barcelona), Spain'
- ' Albert Einstein Center for Fundamental Physics and Laboratory for High Energy Physics, University of Bern, Siedlerstrasse 5, CH-3012 Bern, Switzerland'
- ' Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China'
- ' Department of Physics, University of Liverpool, The Oliver Lodge Laboratory, Liverpool L69 7ZE, UK'
- ' Karlsruhe Institute of Technology, IPE, 76021 Karlsruhe, Germany'
- ' University of Oklahoma, 660 Parrington Oval, Norman, OK 73019, USA'
- ' University of Illinois Urbana Champaign, 1110 W Green St Loomis Laboratory, Urbana, IL 61801, USA'
author:
- Moritz Kiehn
- Francesco Armando Di Bello
- Mathieu Benoit
- Raimon Casanova Mohr
- Hucheng Chen
- Kai Chen
- 'Sultan D.M.S.'
- Felix Ehrler
- Didier Ferrere
- Dylan Frizell
- Sergio Gonzalez Sevilla
- Giuseppe Iacobucci
- Francesco Lanni
- Hongbin Liu
- Claudia Merlassino
- Jessica Metcalfe
- Antonio Miucci
- Ivan Peric
- Mridula Prathapan
- Rudolf Schimassek
- Mateus Vicente Barreto
- Thomas Weston
- Eva Vilella Figueras
- Alena Weber
- Michele Weber
- Winnie Wong
- Weihao Wu
- Ettore Zaffaroni
- Hui Zhang
- Matt Zhang
bibliography:
- 'references.bib'
title: Performance of CMOS pixel sensor prototypes in ams H35 and aH18 technology for the ATLAS ITk upgrade
---
ATLAS ITk upgrade, High Luminosity LHC, Silicon pixel sensor, Monolithic active pixel sensor, CMOS, HV-MAPS
Introduction
============
Tracking detectors at future colliders have to fulfil increasingly demanding requirements. Their environment will contain a higher density of tracks, at higher rate, with a higher radiation dose over their lifetime compared to current detectors. This necessitates high granularity detectors even far away from the interaction point and consequently large instrumented surface areas. The planned ATLAS Inner Tracker (ITk) upgrade [@ATLAS-TDR-025; @ATL-COM-ITK-2017-073] is one such detector that is designed to withstand the environment at the planned high luminosity large hadron collider. Sensors for the outer layer of the ATLAS ITk pixel tracker have to cope with an integrated radiation dose of up to over its lifetime.
Silicon pixel detectors are the only technology that can provide the granularity, rate capability, and radiation hardness. Traditionally they are implemented as hybrid pixel sensors. Traversing particles generate free charges in a fully depleted passive sensor diode. Pixelated electrodes on the sensor are connected via bump-bonding to a dedicated readout chip. The readout chip contains both analog amplifiers and digital processing logic and handles the triggering and readout.
While this separation of concerns allows each component to be optimized separately it also introduces additional challenges. A large charge signal is required to generate a large enough voltage signal over the total capacitance of the sensor diode and the readout electronics. This necessitates a thick, fully depleted sensor diode with a high voltage bias to deplete it. As an example, the central modules used in the ATLAS IBL detector are built from thick planar sensors, with bias voltages foreseen to reach , bump-bonded to a thick readout chips [@ATLAS-TDR-19]. The complexity of the production process, i.e. separate sensor and readout production and hybridization, can be a limiting factor to production capabilities. Yield factors compound and due to the many steps involved the production can not easily scale to large number of sensors and large instrumented surface areas.
Pixel sensors based on CMOS technology enable simpler devices with a reduced material budget by integrating some or all of the readout electronics directly into one chip. Commercial production enables cheap sensors with greatly simplified production complexity, suitable to instrument large surface areas with high granularity sensors.
In the remainder of this paper, basic concepts are introduced and different implementations of CMOS-based pixel sensors using ams technology are presented. Then, specific prototypes and the status of ongoing prototype evaluations are discussed.
ams CMOS prototypes
===================
The ams technology is a set of commercial CMOS processes that are available in and structure sizes [@amshvcmos]. All processes can use nMOS and pMOS transistors and support high voltages of up to . High-resistivity substrates up to are also available.
Previous small-scale prototypes based on this technology used integrated amplifiers and comparators in combination with an external readout chip. They could be operated with efficiencies above for fluences up to using both neutron and proton irradiation [@ccpdv4], demonstrating the radiation hardness of this technology. Other small-scale prototypes demonstrated the feasibility of integrating different levels of readout logic onto the same chip [@Augustin:2015mqa; @Augustin:2016hzx].
![Cross-section of the pixel implants for the H35DEMO prototype along the large pitch direction. Three separate sensor diodes (deep n-well labeled as dntub) are connected together to reduce the total sensor capacitance while maintaining a homogeneous depletion. The additional deep p-type implant (labeled as dptub) underneath the nMOS components is optional and only present in some parts of the test matrices.[]{data-label="fig:h35demo_implants"}](h35demo_pixel_implants){width="\linewidth"}
![Cross-section of the pixel implants for ATLASPix1 prototype along the large pitch direction. The additional deep p-type implant (labeled as dptub) is only present in the IsoSimple matrix.[]{data-label="fig:atlaspix1_implants"}](atlaspix1_pixel_implants){width="\linewidth"}
Two large-scale pixel sensor prototypes, H35DEMO and ATLASPix1, have been recently produced to show the viability of this technology for full-sized production sensors as an upgrade option for ATLAS ITk and to test different aspects of the technology. All prototypes are based on so-called large fill-factor designs. The large fill-factor refers to the size of the sensor diode compared to the pixel pitch. In these designs, the pixel electronics are located inside a deep n-type implant in a p-type substrate. The deep n-type implant and the substrate form the sensor diode that is depleted by applying a bias voltage. Usually bias from the top side is used; back-bias could also be employed but requires additional back-side processing. Figure \[fig:h35demo\_implants\] shows a cross-section of the pixel implants for the H35DEMO prototype along the large pitch direction. Here, three separate sensor diodes are connected together to reduce the total sensor capacitance while maintaining a homogeneous depletion. Figure \[fig:atlaspix1\_implants\] shows the equivalent pixel implants cross-section for the ATLASPix1 prototype. Due to a smaller size only a single diode implantation is used.
![The signal path on the H35demo prototype. The two amplifier stages before the glue interface are implemented in ams H35 CMOS technology on the prototype chip. The glue interface capacitively couples the amplified signal to the FE-I4 readout chip.[]{data-label="fig:h35demo_signal_path"}](h35demo_fei4_signal_path){width="\linewidth"}
H35DEMO is produced in H35 technology with a total size of and a pixel pitch of [@h35demo:design]. It comprises four independent matrices: two monolithic matrices with integrated readout and two analog matrices that integrate only per-pixel amplifiers. On the first monolithic matrix each pixel contains amplifiers and comparator while on the second matrix only the amplifiers are located in the pixel and the comparator is located in the chip periphery. Evaluation results for the two monolithic matrices were previously reported by @Cavallaro:2016gmx and @Terzo:2017hlv. The analog matrices provide an amplified analog signal that is then readout by a capacitively coupled FE-I4 readout chip as shown in figure \[fig:h35demo\_signal\_path\]. The two analog matrices differ in the layout of their in-pixel electronics. Within each analog matrix there are different submatrices with additional small variations of the integrated electronics.
![The signal path on the ATLASPix1 prototype. All components are implemented in ams aH18 CMOS technology on the prototype chip. The amplifier and the discriminator are located inside the pixel. Each pixel is connected with dedicated lines to the periphery of the chip which contains the remaining readout components.[]{data-label="fig:atlaspix1_signal_path"}](atlaspix1_signal_path){width="\linewidth"}
ATLASPix1 is a monolithic prototype with integrated readout logic and a total size of approximately produced in aH18 technology. It comprises three independent matrices — M2, Simple and IsoSimple — that differ in readout architecture and pixel pitch. M2 has a pixel pitch of and uses a triggered readout architecture. Simple and IsoSimple have a pixel pitch of and use an untriggered column-drain architecture. The IsoSimple matrix has an additional isolation p-well, as shown in figure \[fig:atlaspix1\_implants\], and use full CMOS transistor in its comparator logic. The Simple matrix only uses nMOS logic and misses the isolation p-well. As shown in figure \[fig:atlaspix1\_signal\_path\], all matrices have the per-pixel amplifier and comparator located inside each pixel which are connected to additional logic in the digital periphery.
Beam tests
==========
The operational performance of both prototypes was tested in beam test setups. Measurements with the H35DEMO were performed in spring 2017 at the Fermilab MTEST facility using the primary proton beam and in summer 2017 at the CERN north area beam facilities using a secondary mixed hadron beam. Measurements with the ATLASPix1 prototype were performed in October 2017 at the CERN north area facilities also using a secondary mixed hadron beam.
Reference tracks were measured using the Geneva beam telescope. It uses six ATLAS IBL modules with a pixel pitch of to measure particle positions [@genevatelescope]. The sensors are arranged in an optimized geometry with rotated and inclined planes to maximize charge sharing and enhance the resolution. The telescope and FE-I4-based devices-under-test are readout using the RCE/HSIO2 data acquisition system [@genevatelescope]. Triggers are provided by hit coincidences on the first and the last plane with a variable integration time between 8 and 16 time bins of .
The H35DEMO prototypes with capacitively coupled FE-I4 readout sensors can be read out directly using the RCE/HSIO2 system. The ATLASPix1 is operated with an independent data acquisition system based on a commercial Xilinx Nexys FPGA board in combination with a custom control and readout board developed at KIT. It is controlled via USB2 using a custom software. Integration into the telescope system is achieved via a trigger-busy scheme. The trigger signal from the telescope is used by the FPGA to write data only for triggered events into a separated stream. Synchronisation between the two systems happens offline based on trigger counting and timestamps.
Particle tracks are reconstructed with the Proteus reconstruction software [@proteus] using only the data from the telescope planes. Proteus performs hit clustering, weighted center-of-gravity position, and detector alignment. Initial coarse alignment is based on hit correlations and subsequent fine alignment uses track residuals. Reconstructed track positions and the full reconstruction covariance matrix are propagated into the local system of the device-under-test and are then used to calculated residuals and efficiency.
H35DEMO results
===============
A systematic evaluation of the performance of the two H35DEMO analog matrices has been performed. In this configuration only analog electronics, i.e. signal amplifiers and shapers, but no digital readout logic is integrated on the CMOS chip. It can therefore be used to test the performance of the sensor and the analog electronics independent of the readout logic by using the well-tested FE-I4 readout chip. Detailed evaluation results for the H35DEMO analog matrices have previously been reported by @h35demo:results. Only a high-level overview is provided here to give the reader a full picture of all ams CMOS prototypes.
![H35demo efficiency measurements for one sub-matrix of the first analog matrix for different resistivities and thresholds. [@h35demo:results][]{data-label="fig:h35demo_ana1"}](h35demo-eff_ana1_comparison_matrix1){width="\linewidth"}
Figure \[fig:h35demo\_ana1\] shows the resulting global efficiency measured for one submatrix of the first analog matrix for different device configurations as a function of the sensor bias voltage. The threshold quoted is the threshold set on the coupled FE-I4 readout chip. Since the initial signal is already amplified and shaped by the integrated amplifiers on the H35DEMO the effective threshold in terms of generated charge in the sensor is smaller. As expected, a clear dependence is seen on both substrate resistivity and threshold. A higher substrate resistivity leads to larger signal and therefore to a higher efficiency for the same threshold and bias. The same is true for lower threshold. For both shown substrate resistivities an operational region with efficiencies above can be found. The measured efficiency is comparable to or better than results from traditional hybrid modules with bump-bonded passive sensors and fulfil the efficiency requirement of for the outer layer of the ATLAS ITk upgrade [@ATL-COM-ITK-2017-073].
ATLASPix1 results
=================
![ATLASPix1 Simple matrix matching residuals for the the long u and the short v pixel axis for a sensor bias of and a threshold of . Residuals are calculated between the extrapolated particle positions on the device-under-test as reconstructed by the beam telescope and the center-of-gravity of the cluster. The fitted function is a box function of width l smeared with a Gaussian of with $\sigma$ and a polynomial background.[]{data-label="fig:atlaspix1_residuals"}](atlaspix1_kit01_simple-hv65_thr840-match_res){width="\linewidth"}
![ATLASPix1 simple matrix efficiency scan as a function of the global threshold for a fixed sensor bias of . The threshold baseline is . The error bars represent only the statistical uncertainties. Preliminary calibrations indicate that the measured threshold range corresponds to an equivalent charge range of approximately .[]{data-label="fig:atlaspix1_efficiency"}](atlaspix1_kit01_simple-scan_threshold-hv65-eff){width="\linewidth"}
Given the availability of prototype samples and time constraints measurements could only be performed for the ATLASPix1 Simple matrix of one unirradiated sample with a substrate resistivity of . These measurements test both the analog performance of the sensor and the integrated readout logic.
Figure \[fig:atlaspix1\_residuals\] shows the residuals between reconstructed track position and estimated cluster position on the device-under-test for one particular configuration. A matching cut of along both dimensions is used to associate tracks to clusters. The fitted function models the expected response from a pixel of width $l$ assuming a telescope resolution of $\sigma$. The vanishing mean $\mu$ along both axis shows that the system is well-aligned. The fitted pixel width along the long u direction is consistent with the pitch. Along the short direction v the fitted width is smaller than the pitch as a result of increased charge sharing. In both cases the background fraction $f_{\text{bkg}}$ due to mismatches or noise hits is negligible.
Using tracks and clusters within the matching cut, the global efficiency can be calculated as the fraction of tracks with an associated cluster and the total number of tracks. Tracks are only considered within a selected region-of-interest in the central part of the sensors. This avoids edge effects, low statistic regions, and some pixels that were incorrectly tuned. No other cuts or masks are used inside the region-of-interest. The resulting efficiencies as a function of the global threshold are shown in figure \[fig:atlaspix1\_efficiency\]. The efficiency stays almost constant and with the majority of the measurements above for the selected threshold range. The error bars represent only the statistical uncertainty and do not include systematic effects e.g. known-bad pixels and tuning effects. Additional measurements indicate that the efficiency drops to approximately at a threshold of .
Summary
=======
Multiple prototype sensors for the ATLAS ITk upgrade using ams H35 and aH18 CMOS technology have been produced. The H35DEMO is a large scale prototype that can be operated both as a monolithic system with an integrated readout and as a hybrid sensor with a capacitively coupled FE-I4 readout chip. The ATLASPix1 is a first large scale, monolithic prototype designed specifically for the ATLAS ITk upgrade. Both prototypes could be operated in a beam test setup with efficiencies above . This shows that large scale, CMOS pixel sensors in ams technology can be build and operated with a variety of readout options and are a suitable option for the outer layer of the ATLAS ITk upgrade. The ATLASPix1 results only show its basic functionality. Additional performance increases due to better tuning and higher substrate resistivity are expected in the future. Systematic measurements of irradiated ATLASPix1 prototypes with a variety of irradiation sources are currently ongoing to verify previous measurements of radiation hardness with small prototypes [@ccpdv4].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors gratefully acknowledge the support by the CERN PS and SPS instrumentation team and Fermilab Test Beam Facilities. The authors thank Andreas Nürnberg and Dominik Dannheim for fruitful discussions and collaboration. The research presented in this paper was supported by the SNSF grants 20FL20\_173601, 200021\_169015 and 200020\_169000. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 675587.
|
---
abstract: 'This paper deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley (Pitt) inequalities in the noncommutative context. We establish Hörmander’s $L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p \leq 2 \leq q<\infty$ as an application of Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conjugacy classes of compact Lie groups and for a class of countable compact hypergroups.'
address:
- 'Vishvesh Kumar Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium'
- ' Michael Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Belgium and School of Mathematics Queen Mary University of London United Kingdom '
author:
- Vishvesh Kumar
- Michael Ruzhansky
title: 'Hardy-Littlewood inequality and $L^p$-$L^q$ Fourier multipliers on compact hypergroups'
---
Introduction
============
The inequalities which involve functions and their Fourier coefficients played a pivotal role in Fourier analysis as well as in its applications to several different areas. This paper contributes to some of the classical inequalities of this nature, namely, Hardy-Littlewood inequality, Paley inequality and Hausdorff-Young-Paley inequality, and their applications to the theory of Fourier multiplier in the non-commutative setting. The first inequality we consider is the Hardy-Littlewood inequality proved by Hardy and Littlewood for the torus $ \mathbb{T}$ ([@Hardy]). They proved that for each $ 1 \leq p \leq 2$ there exist a constant $C_p>0$ such that $$\left( \sum_{n \in \mathbb{Z}} |\widehat{f}(n)|^p\, (1+|n|)^{p-2} \right)^{\frac{1}{p}} \leq C_p \|f\|_{L^p(\mathbb{T})},\,\quad f \in L^p(\mathbb{T}).$$ Hewitt and Ross [@HR74] extended this inequality for compact abelian groups using the structure theory of groups. Recently, the second author with his coauthors explored the non-commmutative version of the Hardy-Littlewood inequality in the setting of compact homogeneous spaces [@ARN1; @ARN] and compact quantum groups [@AMR] (see also [@Youn]). The Hardy-Littlewood inequality also has an application to Sobolev embedding theorems and to the boundedness of Fourier multipliers [@Youn; @Benyi; @ARN]. Compact Riemannian spaces can be viewed as homogeneous spaces of compact Lie groups. It is well-known that the spherical analysis on Riemannian symmetric spaces is interconnected with the analysis on the double coset spaces which are special examples of hypergroups for which a convolution structure can be defined on the space of all bounded Borel measures. Our goal is to investigate Hardy-Littlewood, Paley and Hausdorff-Young-Paley inequalities and their applications to the boundedness of Fourier multipliers in the context of compact hypergroups. The results of this paper are not only applicable to compact double coset spaces but also to the large class of other examples, for instance, the space of group orbits, space of conjugacy classes of compact (Lie) groups and countable compact hypergroups [@Bloom]. In particular, the results of this paper are also true for several interesting examples including Jacobi hypergroups with Jacobi polynomials as characters [@Gasper], compact hypergroup structure on the fundamental alcove with Heckman-Opdam polynomials as characters [@HRos] and multivariant disk hypergroups [@RV; @AnT]
Hewitt and Ross [@HR74] used structure theory of compact abelian groups and in [@ARN], the authors used the eigenvalue counting formula for Laplace operator on compact manifolds to derive Hardy-Littlewood inequality. When working with compact hypergroups, we do not have such luxury. In this case, we obtain the following Hardy-Littlewood inequality.
\[HLGintro\] Let $1 <p \leq 2$ and let $K$ be a compact hypergroup. Assume that a sequence $\{\mu_\pi \}_{\pi \in \widehat{K}}$ grows sufficiently fast, that is, $$\sum_{\pi \in \widehat{K}} \frac{k_\pi^2}{|\mu_\pi|^\beta}<\infty\,\,\,\, \text{for some}\,\beta \geq 0.$$ Then we have $$\sum_{\pi \in \widehat{K}} k_\pi^2 |\mu_\pi|^{\beta (p-2)} \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi}} \right)^p \lesssim \|f\|_{L^p(K)}.$$
In the case when $K$ is the hypergroup of conjugacy classes of compact Lie group $\text{SU(2)}$ then Theorem \[HLGintro\] gives the following Hardy-Littlewood inequality the commutative hypergroup $\textnormal{Conj(SU)(2)},$ which is a natural analogue of Hardy-Littlewood inequality for $\mathbb{T}.$
\[HLineqintro\] If $1 < p \leq 2$ and $f \in L^p(\textnormal{Conj(SU)(2)}),$ then we have $$\label{in4}
\sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^{p} \leq C_p \|f\|_{L^p(\textnormal{Conj(SU)(2)})}.$$
The inequality can be interpreted in the following form similar to the Hardy-Littlewood inequality on $\mathbb{T}$: $$\label{in5}
\sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{5(p-2)} (2l+1)^2 |\widehat{f}(l)|^{p} \leq C_p \|f\|_{L^p(\textnormal{Conj(SU)(2)})}.$$ In contrast to the case of $\mathbb{T},$ an extra term $(2l+1)^2$ appears in the above Inequality . But this is natural as the Plancherel measure $\omega$ on $\frac{1}{2}\mathbb{N}_0,$ the dual of $\textnormal{Conj(SU)(2)},$ is given by $\omega(l)= (2l+1)^{2}$ for $l \in \frac{1}{2} \mathbb{N}_0$ while for $\mathbb{T},$ the Plancherel measure of the dual group $\mathbb{Z}$ is the counting measure.
\[corhl\] If $2 \leq p <\infty$ and $\sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^{p}<\infty$ then $$f \in L^p(\textnormal{Conj(SU)(2)}).$$ Moreover, we have $$\|f\|_{L^p(\textnormal{Conj(SU)(2)})} \leq C_p \sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^{p}.$$
For $p=2,$ Theorem \[HLineqintro\] and Corollary \[corhl\] boil down to the Plancherel theorem for the hypergroup $\textnormal{Conj(SU)(2)}.$ Therefore, these follow the philosophy of Hardy and Littlewood [@Hardy] who argue that Hardy-Littlewood inequality is a suitable extension of the Plancherel theorem in the case of $\mathbb{T}.$
Another set of interesting examples of the commutative infinite hypergroups which we will investigate is the family of countable compact hypergroups studied by Dunkl and Ramirez [@dun2]. Recently, in [@KKA; @KKAadd] first author with Singh and Ross studied classification results of such classes of hypergroups arising from the discrete semigroups with applications to Ramsey theory [@KumarRamsey]. Interestingly, the property of being countable infinite and compact simultaneously is a purely hypergroups theoretical property as any infinite compact group can never be countable. We also obtain the following analogue of the Hardy-Littlewood inequality for this class of hypergroups $H_a$.
The Hardy-Littlewood inequality is obtained by the following Paley-type inequality for compact hypergroups.
Let $K$ be a compact hypergroup and let $1<p \leq 2.$ If $\varphi(\pi)$ is a positive sequence over $\widehat{K}$ such that the quantity $$M_\varphi:= \sup_{y>0} y \sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \geq y}} k_\pi^2$$ is finite, then we have $$\left(\sum_{\pi \in \widehat{K}} k_\pi^2 \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi}} \right)^p \varphi(\pi)^{2-p} \right)^{\frac{1}{p}} \lesssim M_\varphi^{\frac{2-p}{p}} \|f\|_{L^p(K)}.$$
The Paley-type inequality describes the growth of the Fourier transform of a function in terms of its $L^p$-norm. Interpolating the Paley-inequality with the Hausdorff-Young inequality one can obtain the following Hörmander’s version of the Hausdorff-Young-Paley inequality, $$\label{3}
\left(\int\limits_{\mathbb{R}^n}|(\mathscr{F}f)(\xi)\phi(\xi)^{ \frac{1}{r}-\frac{1}{p'} }|^r\, d \xi \right)^{\frac{1}{r}}\leq \Vert f \Vert_{L^p(\mathbb{R}^n)},\,\,\,1<p\leq r\leq p'<\infty, \,\,1<p<2.$$ Also, as a consequence of the Hausdorff-Young-Paley inequality, Hörmander [@Hormander1960 page 106] proves that the condition $$\label{4}
\sup_{t>0}t^b\{\xi\in \mathbb{R}^n:m(\xi)\geq t\}<\infty,\quad \frac{1}{p}-\frac{1}{q}=\frac{1}{b},$$where $1<p\leq 2\leq q<\infty,$ implies the existence of a bounded extension of a Fourier multiplier $T_{m}$ with symbol $m$ from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n).$ Recently, the second author with his collaborator R. Akylzhanov extended Hörmander’s classical results to unimodular locally compact groups and to homogeneous spaces [@ARN; @AR]. In [@AR], the key idea behind the extension of Hörmander theorem is the reformulation of this theorem as follows: $$\|T_m\|_{L^p(\mathbb{R}^n) \rightarrow L^q(\mathbb{R}^n)} \lesssim \sup_{s>0} s\left( \int_{ \{\xi \in \mathbb{R}^n :\, m(\xi) \geq s\} } d\xi \right)^{\frac{1}{p}-\frac{1}{q}} \simeq \|m\|_{L^{r, \infty}(\mathbb{R}^n)} \simeq \|T_m\|_{L^{r, \infty}(\textnormal{VN}(\mathbb{R}^n))},$$ where $\frac{1}{r}=\frac{1}{p}-\frac{1}{q},$ $\|m\|_{L^{r, \infty}(\mathbb{R}^n)}$ is the Lorentz norm of $m,$ and $\|T_m\|_{L^{r, \infty}(\textnormal{VN}(\mathbb{R}^n))}$ is the norm of the operator $T_m$ in the Lorentz space on the group von Neumann algebra $\textnormal{VN}(\mathbb{R}^n)$ of $\mathbb{R}^n.$ Then one can use the Lorentz spaces and group von Neumann algebra technique for extending it to general locally compact unimodular groups. The unimodularity assumption has its own advantages such as existence of the canonical trace on the group von Neumann algebra and consequently, Plancherel formula and the Hausdorff-Young inequality. It was also pointed out that the unimodularity can be avoided by using the Tomita-Takesaki modular theory and the Haagerup reduction technique.
By interpolating the Hausdorff-Young inequality and Paley-type inequality we get the following Hausdorff-Young-Paley inequality for compact hypergroups.
Let $K$ be a compact hypergroup and let $1<p \leq b \leq p'<\infty.$ If a positive sequence $\varphi(\pi), \pi \in \widehat{K},$ satisfies the condition $$M_\varphi:= \sup_{y>0} y \sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \geq y}} k_\pi^2 <\infty$$ then we have $$\left(\sum_{\pi \in \widehat{K}} k_\pi^2 \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi}}\varphi(\pi)^{\frac{1}{b}-\frac{1}{p'}} \right)^b \right)^{\frac{1}{b}} \lesssim M_\varphi^{\frac{1}{b}-\frac{1}{p'}} \|f\|_{L^p(K)}.$$
Throughout the paper, we denote by $\mathbb{N}$ the set of natural numbers and set $\mathbb{N}_0=\mathbb{N} \cup \{0\}.$ For notational convenience, we take empty sums to be zero.
Preliminaries
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For the basics of compact hypergroups one can refer to standard books, monographs and research papers [@Dunkl; @Jewett; @Bloom; @Ken; @Ken2; @VremD; @Vrem]. However we mention here certain results we need.
Definitions and representations of compact hypergroups
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In [@Jewett], Jewett refers to hypergroups as convos. We begin this section with the definition of a compact hypergroup.
A compact [*hypergroup*]{} is a non empty compact Hausdorff space $K$ with a weakly continuous, associative convolution $*$ on the Banach space $M(K)$ of all bounded regular Borel measures on $K$ such that $(M(K), *)$ becomes a Banach algebra and the following properties hold:
1. For any $x,y \in K,$ the convolution $\delta_x*\delta_y$ is a probability measure with compact support, where $\delta_x$ is the point mass measure at $x.$ Also, the mapping $(x,y)\mapsto {\textnormal{supp}}(\delta_x*\delta_y)$ is continuous from $K\times K$ to the space $\mathcal{C}(K)$ of all nonempty compact subsets of $K$ equipped with the Michael (Vietoris) topology (see [@mi] for details).
2. There exists a unique element $e \in K $ such that $\delta_x*\delta_e=\delta_e*\delta_x=\delta_x$ for every $x\in K.$
3. There is a homeomophism $x \mapsto \check{x}$ on $K$ of order two which induces an involution on $M(K)$ where $\check{\mu}(E)= \mu (\check{E})$ for any Borel set $E,$ and $e \in {\textnormal{supp}}(\delta_x*\delta_y)$ if and only if $x = \check{y}.$
Note that the weak continuity assures that the convolution of bounded measures on a hypergroup is uniquely determined by the convolution of point measures. A compact hypergroup is called a commutative compact hypergroup if the convolution is commutative. A compact hypergroup $K$ is called [*hermitian*]{} if the involution on $K$ is the identity map, i.e., $\check{x}=x$ for all $x \in K.$ Note that a hermitian hypergroup is commutative. Every compact group is a trivial example of a compact hypergroup. Other essential and non-trivial examples are double coset hypergroups $G//H$ asing from a Gelfand pair $(G, H)$ for a compact group $G$ and a closed subgroup $H$ [@Jewett], conjugacy classes of compact Lie groups [@Vrem; @Bloom], countable compact hypergroups [@dun2; @Bloom], Jacobi hypergroups [@Gasper4; @Bloom], hypergroup joins [@Vrem2] of compact hypergroups by finite hypergroups [@Alaga; @Bloom].
A [*left Haar measure*]{} $\lambda$ on $K$ is a non-zero positive Radon measure such that $$\int_K f(x*y) d\lambda(y)=\int_K f(y)\, d\lambda(y)\quad (\forall x \in K, \, f \in C_c(K)),$$ where we used the notation $f(x*y)=(\delta_x*\delta_y)(f)$. It is well known that a Haar measure is unique if it exists [@Jewett]. Throughout this article, a left Haar measure is simply called a Haar measure. We would like to make a remark here that it still not known if a general hypergroup has a Haar measure but several important class of hypergroups including commutative hypergroups, compact hypergroups, discrete hypergroups, nilpotent hypergroups possess a Haar measure [@Jewett; @Bloom; @Willson; @Amini].
An [*irreducible representation*]{} $\pi$ of $K$ is an irreducible $*$- algebra representation of $M(K)$ into $\mathcal{L}(\mathcal{H}_\pi),$ the algebra of all bounded linear operators on some Hilbert space $\mathcal{H}_\pi,$ such that
- $\pi(\delta_e)=I$ and
- for every $u,v\in\mathcal{H}_\pi,$ the mapping $\mu\mapsto\langle \pi(\mu)u,v \rangle$ is continuous from $M(K)^+$ to $\mathbb{C},$ where $M(K)^+$ is equipped with the weak (cone) topology.
In [@Jewett] it was also included in the definition of a representation that $\pi$ must be norm decreasing, that is, $\|\pi(\mu)\|_{\text{op}} \leq \|\mu\|,$ but it follows as a consequence of the above definition. For any $x \in K,$ we also write $\pi(\delta_x)$ as $\pi(x).$ Therefore, we get $ \|\pi(x)\|_{\text{op}} \leq \|\delta_x\|=1,$ where $\|\cdot\|_{\text{op}}$ denotes the operator norm on $\mathcal{L}(\mathcal{H}_{\pi}).$
Fourier analysis on compact hypergroups
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Let $K$ be a compact hypergroup with the normalized Haar measure $\lambda$ and let $ \widehat{K}$ be the set of irreducible inequivalent continuous representations of $K.$ Throughout this paper we will assume that $K$ is metrizable which is equivalent the condition that $\widehat{K}$ is countable [@FR]. The set $\widehat{K}$ equipped with the discrete topology is called the dual space of $K$. Vrem [@Vrem] showed that every irreducible representation $(\pi, \mathcal{H}_{\pi})$ of a compact hypergroup is finite dimensional. For any $\pi \in \widehat{K},$ the map $x \mapsto \langle \pi(x)u, v \rangle$ for $u, v \in \mathcal{H}_{\pi}$ is called matrix coefficient function and is denoted by $\pi_{u,v}.$ Let $\pi(x)= [\pi_{i,j}]_{d_{\pi} \times d_{\pi}}$ be the matrix representation of any $(\pi, \mathcal{H}_{\pi})$ of dimension $d_{\pi}$ with respect to an orthonormal basis $\left\lbrace e_i \right\rbrace_{i=1}^{d_{\pi}}$ of $\mathcal{H}_{\pi}.$ For each pair $\pi, \pi \in \widehat{K}$ there exists a constant $k_{\pi} \geq d_{\pi}$ such that $$\label{ortho}
\int_K \pi_{i,j}(x) \overline{\pi'_{m,l}(x)}\, d\lambda(x) = \begin{cases} \frac{1}{k_{\pi}} & \text{when}\, i=m, j=l, \, \text{and}\, \pi = \pi', \\ 0 & \text{otherwise}.
\end{cases}$$ If $K$ is a compact group then $k_{\pi} = d_{\pi}$ [@Vrem Theorem 2.6]. The constant $k_\pi$ is called the hyperdimension of the representation $\pi$ [@Alaga]. The function $x \mapsto \chi_\pi(x)=:\text{Tr}(\pi(x))$ is called (hypergroup) [*character*]{} and it is a continuous function. The following relation for characters can be derived from orthogonality relation of matrix coefficients $$\label{orch}
\int_K \chi_{\pi}(x) \overline{\chi_{\pi'}(x)} d\lambda(x) = \begin{cases} \frac{d_\pi}{k_\pi} & \text{if} \,\,\pi =\pi', \\ 0 & \text{otherwise}, \end{cases}$$ for all $\pi, \pi' \in \widehat{K}.$ Therefore, $\|\chi_\pi\|_{L^2(K)}^2 =\frac{d_\pi}{k_\pi}.$
The $\ell^p_{\text{sch}}$-spaces on $\widehat{K}$ can be defined as $\ell^p_{\text{sch}}(\widehat{K})$ defined in [@HR D.37, D. 36(e)]. These spaces are studied by Vrem [@VremD]. First, for the space of Fourier coefficients of functions on $K$ we set $$\Sigma(K)=\{\sigma: \pi \mapsto \sigma(\pi) \in \mathbb{C}^{d_\pi \times d_\pi}: \, \pi \in \widehat{K}\} = \prod_{\pi \in\widehat{K}} \mathbb{C}^{d_\pi \times d_\pi}.$$ The space $\ell^p_{\text{sch}}(\widehat{K}) \subset \Sigma(K)$ is defined by the norm $$\|\sigma\|_{\ell^p_{\text{sch}}(\widehat{K})}:= \left(\sum_{\pi \in \widehat{K}} k_\pi \|\sigma(\pi)\|_{S^p}^p \right)^{\frac{1}{p}},\,\,\, \sigma \in \Sigma(K),\,\, 1\leq p<\infty,$$ and $$\|\sigma\|_{\ell^\infty_{\text{sch}}(\widehat{K})}:= \sup_{\pi \in \widehat{K}} \|\sigma(\pi)\|_{\mathcal{L}(\mathcal{H}_\pi)}\,\,\,\, \sigma \in \Sigma(K).$$
The set of all $\sigma \in \Sigma(K)$ such that $\#\{\pi \in \widehat{K}:\, \sigma(\pi) \neq 0\}<\infty$ denoted by $\Sigma_c(\widehat{K})$ and $\Sigma_0(K)$ is the set of all $\sigma \in \Sigma(K)$ such that $\#\{\pi \in \widehat{K}:\, \|\sigma(\pi)\|_{\mathcal{L}(\mathcal{H}_\pi)} \geq \epsilon \}<\infty$ for all $\epsilon>0.$ For each $\pi \in \widehat{K},$ the Fourier transform $\widehat{f}$ of $f \in L^1(K)$ is defined as $$\widehat{f}(\pi) = \int_K f(x) \bar{\pi}(x)\, d\lambda(x),$$ where $\bar{\pi}$ is the conjugate representation of $\pi.$ Vrem [@Vrem] proved that the map $f \mapsto \widehat{f}$ is a non norm increasing $*$-isomorphism of $L^1(K)$ onto a dense subalgebra of $\Sigma_0(K).$ For $f \in L^2(K),$ we have $$\label{Fseries}
f = \sum_{\pi \in \widehat{K}} k_{\pi} \sum_{i,j=1}^{d_{\pi}} \widehat{f}(\pi)_{i,j} \pi_{i,j}$$ and the series converges in $L^2(K)$ [@Vrem Corollary 2.10]. Hence, we have the following Plancherel identity $$\|f\|_2^2= \sum_{\pi \in \widehat{K}} k_{\pi} \sum_{i,j=1}^{d_{\pi}} |\widehat{f}(\pi)_{i,j}|^2 = \sum_{\pi \in \widehat{K}} k_{\pi} \|\widehat{f}(\pi)\|_{\textnormal{HS}}^2 =\|\widehat{f}\|_{\ell^2_{\text{sch}}(\widehat{K})}^2.$$
The following Hausdorff-Young inequality holds for Fourier transform on compact hypergroups [@VremD].
\[HYsch\] Let $1 \leq p \leq 2$ with $\frac{1}{p}+\frac{1}{p'}=1.$ For any $f \in L^p(K)$ we have the following inequality $$\label{HY1sch}
\left( \sum_{\pi \in \widehat{K}} k_\pi \|\widehat{f}(\pi)\|^{p'}_{S^p} \right)^{\frac{1}{p'}} = \|\widehat{f}\|_{\ell^{p'}_{\text{sch}}(\widehat{K})} \leq \|f\|_{L^p(K)}.$$
Recently, the first author with R. Sarma [@KR] also obtained Hausdorff-Young inequality using different norm which was useful to study Hausdorff-Young inequality for Orlicz spaces [@KR]. We will discuss it in the next section in more details.
Commutative compact hypergroups
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In this section we assume that compact hypergroup $K$ is commutative. Then every representation of $K$ is one dimensional. The dual space of $K$ defined as follows $$\widehat{K}= \left\lbrace\chi \in C^b(K): \chi \neq 0,\,\chi(\check{m})= \overline{\chi(m)},\, (\delta_m*\delta_n)(\chi)= \chi(m) \chi(n) \, \text{for all}\,\, m,n \in K \right\rbrace.$$
An element in $\widehat{K}$ will be called a [*character*]{}. Equip $\widehat{K}$ with the uniform convergence on the compact sets. In case of a compact hypergroups $K$ the dual space $\widehat{K}$ is discrete. In general, $\widehat{K}$ may not have a dual hypergroup structure with respect to the pointwise product [@Jewett Example 9.1 C] but it holds for most “natural" hypergroups including the conjugacy classes of compact groups. Then the Fourier transform on $L^1(K, \lambda)$ is defined by $$\widehat{f}(\chi):= \int_K f(x)\, \overline{\chi(x)}\, d\lambda(x),\quad \chi \in \widehat{K}.$$ The Fourier transform is injective and there exists a Radon measure $\omega$ on $\widehat{K},$ called the Plancherel measure on $\widehat{K}$ such that the map $f \mapsto \widehat{f}$ extends to an isometric isomorphism from $L^2(K, \lambda)$ onto $L^2(\widehat{K}, \omega),$ that is, $$\label{pabel}
\sum_{\chi \in \widehat{K}} |\widehat{f}(\chi)|^2 d\omega(\chi)= \int_K |f(x)|^2\, d\lambda(x).$$ In this case, the Fourier series of $f$ given by takes the form $$f= \sum_{\chi \in \widehat{K}} k_{\chi}\, \widehat{f}(\chi)\, \chi.$$ It follows from the orthogonality relation of characters that the set $\{k_\chi^{\frac{1}{2}} \chi\}_{\chi \in \widehat{K}}$ forms an orthonormal basis of $L^2(K).$ It is also known that for each $\chi \in \widehat{K}$ we have that $\omega(\chi)=k_{\chi}$ [@Alaga Proposition 1.2]. If $K$ is a compact commutative group then $k_\chi=d_\chi=1$ for all $\chi \in \widehat{K};$ and therefore Plancherel measure on $\widehat{K}$ is constant $1.$
Hausdorff-Young-Paley and Hardy-Littlewood inequalities on compact hypergroups
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In this section, we will study Paley inequality, Hausdorff-Young-Paley inequality and Hardy Littlewood inequality for compact hypergroups. At times, we will denote $L^p(K, \lambda)$ by $L^p(K)$ for simplicity.
Paley inequality on compact hypergroups
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In this subsection, we prove Paley inequality for compact hypergroups. Paley inequality is an important inequality in itself but also plays a vital role to obtain Hardy-Littlewood inequality and Hausdorff-Young-Paley inequality for compact hypergroups.
\[Paley\] Let $K$ be a compact hypergroup and let $1<p \leq 2.$ If $\varphi:\widehat{K} \rightarrow (0, \infty)$ is a function such that $$\label{Paleycondi}
M_\varphi:= \sup_{y>0} y \sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \geq y}} k_\pi^2<\infty.$$ Then, for all $f \in L^p(K),$ we have $$\label{Paley1}
\left(\sum_{\pi \in \widehat{K}} k_\pi^2 \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi}} \right)^p \varphi(\pi)^{2-p} \right)^{\frac{1}{p}} \lesssim M_\varphi^{\frac{2-p}{p}} \|f\|_{L^p(K)}.$$
Let us consider the measure on $\nu$ on the dual space $\widehat{K}$ on $K$ given by $$\nu(\{\pi\})= \varphi(\pi)^2 k_{\pi}^2,\,\,\,\,\, \pi \in \widehat{K}.$$ Define the space $L^p(\widehat{K}, \nu), 1 \leq p <\infty,$ as the space of all real or complex sequences $a: \pi \mapsto a_\pi $ such that $$\|a\|_{L^p(\widehat{K}, \nu)}= \left( \sum_{\pi \in \widehat{K}} |a_\pi|^p \varphi(\pi)^2 k_{\pi}^2 \right)^{\frac{1}{p}}<\infty.$$ We will show that the sublinear operator $A:L^p(K, \lambda) \rightarrow L^p(\widehat{K}, \nu)$ defined by $$Af:= \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi}\, \varphi(\pi)}\right)_{\pi \in \widehat{K}}$$ is well defined and bounded for $1<p \leq 2.$ In other words, we will get the following estimate which will eventually give us the required estimate , $$\label{vis}
\|Af\|_{L^p(\widehat{K}, \nu)} = \left( \sum_{\pi \in \widehat{K}} \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)} \right)^p \varphi(\pi)^2 k_\pi^2 \right)^{\frac{1}{p}} \lesssim M_\varphi^{\frac{2-p}{p}} \|f\|_{L^p(K)},$$ where $M_\varphi:= \sup_{y>0} y \sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \geq y}} k_\pi^2. $ To prove the above estimate it is enough to show that $A$ is weak type $(1,1)$ and weak type $(2,2),$ thanks to Marcinkiewicz interpolation theorem. In fact, we show that, with the distribution function $\nu_{\widehat{K}},$ that $$\label{Vish11}
\nu_{\widehat{K}}(y; Af) \leq \frac{M_1 \|f\|_{L^1(K)}}{y} \,\,\,\, \text{with the norm}\,\, M_1= M_\varphi,$$ $$\label{Vish22}
\nu_{\widehat{K}}(y; Af) \leq \left( \frac{M_2 \|f\|_{L^2(K)}}{y} \right)^{2}\,\,\,\, \text{with the norm}\,\, M_2=1,$$ where $\nu_{\widehat{K}}(y; Af)$ is defined by $\nu_{\widehat{K}}(y; Af):= \sum_{\overset{\pi \in \widehat{K}}{|(Af)(\pi)| \geq y}} \nu(\pi), \,\,\, y>0.$
First, we show that $A$ is of type $(1,1)$ with norm $M_1=M_\varphi;$ more precisely we show that $$\label{vis8}
\nu_{\widehat{K}}(y; Af)= \nu \left\{ \pi \in \widehat{K}: \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)}>y \right\} \lesssim \frac{M_\varphi \|f\|_{L^1(K)}}{y},$$ where $\nu \left\{ \pi \in \widehat{K}: \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)}>y \right\}$ can be interpreted as the weighted sum $\sum \varphi(\pi)^2 k_\pi^2$ taken over those $\pi \in \widehat{K}$ such that $\frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)}>y.$ By the defintion of the Fourier transform and the fact that $\pi$ is a norm decreasing $*$-homomorphism, i.e., $\|\pi(\check{x})\|_{\text{op}} \leq 1$ for all $x \in K,$ we have $$\|\widehat{f}(\pi)\|_{\textnormal{HS}} \leq \|f\|_{L^1(K)} \|\pi(\check{x})\|_{\textnormal{HS}} \leq \|f\|_{L^1(K)} \sqrt{d_\pi} \|\pi(\check{x})\|_{\text{op}} \leq \sqrt{d_\pi} \|f\|_{L^1(K)}.$$ Therefore, by using $d_\pi \leq k_\pi,$ we get $$y < \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)} \leq \frac{\sqrt{d_\pi} \|f\|_{L^1(K)}}{\sqrt{k_\pi} \varphi(\pi)} \leq \frac{\|f\|_{L^1(K)}}{ \varphi(\pi)}.$$
This inequality yields that $$\left\{ \pi \in \widehat{K}: \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)}>y \right\} \subset \left\{ \pi \in \widehat{K}: \frac{\|f\|_{L^1(K)}}{ \varphi(\pi)} >y \right\}$$ for any $y >0.$ So $$\nu \left\{ \pi \in \widehat{K}: \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)}>y \right\} \leq \nu \left\{ \pi \in \widehat{K}: \frac{\|f\|_{L^1(K)}}{ \varphi(\pi)} >y \right\}.$$ Setting $w= \frac{\|f\|_{L^1(K)}}{y},$ we have $$\nu \left\{ \pi \in \widehat{K}: \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)}>y \right\} \leq \sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \leq w}} \varphi(\pi)^2 k_\pi^2.$$ We claim that $$\sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \leq w}} \varphi(\pi)^2 k_\pi^2 \lesssim M_\varphi w.$$ In fact, we have $$\sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \leq w}} \varphi(\pi)^2 k_\pi^2 = \sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \leq w}} k_\pi^2 \int_{0}^{\varphi^2(\pi)} d\tau.$$ By interchanging sum and integration we have $$\sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \leq w}} k_\pi^2 \int_{0}^{\varphi^2(\pi)} d\tau = \int_{0}^{w^2} d \tau \sum_{\overset{\pi \in \widehat{K}}{\tau^{\frac{1}{2}} \leq \varphi(\pi) \leq w}} k_\pi^2.$$ Next, by making substitution $\tau= t^2$ it yields to $$\int_{0}^{w^2} d \tau \sum_{\overset{\pi \in \widehat{K}}{\tau^{\frac{1}{2}} \leq \varphi(\pi) \leq w}} k_\pi^2 = 2 \int_{0}^w t dt \sum_{\overset{\pi \in \widehat{K}}{ t \leq \varphi(\pi) \leq w}} k_\pi^2 \leq 2 \int_0^w t \, dt \sum_{\overset{\pi \in \widehat{K}}{ t \leq \varphi(\pi)}} k_\pi^2.$$
Since $$t \sum_{\overset{\pi \in \widehat{K}}{ t \leq \varphi(\pi)}} k_\pi^2 \leq \sup_{t>0} t \sum_{\overset{\pi \in \widehat{K}}{ t \leq \varphi(\pi)}} k_\pi^2=M_\varphi$$ is finite by the assumption, we get $$2 \int_0^w t \, dt \sum_{\overset{\pi \in \widehat{K}}{ t \leq \varphi(\pi)}} k_\pi^2 \lesssim M_\varphi w.$$ Therefore, we get the required estimate $$\nu_{\widehat{K}}(y; Af)= \nu \left\{ \pi \in \widehat{K}: \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)}>y \right\} \lesssim \frac{M_\varphi \|f\|_{L^1(K)}}{y}.$$
Now, we will prove that $A$ is weak type $(2,2),$ that is, the equality . By using Plancherel’s identity we get $$\begin{aligned}
y^2 \nu_{\widehat{K}}(y; Af) \leq \|Af\|_{L^2(\widehat{K}, \nu)}^2 &= \sum_{\pi \in \widehat{K}} k_\pi^2 \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)} \right)^2 \varphi(\pi)^2 \\&= \sum_{\pi \in \widehat{K}} k_\pi \|\widehat{f}(\pi)\|_{\textnormal{HS}}^2 = \|f\|_{L^2(K)}^2.
\end{aligned}$$Thus $A$ is of type $(2,2)$ with norm $M_2 \leq 1.$ Thus we have proved and . Thus, by using the Marcinkiewicz interpolation theorem with $p_1=1,\, p_2=2$ and $\frac{1}{p}=1- \theta+\frac{\theta}{2}$ we now obtain $$\left( \sum_{\pi \in \widehat{K}} \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi} \varphi(\pi)} \right)^p \varphi(\pi)^2 k_\pi^2 \right)^{\frac{1}{p}} = \|Af\|_{L^p(\widehat{K}, \nu)} \lesssim M_\varphi^{\frac{2-p}{p}} \|f\|_{L^p(K)}.$$ This completes the proof.
One may notice that instead on Schatten $p$-norm we used Hilbert-Schimdt norm in Theorem \[Paley\]. This is because Hilbert-Schmidt norm gives sharp inequality in Paley-type theorem as already noticed in [@ARN] for compact homogeneous spaces and in [@Youn] for compact quantum groups. We will see this for compact hypergroups from the discussion below.
Now, we will define and discuss an another important family of Lebesgue spaces $\ell^p$ on $\widehat{K}$ defined using the Hilbert-Schmidt norm $\|\cdot\|_{\textnormal{HS}}$ instead of Schatten $p$-norm $\|\cdot\|_{S^p}$ on the space of $(d_\pi \times d_\pi)$-dimensional matrices. Recently, these $\ell^p$-space have been studied in more details by the second author and his collaborators in the context of compact Lie groups and compact homogeneous spaces [@RuzT; @ARN1; @AMR; @KM; @FR; @AR14]. In particular, it was shown in [@AR14] that the space $\ell^p(\widehat{G})$ and the Hausdorff-Young inequality for it become useful for convergence of Fourier series and characterization of Gevrey-Roumieu ultradifferentiable functions and Gevrey-Beurling ultradifferentiable functions on compact homogeneous manifolds.
Next, we define the Lebesgue spaces $\ell^p(\widehat{K}) \subset \Sigma(K)$ by the condition $$\|\sigma\|_{\ell^p(\widehat{K})}:= \left( \sum_{\pi \in \widehat{K}} k_\pi^{(2-\frac{p}{2})} \|\sigma(\pi)\|_{\textnormal{HS}}^p \right)^{\frac{1}{p}},\,\,\,\, 1 \leq p <\infty,$$ and $$\|\sigma\|_{\ell^\infty(\widehat{K})}:= \sup_{\pi \in \widehat{K}} k_\pi^{-\frac{1}{2}} \|\sigma(\pi)\|_{\textnormal{HS}}.$$
We note here that for compact groups such spaces were introduced in [@RuzT Chapter 10].
The following proposition presents the relation between both norms on Lebesgue spaces on $\widehat{K}.$
\[Estisch\] For $1 \leq p \leq 2,$ we have the following continuous embeddings as well as the estimates: $\ell^p(\widehat{K}) \hookrightarrow \ell^p_{sch}(\widehat{K})$ and $\|\sigma\|_{\ell^p_{sch}(\widehat{K})} \leq \|\sigma\|_{\ell^p(\widehat{K})}$ for all $\sigma \in \Sigma(K).$ For $2 \leq p \leq \infty,$ we have $\ell^p_{sch}(\widehat{K}) \hookrightarrow \ell^p(\widehat{K}) $ and $ \|\sigma\|_{\ell^p(\widehat{K})} \leq \|\sigma\|_{\ell^p_{sch}(\widehat{K})} $ for all $\sigma \in \Sigma(K).$
For $p=2,$ the norms coincide since $S^2=\textnormal{HS}.$ Let $1 \leq p <2.$ Since $\sigma(\pi) \in \mathbb{C}^{d_\pi \times d_\pi},$ denoting $s_j$ its singular number, by H$\ddot{\text{o}}$lder inequality we have $$\label{Scht}
\|\sigma(\pi)\|_{S^p}^p = \sum_{j=1}^{d_\pi} s_j^p \leq \left( \sum_{j=1}^{d_\pi} 1 \right)^{\frac{2-p}{2}} \left( \sum_{j=1}^{d_\pi} s_{j}^{p \frac{2}{p}} \right)^{\frac{p}{2}} = d_{\pi}^{\frac{2-p}{2}} \|\sigma(\pi)\|_{\textnormal{HS}}^p.$$ Consequently, it follows that $$\|\sigma\|^p_{\ell_{sch}^p(\widehat{K})}= \sum_{\pi \in \widehat{K}} k_\pi \|\sigma(\pi)\|_{S^p}^p \leq \sum_{\pi \in \widehat{K}} k_\pi d_{\pi}^{\frac{2-p}{2}} \|\sigma(\pi)\|_{\textnormal{HS}}^p \leq \sum_{\pi \in \widehat{K}} k_\pi k_{\pi}^{\frac{2-p}{2}} \|\sigma(\pi)\|_{\textnormal{HS}}^p = \|\sigma\|_{\ell^p(\widehat{K})}^p.$$ Now, for $2<p <\infty,$ we have $$\|\sigma(\pi)\|_{\textnormal{HS}}^2 = \sum_{j=1}^{d_\pi} s_j^2 \leq \left( \sum_{j=1}^{d_\pi} 1 \right)^{\frac{p-2}{p}} \left( \sum_{j=1}^{d_\pi} s_j^{2 \frac{p}{2}}\right)^{\frac{2}{p}} = d_\pi^{\frac{p-2}{p}} \|\sigma(\pi)\|_{S^p}^2,$$ implying $$\|\sigma(\pi)\|_{\textnormal{HS}} \leq d_\pi^{\frac{p-2}{2p}} \|\sigma(\pi)\|_{S^p}.$$ Therefore, we have $$\|\sigma\|_{\ell^p(\widehat{K})}= \sum_{\pi \in \widehat{k}} k_\pi^{(2-\frac{p}{2})} \|\sigma(\pi)\|_{\textnormal{HS}}^p \leq \sum_{\pi \in \widehat{K}} k_\pi^{(2-\frac{p}{2})} d_\pi^{\frac{p-2}{2}} \|\sigma(\pi)\|^p_{S^p} \leq \sum_{\pi \in \widehat{k}} k_\pi\|\sigma(\pi)\|^p_{S^p} = \|\sigma\|_{\ell^p_{sch}(\widehat{K})}^p.$$ Finally, for $p=\infty,$ the inequality $$\|\sigma(\pi)\|_{\textnormal{HS}} \leq k_{\pi}^{\frac{1}{2}}\|\sigma(\pi)\|_{\mathcal{L}(\mathcal{H}_\pi)}$$ implies $$\|\sigma\|_{\ell^\infty(\widehat{K})} = \sup_{\pi \in \widehat{K}} k_\pi^{\frac{1}{2}} \|\sigma(\pi)\|_{\textnormal{HS}} \leq \sup_{\pi \in \widehat{K}} \|\sigma(\pi)\|_{\mathcal{L}(\mathcal{H}_\pi)}= \|\sigma\|_{\ell^\infty_{\text{sch}}(\widehat{K})}.$$
The following Hausdorff-Young inequality for Fourier transform on compact hypergroups was recently obtained by the first author and R. Sarma [@KR].
\[HY\] Let $1 \leq p \leq 2$ with $\frac{1}{p}+\frac{1}{p'}=1.$ For any $f \in L^p(K)$ we have the following inequality $$\label{HY1}
\left( \sum_{\pi \in \widehat{K}} k_\pi^{2-\frac{p'}{2}} \|\widehat{f}(\pi)\|^{p'}_{\textnormal{HS}} \right)^{\frac{1}{p'}} = \|\widehat{f}\|_{\ell^{p'}(\widehat{K})} \leq \|f\|_{L^p(K)}.$$
In the view of Proposition \[Estisch\] one can see that the Hausdorff-Young inequality using Schatten $p$-norm is sharper than inequality . In [@KR], Theorem \[HY\] is further used to define Orlicz space on dual of compact hypergroups and to obtained Hausdorff-Young inequality for Orlicz spaces on compact hypergroup.
The Paley inequality can be reduced to the familiar form using Schatten $p$-norm. The proof of it is immediate from the inequality and the fact that $d_\pi \leq k_\pi.$
Let $K$ be a compact hypergroup and let $1<p \leq 2.$ If $\varphi:\widehat{K} \rightarrow (0, \infty)$ is a function satisfying condition of Theorem \[Paley\] then there exist a universal constant $C=C(p)$ such that $$\left(\sum_{\pi \in \widehat{G}} k_\pi \|\widehat{f}(\pi)\|^{p}_{S^p}\, \varphi(\pi)^{2-p} \right)^{\frac{1}{p}} \leq C \|f\|_{L^p(K)}.$$
Hardy-Littlewood inequality on compact hypergroups
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In this section, we apply Paley inequality to get the Hardy-Littlewood inequality on compact hypergroups. This approach has been recently considered to prove the Hardy-Littlewood inequality in the context of compact Lie groups $\text{SU(2)}$ [@ARN1], compact homogeneous manifolds [@ARN] and compact quantum groups [@AMR; @Youn]. The philosophy to derive Hardy-Littlewood inequality is to choose the function $\varphi$ suitably such that condition of Theorem \[Paley\] is satisfied. In the case of a compact Lie group $G$ of dimension $n$, in [@ARN] the authors took $\varphi(\pi)= \langle \pi \rangle^{-n},$ where $\langle \pi \rangle$ denote the eigenvalues of the operator $(1-\Delta_G)^{\frac{1}{2}}$ corresponding to the representation $\pi$ for a Laplacian $\Delta_G$ on $G.$ Although, for $\text{SU(2)}$ this was proved by repeating the proof of Paley inequality and estimating the bound explicitly ([@ARN1]). In the case of compact quantum groups, the proof of this inequality has been achieved by using the geometric informations of compact quantum groups like spectral triples [@AMR] and the natural length function on the dual of compact quantum groups [@Youn]. The compact hypergroups in general are not equipped with any geometric and differential structure so we prove the following Hardy-Littlewood inequality for compact hypergroups.
\[HLG\] Let $1 <p \leq 2$ and let $K$ be a compact hypergroup. Assume that a positive function $\pi \mapsto \mu_\pi$ on $\widehat{K}$ grows sufficiently fast, that is, $$\label{HLGcondi}
\sum_{\pi \in \widehat{K}} \frac{k_\pi^2}{|\mu_\pi|^\beta}<\infty\,\,\,\, \text{for some}\,\beta \geq 0.$$ Then we have $$\label{HL}
\sum_{\pi \in \widehat{K}} k_\pi^2 |\mu_\pi|^{\beta (p-2)} \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi}} \right)^p \lesssim \|f\|_{L^p(K)}.$$
By the assumption, we know that $$C:= \sum_{\pi \in \widehat{K}} \frac{k_\pi^2}{|\mu_\pi|^\beta}< \infty.$$ Then we have $$C \geq \sum_{\overset{\pi \in \widehat{K}}{|\mu_\pi|^\beta \leq \frac{1}{t}}} \frac{k_\pi^2}{|\mu_\pi|^\beta} \geq t \sum_{\overset{\pi \in \widehat{K}}{|\mu_\pi|^\beta \leq \frac{1}{t}}} k_\pi^2 = t \sum_{\overset{\pi \in \widehat{K}}{ \frac{1}{|\mu_\pi|^\beta}} \geq t} k_\pi^2$$ and consequently we have
$$\sup_{t>0}t \sum_{\overset{\pi \in \widehat{K}}{ \frac{1}{|\mu_\pi|^\beta}} \geq t} k_\pi^2 \leq C <\infty.$$ Then, as an application of Theorem \[Paley\] with $\varphi(\pi)= \frac{1}{|\mu_\pi|^\beta},\,\, \pi \in \widehat{K},$ we get the required estimate .
In the case when $K$ is abelian, the Hardy-Littlewood inequality takes the following form.
\[HLabe\] Let $1<p \leq 2$ and let $K$ be a compact abelian hypergroup. Assume that a positive function $\chi \mapsto \mu_\chi$ on $\widehat{K}$ satisfies the condition $$\label{HLabecon}
\sum_{\chi \in \widehat{K}} \frac{k_\chi^2}{|\mu_\chi|^\beta} <\infty\quad \text{for some}\,\, \beta \geq 0.$$ Then we have $$\sum_{\chi \in \widehat{K}} k_\chi^{2-\frac{p}{2}} |\mu_\chi|^{\beta(p-2)} |\widehat{f}(\chi)|^p \lesssim \|f\|_{L^p(K)}.$$
We would like to note here that in the case when $K$ is a compact Lie group, the natural choices of $\pi \mapsto \mu_\pi$ is $\pi \mapsto \langle \pi \rangle.$ But for this choice of $ \mu_\pi$ the quantity $\sum_{\pi \in \widehat{K} }\frac{k_\pi}{|\mu_\pi|^{\beta}},$ which turns out to be $\sum_{\pi \in \widehat{K} }\frac{d_\pi}{\langle \pi \rangle^{\beta}}$ in this case, is not finite for $\beta=n:=\dim(G)$ as proved by the second author and Dasgupta [@AR14]. So this does not give the Hardy-Littlewood inequality for compact Lie groups, in particular, for $\mathbb{T}^n$ ([@ARN]). Surprisingly, the quantity $\sum_{\pi \in \widehat{K} }\frac{k_\pi}{|\mu_\pi|^{\beta}}$ is finite with a natural choice of $\pi \mapsto \mu_\pi$ and $\beta$ for (pure) hypergroups including conjugacy classes of compact Lie groups and countable compact hypergroups as shown in the last section and consequently, provides the Hardy-Littlewood inequality for compact hypergroups.
Hausdorff-Young-Paley inequality on compact hypergroups
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In this subsection, we prove the Hausdorff-Young-Paley inequality for compact hypergroups. The Hausdorff-Young-Paley inequality is an important inequality in itself but it serves as an essential tool to prove $L^p$-$L^q$ Fourier multiplier for compact hypergroups.
The following theorem [@BL] is useful in the proof of the Hausdorff-Young-Paley inequality.
\[interpolationoperator\] Let $d\mu_0(x)= \omega_0(x) d\mu(x),$ $d\mu_1(x)= \omega_1(x) d\mu(x).$ Suppose that $0<p_0, p_1< \infty.$ If a continuous linear operator $A$ admits bounded extensions, $A: L^p(Y,\mu)\rightarrow L^{p_0}(\omega_0) $ and $A: L^p(Y,\mu)\rightarrow L^{p_1}(\omega_1) ,$ then there exists a bounded extension $A: L^p(Y,\mu)\rightarrow L^{b}(\tilde{\omega}) $ of $A$, where $0<\theta<1, \, \frac{1}{b}= \frac{1-\theta}{p_0}+\frac{\theta}{p_1}$ and $\tilde{\omega}= \omega_0^{\frac{b(1-\theta)}{p_0}} \omega_1^{\frac{b\theta}{p_1}}.$
Now, we are ready to state the Hausdorff-Young-Paley inequality for compact hypergrous. Sometimes, it is also known as Pitt’s inequality in the literature on classical harmonic analysis.
\[HYP\] Let $K$ be a compact hypergroup and let $1<p \leq b \leq p'<\infty.$ If a function $\varphi: \widehat{K} \rightarrow (0, \infty)$ satisfies the condition $$M_\varphi:= \sup_{y>0} y \sum_{\overset{\pi \in \widehat{K}}{\varphi(\pi) \geq y}} k_\pi^2 <\infty$$ then we have $$\left(\sum_{\pi \in \widehat{G}} k_\pi^2 \left( \frac{\|\widehat{f}(\pi)\|_{\textnormal{HS}}}{\sqrt{k_\pi}}\varphi(\pi)^{\frac{1}{b}-\frac{1}{p'}} \right)^b \right)^{\frac{1}{b}} \lesssim M_\varphi^{\frac{1}{b}-\frac{1}{p'}} \|f\|_{L^p(K)}.$$
We consider a sublinear operator $A$ which takes a function $f$ to its Fourier coefficient $\widehat{f}(\pi) \in \mathbb{C}^{d_\pi \times d_\pi}$ divided by $\sqrt{k_\pi}$, that is, $$f \mapsto Af:= \left\{ \frac{\widehat{f}(\pi)}{\sqrt{k_\pi}} \right\}_{\pi \in \widehat{K}},$$ from $L^p(K)$ into the weighted space $\ell^p(\widehat{K}, \tilde{\omega}).$ The space $\ell^p(\widehat{K}, \tilde{\omega})$ is defined by the norm $$\|a\|_{\ell^p(\widehat{K}, \tilde{\omega})}:= \left( \sum_{\pi \in \widehat{K}} \|a(\pi)\|^p_{\textnormal{HS}}\,\, \tilde{\omega}(\pi) \right)^\frac{1}{p},$$ and $\tilde{\omega}$ is a scalar sequence defined on $\widehat{K}$ to be determined. Then the proof of the theorem follows from Theorem \[interpolationoperator\] if we consider the left hand side of the inequalities and as $\|Af\|_{\ell^p(\widehat{K}, \tilde{\omega})}$-norm of the operator $A$ in the weighted sequence spaces over $\widehat{K}$ with the weights given by $\omega_0(\pi)= k_\pi^2 \varphi(\pi)^{2-p}$ and $\omega_1(\pi)=k_\pi^2,$ $\pi \in \widehat{K},$ respectively.
$L^p$-$L^q$-boundedness of Fourier multipliers on compact hypergroups
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In this section, we prove $L^p$-$L^q$ boundedness of Fourier multipliers on compact hypergroups as a natural analogue of Hörmander’s theorem [@Hormander1960] on compact hypergroups. We will apply the Hausdorff-Young-Paley inequality in Theorem \[HYP\] to provide a sufficient condition for the $L^p$-$L^q$ boundedness of Fourier multipliers for the range $1<p \leq 2 \leq q <\infty.$ This approach was developed by the second author with R. Akylzhanov to prove the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact groups [@AR] by using the von-Neumann algebra machinery. In [@NT], this theorem was proved for the torus $\mathbb{T}$ using a different method. We begin this section by recalling the definition of Fourier multipliers on compact hypergroups.
An operator $A$ which is invariant under the left translations will be called a left Fourier multiplier. The left invariant operators can be characterized using the Fourier transform [@VremD; @SKK]. Indeed, if $A$ is a left Fourier multiplier then there exists a function $\sigma_{A}:\widehat{K} \rightarrow \mathbb{C}^{d_\pi \times d_\pi},$ known as the symbol associated with $A,$ such that $$\widehat{Af}(\pi)=\sigma_{A}(\pi) \widehat{f}(\pi),\,\,\,\, \pi \in \widehat{K},$$ for all $f$ belonging to a suitable function space on $K.$ In the next result, we show that if the symbol $\sigma_{A}$ of a Fourier multipliers $A$ defined on $C_c(K)$ satisfies certain Hörmander’s condition then $A$ can be extended as a bounded linear operator from $L^p(K)$ to $L^q(K)$ for the range $1<p \leq 2 \leq q <\infty.$
\[Lp-Lqmulti\] Let $K$ be a compact hypergroup and let $1<p\leq 2 \leq q<\infty.$ Let $A$ be a left Fourier multiplier with symbol $\sigma_{A}$. Then we have $$\label{hocon}
\|A\|_{L^p(K) \rightarrow L^q(K)} \lesssim \sup_{y>0} y \left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{op}\geq y}} k_\pi^2 \right)^{\frac{1}{p}-\frac{1}{q}}.$$
Let us first consider the case when $p \leq q'$ (where $\frac{1}{q}+\frac{1}{q'}=1$). Since $q'\leq 2,$ for $f \in C_c(K),$ the Hausdorff-Young inequality gives $$\begin{aligned}
\|Af\|_{L^q(K)} \leq \|\widehat{Af}\|_{\ell^{q'}(\widehat{K})} &= \|\sigma_A \widehat{f}\|_{\ell^{q'}(\widehat{K})}= \left( \sum_{\pi \in \widehat{K}} k_\pi^2 \left( \frac{\|\sigma_A(\pi) \widehat{f}(\pi)\|_{\textnormal{HS}}}{ \sqrt{k_\pi}} \right)^{q'} \right)^{\frac{1}{q'}} \\&\leq \left( \sum_{\pi \in \widehat{K}} k_\pi^2 \|\sigma_A(\pi)\|_{\text{op}}^{q'}\left( \frac{ \|\widehat{f}(\pi)\|_{\textnormal{HS}}}{ \sqrt{k_\pi}} \right)^{q'} \right)^{\frac{1}{q'}}.
\end{aligned}$$ The case $q'\leq p = (p')'$ can be reduced to the case $p \leq q'$ as follows. The $L^p$-duality (see [@ARN1 Theorem 4.2]) yields $$\|A\|_{L^p(K) \rightarrow L^q(K)} = \|A^* \|_{L^{q'}(K) \rightarrow L^{p'}(K)}.$$ Also, the symbol $\sigma_{A^*}(\pi)$ of the adjoint operator $A^*$ is equal to $\sigma_{A}^*,$ i.e., $$\sigma_{A^*}(\pi)= \sigma_A(\pi)^*,\,\,\,\,\,\pi \in \widehat{K},$$ and its operator norm $\|\sigma_{A^*}(\pi)\|_{\text{op}}$ is equal to $\|\sigma_A(\pi)\|_{\text{op}}.$ We set $\sigma(\pi)= \|\sigma_A(\pi)\|_{\text{op}}^r I_{d_\pi }, \pi \in \widehat{K},$ where $r= \frac{q-p}{pq},$ and it is easy to see that $$\|\sigma(\pi)\|_{\text{op}}=\|\sigma_A(\pi)\|_{\text{op}}^r.$$ Now, its time to apply Theorem \[HYP\]. We observe that with $\varphi(\pi)= \|\sigma(\pi)\|_{\text{op}}, \,\, \pi \in \widehat{K},$ and $b=q',$ the assumption of Theorem \[HYP\] is satisfied, and since $\frac{1}{q'}-\frac{1}{p'}=\frac{1}{p}-\frac{1}{q}=\frac{1}{r},$ we obtain $$\begin{aligned}
\left( \sum_{\pi \in \widehat{K}} k_\pi^2 \|\sigma_A(\pi)\|_{\text{op}}^{q'}\left( \frac{ \|\widehat{f}(\pi)\|_{\textnormal{HS}}}{ \sqrt{k_\pi}} \right)^{q'} \right)^{\frac{1}{q'}} \lesssim \left( \sup_{y>0} y \sum_{\overset{\pi \in \widehat{K}}{\|\sigma(\pi)\|_{op}\geq y}} k_\pi^2 \right)^{\frac{1}{r}} \|f\|_{L^p(K)},\,\, f \in L^p(K).
\end{aligned}$$ Further, it can be easily checked that $$\begin{aligned}
\left( \sup_{y>0} y \sum_{\overset{\pi \in \widehat{K}}{\|\sigma(\pi)\|_{\text{op}}\geq y}} k_\pi^2 \right)^{\frac{1}{r}} &= \left( \sup_{y>0} y \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{\text{op}}^r \geq y}} k_\pi^2 \right)^{\frac{1}{r}} = \left( \sup_{y>0} y^r \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{\text{op}} \geq y}} k_\pi^2 \right)^{\frac{1}{r}} \\&= \sup_{y>0} y \left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{op} \geq y}} k_\pi^2 \right)^{\frac{1}{r}}.
\end{aligned}$$ Therefore, $$\|Af\|_{L^q(K)} \lesssim \sup_{y>0} y \left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{\text{op}} \geq y}} k_\pi^2 \right)^{\frac{1}{r}} \|f\|_{L^p(K)}$$ and hence $$\|A\|_{L^p(K) \rightarrow L^q(K)} \lesssim \sup_{y>0} y \left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{\text{op}}>y}} k_\pi^2 \right)^{\frac{1}{p}-\frac{1}{q}},$$ completing the proof.
Recall that if $\omega(M):= \sum_{\pi \in M} k_\pi^2,$ $M \subseteq \widehat{K},$ is the Plancherel measure on $\widehat{K}$ then we can interpret the condition in a similar form as in Hörmander’s theorem for $\mathbb{R}^n$ ([@Hormander1960]) as follow: $$\label{alt. Hor}
\|A\|_{L^p(K) \rightarrow L^q(K)} \leq \sup_{s>0} \left\{s\,\, \omega \{\pi \in \widehat{K}\,:\, \|\sigma_{A}(\pi)\|_{\text{op}}>s \}\right\}^{\frac{1}{p}-\frac{1}{q}}.$$ We note that condition is sharp for $p=q=2.$ Indeed, first note that, using the Plancherel identity, we have $$\begin{aligned}
\label{l2}
\|A\|_{L^2(K) \rightarrow L^2(K)} &= \sup_{\overset{f \in L^2(K)}{\|f\|_2=1}} \|Af\|_{L^2(K)}=\sup_{\overset{f \in L^2(K)}{\|f\|_2=1}} \|\widehat{Af}\|_{\ell^2(\widehat{K})} =\sup_{\overset{f \in L^2(K)}{\|f\|_2=1}} \left(\sum_{\pi \in \widehat{K}} k_\pi \|\sigma_A(\pi) \widehat{f}(\pi)\|_{\textnormal{HS}}^2 \right)^{\frac{1}{2}} \nonumber
\\ & \leq \sup_{\pi \in \widehat{K}} \|\sigma_{A}(\pi)\|_{\text{op}} \sup_{\overset{f \in L^2(K)}{\|f\|_2=1}} \left( \sum_{\pi \in \widehat{K}} k_\pi \|\widehat{f}(\pi)\|_{\textnormal{HS}}^2 \right)^{\frac{1}{2}} = \sup_{\pi \in \widehat{K}} \|\sigma_{A}(\pi)\|_{\text{op}}.
\end{aligned}$$ Now, observe that the set $\{\pi \in \widehat{K}: \|\sigma_{A}(\pi)\|_{\text{op}} \geq s \}$ is empty for $s>\|A\|_{L^2(K) \rightarrow L^2(K)}$ in view of and, therefore, we have $$\begin{aligned}
\|A\|_{L^2(K) \rightarrow L^2(K)} &\leq \sup_{s>0} s \left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{op}>y}} k_\pi^2 \right)^{\frac{1}{2}-\frac{1}{2}} \\& = \sup_{0<s \leq \|A\|_{L^2(K) \rightarrow L^2(K)}} s \cdot\, 1 \leq \|A\|_{L^2(K) \rightarrow L^2(K)}.\end{aligned}$$ Therefore, we obtained an equality in for $p=q=2.$
Let $1<p , q <\infty$ and suppose that $A$ is a Fourier multiplier with symbol $\sigma_{A}$ on a compact hypergroup $K.$ If $1<p,q\leq 2,$ then $$\begin{aligned}
\|A\|_{L^p(K) \rightarrow L^{q}(K)} \lesssim \sup_{y>0} y\left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{op}\geq y}} k_\pi^2 \right)^{\frac{1}{p}-\frac{1}{2}}, \end{aligned}$$ while for $2\leq p,q<\infty$ we have $$\begin{aligned}
\|A\|_{L^p(K) \rightarrow L^{q}(K)}\lesssim \sup_{y>0} y\left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{op}\geq y}} k_\pi^2 \right)^{\frac{1}{q'}-\frac{1}{2}}.\end{aligned}$$
Let us assume that $1<p,q \leq 2.$ Using the compactness of $K,$ we have $\|A\|_{L^p(K) \rightarrow L^{q}(K)}\lesssim \|A\|_{L^p(K) \rightarrow L^{2}(K)}$ and therefore, Theorem \[Lp-Lqmulti\] gives $$\begin{aligned}
\|A\|_{L^p(K) \rightarrow L^{q}(K)}\lesssim \|A\|_{L^p(K) \rightarrow L^{2}(K)} \lesssim \sup_{y>0} y\left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{op}\geq y}} k_\pi^2 \right)^{\frac{1}{p}-\frac{1}{2}}.\end{aligned}$$ Now, let us assume that $2\leq p,q<\infty.$ Then $1<p',q'\leq 2,$ and using the first part of the proof we deduce $$\begin{aligned}
\|A\|_{L^p(K) \rightarrow L^{q}(K)}= \|A^*\|_{L^{q'}(K) \rightarrow L^{p'}(K)} \lesssim \sup_{y>0} y\left( \sum_{\overset{\pi \in \widehat{K}}{\|\sigma_A(\pi)\|_{op}\geq y}} k_\pi^2 \right)^{\frac{1}{q'}-\frac{1}{2}}.\end{aligned}$$Thus, we finish the proof.
Examples of hypergroups
=======================
In this section we discuss the results obtained in previous sections and prove some new results for two important classes of hypergroups, namely, the conjugacy classes of the compact non-abelian Lie group $\text{SU(2)}$ and countable compact hypergroups introduced and studied by Dunkl and Ramirez [@dun2].
Conjugacy classes of compact Lie groups
---------------------------------------
Let $G$ be a compact non abelian (Lie) group. Denote the set of all conjugacy classes of $G$ by $\text{Conj}(G),$ that is, $\text{Conj}(G):=\{C_x: x \in G\},$ where for each $x \in G$ the conjugacy class of $x$ is given by $C_x:=\{yxy^{-1}: y \in G\}.$ The set $\text{Conj}(G)$ equipped with the topology induced by the natural map $q: x \mapsto C_x,$ is a compact Hausdorff space. The compact Hausdorff space $\text{Conj}(G)$ becomes a commutative hypergroup [@Jewett Section 8] with respect to the convolution defined by, for $x, y \in G,$ $$\delta_{C_x}*\delta_{C_y}= \int_G \int_G \delta_{C_{txt^{-1}sys^{-1}}}\, dt\,ds.$$ Let $\widehat{G}$ be the unitary dual of $G.$ Suppose that each $\pi \in \widehat{G}$ has dimension $d_{\pi}$ and trace $\psi_\pi.$ The fuctions $\psi_\pi$ are called the characters of $G$ but the hypergroup characters are normalized by dividing $\psi_\pi$ by $d_{\pi}.$ More precisely, the hypergroup characters $\chi_\pi$ are obtained by the following relation: $\chi_\pi \circ q= d_{\pi}^{-1} \psi_\pi,$ where $q$ is the natural map $x \mapsto C_x.$ Then the dual $\widehat{\text{Conj}(G)}$ of the commutative hypergroup $\text{Conj}(G)$ is given by: $\widehat{\text{Conj}(G)}:= \{\chi_\pi: \pi \in \widehat{G} \}.$ In fact, the map $\pi \mapsto d_\pi^2 \psi_\pi$ is a bijection between $\widehat{G}$ and $\widehat{\text{Conj}(G)}.$ The Haar measure $\lambda$ of $\widehat{\text{Conj}(G)}$ is induced from the Haar measure of $G$ by the map $q.$ The Haar measure on $\widehat{\text{Conj}(G)}$ is given by $$\omega(\chi_\pi):= k_{\chi_\pi}= d_{\pi}^{2}.$$
In the sequel of the paper we will consider the case when $G=\text{SU}(2),$ the compact group of all $2 \times 2$ special unitary matrices. The representation theory of $\text{SU(2)}$ is well established. One can refer to [@HR; @Vilenkin; @RuzT] for more details. Denote the commutative hypergroup $\text{Conj(\text{SU}(2))}$ by $K.$ We identify $K$ with $[0, 1]$ where $t$ in $[0, 1]$ corresponds to the conjugacy class containing the matrix $$\begin{bmatrix}
\exp{(i \pi t)} & 0 \\
0 & \exp{(-i \pi t)}
\end{bmatrix},$$ see [@Jewett 15.4]. The dual of $\text{SU}(2)$ can be represented by $$\{\pi_l \in \text{Hom}(\text{SU}(2), \text{U}(2l+1)) : l \in \frac{1}{2}\mathbb{N}_0 \},$$ where $\text{U}(d)$ is the unitary matrix group. The number $l \in \frac{1}{2}\mathbb{N}_0$ is called the quantum number. The character $\psi_{l},$ defined as the trace of $\pi_l,$ is computed at $t \in [0, 1]$ given by $$\psi_l(t)= \frac{\sin(2l+1) \pi t}{\sin \pi t}.$$ Therefore, the dual $\widehat{K}$ is given by $\{(2l+1)^{-1} \psi_l: l \in \frac{1}{2} \mathbb{N}_0\}$ and $k_{\chi_l}= (2l+1)^2.$
The Paley inequality in Theorem \[Paley\] takes the following form in the setting of the compact abelian hypergroup $\text{Conj(SU(2))}.$
Let $1 < p \leq 2$ and let $\{\varphi(l)\}_{l \in \frac{1}{2}\mathbb{N}_0}$ be a positive sequence such that $$M_\varphi:= \sup_{y>0} y \sum_{\overset{l \in \frac{1}{2}\mathbb{N}_0}{\varphi(l) \geq y}} (2l+1)^4<\infty.$$ Then we have $$\sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{4-p} \widehat{f}(l) \varphi(l)^{2-p} \lesssim M_\varphi^{2-p} \|f\|_{L^p(\text{Conj}(SU(2)))}^p.$$
We have the following Hardy-Littlewood inequality for the commutative hypergroup $\text{Conj(SU)(2)}.$
\[HLineq\] If $1 < p \leq 2$ and $f \in L^p(\textnormal{Conj(SU)(2)}),$ then there exists a universal constant $C=C(p)$ such that $$\label{37}
\sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^{p} \leq C \|f\|_{L^p(\textnormal{Conj(SU)(2)})}.$$
Take $\beta=3=\dim(\text{SU(2)})$ and $\{\mu_{\chi_\pi}\}_{\pi \in \widehat{\textnormal{Conj(SU)(2)}}}:=\{(2l+1)^2\}_{l \in \frac{1}{2} \mathbb{N}_0}.$ Then the condition turns out to be $$\sum_{l \in \frac{1}{2}\mathbb{N}_0} \frac{(2l+1)^4}{(|(2l+1)^2|)^3}= \sum_{l \in \frac{1}{2}\mathbb{N}_0} \frac{1}{(2l+1)^2}=\frac{\pi^2}{6}$$ which is finite. Therefore, by Theorem \[HLabe\] the proof of inequality follows.
We would like to recall here the Hardy-Littlewood inequality on the compact Lie group $\text{SU}(2)$ obtained by the second author and R. Akylzhanov in [@ARN1], which says that for $1< p \leq 2$ and $f \in L^p(\text{SU}(2))$ we have $$\sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{\frac{5}{2}p-4} \|\widehat{f}(l)\|_{\textnormal{HS}} \leq C_p \|f\|_{L^p(\text{SU}(2))}.$$ In view of this inequality the Hardy-Littlewood inequality for the compact commutative hypergroup $\text{Conj(SU(2))}$ above is a suitable analogue because in $\text{Conj(SU(2))}$ the dimension $(2l+1)$ of the representation $\pi_l$ is replaced by hyperdimension $(2l+1)^2$ of $\pi_l$ and Fourier transform $f$ at $l \in \frac{1}{2}\mathbb{N}_0$ is scalar so $\|\widehat{f}(l)\|_{\textnormal{HS}}$ is just $|\widehat{f}(l)|.$
Using the duality, we get the following corollary.
\[5.3cor\] If $2 \leq p <\infty$ and $\sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^{p}<\infty$ then $$f \in L^p(\textnormal{Conj(SU)(2)}).$$ Moreover, we have $$\|f\|_{L^p(\textnormal{Conj(SU)(2)})} \leq C(p) \sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^{p}.$$
Using the duality of $L^p$-spaces, we have $$\|f\|_{L^p(\textnormal{Conj(SU)(2)})} = \sup_{\underset{\|g\|_{L^{p'}(\textnormal{Conj(SU)(2)})} \leq 1}{g \in L^{p'}(\textnormal{Conj(SU)(2)})}} \left|\int_{\textnormal{Conj(SU)(2)}} f(x)\, \overline{g(x)}\, d\lambda(x)\right|.$$ Now, by the Plancherel identity , we get $$\int_{\textnormal{Conj(SU)(2)}} f(x) \overline{g(x)}\, d\lambda(x) = \sum_{l \in \frac{1}{2}\mathbb{N}_0} (2l+1)^2 \widehat{f}(l)\,\overline{\widehat{g}(l)}.$$ By noting that $(2l+1)^2=(2l+1)^{2\left(\frac{5}{2}-\frac{4}{p}+\frac{5}{2}-\frac{4}{p'} \right)}$ and applying the Hölder inequality, for any $g \in L^{p'}(\textnormal{Conj(SU)(2)}),$ we have $$\begin{aligned}
\left| \sum_{l \in \frac{1}{2} \mathbb{N}_0} (2l+1)^2 \widehat{f}(l)\, \overline{\widehat{g}(l)} \right| &\leq \sum_{l \in \frac{1}{2} \mathbb{N}_0} (2l+1)^{5-\frac{8}{p}} |\widehat{f}(l)| (2l+1)^{5-\frac{8}{p}} |\widehat{g}(l)| \\& \leq \left( \sum_{l \in \frac{1}{2} \mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^p \right)^{\frac{1}{p}} \left( \sum_{l \in \frac{1}{2} \mathbb{N}_0} (2l+1)^{5p'-8} |\widehat{g}(l)|^{p'} \right)^{\frac{1}{p'}} \\&\leq C(p) \left( \sum_{l \in \frac{1}{2} \mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^p \right)^{\frac{1}{p}} \,\, \|g\|_{L^{p'}(\textnormal{Conj(SU)(2)}},
\end{aligned}$$ where we have used Theorem \[HLineq\] in the last inequality. Therefore, by we have
$$\left|\int_{\textnormal{Conj(SU)(2)}} f(x)\, \overline{g(x)}\, d\lambda(x)\right| \leq C(p) \left( \sum_{l \in \frac{1}{2} \mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^p \right)^{\frac{1}{p}} \,\, \|g\|_{L^{p'}(\textnormal{Conj(SU)(2)}}.$$ Thus, by taking supremum over all $g \in L^{p'}(\textnormal{Conj(SU)(2)})$ with $\|g\|_{L^{p'}(\textnormal{Conj(SU)(2)})} \leq 1,$ we get $$\|f\|_{L^p(\textnormal{Conj(SU)(2)})} \leq C(p) \left( \sum_{l \in \frac{1}{2} \mathbb{N}_0} (2l+1)^{5p-8} |\widehat{f}(l)|^p \right)^{\frac{1}{p}},$$ completing the proof.
Countable compact hypergroups
-----------------------------
Dunkl and Ramirez [@dun2] studied an interesting class of countable hypergroups. Let $\mathbb{N}_0^*=\{0,1,2, \ldots, \infty\}$ be the one-point compactification of $\mathbb{N}_0.$ Dunkl and Ramirez [@dun2] defined a convolution structure $*$ on $\mathbb{N}_0^*$ for every $ 0<a \leq \frac{1}{2},$ denoted by $H_a,$ to make it a (hermitian) countable compact hypergroup . For a prime $p,$ let $\Delta_p$ be the ring of p-adic integers and $\mathcal{W}$ be its group of units, that is, $\{x=x_0+x_1p+ \ldots+ x_np^n+ \ldots \in \Delta_p : x_j = 0,1, \ldots,p-1 \, \text{for}\, j \geq 0 \, \text{and} \, x_0 \neq 0 \}$. For $a=\frac{1}{p},$ $H_{\frac{1}{p}}$ derives its structure from $\mathcal{W}$-orbits of the action of $\mathcal{W}$ on $\Delta_p$ by multiplication in $\Delta_p.$ In fact, the convolution is given as follows: for $m, n \in \mathbb{N}_0,$ define $$\delta_m*\delta_n= \delta_{\text{min}\{ m,n\}}\,\,\,\,\,\, \text{if}\, m \neq n,$$ $\delta_m*\delta_\infty= \delta_\infty*\delta_m=\delta_m$, $\delta_\infty*\delta_\infty=\delta_\infty,$ and for $m=n,$ $$\delta_m* \delta_m(t)= \begin{cases} 0 & t<m, \\ \frac{1-2a}{1-a} & t=m, \\ a^k & t=m+k>m, \\ 0 & t= \infty. \end{cases}$$ The Haar measure $\lambda$ on $H_a$ is given by $$\lambda(\{k\})=a^k(1-a)\quad \text{for}\,\, k<\infty, \quad \lambda(\{\infty\})=0.$$ The elements of $\widehat{H_a}$ are given by $\{\chi_n : n \in \mathbb{N}_0\},$ where, for $k \in H_a,$ $$\begin{aligned}
\chi_n(k)= \begin{cases} 0 & \text{if}\,\,k <n-1, \\ \frac{a}{a-1} & \text{if}\,\,k=n-1, \\ 1 &\text{if}\,\, k \geq n \,\,\, (\text{or}\,\, k = \infty).\end{cases}
\end{aligned}$$ Then the convolution ‘$*$’ on $\mathbb{N}_0$ identified with $\widehat{H_a}=\{\chi_n : n \in \mathbb{N}_0\}$ is dictated by pointwise product of functions in $\widehat{H_a},$ that is: $$\begin{aligned}
\delta_{\chi_m}*\delta_{\chi_n} &=& \delta_{\chi_{\text{max}\{m,n\}}} \,\,\, \text{for}\,\,\, m \neq n, \\
\delta_{\chi_0}*\delta_{\chi_0}&=&\delta_{\chi_0}, \,\,\,\, \delta_{\chi_1}*\delta_{\chi_1} = \frac{a}{1-a} \delta_{\chi_0}+ \frac{1-2a}{1-a} \delta_{\chi_1},\\
\delta_{\chi_n}*\delta_{\chi_n}&=& \frac{a^n}{1-a} \delta_{\chi_0}+ \sum_{k=1}^{n-1} a^{n-k} \delta_{\chi_k}+\frac{1-2a}{1-a} \delta_{\chi_n} \,\,\,\,\, \text{for}\, n \geq 2.
\end{aligned}$$ The dual space $\widehat{H_a}$ of $H_a$ turns into a hermitian discrete hypergroup with respect to the above convolution. The Plancherel measure $\omega$ on $\widehat{H}_a$ is given by $$\omega(\chi_0)=1\quad \text{and} \quad \omega(\chi_n)=(1-a)a^{-n}\quad\text{for}\,\, n \geq 1.$$
The Paley-type inequality for Dunkl-Ramirez hypergroup is then given by the following theorem.
Let $1 < p \leq 2$ and let $\{\varphi(n)\}_{n \in \mathbb{N}_0}$ be a positive sequence such that $$M_\varphi:= \sup_{y>0} y \sum_{\overset{n \in \mathbb{N}}{\varphi(n) \geq y}} (1-a)^2a^{-2n}+\varphi(0)<\infty.$$ Then we have $$\sum_{n \in \mathbb{N}} (a^{-n}(1-a))^{2-\frac{p}{2}} \widehat{f}(n) \varphi(n)^{2-p} \lesssim M_\varphi^{2-p} \|f\|_{L^p(H_a)}^p.$$
We have the following Hardy-Littlewood inequality for the compact countable commutative hypergroups $H_a.$
\[HLineqduk\] If $1 < p \leq 2$ then there exists a constant $C=C(p)$ such that $$\label{39}
f(0)+\sum_{n \in \mathbb{N}} ((1-a)a^{-n})^{p(\frac{5}{2}-\frac{4}{p})} |\widehat{f}(n)|^{p} \leq C \|f\|_{L^p(H_a)}.$$
We apply Theorem \[HLabe\] to get inequality above. The condition for $\beta=3$ by choosing the sequence $\{\mu_{\chi_n}\}_{n \in \mathbb{N}}:= \{(1-a)a^{-n}\}_{n \in \mathbb{N}}$ with $\mu_{\chi_0}=1$ turns out to be $$\sum_{n \in \mathbb{N}_0} \frac{k_{\chi_n}^2 }{|\mu_{\chi_n}|^\beta}=\sum_{n \in \mathbb{N}_0} \frac{(1-a)^2 a^{-2n}}{(1-a)^3 a^{-3n}}=1+ \frac{1}{1-a} \sum_{n \in \mathbb{N}} a^n= \frac{1-a+a^2}{(1-a)^2}=\frac{(1-a)^2-a}{(1-a)^2},$$ which is finite. Therefore, by Theorem \[HLabe\] the proof of inequality follows.
The proof of the following corollary is exactly similar to Corollary \[5.3cor\] in the previous subsection.
If $2 \leq p <\infty$ and $f(0)+\sum_{n \in \mathbb{N}} ((1-a)a^{-n})^{p(\frac{5}{2}-\frac{4}{p})} |\widehat{f}(n)|^{p}<\infty,$ then $$f \in L^p(H_a).$$ Moreover, we have $$\|f \|_{ L^p(H_a)} \leq C_p \left(f(0)+\sum_{n \in \mathbb{N}} ((1-a)a^{-n})^{p(\frac{5}{2}-\frac{4}{p})} |\widehat{f}(n)|^{p} \right).$$
Acknowledgement {#acknowledgement .unnumbered}
===============
Vishvesh Kumar thanks Prof. Ajit Iqbal Singh and Prof. Kenneth A. Ross for their suggestions and comments. The authors are supported by FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations. Michael Ruzhansky is also supported in parts by the EPSRC Grant EP/R003025/1 and by the Leverhulme Research Grant RPG-2017-151.
[aaa]{}
R. Akylzhanov, E. Nursultanov and M. Ruzhansky, Hardy-Littlewood-Paley inequalities and Fourier multipliers on $SU(2).$ [*Studia Math.*]{} 234 (2016), no. 1, 1-29. R. Akylzhanov, S. Majid and M. Ruzhansky, Smooth dense subalgebras and Fourier multipliers on compact quantum groups, [*Comm. Math. Phys.*]{} 362(3) (2018) 761–799.
R. Akylzhanov, E. Nursultanov and M. Ruzhansky. Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and $L^p$-$L^q$ Fourier multipliers on compact homogeneous manifolds. [*J. Math. Anal. Appl.*]{} 479 (2019), no. 2, 1519–1548.
R. Akylzhanov, and M. Ruzhansky, $L^p-L^q$ multipliers on locally compact groups, [*J. Func. Anal.*]{}, 278(3) (2019), DOI: https://doi.org/10.1016/j.jfa.2019.108324
J-P. Anker, Fourier multipliers on Riemannian symmetric space of the noncompact type, *Ann. of Math. (2)*, 132(3) (1990) 597-628. M. Alaghmandan and M. Amini, Dual space and hyperdimension of compact hypergroups, *Glasg. Math. J.* 59(2) (2017), 421-435. M. Amini and C.-H. Chu, Harmonic functions on hypergroups, *J. Funct. Anal.* 261(7) (2011), 1835-1864.
H. Annabi and K. Trimeche, Convolution généralisée sur le disque unité, [*C. R. Acad. Sci. Ser. A*]{} 278 (1974) 21-24.
J. Bergh and J. Lofstrom, [*Interpolation spaces*]{}, Grundlehren der mathematischen Wissenschaften, (1976).
A. Bényi and T. Oh, The Sobolev inequality on the torus revisited, [*Publ. Math. Debrecen*]{} 83(3) (2013) 359-374. W. R. Bloom and Herbert Heyer, *Harmonic analysis on probability measures on hypergroups*, De Gruyter, Berlin (1995) (Reprint: 2011). A. P. Caldron and A. Zygmund, A note on the interpolation of sublinear operator, *Amer. J. Math.*, 78(2) (1956), 282-288. A. Dasgupta and M. Ruzhansky, Gevrey functions and ultra distributions on compact Lie groups and homogeneous spaces, *Bull. Sci. Math.* 138(6), 756-782 (2014). S. Degenfeld-Schonburg, On the Hausdorff-Young theorem for commutative hypergroups, *Colloq. Math.*, 131(2) (2013) 219-231. C. F. Dunkl, The measure algebra of a locally compact hypergroup, *Trans. Amer. Math. Soc.*, 179 (1973) 331-348. C. F. Dunkl and D. E. Ramirez, A family of countable compact $P_*$-hypergroups, *Trans. Amer. Math. Soc.*, [**202**]{} (1975), 339–356. J. J. F. Fournier and K. A. Ross, Random Fourier series on compact abelian hypergroups, [*J. Austral. Math. Soc. Ser. A*]{} 37(1) (1984) 45-81. V. Fischer and M. Ruzhansky, *Quantization on nilpotent Lie groups*, Progress in Mathematics, 314. Birkhäuser/Springer, \[Cham\], (2016) xiii+557 pp.
G. Gasper and W. Trebels, Multiplier criteria of Marcinkiewicz type for Jacobi expansions, [*Trans. Amer. Math. Soc.*]{} 231(1) (1977) 117-132.
G. Gasper and W. Trebels, Jacobi and Hankel multipliers of type $(p,q), 1<p<q<\infty$, [*Math. Ann.*]{} 237(3) (1978) 243-251. G. Gasper and W. Trebels, Multiplier criteria of Hörmander type for Fourier series and applications to Jacobi series and Hankel transforms,[*Math. Ann.*]{} 242(3) (1979) 225-240.
G. Gasper and W. Trebels, Multiplier criteria of Hörmander type for Jacobi expansions, [*Studia Math.*]{} 68(2) (1980) 187-197.
G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constant, *Math. Annalen* 97 (1927) 159-209.
E. Hewitt and K. A. Ross, Rearrangement of $L^r$ Fourier series on compact abelian groups, *Proc. Lond. Math. Soc.*, 29(3) (1974) 519-540. E. Hewitt and K. A. Ross, *Abstract Harmonic analysis Vol. II: Structure and analysis for compact groups Analysis on locally compact Abelian groups.* Die Grundlehren der mathematischen Wissenschaften, Band 152 Springer-Verlag, New York-Berlin (1970).
L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces. [*Acta Math.*]{}, 104 (1960) 93-140
R. I. Jewett, Spaces with an abstract convolution of measures, *Adv. Math.*, 18 (1975) 1-101. V. Kumar, K. A. Ross and A. I. Singh, Hypergroup deformations of semigroups, [*Semigroup Forum*]{} 99(1) (2019), 169-195.
V. Kumar, K. A. Ross and A. I. Singh, An addendum to “Hypergroup deformations of semigroups", [*Semigroup Forum*]{} 99(1) (2019), 196-197.
V. Kumar, K. A. Ross and A. I. Singh, Ramsey theory for hypergroups. [*Semigroup Forum*]{} (2019). https://doi.org/10.1007/s00233-019-10009-0
V. Kumar and R. Sarma, The Hausdorff-Young inequality for Orlicz spaces on compact hypergroups, [*Colloquium Mathematicum*]{} 160 (2020), 41-51.
V. Kumar and M. Ruzhansky, Hausdorff-Young inequality for Orlicz spaces on compact homogeneous manifolds, [*Indag. Math. (N.S.)*]{} 31(2) (2020) 266-276.
R. Lasser, Orthogonal polynomials and hypergroups, *Rend. Math.*, [**3**]{} (1983), 185–209.
J. E. Littlewood and R. E. A. Paley, Theorems on Fourier series and power series, [*J. London Math. Soc.*]{} 6 (1931), 230–233 J. E. Littlewood and R. E. A. Paley, Theorems on Fourier series and power series (II). [*Proc. London Mat. Soc.*]{}, 42 (1937), 52–89.
E. Michael, Topologies on spaces of subsets, *Trans. Amer. Math. Soc.*, 71 (1951) 152-182.
E. Nursultanov and N. T. Tleukhanova, Lower and upper bounds for the norm of multipliers of multiple trigonometric Fourier series in Lebesgue spaces, [*Funktsional. Anal. i Prilozhen.*]{}, 34(2) (2000) 86-88.
H. Remling and M. Rösler, Convolution algebras for Heckman-Opdam polynomials derived from compact Grassmannians, [*J. Approx. Theory*]{} 197 (2015), 30-48.
M. Rösler and M. Voit, A multivariate version of the disk convolution, [*J. Math. Anal. Appl.*]{} 435(1) (2016), 701-717.
K. A. Ross, Centers of hypergroups, *Trans. Amer. Math. Soc.*, 243 (1978) 251-269.
K. A. Ross, Hypergroups and centers of measure algebras, *Symposia Mathematica, Vol. XXII (Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976), Academic Press, London*, (1977) 189–203. M. Ruzhansky and V. Turunen, *Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics*. Birkhaüser-Verlag, Basel (2010).
R. Sarma, N. S. Kumar and V. Kumar, Multipliers on vector-valued $L^1$-spaces for hypergroups, [*Acta Math. Sin. (Engl. Ser.)*]{} 34(7) (2018) 1059-1073.
N. Ja. Vilenkin and A. U. Klimyk, *Representation of Lie Groups and Special Functions. Vol. 1. Simplest Lie Groups, Special Functions and Integral Transforms,* Kluwer Academic Publishers Group, Dordrecht (1991).
R. C. Vrem, *Representations and Harmonic analysis of compact hypergroups*, Doctoral Dissertation, University of Oregon (1978). R. C. Vrem, Harmonic analysis on compact hypergroups, *Pacific J. Math.*, 85(1) (1979) 239-251. R. C. Vrem, Hypergroup joins and their dual objects, [*Pacific J. Math.*]{} 111(2) (1984), 483-495. B. Willson, Configurations and invariant nets for amenable hypergroups and related algebras, *Trans. Amer. Math. Soc.* 366(10) (2014), 5087-5112. S.-G. Youn, Hardy-Littlewood inequalities on compact quantum groups of Kac type, [*Anal. PDE*]{} 11(1) (2018) 237–261.
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---
author:
- Paul Sobaje
title: Springer Isomorphisms In Characteristic $p$
---
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be a simple algebraic group over $k$, by which we mean that $G$ is non-commutative and that the trivial group is the only connected normal reduced algebraic subgroup (this is sometimes referred to as quasi-simple or almost-simple). Assume that $p$ is *good* for $G$. Let $P$ be a parabolic subgroup whose unipotent radical $U_P$ has nilpotence class less than $p$, and let $\mathfrak{g}, \mathfrak{u}_P$ be the Lie algebras of $G$ and $U_P$ respectively. Denote by $\mathcal{N}(\mathfrak{g})$ the nilpotent variety of $\mathfrak{g}$, and by $\mathcal{U}(G)$ the unipotent variety of $G$.
An argument due to J.-P. Serre, given in [@Ser] and more elaborately in [@Sei], demonstrates that there is a canonical $P$-equivariant isomorphism $\varepsilon_P: \mathfrak{u}_P \xrightarrow{\sim} U_P$ which is uniquely determined by a few desirable properties which we will detail below. If the prime $p$ is also *very good* for $G$, which means that is it good and that it does not divide the order of the fundamental group of $G$, then in this case T.A. Springer proved that there exists a $G$-equivariant isomorphism between $\mathcal{N}(\mathfrak{g})$ and $\mathcal{U}(G)$ [@Sp2]; such a map is known as a ‘Springer isomorphism.’ There are in general many Springer isomorphisms for a given $G$; Serre has shown in the appendix to [@M2] that they can be parameterized by a variety of dimension equal to the rank of $G$. However, every Springer isomorphism for $G$ will restrict to a $P$-equivariant isomorphism between $\mathfrak{u}_P$ and $U_P$ [@M2 Remark 10], and G. McNinch observed in Remark 27 of *loc. cit.* that if $p \ge h$, the Coxeter number of $G$, then there is always some Springer isomorphism whose restriction yields $\varepsilon_P$ for any parabolic subgroup $P$, each of which has unipotent radical with nilpotence class less than $p$. The author then asks if this remains true when $p<h$. This same question also appears in work by J. Carlson, Z. Lin, and D. Nakano [@CLN §2.7], where the question is posed because an answer in the affirmative would give an immediate proof of, and in fact extend, [@CLN Theorem 3] (see Remark \[CLNexplained\] below).
In this paper we show that when $p$ is very good for $G$ there is indeed a Springer isomorphism which restricts to $\varepsilon_P$ on every parabolic subgroup $P$ whose unipotent radical has nilpotence class less than $p$. Thanks to a result due to McNinch in [@M], we show that if $G$ is a classical simple subgroup of $GL_n$ then a particular Springer isomorphism with this property can be given explicitly by the Artin-Hasse exponential series. On the other hand, proving that these isomorphisms exist in general relies on work by G. Seitz in [@Sei] and [@Sei2] on abelian unipotent overgroups of unipotent elements in $G$. These papers in turn depend on and sharpen earlier results by D. Testerman [@T] and R. Proud [@P].
To further highlight the relevance of these results, we point out that if $H$ is a simple algebraic group over an algebraically closed field of characteristic $0$, then there is always a “preferred" Springer isomorphism which is given by the exponential map. That is, $H$ may be embedded in some $GL_n$, and the exponential map on nilpotent matrices in $\mathfrak{gl}_n$ restricts to a Springer isomorphism for $H$ (as follows from [@M Proposition 7.1]). Theorem \[main\] can therefore be seen as an attempt to find a suitable analogue of the exponential map in prime characteristic (although there is no claim that when $p<h$ the properties in the theorem uniquely specify an isomorphism). For instance, if $0 \ne X \in \mathcal{N}(\mathfrak{h})$ then the exponential map takes the line $kX$ to a one-parameter additive subgroup of $H$, and Theorem \[main\](3) gives a characteristic $p$ generalization of this, replacing one-parameter subgroups with Witt groups. The exponential map also defines a group isomorphism between the Lie algebra of a Borel subgroup $B$ (as a group under the Baker-Campbell-Hausdorff formula) and $B$, while Theorem \[main\](1) effectively gives the strongest version of this in positive characteristic.
Preliminaries
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Notation and Conventions
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Throughout $G$ will denote a simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. Fix a maximal torus of $G$, and let $\Phi$ denote the root system of $G$ with respect to this choice. We say that $p$ is a *good* prime for $G$ if $p$ does not divide the coefficients of the highest root of $\Phi$ with respect to some choice of simple roots. This means that $p>2$ if $G$ is of type $B,C$ or $D$; $p>3$ if $G$ is of type $E_6,E_7,F_4$ or $G_2$; and $p>5$ if $G=E_8$ . We further say that $p$ is *very good* if it does not divide the order of the fundamental group of $G$. This is already satisfied by good primes in all types except for type $A$. We note that $p$ is very good if and only if $p$ is good for $G$ and the covering $G_{sc} \rightarrow G$ is a separable morphism, where $G_{sc}$ denotes the simply-connected group isogenous to $G$.
We denote by $\mathcal{U}(G)$ the unipotent variety of $G$, and by $\mathcal{N}(\mathfrak{g})$ the nilpotent variety of its Lie algebra. The conjugation action of $G$ on itself induces an action on both $\mathcal{U}(G)$ and $\mathcal{N}(\mathfrak{g})$. Each variety is irreducible, and each has a unique open orbit under the action of $G$, referred to in both contexts as the *regular* orbit. An element in the regular nilpotent (resp. unipotent) orbit is called a regular nilpotent (resp. unipotent) element. The subvariety of $p$-unipotent elements in $G$ will be denoted by $\mathcal{U}_1(G)$, while $\mathcal{N}_1(\mathfrak{g})$ denotes the $[p]$-nilpotent variety of $\mathfrak{g}$, where $x \mapsto x^{[p]}$ is the restriction map on $\mathfrak{g}$. We also will refer to $\mathcal{N}_1(\mathfrak{g})$ as the restricted nullcone. An element $X$ will be said to have nilpotent order $p^m$ if $X^{[p^m]}=0$ and $X^{[p^{m-1}]} \ne 0$.
Let $P$ be a parabolic subgroup of $G$ whose unipotent radical $U_P$ has nilpotence class less than $p$. We will follow [@CLNP] in referring to $P$ as a *restricted* parabolic subgroup of $G$ (we note that this formulation is stated differently than in *loc. cit.*, though it is equivalent in the cases we are considering).
For any affine algebraic group $H$ over $k$, we denote by $k[H]$ its coordinate algebra. This is a commutative Hopf algebra over $k$. The subgroup $H^0$ denotes the identity component of $H$.
We recall that an abstract group $\Gamma$ is called nilpotent if its descending central series has finite length, in which case this length is known as the nilpotence class of $\Gamma$.
Springer Isomorphisms
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For $p$ very good for $G$, Springer [@Sp2] first proved that there exists a $G$-equivariant homeomorphism between $\mathcal{N}(\mathfrak{g})$ and $\mathcal{U}(G)$. This was later shown to be an isomorphism due to the normality of both varieties. Springer’s method will not be recounted here (but is presented clearly in [@H §6.21]). Rather, we aim to justify the claim that finding a Springer isomorphism for $G$ (in very good characteristic) reduces to finding a regular nilpotent element $X$ and a regular unipotent element $u$ whose centralizers $C_G(X)$ and $C_G(u)$ are equal.
First, we noted earlier that the regular unipotent and the regular nilpotent orbits are open, and in fact both have the property that their complements are of codimension at least $2$ in their respective varieties. The normality of $\mathcal{N}(\mathfrak{g})$ and $\mathcal{U}(G)$ then allows for any isomorphism between these orbits to be extended uniquely to an isomorphism between $\mathcal{N}(\mathfrak{g})$ and $\mathcal{U}(G)$. Thus, finding a Springer isomorphism reduces to finding a $G$-equivariant isomorphism between the regular orbits. This, however, can be reduced to finding $X$ and $u$ as above. The key result used in this last step is that the $G$-orbit of $X$ is isomorphic to the quotient $G/C_G(X)$ (see §2.2 and §2.9 of [@J], and note that in the case of type $A$, we obtain the result for $SL_n$ by instead working over $GL_n$, as the unipotent and nilpotent varieties of $SL_n$ are the same as those of $GL_n$).
We now gather some important results about Springer isomorphisms which are found in Remark 27 and the Appendix of [@M2], and [@MT Theorem E].
\[differential\] [@M2] [@MT] Let $\phi$ be a Springer isomorphism from $\mathcal{N}(\mathfrak{g})$ to $\mathcal{U}(G)$.
1. For any parabolic subgroup $P \le G$ with unipotent radical $U_P$, $\phi$ restricts to an isomorphism between $\mathfrak{u}_P$ and $U_P$.
2. The restriction of $\phi$ to $\mathfrak{u}_P$ has differential sending $\mathfrak{u}_P$ to $\mathfrak{u}_P$, and this map is a scalar multiple of the identity.
3. If $\phi^{\prime}$ is any other Springer isomorphism for $G$, then $\phi$ and $\phi^{\prime}$ give the same bijection between nilpotent and unipotent orbits.
\[sameorder\] McNinch showed in [@M3 Theorem 35] that there is always some Springer isomorphism $\rho$ which satisfies $\rho(X^{[p]})=\rho(X)^p$, hence for any Springer isomorphism $\phi$ Theorem \[differential\](3) implies that $X$ has nilpotent order $p^m$ if and only if $\phi(X)$ has unipotent order $p^m$.
\[tangentmap\] If $\phi$ is a Springer isomorphism and $B$ a Borel subgroup of $G$, then in particular $\phi$ restricts to a $B$-equivariant isomorphism between the smooth varieties $\mathfrak{u}_B$ and $U_B$, and the tangent map at $0$ of this restriction is given by multiplication by some $c \in k^{\times}$ (see [@MT §5.5] for more). Note that $c$ is independent of the choice of $B$, and by abuse of terminology we shall refer to this scalar map as “the tangent map of $\phi$."
A Canonical Exponential Map For Restricted Parabolics
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Let $p$ be good for $G$, and suppose that $P$ is a restricted parabolic subgroup of $G$. In [@Sei Proposition 5.3] (credited by the author to Serre), a $P$-equivariant isomorphism $\varepsilon_P: \mathfrak{u}_P \xrightarrow{\sim} U_P$ is obtained by base-changing the usual exponential isomorphism in characteristic $0$. More specifically, we may assume that $P$ is a standard parabolic subgroup of $G$. Then $G,P,$ and $U_P$ are defined over $\mathbb{Z}$, and one can show that the exponential isomorphism $\mathfrak{u}_{P,\mathbb{Q}} \xrightarrow{\sim} U_{P,\mathbb{Q}}$ is defined over $\mathbb{Z}_{(p)}$, and hence can be base-changed to $k$. This isomorphism can be identified over $k$ according to the following properties (see [@B Ch. 2, §6] for Baker-Campbell-Hausdorff formula):
1. It is $P$-equivariant.
2. There is a group structure on $\mathfrak{u}_P$ given by the Baker-Campbell-Hausdorff formula, and $\varepsilon_P$ is an isomorphism of algebraic groups with respect to this structure on $\mathfrak{u}_P$.
3. The tangent map is the identity.
We note that Theorem \[differential\](2) indicates that this last condition, on its own, is not all that unique. However, when coupled with the first property it specifies uniquely the isomorphism between $\textup{Lie}(U_{\alpha})$ and $U_{\alpha}$, where $U_{\alpha}$ is a root subgroup of $U_P$. Since the root subgroups generate $U_P$, there is at most one map satisfying properties (1)-(3). This essentially recapitulates an argument given in the proof of [@Sei Proposition 5.2].
The Artin-Hasse Exponential
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For a given prime $p$, the Artin-Hasse exponential is the power series $E_p(t)$ defined by
$$E_p(t) = \textup{exp}\left(t + \frac{t^p}{p} + \frac{t^{p^2}}{p^2} + \cdots \right)$$
This power series evidently lies in $\mathbb{Q}\llbracket t \rrbracket$, however one can actually prove that $E_p(t) \in \mathbb{Z}_{(p)}\llbracket t \rrbracket$ (see [@D Proposition 1] for a more general fact). Let $C_i$ denote the coefficient of $t^i$ in $E_p(t)$, and $c_i$ its image in $\mathbb{F}_p$ under the unique homomorphism from $\mathbb{Z}_{(p)}$ to $\mathbb{F}_p$. We obtain in this way an element $e_p(t) \in \mathbb{F}_p \llbracket t \rrbracket \subseteq k\llbracket t \rrbracket$, where the coefficient of $t^i$ in $e_p(t)$ is $c_i$.
We note that as elements in $\mathbb{Q}\llbracket t \rrbracket$, the series $E_p(t)$ will agree with the series $\textup{exp}(t)$ over its first $p$ coefficients. Thus $C_i = 1/i!$ for $i<p$. We also must point out that some sources, for example [@Ser2], define the Artin-Hasse exponential to be the series
$$F_p(t) = \textup{exp}\left(-\left(t + \frac{t^p}{p} + \frac{t^{p^2}}{p^2} + \cdots \right)\right)$$
In particular, this definition is the one employed by McNinch in [@M Proposition 7.5], a result which we will later use. As observed in [@D], this series is just the inverse of $E_p(t)$, in the sense that $F_p(t)E_p(t) = 1 \in \mathbb{Z}_{(p)}\llbracket t \rrbracket$.
Witt Groups and Connected Abelian Unipotent Groups {#witt}
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We will need some basic information about Witt groups, and more generally about connected abelian unipotent groups. A standard source is [@Ser2], we have also benefited from the exposition in [@P] and [@M §3].
Let $\mathcal{W}_m$ denote the group of Witt vectors of length $m$ over $k$. This is a connected abelian unipotent group which is isomorphic as a variety to $\mathbb{A}^m$. We can therefore put coordinates on $\mathcal{W}_m$ so that an element can be written as $(a_0,a_1,\ldots,a_{m-1})$, and accordingly $k[\mathcal{W}_m] \cong k[t_0,t_1,\ldots,t_{m-1}]$.
The following theorem uses the Artin-Hasse exponential series to explicitly describe $\mathcal{W}_m$ as a matrix group.
\[wittmorphism\][@P Theorem 7.4] [@Ser2 §V.16] Let $X \in \mathfrak{gl}_n$ be such that $X^{p^m}=0$ and $X^{p^{m-1}}\ne0$. Then the map $f: \mathcal{W}_m \rightarrow GL_n$ given by $$(a_0,a_1,\ldots,a_{m-1}) \mapsto e_p(a_0X)e_p(a_1X^p)\cdots e_p(a_{m-1}X^{p^{m-1}})$$ is an isomorphism of algebraic groups onto its image.
This realization of $\mathcal{W}_m$ makes a few of its properties clear. First, writing the group operation of $\mathcal{W}_m$ in multiplicative notation, we have $$(a_0,a_1,\ldots,a_{m-1})^p = (0,a_0^p,a_1^p,\ldots,a_{m-2}^p).$$ Second, let $\frac{d}{dt_i} \in \textup{Lie}(\mathcal{W}_m)$ be dual to $t_i$. Then we observe, as is also done in [@M Lemma 3.3(2)], that $$df\left(\frac{d}{dt_i}\right)=X^{p^i}$$ thus $$\left(\frac{d}{dt_i}\right)^{[p]}=X^{p^{i+1}}= \frac{d}{dt_{i+1}}.$$ Finally, we observe that the elements of order $p^j$ are those of the form $$(0,\ldots,0,a_{m-j},\ldots,a_{m-1}), \text{ where } a_{m-j} \ne 0.$$
Let $H$ now be an arbitrary connected abelian unipotent group over $k$. For each $j \ge 1$, let $H^{p^j}$ denote the subgroup generated by all $p^j$-th powers of elements in $H$, and $H_{p^j}$ the subgroup of all elements in $H$ having order dividing $p^j$. By [@Ser2 VII.10] we know that $H$ is isogenous to a direct product of Witt groups, and in the special case that $H$ has dimension $m$ and $m$ is also the smallest integer for which $H^{p^m} = 0$ then $H$ is isogenous to $\mathcal{W}_m$.
Unipotent Overgroups And Centralizers
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Again let $G$ be simple and $p$ good for $G$. Let $u \in G$ be unipotent of order $p^r$. When $r=1$ it was shown by Testerman [@T] that $u$ lies in a closed simple subgroup of $G$ of type $A_1$ (isomorphic to either $PSL_2$ or $SL_2$). In particular this shows that $u$ is always contained in a one-parameter additive subgroup of $G$. Seitz then extended this in [@Sei], showing that that there is a canonical one-parameter additive subgroup of $G$ containing $u$. This one-parameter subgroup is referred to as the saturation of $u$.
Saturation is achieved (or specified) as follows. Let $A$ be a simple subgroup of $G$ of type $A_1$, and let $T_A$ be a maximal torus of $A$. Then $A$ is said to be a *good* $A_1$ subgroup if $\mathfrak{g}$, as a $T_A$-module, only has weights which are $\le 2p-2$. We then have the following:
\[mono\][@Sei] Let $u \in G$ be unipotent of order $p$. Then there is a unique monomorphism $\varphi_u: \mathbb{G}_a \rightarrow G$ with image contained in a good $A_1$ and satisfying $\varphi_u(1) = u$.
When $r>1$, it is clear that $u$ cannot be contained in a subgroup isomorphic to $\mathbb{G}_a$. However, it was shown by Proud [@P] that $u$ can be embedded in a subgroup of $G$ isomorphic to $\mathcal{W}_r$. This result was again refined by Seitz in [@Sei2], and relies on the following result about centralizers of unipotent elements which was first established by Proud in an unpublished manuscript.
\[center\] [@Sei2] Let $u$ be a unipotent element of $G$. Then $$Z(C_G(u))=Z(G) \times Z(C_G(u))^0,$$ and $Z(C_G(u))^0$ is the unipotent radical of $Z(C_G(u))$.
Following Seitz, we call a one-dimensional torus $T$ of $G$ $u$-distinguished if there is a nilpotent element $X \in \mathfrak{g}$ such that $X$ is a weight vector for $T$ of weight $2$, $C_G(X) = C_G(u)$, and $T$ is contained in the derived subgroup of a Levi subgroup of $G$ for which $u$ is distinguished. Such a torus is also the image of an *associated cocharacter* of the nilpotent element $X$ (see [@J §5.3] for an explanation of this terminology). Seitz then proved the following:
\[overgroup\][@Sei2] For a fixed $u$-distinguished torus $T$, there is a unique subgroup $W \le Z(C_G(u))^0$ containing $u$ which is isogenous to $\mathcal{W}_r$ and such that $T$ acts on $W$ without fixed points. The action of $T$ on $W/W^p$ is by weight $2$, and $W^{p^{r-1}}$ is the saturation of $u^{p^{r-1}}$.
We highlight a few further details about this result:
1. According to [@Sei2 Lemma 2.7] we may assume that the $X$ above is an element of $\textup{Lie}(W)$.
2. When $u$ has order $p$ then $W$ is the saturation of $u$ and is therefore canonical. In general the overgroup $W$ depends on the choice of $T$ [@Sei2 §4.3].
3. Any group $W$ which is isogenous to a Witt group and on which a one-dimenional torus $T$ acts without fixed points is referred to by Seitz as being *$T$-homocyclic* group. If $W$ is $T$-homocyclic and isogenous to $\mathcal{W}_r$, then for each $1 \le j \le r$ we have $W^{p^j} = W_{p^{r-j}}$ (see remarks just above [@Sei2 Theorem 1]). There are groups isogenous to $\mathcal{W}_r$ which do not have this property [@Ser2 VII.11].
Finally, we prove a useful lemma for groups of exceptional type which will be needed in proving the “main theorem" found in Section 4.
\[isoismorphic\] Suppose that $G$ is of exceptional type and that $u$ is a unipotent element of order $p^2$. Fix a $u$-distinguished torus $T$ and let $W$ be as in Theorem \[overgroup\]. Then $W$ is isomorphic to $\mathcal{W}_2$.
In [@Ser2 VII.11] it is shown that there are two invariants which determine all connected abelian unipotent groups of dimension $2$ up to isomorphism. The first invariant of $W$ is the isomorphism class of the finite subgroup $W^p/W_p$, which from the comments above must be the trivial group. The second invariant comes from the bijective algebraic group homomorphism $W/W^p \rightarrow W^p$ given by sending $w$ to $w^p$. Putting coordinates on $W$, this $p$-th power map takes the form $(a,b)^p = (0,a^{p^h})$, where $h \ge 1$. The integer $h$ is then the second invariant of $W$.
Seitz proves in [@Sei2 Lemma 4.3] that $T$ acts with weight 2 on $W/W^p$, and with weight $2p$ on $W^p$. We claim that this implies that $h=1$ for $W$. Indeed, if $t \in T$ and $w \in W$, then it is clear that $t.w^p = (t.w)^p$. Now put coordinates on $W$ so that $w = (a,b)$, and fix an isomorphism from $k^{\times}$ to $T$ so that if $t$ is the image of $c \in k^{\times}$ then $t.w = (c^2a,b^{\prime})$. We then have that $$t.(0,a^{p^h}) = t.w^p = (t.w)^p = (0,(c^2a)^{p^h}) = (0,c^{2p^h}a^{p^h}).$$ On the other hand, if $T$ acts by weight $2p$ on $W^p$, then we see that $t.(0,a^{p^h}) = (0,c^{2p}a^{p^h})$. Thus we must have that $2p^h = 2p$, so that $h=1$.
From the explicit description of Witt groups given in the previous section it is clear that $\mathcal{W}_2$ has these same invariants, therefore that $W \cong \mathcal{W}_2$.
Existence In General Type
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Let $G$ be simple and $p$ very good for $G$. Suppose that $\phi$ is a Springer isomorphism which restricts to $\varepsilon_P$ on all unipotent radicals of restricted parabolic subgroups. Then it follows that for every $X \in \mathcal{N}_1(\mathfrak{g})$ this isomorphism $\phi$ maps the line $kX \subseteq \mathcal{N}_1(\mathfrak{g})$ to a one-parameter additive subgroup of $G$. As it turns out, up to scalar multiplication this property is also a sufficient condition for $\phi$ to restrict to $\varepsilon_P$ (see Remark \[tangentmap\] for an explanation of the terminology “the tangent map of $\phi$").
\[sufficient\] If there exists a Springer isomorphism $\phi$ with tangent map the identity and having the property that for every $X \in \mathcal{N}_1(\mathfrak{g})$ the one-dimensional closed subset $\phi(kX)$ is a one-parameter additive subgroup of $G$, then for every restricted parabolic subgroup $P \le G$ the map $\phi$ restricts to $\varepsilon_P$ on $\mathfrak{u}_P$.
Let $P$ be restricted, and $X \in \mathfrak{u}_P$. We have that $\phi$ restricts to a variety isomorphism between $kX$ and its image in $G$, the latter a one-parameter subgroup by assumption, so there exists a group isomorphism $\varphi$ from $\mathbb{G}_a$ to $\phi(kX)$ for which the map $\gamma:\mathbb{G}_a \rightarrow \mathbb{G}_a$ given by $\gamma(s) = \varphi^{-1}(\phi(sX))$ defines a variety automorphism of $\mathbb{G}_a$. Since $\phi(0\cdot X)=1 = \varphi(0)$, it follows that $\gamma(0)=0$. But a variety automorphism of $\mathbb{G}_a$ is of the form $s \mapsto b\cdot s+c$ for some $b,c \in k$ where $b \ne 0$, thus if $\gamma(0)=0$ it must in fact be a group automorphism. Therefore, we see that the map sending $s$ to $\phi(sX)$ for all $s \in \mathbb{G}_a$ defines a monomorphism from $\mathbb{G}_a$ to $G$.
Let $T$ be a one-dimensional torus of $G$ which is the image of an associated cocharacter of $X$. Then $u=\phi(X)$ is a $p$-unipotent element in $G$, and as $C_G(X)=C_G(u)$, we have that $T$ is $u$-distinguished. For every $0\ne s\in k$ we have that $C_G(sX)=C_G(X)=C_G(u)$, hence $\phi(kX)$ is a one-parameter subgroup of $Z(C_G(u))$, and by Theorem \[center\] it follows that $\phi(kX) \subseteq Z(C_G(u))^0$. But this one-parameter subgroup is $T$-stable since $T$ stabilizes $kX$ and $\phi$ is $T$-equivariant. By Theorem \[overgroup\], $\phi(kX)$ is therefore the saturation of $u$, and since $u = \phi(X)$, it follows that the unique monomorphism $\varphi_u$ in Theorem \[mono\] is given by $\varphi_u(s) = \phi(sX)$. Since $\phi$ is a Springer isomorphism and $X \in \mathfrak{u}_P$, then by Theorem \[differential\](1) the saturation of $u$ is a subgroup of $U_P$. The argument in the proof of [@Sei Proposition 5.5] then applies and shows that there is some $Y \in \mathfrak{u}_P$ such that $\varphi_u(s) = \varepsilon_P(sY)$. As both $\phi$ and $\varepsilon_P$ have tangent map the identity, the equality $\varphi_u(s) = \varepsilon_P(sY)$ implies on the one hand that $d\varphi_u(\frac{d}{dt})=Y$, while we see that $d\varphi_u(\frac{d}{dt})=X$ from the fact that $\varphi_u(s)=\phi(sX)$. Therefore $Y=X$ and $\varepsilon_P(X)=\varphi_u(1)=\phi(X)$. Since $P$ and $X$ were arbitrary, this finishes the proof.
We now use Theorem \[overgroup\] to construct a Springer isomorphism which will satisfy the hypotheses in the previous proposition. We remind the reader that as pointed out in the remarks following Theorem \[overgroup\], if $T$ is a $u$-distinguished torus and $W$ is the unique $T$-homocyclic subgroup of $C_G(U)^0$ containing $u$, then $\textup{Lie}(W)$ contains a $T$-weight vector of weight $2$ having the same centralizer in $G$ as does $u$.
\[additive\] Let $u$ be a regular unipotent element in $G$, let $T$ be a $u$-distinguished torus, and let $W \subseteq C_G(U)^0$ be the unique $T$-stable subgroup containing $u$. Let $X \in \textup{Lie}(W)$ be a $T$-weight vector of weight $2$ such that $C_G(X)=C_G(u)$, and let $\phi$ be the Springer isomorphism for $G$ defined by $\phi(X)=u$. Then if $Y \in \mathcal{N}(\mathfrak{g})$ is of nilpotent order $p^m$, there is for every $a,b \in k$ some $g \in \mathcal{U}(G)$ of order $<p^m$ such that $\phi(aY+bY)=\phi(aY)\phi(bY)g$.
Write $|u| = p^r$, and let $a,b \in k$. Since $kX = T.X \cup \{0\}$ and $W$ is stablized by $T$, $\phi$ maps the line $kX$ to the closed subspace $T.u \cup \{1\} = \overline{T.u} \subseteq W$. We have that $W/W^p \cong \mathbb{G}_a$, and $\overline{T.u}$ clearly maps isomorphically (as a variety) onto this quotient. We therefore have an isomorphism of varieties from $kX$ to $\mathbb{G}_a$ which sends $0$ to $0$, hence an isomorphism of algebraic groups. This shows that $\phi(aX)\phi(bX)\phi(-aX-bX) \in W^p$. As every element in $W$ of order less than $p^r$ is the $p^i$-th power of an element of maximal order for some $i$ (this is noted in the remarks following Theorem \[overgroup\]), there is some $w \in W$ having the same order as $u$ and $i>0$ such that $$\phi(aX)\phi(bX)\phi(-aX-bX) = w^{p^i}$$ We observe that $w$ is sent to a non-identity element in $W/W^p$, so that there is some $s \in T$ such that $w \in s.uW^p$. This implies by [@Sei2 Lemma 2.4] that $w$ is in the $G$-orbit of $s.u$ hence in the $G$-orbit of $u$, so by [@Sei2 Lemma 2.2(iii)] we have $C_G(u)=C_G(w)$. Thus there is a Springer isomorphism $\psi$ with $\psi(X)=w$.
Now let $\widetilde{\phi}$ be the map from $\mathcal{N}(\mathfrak{g})$ to $G$ defined by $$\widetilde{\phi}(Y) = \phi(aY)\phi(bY)\phi(-aY-bY).$$ It is not hard to see that $\widetilde{\phi}$ defines a $G$-equivariant morphism of varieties. Indeed, viewing $\phi$ as a morphism to $G$ via inclusion, $\widetilde{\phi}$ can be factored as $$\mathcal{N}(\mathfrak{g}) \xrightarrow{f} \mathcal{N}(\mathfrak{g}) \times \mathcal{N}(\mathfrak{g}) \times \mathcal{N}(\mathfrak{g}) \xrightarrow{\phi \times \phi \times \phi} G \times G \times G \xrightarrow{mult.} G$$ with $f(Y) = \left(aY,bY,-aY-bY\right)$, and $G$ acting diagonally on the product varieties.
Let $\psi^{p^i}$ be the morphism from $\mathcal{N}(\mathfrak{g})$ to $G$ given by $\psi^{p^i}(Y) = \psi(Y)^{p^i}$. In a similar way this is seen to be a $G$-equivariant morphism. Since $\widetilde{\phi}(X) = w^{p^i} = \psi^{p^i}(X)$ and both maps are $G$-equivariant morphisms, they must be equal on the regular nilpotent orbit, hence by density on all of $\mathcal{N}(\mathfrak{g})$. Thus, for all $Y \in \mathcal{N}(\mathfrak{g})$, we have that $\phi(aY)\phi(bY)\phi(-aY-bY) = \psi(Y)^{p^i}$. By Remark \[sameorder\] if $Y$ has nilpotent order $p^m$ then $\psi(Y)^{p^i}$ is a unipotent element of order $<p^m$. As the choice of $a,b$ was arbitrary, this proves the proposition.
These two propositions now prove the following:
\[answer\] If $G$ is simple and $p$ is very good for $G$, then there exists a Springer isomorphism $\phi: \mathcal{N}(\mathfrak{g}) \rightarrow \mathcal{U}(G)$ such that $\phi$ restricts to $\varepsilon_P$ for every restricted parabolic subgroup $P \le G$.
We may take $\phi$ to be as in Proposition \[additive\], possibly composing with a scalar map on $\mathcal{N}$ if needed to ensure that the tangent map is the identity thanks to property (2) in Theorem \[differential\]. It follows that if $Y^{[p]}=0$, then $\phi(aY+bY)=\phi(aY)\phi(bY)$, therefore $\phi(kY)$ is a one-parameter additive subgroup of $G$. As the tangent map of $\phi$ is the identity, we may now apply Proposition \[sufficient\] which completes the proof.
\[expomono\] Though it is clear from the arguments in this section, we highlight for later use that if $\phi$ restricts to $\varepsilon_P$ for all restricted $P$, then $\phi$ “exponentiates" the one-parameter subgroups in Theorem \[mono\]. That is, if $Y=d\varphi_u(\frac{d}{dt})$, then $\varphi_u(a)=\phi(aY)$.
An Explicit Isomorphism For Classical Groups
============================================
In this section we show that for classical matrix groups, the existence of a Springer isomorphism restricting to $\varepsilon_P$ can be given explicitly by the Artin-Hasse exponential series.
Let $a = \{a_i\}_{i=1}^{n-1}$ be any sequence of elements in $k$, and consider the map
$$\phi_a: \mathcal{N}(\mathfrak{gl}_n) \rightarrow \mathcal{U}(GL_n), \quad \phi_a(Y) = 1 + \sum_{i=1}^{n-1} a_i Y^i$$
This map is algebraic, respects the conjugation action of $GL_n$, and thus defines a $GL_n$-equivariant morphism from $\mathcal{N}(\mathfrak{gl}_n)$ to $\mathcal{U}(GL_n)$. Moreover if $a_1 \ne 0$ and if $X$ is regular nilpotent, then it follows from [@J 6.7(1)] that $a_1X + \sum_{i=2}^{n-1} a_i X^i$ will also be regular nilpotent, so that $\phi_a(X)$ is a regular unipotent element. This is most easily seen when $X$ is the nilpotent matrix which is a Jordan block of size $n$, and it is then true for any conjugate of $X$.
We see that $C_{GL_n}(X) \subseteq C_{GL_n}(\phi_a(X))$. By the existence of a Springer isomorphism for $GL_n$, $C_{GL_n}(\phi_a(X))$ is equal to $gC_{GL_n}(X)g^{-1}$ for some $g \in GL_n$. The inclusion $C_{GL_n}(X) \subseteq gC_{GL_n}(X)g^{-1}$ implies that they are equal as they have the same dimension and are both connected, so we have $C_{GL_n}(X) = gC_{GL_n}(X)g^{-1} = C_{GL_n}(\phi_a(X))$. Thus there is a Springer isomorphism $\phi$ which maps $X$ to $\phi_a(X)$, and it must in fact be given by $\phi_a$, since $\phi$ and $\phi_a$ are equal on the regular nilpotent orbit which is open in the irreducible variety $\mathcal{N}(\mathfrak{gl}_n)$. In this way any sequence $a_1, \ldots, a_{n-1}, a_1 \ne 0$, defines a Springer isomorphism for $GL_n$ (compare with [@M2 §10]).
In particular, we may choose a sequence such that $a_i = 1/i!$ for $i<p$. If $\phi$ is the resulting Springer isomorphism, then for a $[p]$-nilpotent matrix $Y$ we have $$\phi(Y) = 1 + Y + \frac{Y^2}{2} + \cdots + \frac{Y^{p-1}}{(p-1)!}$$ hence $\phi(aY+bY)=\phi(aY)\phi(bY)$ for all $a,b \in k$. By Proposition \[sufficient\], such a sequence will define a Springer isomorphism for $GL_n$ which has our desired restriction property. If $G$ is one of the classical subgroups of $GL_n$ listed above, however, it is not true in general that $\phi$ will restrict to a Springer isomorphism for $G$. To ensure this latter property holds, we will work with the sequence given by the Artin-Hasse exponential series.
\[restricts\] Let $G$ be either $SO_n$ or $Sp_{n}$, $n=2n^{\prime}$ in the latter case, and identify $G \le GL_n$ via its natural embedding. Let $\phi$ be the Springer isomorphism for $GL_n$ given by the sending $X \in \mathcal{N}(\mathfrak{gl}_n)$ to $e_p(X)$, where $e_p(t)$ is the image of the Artin-Hasse exponential series in $k\llbracket t \rrbracket$. Then $\phi$ restricts to a Springer isomorphism for $G$.
Let $0 \ne X \in \mathcal{N}(\mathfrak{gl}_n)$ be of nilpotent degree $p^m$. In [@M Proposition 7.5], McNinch proves that if $X \in \mathfrak{g}$, then the injective morphism of Theorem \[wittmorphism\] has image in $G$. In particular, $e_p(X)$ is the image of $(1,0,\ldots,0)$ under this map, proving the claim.
As noted earlier, the definition of the Artin-Hasse exponential used in [@M] is inverse to the one we are using. Thus, the definition of the map $E_X$ given here would correspond in McNinch’s work to the map $E_{-X}$. As $X \in \mathcal{N}(\mathfrak{g}) \iff -X \in \mathcal{N}(\mathfrak{g})$, the proof holds regardless.
Statement of Main Result
========================
\[main\] Let $G$ be a simple algebraic group, and suppose that $p$ is very good for $G$. Then there is a Springer isomorphism $\phi: \mathcal{N}(\mathfrak{g}) \xrightarrow{\sim} \mathcal{U}(G)$ such that:
1. For any restricted parabolic $P \le G$, $\phi$ restricted to $\mathfrak{u}_P$ is $\varepsilon_P$.
2. For all $X \in \mathcal{N}(\mathfrak{g})$, $\phi(X^{[p]}) = \phi(X)^p$.
3. If $X \ne 0$, and $m$ is the least integer such that $X^{[p^{m}]}=0$, then $\phi$ defines an injective morphism $\mathcal{W}_m \rightarrow G$ given by
$(a_0,a_1,\ldots,a_{m-1}) \mapsto \phi(a_0X)\phi(a_1X^{[p]})\cdots \phi(a_{m-1}X^{[p^{m-1}]})$
We know by Theorem \[answer\] that there is some Springer isomorphism satisfying (1) for all such $G$. However, to show that one exists which satisfies all of the properties above we will split the proof into classical and exceptional cases.
First suppose that $G$ is one of the groups $SL_n, SO_n$, or $Sp_{2n}$, with its natural embedding in $GL_n$ or $GL_{2n}$, and with corresponding Springer isomorphism given by the Artin-Hasse exponential series. Property (1) then follows from Proposition \[sufficient\], while (3) holds for $\phi$ thanks to [@M Proposition 7.5]. To see that (2) holds we note that since the coefficients of $e_p(t)$ lie in $\mathbb{F}_p$ we get $e_p(X)^p = e_p(X^p)=e_p(X^{[p]})$ (in this last equality we are using the fact that the embedding of $G$ guarantees that $X^p$ as an element of $\mathfrak{gl}_n$ is equal to the image of $X^{[p]}$).
The assumption that $p$ is very good ensures that these results will also apply to any group isogenous to one of these classical groups above, so this proves (2) and (3) for classical types.
If $G$ is of exceptional type, let $\phi$ be as in Proposition \[additive\] (and again, adjusting if necessary by a scalar map on $\mathcal{N}$ so that tangent map is identity). As $p$ is good for $G$, [@T 0.4] shows that all unipotent elements either have order $p$ or $p^2$, hence all nilpotent elements have nilpotent order $p$ or $p^2$. However, it is clear that $\phi$ satisfies (2) and (3) if every element is $[p]$-nilpotent, thus we suppose we are in the second case.
Let $X$ and $T$ be as in the construction of $\phi$ in Proposition \[additive\]. So $\phi(X)$ is contained in a unique $W \le Z(C_G(\phi(X)))^0$ which by Lemma \[isoismorphic\] is isomorphic to $\mathcal{W}_2$ and on which $T$ acts without fixed points. We have $\phi(kX) \subseteq W$, and we may put coordinates $(a_0,a_1)$ on $\mathcal{W}_2$ so that there is an isomorphism $f: \mathcal{W}_2 \rightarrow W$ with $f((a_0,0)) = \phi(a_0X)$. As in §\[witt\], we have $f((0,a_1)) = \phi(F^{-1}(a_1)X)^p$, where $F^{-1}$ is the inverse of the Frobenius map (and, we note, not an algebraic map). We also see that if $k[\mathcal{W}_2] = k[t_0,t_1]$, then the differential $df$ maps $\frac{d}{dt_0} \mapsto X$, and therefore sends $\frac{d}{dt_1} \mapsto X^{[p]}$.
Let $u=\phi(X)^p$. Because $W^p$ is the saturation of $u$, it follows that the unique monomorphism $\varphi_u$ of Theorem \[mono\] can be given by $\varphi_u(a)=f(0,a)$. We see then that $$d\varphi_u\left(\frac{d}{dt}\right) = df\left(\frac{d}{dt_1}\right)=X^{[p]}.$$ By Remark \[expomono\] we have that $\varphi_u(a) = \phi(aX^{[p]})$. Thus $$\phi(X^{[p]})=\varphi_u(1)=f(0,1)=\phi(X)^p.$$ This shows that (2) and (3) hold for regular elements, and by arguments similar to those in the proof of Proposition \[additive\], must hold for all nilpotent elements.
We now establish a simple lemma about Springer isomorphisms (possibly shown elsewhere), after which we have as a corollary that [@CLN Theorem 3] extends to all very good primes.
If $\phi$ is any Springer isomorphism for $G$, then for any $X,Y \in \mathcal{N}(\mathfrak{g})$, $[X,Y]=0$ if and only if $\phi(X)$ commutes with $\phi(Y)$.
Under our conditions on $p$, we have by [@J §2.5,2.6] that $C_{\mathfrak{g}}(Y)=\textup{Lie}(C_G(Y))$. If $[X,Y]=0$, then $X \in C_{\mathfrak{g}}(Y)=\textup{Lie}(C_G(Y))=\textup{Lie}(C_G(\phi(Y))$, therefore $\phi(Y)$ acts trivially via the adjoint action on $X$, and hence $\phi(Y)$ commutes with $\phi(X)$. Conversely, by the remarks preceding [@MT Theorem E], if $\phi(X) \in C_G(\phi(Y))$ then $X \in \textup{Lie}(C_G(\phi(Y))$, hence $X \in \textup{Lie}(C_G(Y)) = C_{\mathfrak{g}}(Y)$.
Let $G$ and $\phi$ be as in Theorem \[main\]. Then $\phi$ restricts to a unique $G$-equivariant isomorphism $\phi_1: \mathcal{N}_1(\mathfrak{g}) \xrightarrow{\sim} \mathcal{U}_1(G)$ of algebraic varieties with the following properties:
1. For all $0 \ne X \in \mathcal{N}_1(\mathfrak{g})$, $\phi_1$ restricted to $kX$ is a one-parameter subgroup of $G$ which can be extended to a good $A_1$ subgroup.
2. For any $X,Y \in \mathcal{N}_1(\mathfrak{g})$, $[X,Y]=0$ if and only if $\phi_1(X)$ commutes with $\phi_1(Y)$.
3. The isomorphism $\phi_1$ is defined over $\mathbb{F}_p$.
\[CLNexplained\] In [@CLN] this result was established for all simple groups $G$ in very good characteristic provided that $\mathcal{N}_1(\mathfrak{g})$ is a normal variety. It is not known in general whether this normality condition holds when $p<h$, and the authors observed that this condition could be dropped if, in our notation, $\varepsilon_P$ came from restricting a Springer isomorphism (see the final paragraph of §2.7 of *loc. cit.*). The authors demonstrate the importance of $\phi_1$ (which they call “exp") in Sections 3 and 4 of *loc. cit.*, using it in a critical way to prove significant results about cohomological support varieties of rational $G$-modules. This corollary is then a first step in extending their results to smaller primes.
**Acknowledgements:** We wish to acknowledge helpful discussions and comments on various versions of this paper from Zongzhu Lin, George McNinch, Dan Nakano, Arun Ram, and Craig Westerland. We thank the referees for many good suggestions which have surely improved the exposition in this paper. This research was partially supported by grants from the Australian Research Council (DP1095831, DP0986774 and DP120101942).
[20]{}
N. Bourbaki, [*Lie Groups and Lie Algebras*]{}, Springer, Berlin, 1989 (Chapters 1-3).
J. Carlson, Z. Lin, and D. Nakano, [*Support Varieties for modules over Chevalley groups and classical Lie algebras*]{}, Trans. A.M.S. [**360**]{} (2008), 1870-1906.
J. Carlson, Z. Lin, D. Nakano, and B. Parshall, [*The restricted nullcone*]{}, Contemp. Math., [**325**]{} (2003), 51-75.
J. Dieudonné, [*On the Artin-Hasse exponential series*]{}, Proc. Amer. Math. Soc. [**8**]{} (1957), 210-214.
J.E. Humphreys, [*Conjugacy Classes In Semisimple Algebraic Groups*]{}, Mathematical Surveys and Monographs [**43**]{}, Amer. Math. Soc., Providence, 1995.
J.C. Jantzen, [*Nilpotent Orbits in Representation Theory*]{}, Progr. Math. 228, Birkháuser, Boston, 2004.
G. McNinch, [*Abelian unipotent subgroups of reductive groups*]{}, J. Pure Appl. Algebra, [**167**]{} (2002), no. 2-3, 269-300.
G. McNinch, [*Optimal $SL(2)$-homomorphisms*]{}, Comment. Math. Helv., [**80**]{} (2005), no. 2, 391-426.
G. McNinch, [*Sub-principal homomorphisms in positive characteristic*]{}, Math. Z. [**244**]{} (2003), 433-455.
G. McNinch, D. Testerman, [*Nilpotent centralizers and Springer isomorphisms*]{}, J. Pure Appl. Algebra [**213**]{} (2009), no. 7, 1346-1363.
R. Proud, [*Witt groups and unipotent elements in algebraic groups*]{}, Proc. London Math. Soc. (3) [**82**]{} (2001), 647-675.
G. Seitz, [*Unipotent elements, tilting modules, and saturation*]{}, Invent. Math. [**141**]{} (2000) 3, 467-502.
G. Seitz, [*Unipotent centralizers in algebraic groups*]{}, Journal of Algebra [**279**]{} (2004), 226-259.
J.-P. Serre, [*Exemples de plongements des groupes $\mathbb{PSL}_2(\mathbb{F}_p)$ dans des groupes de Lie simples*]{}, Invent. Math. [**124**]{} (1-3) (1996) 525-562.
J.-P. Serre, [*Algebraic Groups and Class Fields*]{}, Graduate Texts in Mathematics, vol. 117, Springer, New York, 1988.
T. A. Springer, The unipotent variety of a semi-simple group, in: Algebraic Geometry (International Colloquium, Tata Institute of Fundamental Research, Bombay, 1968), Oxford University Press, London, 1969, pp. 373-391.
A. Suslin, E. Friedlander, and C. Bendel, [*Infinitesimal 1-parameter subgroups and cohomology*]{}, Journal of the A.M.S. [**10**]{} (1997), 693-728.
D. Testerman, [*$A_1$-type overgroups of elements of order $p$ in semisimple algebraic groups and the associated finite groups*]{}, J. Algebra [**177**]{} (1995), 34-76.
\
paul.sobaje@unimelb.edu.au\
Phone: 61 401769982
|
---
abstract: 'We calculate the probability distribution function (PDF) of the expected annihilation luminosities of dark matter subhalos as a function of subhalo mass and distance from the Galactic center using a semi-analytical model of halo evolution. We find that the PDF of luminosities is relatively broad, exhibiting a spread of as much as an order of magnitude at fixed subhalo mass and halo-centric distance. The luminosity PDF allows for simple construction of mock samples of $\gamma$-ray luminous subhalos and assessment of the variance in among predicted $\gamma$-ray signals from dark matter annihilation. Other applications include quantifying the variance among the expected luminosities of dwarf spheroidal galaxies, assessing the level at which dark matter annihilation can be a contaminant in the expected $\gamma$-ray signal from other astrophysical sources, as well as estimating the level at which nearby subhalos can contribute to the antimatter flux.'
author:
- 'Savvas M. Koushiappas'
- 'Andrew R. Zentner'
- 'Andrey V. Kravtsov'
bibliography:
- 'manuscript.bib'
title: The distribution of annihilation luminosities in dark matter substructure
---
Introduction {#sec:introduction}
============
Multiple lines of observational evidence have established the existence of a form of non-baryonic dark matter binding galaxies. Dark matter is commonly considered a yet-to-be discovered elementary particle. A promising candidate is a Weakly-Interacting Massive Particle (WIMP) that arises in extensions to the standard model of particle physics. Examples include the lightest supersymmetric particle [@Jungman:1995df; @BHS05] and particle excitations in theories of Universal Extra Dimensions [@Hooper:2007qk]. WIMPs interact via the Weak interaction and were in thermal equilibrium in the early Universe. If the dark matter is a thermal relic WIMP, the WIMP annihilation cross section can be constrained by requiring that the present dark matter density is $\Omega_{\mathrm{c}} \approx 0.23$ [@Komatsu:2008hk]. Implied values for this cross section are of order $\langle \sigma v \rangle \sim 10^{-26}$ cm$^3$/s, and candidate WIMPs can annihilate to many observable states, such as $\gamma$-rays and high-energy neutrinos.
The Cold Dark Matter (CDM) model of cosmological structure formation predicts that dark matter is distributed in “halos" in a hierarchical fashion. The host halo of the Milky Way is expected to have mass of $M \approx 10^{12}\ h^{-1}$M$_{\odot}$ [@Klypin:2001xu; @Li:2007eg]. Within the CDM model such halos are also expected to contain numerous smaller dark matter “subhalos" which, in turn, contain subhalos of their own, perhaps with masses all the way down to the cutoff scale in the primordial density fluctuation power spectrum [@Schmid:1998mx; @HSS01; @Green:2003un; @Green:2005fa].
A well-studied avenue for possible identification of the dark matter is to search for unique products of dark matter annihilation at the center of our Galactic halo, where densities are highest [@Dodelson:2007gd; @Serpico:2008ga; @Gondolo:1999ef; @Horns:2004bk; @Merritt:2002vj; @Bergstrom:1997fj; @Aloisio:2004hy; @Zaharijas:2006qb]. However, astrophysical backgrounds are also highest toward the Galactic center, so an alternative approach is to search for annihilation within subhalos [@Baltz:1999ra; @Tyler:2002ux; @Evans:2003sc; @Profumo:2005xd; @Bergstrom:2005qk; @Calcaneo-Roldan:2000yt; @Tasitsiomi:2002vh; @Stoehr:2003hf; @Koushiappas:2003bn; @Baltz:2006sv; @Strigari:2006rd; @Strigari:2007at; @Diemand:2006ik] (including potentially sub-solar mass halos [@Bringmann:2009vf; @Koushiappas:2009du; @Bringmann:2006mu; @Pieri:2005pg; @Ando:2008br; @Diemand:2006ik]), or to infer annihilation within subhalos statistically through the angular distribution of diffuse $\gamma$-ray emission [@Ando:2009fp; @Ando:2006cr; @Cuoco:2006tr; @Cuoco:2007sh; @Fornasa:2009qh; @Hooper:2007be; @Lee:2008fm; @SiegalGaskins:2009pz; @SiegalGaskins:2009ux; @SiegalGaskins:2008ge; @Taoso:2008qz]. More recently, the one-point $\gamma$-ray flux probability distribution function (PDF) was proposed as another way to quantify the expected annihilation signal [@Lee:2008fm] . The main idea behind this approach is that the pixel-to-pixel flux variation from $\gamma$-ray emission from small dark matter subhalos would deviate from the Poisson fluctuations that would be expected from a smoothly-distributed dark matter halo and smoothly-distributed backgrounds.
A calculation of the contribution of annihilation products from subhalos to the flux along any sight line consists of the following two ingredients. The first is the number of subhalos intercepted along the line of sight, assuming subhalos to be small compared to the angular resolution of the instrument as is the case with contemporary detectors. The second is the annihilation luminosity of each intercepted subhalo. The latter depends on the distribution of dark matter within subhalos, which reflects the underlying process of nonlinear mass assembly individual to each subhalo. Although two subhalos may have the same mass and be located at the same Galacto-centric distance today, they may have different annihilation luminosities because they may have different formation times, mass assembly histories, and different orbits in the Milky Way potential. Each of these factors affects the internal densities of subhalos. Therefore, we expect a distribution of luminosities at each subhalo mass and Galacto-centric distance.
In this paper, we estimate the probability distribution function (PDF) of the subhalo luminosity as a function of the subhalo mass and Galacto-centric distance from a large ensemble of subhalo populations generated using a semi-analytic model of halo and subhalo evolution [@Zentner:2004dq]. Our aim is to use a statistically-large sample of subhalo properties to provide a useful tool for computing subhalo annihilation signals in a way that captures some of the complexity of nonlinear evolution. We show that the luminosity PDF of subhalos is well fit by a log-normal distribution, reflecting the underlying distribution in formation times (or concentrations, [@Bullock:1999he; @Wechsler:2001cs]) and provide simple, empirical fits to the luminosity PDFs as a function of mass and distance. Applications for the derived PDF range from quantifying the variance in the expected luminosities of dwarf spheroidal galaxies to generating mock synthetic $\gamma$-ray sky maps, to understanding the level at which dark matter annihilation can be a contaminant in the expected $\gamma$-ray signal from other astrophysical sources [@Ando:2006mt; @Miniati:2007ke; @Zhang:2004tj; @Pavlidou:2002va], and to ascertain the level at which a nearby subhalo can contribute to the measured flux of antimatter [@Hooper:2008kv].
The importance of the subhalo luminosity PDF {#section:importance}
============================================
The effects of a distribution of luminosities on any calculation that involves the diffuse emission is only important if most of the diffuse flux originates from a large number of sources. If, for example, the diffuse flux is due to very few sources with high luminosities, then the properties of the diffuse background would be dominated by the Poisson statistics of the emission of photons from these sources. If, on the other hand, there are many dim sources along the line of sight, the intrinsic variation in the luminosities of these sources will have an effect on the flux PDF. This is due to the fact that the flux PDF will deviate from Poisson statistics as it will depend not only on the flux-density distribution but also on the mean number of sources (see the ’$P(D)$ analysis’ discussion in the Appendix of @Lee:2008fm).
{height="7cm"} {height="7cm"}
We can quantify this argument as follows. Suppose the number density of subhalos of luminosity $L$ at position $\ell$ along a particular line of sight is given by $dN/dMdV \propto M^{-\alpha} n[\ell(r)]$, where $n[\ell(r)]$ is the number density of subhalos at Galacto-centric distance $r$ along line of sight distance $\ell$. Additionally, assume a mapping between the luminosity and the mass of a subhalo, $L\propto M^\beta$. Then the contribution to the received flux that is produced by subhalos in a given logarithmic mass and line of sight interval is given by $$\tilde{F}_{M,\ell} \equiv \frac{d F}{d \ln M d \ln \ell} \propto M^{-\alpha + \beta + 1} \ell \, n[r(\ell)],
\label{eq:Ftilde}$$ Taking $\alpha \approx 1.9$ and $\beta \approx 0.8$ (consistent with analytical arguments [@Strigari:2006rd], and numerical simulations [@Stoehr:2003hf; @Diemand:2006ik; @kuhlen:2008aw]), the flux per logarithmic interval in mass and line of sight distance has a weak dependence on mass $\tilde{F}_{M,\ell} \propto M^{-0.1}$.
In order for the low mass subhalo contribution to the annihilation flux to be roughly the same as high mass halos, their abundance must be larger. The mean number of subhalos per logarithmic line of sight interval and logarithmic mass interval is $$\tilde{N}_{M,\ell} \equiv \frac{ dN}{d \ln M d \ln \ell} \propto M^{-\alpha + 1} \ell^3 \, n[r(\ell)].
\label{eq:Ntilde}$$ Assuming that the distribution of subhalos traces that of dark matter, $n[\tilde{r}(\ell)] \propto [\tilde{r}( 1 + \tilde{r})]^{-1}$ [@NFW96; @Navarro:1996gj], where $\tilde{r} = r/r_s$ and $r_s$ is the scale radius of the NFW dark matter profile. For $\alpha \approx 1.9$ and small distances ($\ell \ll r \ll r_s$), $n[\tilde{r}(\ell)] \propto
1/\tilde{r}(\ell)$ so that $\tilde{N}_{M,\ell} \propto M^{-0.9}
\ell^2$. For large distances ($\ell \approx r \gg r_s$), $n[\tilde{r}(\ell)] \propto 1/\tilde{r}^3(\ell)$ and $\tilde{N}_{M,\ell} \propto M^{-0.9}$, roughly independent of $\ell$ for all masses.
The previous two paragraphs show that we should expect the majority of the diffuse annihilation flux to be due to the presence of numerous low-mass subhalos. In the left panel of Fig. \[fig:fig0\] we show the quantity $\tilde{F}_{M,\ell}$ in units of ${\rm cm}^{-2}{\rm
s}^{-1}{\rm sr}^{-1}$, as a function of subhalo mass and position along the line of sight, Eq. (\[eq:Ftilde\]), at an angle $\psi =
90^\circ$ with respect to the Galactic center. We assume a Milky Way halo with a radius $R_{MW} = 250 \, {\rm kpc}$, and a scale radius $r_s \approx 20 \, {\rm kpc}$. Note that at any fixed mass, most of the flux comes from a region at $\ell \approx 10 \, {\rm kpc}$. This is because the number of objects declines rapidly with distance at Galacto-centric distances signficantly larger than a scale radius. Moreover, low-mass subhalos contribute marginally more flux than their high-mass counterparts. In the right panel of Fig. \[fig:fig0\] we show the mean number per logarithmic mass and line-of-sight interval, $\tilde{N}_{M,\ell}$. The two panels of Fig. \[fig:fig0\] illustrate a basic conclusion that the mean number of objects along a line of sight increases with decreasing subhalo mass (right panel), and as all intervals of subhalo mass have comparable flux contributions (left panel), the signal is set by relatively low-mass subhalos that are close to the observer. This is in qualitative agreement with the result of @Lee:2008fm who found that substructures give rise to photon counts that deviate from a Poisson distribution.
Our simple demonstration neglects the presence of a baryonic disk in the Milky Way halo, and its effect on the subhalo population in the inner regions of the halo. Recent studies find that the inner 30 kpc of a Milky Way-sized halo may be deficient in subhalos due to interactions with the disk [@D'Onghia:2009pz]. This should have an effect on the flux contributions we derived, but the approximate mass and distance dependence that we describe should be maintained. Nevertheless, it is important to keep in mind that a depleted subhalo population in the inner regions of the Galactic disk may have an effect on the flux from substructure. The maximal net suppression in annihilation flux near the disk due to this suppression is expected to be a factor of a few [@D'Onghia:2009pz].
The subhalo annihilation luminosity PDF {#section:pdf_methods}
=======================================
In order to derive the subhalo luminosity PDF, we model the accretion and dynamical evolution of subhalos in the Milky Way potential well using an approximate semi-analytic technique [@Zentner:2004dq]. This approximate approach greatly reduces the computational cost of modeling substructure by treating subhalo density profiles as continuously-evolving functions that can be described by a small number of parameters, rather than a collection of a very large number of individual particles. This enables calculations of the properties of subhalo populations in a large set of Milky Way-sized halos, so that object-to-object variance can be estimated. This is not yet feasible in simulations directly. This procedure also provides a physically-motivated method to extrapolate the results of numerical simulations to masses below their resolution limits, masses which are not negligible from the annihilation perspective. Of course, the cost of this method is that it is approximate and the approximations implemented cannot be validated outside the range of scales that are resolved by $N$-body simulations. This method is described in detail in [@Zentner:2004dq], which also shows a number of non-trivial comparisons with $N$-body simulations that validate this treatment of halo substructure.
We use an ensemble of 200 realizations of subhalo populations within a dark matter halo of mass $M_{\rm MW} = 1.26 \times 10^{12} \,
h^{-1}$M$_\odot$ in a flat cosmological model with $\Omega_{dm} =
0.228$, $\Omega_b h^2 = 0.0227$, $h=0.71$, and $\sigma_8=0.81$, favored by the five-year Wilkinson Microwave Anisotropy Probe results [@Komatsu:2008hk]. Each realization represents a possible subhalo population within a Milky Way-like halo. The populations differ because only some statistical properties of the initial density field in the local neighborhood are known, not the precise initial conditions for collapse. We use this large number of distinct populations to quantify the predicted variation from one Milky Way-sized halo to another. The result of the calculation is a list of all subhalos, complete with their structural parameters such as bound mass, scale radius, and tidal radius. These quantities all evolve with time, and we study them at the present epoch ($z=0$).
Figure \[fig:fig1\] shows the cumulative velocity function of Galactic subhalos. The power-law behavior is similar to what is found in numerical simulations, with the number of subhalos increasing as $N(>V_{\rm max}) \propto V_{\rm max}^{-3}$. Our model also reproduces the abundance of subhalos in cosmological simulations reasonably well (see also Ref. [@Zentner:2004dq] for a discussion). For example, the average number of subhalos with $v_{\mathrm{max}} > 4$ km/s within the inner 200 kpc of the Milky Way in our models is 2481 with a 68 percentile range of \[1964-3007\]. This is consistent with 2469 subhalos of $v_{\mathrm{max}} > 4$ km/s within the same radius in the Via Lactea II simulation of a Milky Way-sized halo [@Kuhlen:2008qj]. Moreover, the dispersion in the number of subhalos is consistent with the empirical finding of a Poisson scatter added in quadrature to an intrinsic scatter of about $20\%$ derived from recent numerical simulations [@BoylanKolchin:2009an]. We note that the validation exercises in Ref. [@Zentner:2004dq] show that this model may over-estimate halo-to-halo variance mildly (see their Fig. 7), though it is difficult to assess through a comparison with simulations with greater precision at this point.
Note that in general, the slope and normalization of the velocity function depend on the concentration of the host dark matter halo [@Zentner:2004dq]. As shown in Fig. \[fig:fig1\], the normalization of the velocity function of subhalos is reduced. This is to be expected as host dark matter halos with high concentrations are formed earlier than low mass halos. As a result, there is more available time for evolution of the subhalo population leading to a decrease in the total number of subhalos. A subsidiary effect that also contributes to the decrease of the subhalo population of high-concentration halos is the efficient tidal disruption due to the higher central densities. On the other hand, subhalos in host halos with low concentrations will, on average, have spent less time in the host and have a higher rate of survival due to the decreased tidal forces.
We model the final dark matter distribution in a subhalo of mass $M$ as a Navarro-Frenk-White (NFW) profile [@NFW96; @Navarro:1996gj], $$\label{eq:nfw} \rho(r) = \frac{M}{4 \pi r_s^3}
\frac{1}{f(\tilde{r}_t)} \frac{1}{\tilde{r} ( 1 + \tilde{r} )^2}.$$ In Eq. (\[eq:nfw\]), $\tilde{r} = r / r_s$, where $r_s$ is the evolved scale radius. $f(x) = \ln ( 1 + x ) - x / ( 1 + x )$, and $\tilde{r}_t = r_t / r_s$ is the ratio of a tidal truncation radius $r_t$, to the scale radius. We treat tidal truncation by assuming an abrupt limit to the extent of the subhalo density profile for simplicity. Isolated high-resolution simulations of subhalo evolution support a very sharp truncation (e.g., Ref. [@Kazantzidis:2003hb; @Kazantzidis:2009zq]) and, furthermore, the annihilation luminosity emanating from the transition region should be very small in comparison to the total annihilation luminosity. Our simplifying assumption of a truncated NFW profile is not crucial to our analysis. Deviations in the assumed power law in the inner regions of the profile do not result in large changes in the total luminosity of the halo. Profiles with different inner slopes also require different normalizations to maintain the same bound mass against the tidal forces of the host. Moreover, much of the total luminosity of a nearly-NFW halo arises from the region near the scale radius, $r_s$. As such, changes in the inner slope do not lead to large changes in the annihilation luminosity in most circumstances (see Refs. [@Koushiappas:2003bn; @Bergstrom:2005qk; @Strigari:2006rd; @Robertson:2009bh; @Reed:2010]).
![The cumulative velocity function of subhalos. The solid black line shows the average velocity function over all 200 realizations of the formation of a Milky Way-size halo. The short-dashed red line shows the velocity function derived for Galactic halos which have a concentration $c > 13.4$, while the long-dashed blue line shows the velocity function for Galactic halos with $c <
6.7$. The dotted line shows the behavior of a $N(>V_{\rm max}) \propto
V_{\rm max}^{-3}$ power law, which describes the subhalo velocity function in cosmological simulations.[]{data-label="fig:fig1"}](fig2.ps){height="8cm"}
The dark matter luminosity of a subhalo is then obtained from $$\begin{aligned}
\label{eq:luminosity} L &=& 4 \pi \frac{ \langle \sigma v \rangle \,
N_{\gamma}^{\rm tot}}{M_{\chi}^2} \int_0^{{r_t}} \rho^2 r^2 dr \nonumber \\
& = & \frac{3.32 \times 10^{37}\, \mathrm{ph}}{\mathrm{s}} \ \frac{ \langle \sigma v
\rangle_{-26} \, N_{\gamma,30}^{\rm tot}}{M_{\chi,100}^2} \nonumber \\
&\times& \left(\frac{r_s}{{\mathrm{kpc}}} \right)^3
\int_0^{\tilde{r}_t} \left( \frac{
\rho(\tilde{r})}{{\mathrm{GeV/cm}}^{3}} \right)^2 \tilde{r}^2 d
\tilde{r} .\end{aligned}$$ Here, $\langle \sigma v \rangle_{-26}$ is the annihilation cross section in units of $3 \times 10^{-26} {\rm cm}^3 {\rm s}^{-1}$, $M_{\chi,100}$ is the mass of the dark matter particle in units of 100 GeV, and $N_{\gamma,30}^{\rm tot}$ is the total number of photons emitted above a threshold of 1 GeV, in units of 30. This fiducial choice of parameters is representative of optimistic scenarios in the Minimal Supersymmetric Model. We emphasize that we defined luminosity as [*number of photons per unit time*]{}, and not energy per time. Similarly, when we discuss flux, we implicitly mean photon flux, and not the energy flux. Note also that as the goal of this work is to quantify the spread in luminosities in the subhalo population, the choice of particle physics parameters in Eq. \[eq:luminosity\] is intended only to provide a useful representation of the magnitude of the luminosity. However, the PDFs of luminosities derived below are affected by the choice of particle physics parameters only in their normalization.
Results {#section:pdf_results}
=======
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We compute subhalo luminosities as a function of radial position and mass as follows. We first determine the minimum and maximum of the radial distribution (typically $r_{\rm min} \sim {\rm kpc}$, $r_{\rm max} \sim 250 {\rm kpc}$) of subhalos as well as the minimum and maximum subhalo mass at the present epoch (typically $M_{\rm min} \sim 10^4 M_\odot$, $M_{\rm max} \sim 10^{11} M_\odot$) . We then divide the radial distribution of subhalos into $N_r$ bins. Each radial bin is then subdivided into $N_m$ mass bins of equal logarithmic size in mass. We then fit the [*distribution*]{} of luminosities for different values of $N_r$ and $N_m$ until the maximum deviation of the fit is less than 10% of the true value. We find that $N_m=N_r=50$ with at least 200 subhalos per bin has errors of at most 10% of the true value.
The luminosity of a subhalo is a function of its mass as well as its position within the host halo. We find that the subhalo luminosity PDF is well fit by a log-normal distribution, as $$\label{eq:PofL} P[\ln L_{M,r}]= \frac{1}{\sqrt{2 \pi}}
\frac{1}{\sigma_{M,r}} \exp \left[ - \frac{[ \ln L_{M,r} - \langle \ln
L_{M,r} \rangle ]^2 }{2 \sigma_{M,r}^2}\right]$$ where, $$\langle \ln L_{M,r} \rangle = a_1+ a_2 \ln \left(
\frac{M}{10^5M_\odot} \right) +a_3 \ln \left( \frac{r} {{\rm 50 kpc}}
\right) ,
\label{eq:lum}$$ and, $$\sigma_{M,r} = b_1 + b_2 \ln \left(
\frac{M}{10^5 M_\odot} \right) +b_3 \ln \left( \frac{r} {{\rm 50 kpc}}
\right) .
\label{eq:sigma}$$ Here, we implicitly assume that the luminosity $L_{M,r}$ is expressed in units of photons per second.
Assuming the fiducial particle physics parameters as shown in Eq. \[eq:luminosity\], the best fit parameters for the whole population of subhalos in all 200 realizations are $a_1=77.5$, $a_2=0.87$, $a_3=-0.22$, $b_1=0.75$, $b_2=-0.0026$, and $b_3=0.0061$ (see “All" in Table \[table:fitparam\]). This result can be scaled to any assumed particle physics parameters, by simply adding the term $$\ln \left( \frac{ N_\gamma^{\rm tot} \langle \sigma v \rangle M_\chi^{-2}}{9 \times 10^{-29} {\rm cm}^3 {\rm s}^{-1} {\rm GeV}^{-2}} \right)$$ to the parameter $a_1$ of Eq. \[eq:lum\]
The fitting function \[Eq. \[eq:PofL\] with Eq. (\[eq:lum\]) & Eq. (\[eq:sigma\])\] is good to within $\sim$ a few $\%$ for $P[\ln L_{M,r}]$ as a function of $\ln
L_{M,r}$ over the range of masses and radii we have examined with sufficient statistics, $M \approx[10^4 - 10^{10}] M_\odot$, and $r\approx [ 5 - 250]\, {\rm kpc}$. The mean and variance of the distribution are functions of mass and radius, reflecting the fact that the PDF of luminosity is set by the interplay between the mass function of accreted objects, the redshift of accretion, and the orbital evolution of the individual subhalos constituting the population.
$\ \ a_1 \ \ $ $\ \ a_2 \ \ $ $\ \ a_3\ \ $ $\ \ b_1\ \ $ $\ \ b_2\ \ $ $\ \ b_3\ \ $
---------------- ---------------- ---------------- --------------- --------------- --------------- ---------------
[All]{} 77.4 0.87 -0.22 0.75 -0.0026 0.0061
$ {\rm C }_0 $ 77.4 0.87 -0.23 0.74 -0.0030 0.011
$ {\rm C}_+$ 77.5 0.87 -0.26 0.76 -0.0021 0.0077
$ {\rm C}_-$ 77.3 0.87 -0.18 0.75 -0.0013 0.0043
: Fitting parameters for the mean and width of the $\gamma$-ray annihilation flux distribution function (see text). []{data-label="table:fitparam"}
The mean luminosity of isolated, field halos that have [*not*]{} experienced strong interactions within the potential of a more massive halo, scales as $L \propto \rho_s^2 r_s^3 \propto M c^3 / f^2(c)$, where $M$ is the halo mass, corresponding to a virial radius radius $R
\propto M^{1/3}$, $c \equiv R / r_s$ is the concentration and $f(c)
\propto c^{0.4}$ near $c \approx 30$ as is relevant for small subhalos [@Koushiappas:2003bn]. If we assume a weak dependence of concentration on mass $c(M) \propto M^{-0.1}$ [@Bullock:1999he; @Wechsler:2001cs; @Neto:2007vq; @Maccio':2006nu; @Klypin:2010qw], the luminosity of a halo scales roughly as $L \propto M^{0.8}$. However, subhalo populations deviate from this scaling somewhat for several reasons. Subhalos merge into the host halo at a variety of times, so they sample the $c(M)$ relation at a variety of redshifts and subhalos of fixed mass at the time of merger exhibit a variety of bound masses at $z=0$ as a result of their distinct orbital evolution histories. Therefore, subhalo luminosities at fixed mass are influenced by the mass and redshift dependence of concentrations and subhalo orbital properties. The effects of these physical changes in the structure and description of subhalos compared to isolated halos are reflected in the best fit parameters of Eq. \[eq:lum\]. In particular, we find that the luminosity of [*evolved*]{} subhalos scales approximately as $L \propto M^{0.87}$.
{height="5.9cm"} {height="5.9cm"} {height="5.9cm"}
In Figure \[fig:fig2\] we show the distribution of luminosities in three examples of radial and mass bins. The figure also shows the fit to the luminosity distribution given in Eq. (\[eq:PofL\]). The choice of mass and radius displayed in Fig. \[fig:fig2\] are only meant to demonstrate schematically the agreement between the fitting functions and the numerical results. Notice that the spread in luminosities at fixed mass and position can be roughly [*an order of magnitude*]{}, and that the peak of the luminosity PDF depends on both, mass as well as radius.
As the structural properties and abundances of subhalos are influenced by the merger history and properties of the host halo, we expect that the luminosity PDF will be affected by the distribution of dark matter within the host dark matter halo. In order to explore the relationship between the host halo properties and the subhalo luminosity PDF we divided the subhalo populations into three groups based upon the concentrations of their host halos. We grouped subhalos with host halo concentrations within the 68% range of the mean concentration, $\bar{c}=9.7$. This group, “$C_0$", consists of subhalos in primaries with concentrations in the range $6.7 \le c \le 13.4$. We then collected all subhalos in hosts in the upper and lower $16\%$ ranges, with group $C_-$ consisting of host concentrations in the range $c < 6.7$ and group $C_+$ consisting of primaries with $c > 13.4$. The fitting parameters of Eq. (\[eq:PofL\]) for these three concentration bins are also given in Table \[table:fitparam\].
Fig. \[fig:fig3\] shows the effects of the host halo concentration on the luminosity PDF of subhalos. At inner radii (left panel of Fig. \[fig:fig3\]), the effects are more pronounced. This is the region where earliest merging subhalos reside. Host halos with high concentrations are formed early, and contain subhalos that on average have formed earlier, and so are also more concentrated [@Wechsler:2001cs]. As the annihilation signal is sensitive to the concentration of subhalos [@Koushiappas:2003bn], the luminosity of subhalos in a high concentration host is slightly higher ($C_+$ curve in Fig. \[fig:fig3\]) than subhalos residing in host halos with lower concentration ($C_-$ curve in Fig. \[fig:fig3\]). At large radii, $r > r_s$ (right panel of Fig. \[fig:fig3\]) the difference between host halos of different concentrations diminishes, reflecting the fact that the outer regions of halos typically contain recently-merged substructure (the host concentration influences subhalo dynamics little beyond the halo scale radius). Fig. \[fig:fig3\] and Table \[table:fitparam\] demonstrate that the maximum shift in typical subhalo luminosities is in the inner regions of host halos, and is relatively small, $\sim 20\%$.
Applications {#section:app}
============
The flux PDFs that we provide have numerous potential applications. As an example, in this section we compute the probability distribution for measuring a flux between $F$ and $F+dF$ from [*any individual subhalo*]{} along a line of sight at an angle $\psi$, from the Galactic center. We compute this “single halo flux PDF" as $$\begin{aligned}
P_1[F | \psi] &\propto&
\int_0^{{\ell_{\rm max}}} \int_{{M_{\mathrm{min}}}}^{{M_{\mathrm{max}}}} \ell^4 \,
\frac{dN[r(\ell,\psi)]}{dMdV} \nonumber \\
&\times& \, P[L_{F,\ell}|M,r(\ell,\psi)] \, dM \, d \ell
\label{eq:PofF}\end{aligned}$$ Here, $\ell$ is the line-of-sight distance, $dN[r(\ell,\psi)]/dMdV$ is the subhalo mass function, and $L_{F,\ell} = 4 \pi \ell^2 F$ ensures a proper flux measurement for a subhalo of luminosity $L$ at a distance $\ell$. The quantity $\ell_{\rm max}$ is the maximum line-of-sight distance we consider, given by $$\ell_{\rm max} = d_\odot \left[ \cos \psi + \sqrt{ (R_{\rm G} / d_\odot)^2 - \sin^2 \psi}\right],$$ where $d_\odot = 8 \, {\rm kpc}$ is the distance of the Sun to the Galactic center, and $R_{\rm G} =250 \, {\rm kpc}$ is the approximate radius of the Galactic halo. As in Sec. \[section:importance\], we assume a mass function of the form $dN/dMdV \propto M^{-\alpha} /
\tilde{r} ( 1 + \tilde{r})$, with $\alpha = 1.9$, as predicted by both N-body simulations [@Springel:2008cc], as well as by the semi-analytic model of subhalo populations we use for this study. Both simulations and our analytic method show little evidence that $\alpha$ varies significantly as a function of radius [@Zentner:2004dq; @Diemand:2006ik; @Springel:2008cc], so any variations are subtle (though they may depend upon global host halo properties). Consequently, it is convenient and informative to couple standard halo mass functions with our luminosity PDFs to estimate relevant observable quantities. However, we emphasize that this particular choice of mass function is not unique and, in principle, the flux PDF can be derived using Eq. (\[eq:PofL\]) with a variant subhalo mass function.
In Figure \[fig:fig4\] we show the normalized single halo flux probability distribution function for two different lines of sight generated from the contributions of subhalos in the range $M_{\rm min} = [ 10^4 - 10^{10}] M_\odot$ (thick lines) and $M_{\rm min} = [ 10^5 -
10^{10}] M_\odot$ (thin lines). This choice of subhalo mass does not affect the overall shape of the mass function. A change in the value of $M_{\rm max}$ does not affect the result as the mass function power law is a steep decreasing function of mass (see also the left panel of Fig. \[fig:fig0\], which shows how the average flux due to high-mass halos is lower than that due to low-mass halos). The high-flux power-law shape of the flux PDF can be understood in the following way. As the mass function is proportional to $dN/dMdV \propto
M^{-1.9}$, and the luminosity of a subhalo scales with mass as $L
\propto M^{0.87}$ (see Table I), then the integrand is proportional to $L^{-2.03}$. However, $F \propto L/\ell^2$, so the flux PDF is a power law with a shape given roughly by $P[F] \propto F^{-2.03}$. This is apparent for both lines of sight in the high-flux regime.
Changes in the value of $M_{\rm min}$ affect the low-flux behavior of the flux PDF. It is easier to understand the low-flux cutoff if we assume a delta function luminosity PDF instead of the log-normal distribution, as in [@Lee:2008fm]. For a line of sight at some angle $\psi$, there is a maximum distance $\ell_{\rm max}$ which is a function of $\psi$ and the radius of the Milky Way halo. If we assume that $L \propto M$, and that the subhalo radial distribution is the same for all masses, then the [*minimum*]{} flux would be given by $F_{\rm min} \propto M_{\rm min}
/ \ell_{\rm max}^2$, i.e., the smaller the minimum mass, the broader the $P_1(F)$. For illustrative purposes we show the $P_1(F)$ flux PDF derived under the assumption of a delta function luminosity PDF as in [@Lee:2008fm] in Fig. \[fig:fig4\]. Assuming the log-normal luminosity PDF (instead of a delta function), results in a tail in the low-flux region of the $P_1(F)$, and thus a broader $P_1(F)$ PDF.
The slope of the flux PDF for low fluxes does depend on the angle between the line of sight and the Galactic center (see the blue solid, and red dashed lines in Fig. \[fig:fig4\]). At small angles, there is flux probability excess in the low flux regime relative to the high flux region. This can be explained using Fig. \[fig:fig0\]. At large angles from the Galactic center, the total number of halos intercepted along a line of sight is smaller than the total number of halos intercepted along a line of sight that passes near the Galactic center. In addition, the fraction of these halos that are close to the Sun is smaller for large $\psi$. Therefore, any effects due to the spread of luminosities will be more pronounced where the fraction of contributing sources is large, and also nearby (due to the $1/\ell^2$ term). The spread of luminosities thus introduces a spread in flux, giving rise to substantial a change in the power-law behavior of the flux PDF.
It is important to draw the distinction between the single halo flux PDF (Eq. \[eq:PofF\]) and the probability distribution function of measuring a [*total*]{} flux $P(F)$, from the contribution of numerous halos along the line of sight. The total flux PDF can be computed in a straightforward manner from the basic quantity $P_1(F)$ [@Scheuer:1957; @Barcons:1992ApJ; @1990MNRAS.243..366B; @1982ApSS..86....3F; @2000MNRAS.319..591W], and more recently [@Lee:2008fm]). A thorough investigation of the total flux PDF using the single halo PDF $P_1(F)$, is presented in [@Baxter:2010], where the authors evaluate the ability of FGST to discover dark matter via $\gamma$-rays from Galactic substructure.
![The flux probability distribution function derived using Eq. \[eq:PofL\] and the $C_0$ parameters of Table \[table:fitparam\]. The solid blue lines is the flux PDF along a line of sight at $20^\circ$ relative to the Galactic center, while the dashed red lines are along $180^\circ$. Thick lines correspond to a minimum subhalo mass of $10^4 M_\odot$, while thin lines correspond to a minimum subhalo mass of $10^5 M_\odot$. The black dotted line depicts the single halo flux PDF derived under the assumption of a delta function luminosity PDF. []{data-label="fig:fig4"}](fig5.ps){height="7.8cm"}
Another possible application of the luminosity PDF is in the intrinsic spread of the expected $\gamma$-ray luminosity signal from dwarf spheroidal galaxies in the Milky Way halo. Dwarf spheroidals are very low surface brightness systems dynamically bound to the dark matter halo of the Milky Way. Their very high mass-to-light ratios make them ideal for $\gamma$-ray studies as their dark matter distributions can be well constrained using the velocity dispersions of their stellar populations. Numerous studies addressed the possibility of detecting dark matter annihilation in dwarf spheroidals using either FGST or ground-based Čerenkov telescopes, such as VERITAS and H.E.S.S. [@Baltz:1999ra; @Tyler:2002ux; @Evans:2003sc; @Profumo:2005xd; @Bergstrom:2005qk; @Strigari:2007at]. Dynamical studies of the velocity dispersion of stars in these systems constrain the dark matter mass (and profile) [@Strigari:2006rd; @Strigari:2007at]. Suppose that a group of dwarf spheroidal galaxies have masses estimated via dynamical measurements. The luminosity PDF presented in Sec. \[section:pdf\_results\] can be used to assess the expected variance of a possible detection of $\gamma$-rays from either Čerenkov telescopes or the FGST. The estimated variance can be used in concert with the measurement of the distribution of dark matter from dynamical studies in the interpretation of any signals or limits from these systems.
Moreover, the use of the luminosity PDF can be important in studies aimed at disentangling the different source contributions to the diffuse gamma-ray background, such as blazars, starburst galaxies, pulsars, supernova remnants, and potentially cataclysmic binary systems, and dark matter [@Ando:2006cr; @Ando:2006mt; @Ando:2005xg; @Miniati:2007ke; @Cuoco:2006tr; @Cuoco:2007sh; @Taoso:2008qz; @Fornasa:2009qh; @SiegalGaskins:2008ge; @Zhang:2004tj; @Pavlidou:2002va; @Hooper:2007be]. A useful tool in these studies is the use of the angular correlation function of flux fluctuations, which is estimated from the convolution of the emissivity as a function of distance and the spatial correlation function of the sources. Knowledge of the expected distribution in luminosities of contributing dark matter substructure can be used to remove the dark matter contamination to the astrophysical background, and thus potentially open the window for the detection of $\gamma$-rays from yet undiscovered sources [@Miniati:2007ke; @Pinzke:2010st; @Pavlidou:2006rb].
Finally, the annihilation luminosity PDF can be used in studies aimed at estimating the likelihood that nearby dark matter subhalos contribute significantly to the measured antimatter flux [@Adriani:2008zr; @Boezio:2009zz]. A dark matter explanation for the anomalous excess in antimatter flux must rely on either non-standard extensions to the standard model of particle physics (prominent annihilation to charged leptons [@Cholis:2008hb; @Zurek:2008qg; @Fox:2008kb; @Chen:2008dh], or some new, long-range force [@ArkaniHamed:2008qn], or potentially the presence of a dark matter halo in the near solar-system neighborhood [@Hooper:2008kv]. The luminosity PDF presented in this paper can be used to assess the minimum spread of the expected flux from a given subhalo and the likelihoods of particular flux measurements from rare, nearby objects. Note however, that propagation effects as well as the degeneracy between luminosity and the square of the distance to the subhalo, and velocity-dependent annihilation [@Robertson:2009bh; @Kuhlen:2009kx; @Cline:2010ag] can all increase the spread considerably.
Conclusions {#sec:conclusions}
===========
We have presented an estimate of the luminosity probability distribution function of dark matter subhalos within a Milky Way-like parent halo. An empirical fit to our numerical calculations suggest that the luminosity PDF can be described well by a log-normal distribution with subhalo mass- and position-dependent mean and variance. This log-normal probability distribution has a width that is determined by the wide distribution of formation times and concentrations for both the host halo and subhalos.
The derived luminosity PDF can be used as an ingredient in a number of interesting calculations regarding predictions for observable dark matter annihilation products. This distillation of a complex set of halo properties should be particularly useful with the impending data from the Fermi Gamma-ray Space Telescope and continued advances of ground-based Air Cerenkov Telescopes as well as neutrino telescopes and antimatter detection instruments. The tool we provide can be used to address such observations, including variance in the predicted signal, in a relatively simple manner.
As a straightforward example of the application of the PDF, we have estimated the distribution of observed $\gamma$-ray fluxes along lines of sight as a function of the angular separation between the line-of-sight and the Galactic center. This may be used as a signature to diagnose unresolved annihilation in a population of Galactic subhalos (see [@Baxter:2010], where the authors utilize this distribution to estimate the robustness of estimating dark matter properties from the diffuse flux measurements of FGST). Additional applications of the annihilation luminosity PDF include estimates in the spread of the angular power spectrum of flux fluctuations as a probe of unresolved substructure in the Milky Way, as well as the antimatter flux distribution from nearby subhalos. These applications make the luminosity PDF a useful tool in the analysis of forthcoming data in the ongoing effort to identify the dark matter.
SMK and AVK thank the Center for Scientific Computation and Mathematical Modeling at the University of Maryland College Park for hospitality. We thank Eric J. Baxter, Scott Dodelson and Louie Strigari for useful comments. ARZ acknowledges the support and hospitality of the Michigan Center for Theoretical Physics at the University of Michigan. SMK is funded by the NSF and by Brown University. ARZ is funded by the University of Pittsburgh, by the NSF through grant AST-0806367, and by the DoE. AVK is supported by the DoE and NSF grant AST-0708154, and by the Kavli Institute for Cosmological Physics at the University of Chicago through the NSF grant PHY-0551142 and an endowment from the Kavli Foundation.
|
---
abstract: |
We derive the one-dimensional optimal system for a system of three partial differential equations which describe the two-dimensional rotating ideal gas with polytropic parameter $\gamma >2.$ The Lie symmetries and the one-dimensional optimal system are determined for the nonrotating and rotating systems. We compare the results and we found that when there is no Coriolis force the system admits eight Lie point symmetries, while the rotating system admits seven Lie point symmetries. Consequently the two systems are not algebraic equivalent as in the case of $\gamma =2~$ which was found by previous studies. For the one-dimensional optimal system we determine all the Lie invariants, while we demonstrate our results by reducing the system of partial differential equations into a system of first-order ordinary differential equations which can be solved by quadratures.
Keywords: Lie symmetries; invariants; shallow water; similarity solutions
author:
- |
Andronikos Paliathanasis[^1]\
[ *Institute of Systems Science, Durban University of Technology* ]{}\
[ *PO Box 1334, Durban 4000, Republic of South Africa*]{}
title: 'One-dimensional optimal system for 2D Rotating Ideal Gas'
---
Introduction {#sec1}
============
A powerful mathematical treatment for the determination of exact solutions for nonlinear differential equations is the Lie symmetry analysis [olver,kumei,ibra]{}. Specifically, Lie point symmetries help us in the simplification of differential equations by means of similarity transformations which reduce the differential equation. The reduction process is based on the existence of functions which are invariant under a specific group of point transformations. When someone uses these invariants as new dependent and independent variables the differential equation is reduced. The reduction process differs between ordinary differential equations (ODEs) and partial differential equations (PDEs). For ODEs Lie point symmetries are applied to reduce the order of ODE by one; while on PDEs Lie point symmetries are applied to reduce by one the number of independent variables, while the order of the PDE remain the same. The solutions which are found with the application of those invariant functions are called similarity solutions. Some applications on the determination of similarity solutions for nonlinear differential equations can be found in [@ref1; @ref2; @ref3; @ref4; @ref5; @ref10] and references therein.
A common characteristic in the reduction process is that the Lie point symmetries are not preserved during the reduction, hence we can say that the symmetries can be lost. That is not an accurate statement, because symmetries are not destroyed or created under point transformations but the nature of the symmetry change. In addition, Lie symmetries can be used to construct new similarity solutions for a given differential equation by applying the adjoint representation of the Lie group [@leach01].
It is possible that a given differential equation to admit more than one similarity solution when the given differential equation admits a large number of Lie point symmetries. Hence, in order for someone to classify a differential equation according to the admitted similarity solutions, all the inequivalent Lie subalgebras of the admitted Lie symmetries should be determined.
The first group classification problem was carried out by Ovsiannikov [Ovsi]{} who demonstrated the construction of the one-dimensional optimal system for the Lie algebra. Since then, the classification of the one-dimensional optimal system has become a main tool for the study of nonlinear differential equations [@opt1; @opt2; @opt3; @opt4].
In this work, we focus on the classification of the one-dimensional optimal system for the two-dimensional rotating ideal gas system for a described by the following system of PDEs [@val1; @cc1; @kk1]
$$\begin{aligned}
h_{t}+\left( hu\right) _{x}+\left( hv\right) _{y} &=&0, \label{sw.01} \\
u_{t}+uu_{x}+vu_{y}+h^{\gamma -2}h_{x}-fv &=&0, \label{sw.02} \\
v_{t}+uv_{x}+vv_{y}+h^{\gamma -2}h_{y}+fu &=&0. \label{sw.03}\end{aligned}$$
where $u$ and $v$ are the velocity components on the $x$ and $y$ directions respectively, $h$ is the density of the ideal gas, $f$ is the Coriolis parameter and $\gamma $ is the polytropic parameter of the fluid. Usually $\gamma $ is assumed to be $\gamma =2$ where equations (\[sw.01\])-([sw.03]{}) reduce to the shallow water system. However in this work we consider that $\gamma >2$. In this work, polytropic index $\gamma $ is defined as $\frac{C_{p}}{C_{v}}=\gamma -1$.
Shallow-water equations describe the flow of a fluid under a pressure surface. There are various physical phenomena which are described by the Shallow-water system with emphasis on atmospheric and oceanic phenomena [oc01,oc02,oc03]{}. Hence, the existence of the Coriolis force it becomes critical in the description of the physical phenomena.
In the case of $\gamma =2$, the complete symmetry analysis of the system (\[sw.01\])-(\[sw.03\]) is presented in [@swr06]. It was found that for $\gamma =2$ the given system of PDEs is invariant under a nine-dimensional Lie algebra. The same Lie algebra but in a different representation is also admitted by the nonrotating system, i.e. $f=0$. One of the main results of [@swr06] is that the transformation which relates the two representations of the admitted Lie algebras for the rotating and nonrotating system, transform the rotating system (\[sw.01\])-(\[sw.03\]) into the nonrotating one. For other applications of Lie symmetries on Shallow-water equation we refer the reader in [@sw1; @sw2; @sw3; @sw4; @sw5; @sw6].
For the case of a ideal gas [@cc1], i.e. parameter $\gamma >1$ from our analysis it follows that this property is lost. The nonrotating system and the rotating one are invariant under a different number of Lie symmetries and consequently under different Lie algebras. For each of the Lie algebras we æther the one-dimensional optimal system and all the Lie invariants. The results are presented in tables. We demonstrate the application of the Lie invariants by determining some similarity solutions for the system ([sw.01]{})-(\[sw.03\]) for $\gamma >2$. The paper is structured as follows.
In Section \[sec2\]. we briefly discuss the theory of Lie symmetries for differential equations and the adjoint representation. The nonrotating system (\[sw.01\])-(\[sw.03\]) is studied in Section \[sec3\]. Specifically we determine the Lie points symmetries which form an eight-dimensional Lie algebra. The commutators and the adjoint representation are presented. We make use of these results and we perform a classification of the one-dimensional optimal system. We found that in total there are twenty-three one-dimensional indepedent Lie symmetries and possible reductions, the corresponding invariants are determined and presented in tables. In Section \[sec4\] we perform the same analysis for the rotating system. There we find that the admitted Lie symmetries form a seven-dimensional Lie algebra while there are twenty independent one-dimensional Lie algebras. We demonstrate the results by reducing the system of PDEs (\[sw.01\])-(\[sw.03\]) into an integrable system of three first-order ODEs which solution is given by quadratures. In Section \[sec5\] we discuss our results and draw our conclusions. Finally, in Appendix \[app1\] we present the tables which includes the results of our analysis.
Lie symmetry analysis {#sec2}
=====================
Let $H^{A}\left( x^{i},\Phi ^{A},\Phi _{i}^{A},...\right) =0$ be a system of partial differential equations (PDEs) where $\Phi ^{A}~$denotes the dependent variables and $x^{i}$ are the indepedent variables, at this point it is important to mention that we make use of the Einstein summation convention. By definition under the action of the infinitesimal one-parameter point transformation (1PPT) $$\bar{x}^{i}=x^{i}\left( x^{j},\Phi ^{B};\varepsilon \right) ,~~\bar{\Phi}^{A}=\Phi ^{A}\left( x^{j},\Phi ^{B};\varepsilon \right) , \label{de.03}$$which connects two different points $P\left( x^{j},\Phi ^{B}\right)
\rightarrow Q\left( \bar{x}^{j},\bar{\Phi}^{B},\varepsilon \right) $, the differential equation $H^{A}=0$ remains invariant if and only if $\bar{H}^{A}=H^{A}$, that is [@kumei]$$\lim_{\varepsilon \rightarrow 0}\frac{\bar{H}^{A}\left( \bar{y}^{i},\bar{u}^{A},...;\varepsilon \right) -H^{A}\left( y^{i},u^{A},...\right) }{\varepsilon }=0. \label{ls.05}$$The latter condition means that the $\Phi ^{A}\left( P\right) $ and $\Phi
^{A}\left( Q\right) $ are connected through the transformation.
The lhs of expression (\[ls.05\]) defines the Lie derivative of $H^{A}$ along the vector field $X$ of the one-parameter point transformation ([de.03]{}), in which $X$ is defined as $$X=\frac{\partial \bar{x}^{i}}{\partial \varepsilon }\partial _{i}+\frac{\partial \bar{\Phi}}{\partial \varepsilon }\partial _{A}.$$
Thus, condition (\[ls.05\]) is equivalent with the following expression [@kumei]$$\mathcal{L}_{X}\left( H^{A}\right) =0, \label{ls.05a}$$where $\mathcal{L}$ denotes the Lie derivative with respect to the vector field $X^{\left[ n\right] }$ which is the $n$th-extension of generator $X~$of the transformation (\[de.03\]) in the jet space $\left\{ x^{i},\Phi
^{A},\Phi _{,i}^{A},\Phi _{,ij}^{A},...\right\} $ given by the expression [@kumei] $$X^{\left[ n\right] }=X+\eta ^{\left[ 1\right] }\partial _{\Phi
_{i}^{A}}+...+\eta ^{\left[ n\right] }\partial _{\Phi
_{i_{i}i_{j}...i_{n}}^{A}}, \label{ls.06}$$in which$$\eta ^{\left[ n\right] }=D_{i}\eta ^{\left[ n-1\right]
}-u_{i_{1}i_{2}...i_{n-1}}D_{i}\left( \frac{\partial \bar{x}^{j}}{\partial
\varepsilon }\right) ~,~i\succeq 1~,~\eta ^{\left[ 0\right] }=\left( \frac{\partial \bar{\Phi}^{A}}{\partial \varepsilon }\right) . \label{de.08}$$
Conditions (\[ls.05a\]) provides a system of PDEs whose solution determine the components of the $X$, consequently the infinitesimal transformation. The vector fields $X$ which satisfy condition (\[ls.05a\]) are called Lie symmetries for the differential equation $H^{A}=0$. The Lie symmetries for a given differential equation form a Lie algebra.
Lie symmetries can be used by different ways [@kumei] in order to study a differential equation. However, their direct application is on the determination of the so-called similarity solutions. The steps which we follow to determine a similarity solution is based on the determination and application of the Lie invariant functions.
Let $X$ be a Lie symmetry for a given differential equation $H^{A}=0$, then the differential equation $X\left( F\right) =0$, where $F$ is a function, provides the Lie invariants where by replacing in the differential equation $H^{A}=0$, we reduce the number of the indepedent variables (in the case of PDEs) or the order of the differential equation (in the case of ordinary differential equations (ODEs)).
Optimal system
--------------
Consider the $n$-dimensional Lie algebra $G_{n}$ with elements $X_{1},~X_{2},~...~X_{n}$. Then we shall say that the two vector fields [kumei]{} $$Z=\sum\limits_{i=1}^{n}a_{i}X_{i}~,~W=\sum\limits_{i=1}^{n}b_{i}X_{i}~,~\text{\ }a_{i},~b_{i}\text{ are constants.} \label{sw.04}$$are equivalent iff there $$\mathbf{W}=lim_{j=i}^{n}Ad\left( \exp \left( \varepsilon _{i}X_{i}\right)
\right) \mathbf{Z} \label{sw.05}$$or$$W=cZ~,~c=const. \label{sw.06}$$where the operator [@kumei]$$Ad\left( \exp \left( \varepsilon X_{i}\right) \right)
X_{j}=X_{j}-\varepsilon \left[ X_{i},X_{j}\right] +\frac{1}{2}\varepsilon
^{2}\left[ X_{i},\left[ X_{i},X_{j}\right] \right] +... \label{sw.07}$$is called the adjoint representation.$~$
Therefore, in order to perform a complete classification for the similarity solutions of a given differential equation we should determine all the one-dimensional indepedent symmetry vectors of the Lie algebra $G_{n}$.
We continue our analysis by calculating the Lie point symmetries for the system (\[sw.01\])-(\[sw.03\]) for the case where the system is rotating$~(f\neq 0)\ $and nonrotating $\left( f=0\right) $.
Symmetries and optimal system for nonrotating shallow water {#sec3}
===========================================================
We start our analysis by applying the symmetry condition (\[ls.05a\]) for the Coriolis free system (\[sw.01\])-(\[sw.03\]) with $f=0$. We found that the system of PDEs admit eight Lie point symmetries as they are presented in the following [@Ovsi] $$\begin{aligned}
X_{1} &=&\partial _{t}~,~X_{2}=\partial _{x}~,~X_{3}=\partial _{y}~, \\
X_{4} &=&t\partial _{x}+\partial _{u}~,~X_{5}=t\partial _{y}+\partial _{v}~,
\\
X_{6} &=&y\partial _{x}-x\partial _{y}+v\partial _{u}-u\partial _{v}, \\
X_{7} &=&t\partial _{t}+x\partial _{x}+y\partial _{y}, \\
X_{8} &=&\left( \gamma -1\right) \left( x\partial _{x}+y\partial
_{y}+u\partial _{u}+v\partial _{v}\right) +2h\partial _{h}.\end{aligned}$$
The commutators of the Lie symmetries and the adjoint representation are presented in tables \[tabl1\] and \[tabl2\] respectively.
---------------------- --------------------------------- ---------------------------------- ------------------ ---------------------------------- ---------------------------------- ------------------ ------------------------- ---------------------------------
$\left[ ~,~\right] $ $\mathbf{X}_{1}$ $\mathbf{X}_{2}$ $\mathbf{X}_{3}$ $\mathbf{X}_{4}$ $\mathbf{X}_{5}$ $\mathbf{X}_{6}$ $\mathbf{X}_{7}$ $\mathbf{X}_{8}$
$\mathbf{X}_{1}$ $0$ $0$ $0$ $X_{2}$ $X_{3}$ $0$ $-\left( $0$
\gamma -1\right) X_{1}$
$\mathbf{X}_{2}$ $0$ $0$ $0$ $0$ $0$ $-X_{3}$ $0$ $\left(
\gamma -1\right) X_{2}$
$\mathbf{X}_{3}$ $0$ $0$ $0$ $0$ $0$ $X_{2}$ $0$ $\left(
\gamma -1\right) X_{3}$
$\mathbf{X}_{4}$ $-X_{2}$ $0$ $0$ $0$ $0$ $-X_{5}$ $\left( $\left( \gamma -1\right) X_{4}$
\gamma -1\right) X_{4}$
$\mathbf{X}_{5}$ $-X_{3}$ $0$ $0$ $0$ $0$ $X_{4}$ $\left( $\left( \gamma -1\right) X_{5}$
\gamma -1\right) X_{5}$
$\mathbf{X}_{6}$ $0$ $X_{3}$ $-X_{2}$ $X_{5}$ $-X_{4}$ $0$ $0$ $0$
$\mathbf{X}_{7}$ $\left( \gamma -1\right) X_{1}$ $0$ $0$ $-\left( $-\left( \gamma -1\right) X_{5}$ $0$ $0$ $0$
\gamma -1\right) X_{4}$
$\mathbf{X}_{8}$ $0$ $-\left( \gamma -1\right) X_{2}$ $-\left( \gamma $-\left( \gamma -1\right) X_{4}$ $-\left( \gamma $0$ $0$ $0$
-1\right) X_{3}$ -1\right) X_{5}$
---------------------- --------------------------------- ---------------------------------- ------------------ ---------------------------------- ---------------------------------- ------------------ ------------------------- ---------------------------------
: Commutators of the admitted Lie point symmetries for the nonrotating 2D Shallow water
\[tabl1\]
We continue by determine the one-dimensional optimal system. Let us consider the generic symmetry vector$$Z^{8}=a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3}+a_{4}X_{4}+a_{5}X_{5}+a_{6}X_{6}+a_{7}X_{7}+a_{8}X_{8}$$From table \[tabl2\] we see that by applying the following adjoint representations$$Z^{\prime 8}=Ad\left( \exp \left( \varepsilon _{5}X_{5}\right) \right)
Ad\left( \exp \left( \varepsilon _{4}X_{4}\right) \right) Ad\left( \exp
\left( \varepsilon _{3}X_{3}\right) \right) Ad\left( \exp \left( \varepsilon
_{2}X_{2}\right) \right) Ad\left( \exp \left( \varepsilon _{1}X_{1}\right)
\right) Z^{8}$$parameters $\varepsilon _{1},~\varepsilon _{2},~\varepsilon _{3}$, $\varepsilon _{4}$ and $\varepsilon _{5}$ can be determined such that$$Z^{\prime 8}=a_{6}^{\prime }X_{6}+a_{7}^{\prime }X_{7}+a_{8}^{\prime }X_{8}$$
Parameters $a_{6},~a_{7},$ and $a_{8}$ are the relative invariants of the full adjoint action. Indeed in order to determine the relative invariants we solve the following system of partial differential equation [@olver]$$\Delta \left( \phi \left( a_{i}\right) \right) =C_{ij}^{k}a^{i}\frac{\partial }{\partial a_{j}}$$where $C_{ij}^{k}$ are the structure constants of the admitted Lie algebra as they presented in Table \[tabl1\]. Consequently, in order to derive all the possible one-dimensional Lie symmetries we should study various cases were non of the invariants are zero, one of the invariants are zero, two of the invariants are zero or all the invariants are zero.
Hence, for the first tree cases infer the following one-dimensional independent Lie algebras$$X_{6}~,~X_{7}~,~X_{8}~,~\xi _{\left( 67\right) }=X_{6}+\alpha X_{7}~,~\xi
_{\left( 68\right) }=X_{6}+\alpha X_{8}~$$$$\xi _{\left( 78\right) }=X_{7}+\alpha X_{8}~,~\xi _{\left( 678\right)
}=X_{6}+\alpha X_{7}+\beta X_{8}.$$
We apply the same procedure for the rest of the possible linear combinations of the symmetry vectors and we find the one-dimensional dependent Lie algebras$$X_{1},~X_{2}~,~X_{3}~,~X_{4}~,~X_{5}~,~\xi _{\left( 12\right) }=X_{1}+\alpha
X_{2}~,~\xi _{\left( 13\right) }=X_{1}+\alpha X_{3}~,~\xi _{\left( 23\right)
}=X_{2}+\alpha X_{3}~,~\xi _{\left( 14\right) }=X_{1}+\alpha X_{4}~,$$$$~\xi _{\left( 15\right) }=X_{1}+\alpha X_{5}~,~\xi _{\left( 16\right)
}=X_{1}+\alpha X_{6},~\xi _{\left( 34\right) }=X_{3}+\alpha X_{4}~,~\xi
_{\left( 25\right) }=X_{2}+\alpha X_{5}~~\xi _{\left( 45\right)
}=X_{4}+\alpha X_{5}~,$$$$~\xi _{\left( 123\right) }=X_{1}+\alpha X_{2}+\beta X_{3}~\xi _{\left(
145\right) }=X_{1}+\alpha X_{4}+\beta X_{5}~,~\xi _{\left( 125\right)
}=X_{1}+\alpha X_{2}+\beta X_{5}~,~\xi _{\left( 134\right) }=X_{1}+\alpha
X_{3}+\beta X_{4},$$in which $\alpha $ and $\beta $ are constants.
Therefore by applying one of the above Lie symmetry vectors we find all the possible reductions from a system of $1+2$ PDEs to a system of $1+1$ PDEs. The reduced system will not admit all the remaining Lie symmetries. The Lie symmetries which survive under a reduction process are given as described in the following example.
Let a PDE admits the Lie point symmetries $\Gamma _{1},~\Gamma _{2}$ which are such that $\left[ \Gamma _{1},\Gamma _{2}\right] =C_{12}^{1}X_{1},~$with $C_{12}^{1}\neq 0$.$~$Reduction with the symmetry vector $\Gamma _{1}$ leads to a reduced differential equation which admits $\Gamma _{2}$ as Lie symmetry. On the other hand, reduction of the mother equation with respect to the Lie symmetry $\Gamma _{2}$ leads to a different reduced differential equation which does not admit as a Lie point symmetry the vector field $\Gamma _{1}.$ In case the two Lie symmetries form an Abelian Lie algebra, i.e. $C_{12}^{1}=0$, then under any reduction process symmetries are preserved by any reduction.
We found that the optimal system admit twenty-three one-dimensional Lie symmetries, and possible independent reductions. All the possible twenty-three Lie invariants are presented in tables \[tabl2a\] and [tabl2b]{}.
An application of the Lie invariants is presented below.
Application of $\protect\xi _{145}$
-----------------------------------
Let us now demonstrate the results of tables \[tabl2a\] and \[tabl2b\] by the Lie invariants of the symmetry vector $\xi _{145}$ and construct the similarity solution for the system.
The application of $\xi _{145}$ in the nonrotating system (\[sw.01\])-([sw.03]{}) reduce the PDEs in the following system $$\begin{aligned}
\left( hu\right) _{z}+\left( hv\right) _{w} &=&0 \label{s.01} \\
\alpha +uu_{z}+vu_{w}+h^{\gamma -2}h_{z} &=&0 \label{s.02} \\
\beta +uv_{z}+vv_{w}+h^{\gamma -2}h_{w} &=&0 \label{s.03}\end{aligned}$$where $z=x-\frac{\alpha }{2}t^{2}~$and $w=~y-\frac{\beta }{2}t^{2}$.
System (\[s.01\])-(\[s.03\]) admits the Lie point symmetries$$\partial _{z}~,~\partial _{w}~,~z\partial _{z}+w\partial _{w}+\frac{2}{\gamma -1}h\partial _{h}+u\partial _{u}+v\partial _{v} \label{s.04}$$
Reduction with the symmetry vector $\partial _{z}+c\partial _{w}$ provides the following system of first-order ODEs$$\begin{aligned}
Fh_{\sigma } &=&\left( c\alpha -\beta \right) h^{2}, \label{s.05} \\
Fv_{\sigma } &=&\frac{\left( \alpha -c\beta \right) ch^{\gamma }-\alpha
h\left( v-cu\right) ^{2}}{v-cu}, \label{s.06} \\
Fh_{\sigma } &=&\frac{\left( \alpha -c\beta \right) cu^{\gamma }-\beta
h\left( v-cu\right) ^{2}}{v-cu}. \label{s.08}\end{aligned}$$where $F=\left( 1+c^{2}\right) h^{\gamma }-h\left( v-cu\right) ^{2}~$and $\sigma =z+cw$.
By performing the change of variable $d\sigma =fd\tau $ function $f$ can be removed from the above system. For $h\left( \tau \right) =0$, system ([s.05]{}), (\[s.06\]), (\[s.08\]) admits a solution $u=u_{0},~v=v_{0}$ which is a critical point. The latter special solutions are always unstable when $\alpha c>\beta .$
We proceed our analysis by considering the rotating system.
---------------------- ------------------ ---------------------------------- ------------------ ------------------ ---------------------------------- ------------------ -------------------------
$\left[ ~,~\right] $ $\mathbf{Y}_{1}$ $\mathbf{Y}_{2}$ $\mathbf{Y}_{3}$ $\mathbf{Y}_{4}$ $\mathbf{Y}_{5}$ $\mathbf{Y}_{6}$ $\mathbf{Y}_{7}$
$\mathbf{Y}_{1}$ $0$ $0$ $0$ $0$ $fY_{6}$ $-fY_{5}$ $0$
$\mathbf{Y}_{2}$ $0$ $0$ $0$ $-Y_{3}$ $0$ $0$ $\left( \gamma
-1\right) Y_{2}$
$\mathbf{Y}_{3}$ $0$ $0$ $0$ $Y_{2}$ $0$ $0$ $\left( \gamma
-1\right) Y_{3}$
$\mathbf{Y}_{4}$ $0$ $Y_{3}$ $-Y_{2}$ $0$ $-Y_{6}$ $Y_{5}$ $0$
$\mathbf{Y}_{5}$ $-fY_{6}$ $0$ $0$ $Y_{6}$ $0$ $0$ $\left(
\gamma -1\right) Y_{5}$
$\mathbf{Y}_{6}$ $fY_{5}$ $0$ $0$ $-Y_{5}$ $0$ $0$ $\left(
\gamma -1\right) Y_{6}$
$\mathbf{Y}_{7}$ $0$ $-\left( \gamma -1\right) Y_{2}$ $-\left( \gamma $0$ $-\left( \gamma -1\right) Y_{5}$ $-\left( \gamma $0$
-1\right) Y_{3}$ -1\right) Y_{6}$
---------------------- ------------------ ---------------------------------- ------------------ ------------------ ---------------------------------- ------------------ -------------------------
: Commutators of the admitted Lie point symmetries for the rotating 2D Shallow water
\[tabl3\]
Symmetries and optimal system for rotating shallow water {#sec4}
========================================================
For the rotating system $\left( f\neq 0\right) $, the Lie symmetries are $$\begin{aligned}
Y_{1} &=&\partial _{t}~,~Y_{2}=\partial _{x}~,~Y_{3}=\partial _{y}~, \\
Y_{4} &=&y\partial _{x}-x\partial _{y}+v\partial _{u}-u\partial _{v}~, \\
Y_{5} &=&\sin \left( ft\right) \partial _{x}+\cos \left( ft\right) \partial
_{y}+f\left( \cos \left( ft\right) \partial _{u}-\sin \left( ft\right)
\partial _{v}\right) \\
Y_{6} &=&\cos \left( ft\right) \partial _{x}-\sin \left( ft\right) \partial
_{y}-f\left( \sin \left( ft\right) \partial _{u}+\cos \left( ft\right)
\partial _{v}\right) \\
Y_{7} &=&\left( \gamma -1\right) \left( x\partial _{x}+y\partial
_{y}+u\partial _{u}+v\partial _{v}\right) +2h\partial _{h}\end{aligned}$$
The commutators and the adjoint representation are given in tables [tabl3]{} and \[tabl4\]. The Lie symmetries for the rotating system form a smaller dimension Lie algebra than the non-rotating system. That is not the case when $\gamma =2$, where the two Lie algebras have the same dimensional and are equivalent under point transformation [@swr06]. Therefore, for $\gamma >2$ the Coriolis force cannot be eliminated by a point transformation as in the $\gamma =2$ case.
As far as the admitted Lie symmetries admitted by the given system of PDEs with or without the Coriolis terms for $\gamma >2$, we remark that the rotating and the nonrotating system have a common Lie subalgebra of one-parameter point transformations consists by the symmetry vectors $Y_{1},~Y_{2},~Y_{3},~Y_{4}$ and $Y_{7}$, or for the nonrotating system $X_{1},~X_{2},~X_{3},~X_{6}$ and $X_{8}$.
We proceed with the determination of the one-dimensional optimal system and the invariant functions. Specifically, the relative invariants for the adjoint representation are calculated to be $a_{1}~,~a_{7}$ and $a_{8}.~$From tables \[tabl3\] and \[tabl4\] we can find the one-dimensional optimal system, which is $$Y_{1},~Y_{2},~Y_{3},~Y_{4},~Y_{5},~Y_{6},~Y_{7},~\chi _{12}=Y_{1}+\alpha
Y_{2},~\chi _{13}=Y_{1}+\alpha Y_{3},$$$$\chi _{14}=Y_{1}+\alpha Y_{4}~,~\chi _{15}=Y_{1}+\alpha Y_{5},~~\chi
_{16}=Y_{1}+\alpha Y_{6},~\chi _{17}=Y_{1}+\alpha Y_{7},$$$$\chi _{23}=Y_{2}+\alpha Y_{3}~,~~\chi _{45}=Y_{4}+\alpha Y_{5},~\chi
_{46}=Y_{4}+\alpha Y_{6},~\chi _{56}=Y_{5}+\alpha Y_{6}$$$$~\chi _{47}=Y_{4}+\alpha Y_{6}~,~\chi _{123}=Y_{1}+\alpha Y_{2}+\beta
Y_{3},~\chi _{147}=Y_{1}+\alpha Y_{4}+\beta Y_{7}.$$
The Lie invariants which correspond to all the above one-dimensional Lie algebras are presented in tables \[tabl4a\] and \[tabl4b\].
Let us demonstrate the application of the Lie invariants by the following applications, from where we can see that the Lie invariants reduce the nonlinear field equations into a system of integrable first-order ODE which can be solved with quadratures.
Application of $\protect\chi _{12}$
-----------------------------------
We consider the travel-wave similarity solution in the $x-$plane provided by the symmetry vector $\chi _{12}$, and the vector field $Y_{3}$. The resulting equations is described by the following system of first order ODEs$$\begin{aligned}
v_{z} &=&f\frac{u}{\alpha -u}~ \label{se.01} \\
\bar{F}u_{z} &=&f\left( \alpha -u\right) vh \label{se.02} \\
\bar{F}h_{z} &=&fvh^{2} \label{se.04}\end{aligned}$$where $\bar{F}=h^{\gamma }-\left( a-u\right) ^{2}h$ and $z=t-\alpha x$. Because we performed reduction with a subalgebra admitted by the nonrotating system, by setting $f=0$ in (\[se.01\]), (\[se.02\]), (\[se.04\]) we get the similarity solution for the nonrotating system where in this case is found to be $h\left( z\right) =h_{0},~u\left( z\right) =u_{0}$ and $v\left(
z\right) =v_{0}.$
We perform the substitution $dz=\frac{\bar{F}}{fv}d\tau $ and the latter system is simplified as follows$$\begin{aligned}
\frac{v}{\bar{F}}v_{\tau } &=&\frac{u}{\alpha -u}~ \\
u_{\tau } &=&\left( \alpha -u\right) h \\
h_{\tau } &=&h^{2}\end{aligned}$$from where we get the solution$$h\left( \tau \right) =\left( h_{0}-\tau \right) ^{-1}~,~u\left( \tau \right)
=\alpha +u_{0}-\frac{u_{0}}{h_{0}}\tau$$and$$v\left( t\right) ^{2}=2\int \frac{\left( a+u_{0}-\frac{u_{0}}{h_{0}}\tau
\right) }{\frac{u_{0}}{h_{0}}\left( h_{0}-\tau \right) }\left( \left(
h_{0}-\tau \right) ^{-\gamma }+\left( \frac{u_{0}}{h_{0}}\right) ^{2}\tau -\frac{\left( u_{0}\right) ^{2}}{h_{0}}\right) d\tau .$$
Application of $\protect\chi _{23}$
-----------------------------------
Consider now the reduction with the symmetry vector fields $\chi _{23}$. The resulting system of 1+1 differential equations admit five Lie point symmetries, they are$$\begin{aligned}
&&\partial _{t},~\partial _{w}~,~\left( \sin \left( ft\right) +\alpha \cos
\left( ft\right) \right) \partial _{w}+f\left( \sin \left( ft\right)
\partial _{u}+\cos \left( ft\right) \partial _{v}\right) \\
&&\left( \alpha \sin \left( ft\right) -\cos \left( ft\right) \right)
\partial _{w}-f\left( \cos \left( ft\right) \partial _{u}-\sin \left(
ft\right) \partial _{v}\right) ~,~\left( \gamma -1\right) \left( \partial
_{w}+u\partial _{u}+v\partial _{v}\right) +2h\partial _{h}.\end{aligned}$$where $w=y-\alpha x$. For simplicity on our calculations let us assume $\gamma =3$.
Reduction with the scaling symmetry provides the following system of first order ODEs$$\begin{aligned}
H_{t} &=&2H\left( \alpha U-V\right) , \label{se.05} \\
U_{t} &=&\alpha H^{2}+u\left( \alpha U-V\right) +fV, \label{se.06} \\
V_{t} &=&-H^{2}-v\left( \alpha U-V\right) -fU, \label{se.08}\end{aligned}$$where $h=wH,~u=wU$ and $v=wU.~$The latter system is integrable and can be solved with quadratures.
Reduction with respect the symmetry vector $\left( \alpha \sin \left(
ft\right) -\cos \left( ft\right) \right) \partial _{w}-f\left( \cos \left(
ft\right) \partial _{u}-\sin \left( ft\right) \partial _{v}\right) $ we find the reduced system$$\begin{aligned}
\frac{H_{t}}{H} &=&-\frac{\alpha \cos \left( ft\right) +\sin \left(
ft\right) }{\cos \left( ft\right) -\alpha \sin \left( ft\right) },
\label{se.10} \\
U_{t} &=&-\alpha f\frac{\sin \left( ft\right) V-\cos \left( ft\right) U}{\cos \left( ft\right) -\alpha \sin \left( ft\right) }, \label{se.11} \\
V_{t} &=&-f\frac{\sin \left( ft\right) V-\cos \left( ft\right) U}{\cos
\left( ft\right) -\alpha \sin \left( ft\right) }, \label{se.12}\end{aligned}$$where now $$\begin{aligned}
h &=&H\left( t\right) ~,~ \\
u &=&\frac{\cos \left( ft\right) }{\cos \left( ft\right) -\alpha \sin \left(
ft\right) }fw+U\left( t\right) , \\
v &=&-\frac{\sin \left( ft\right) }{\cos \left( ft\right) -\alpha \sin
\left( ft\right) }fw+V\left( t\right) .\end{aligned}$$
System (\[se.10\]), (\[se.11\]), (\[se.12\]) is integrable and the solution is expressed in terms of quadratures.
Conclusions {#sec5}
===========
In this work, we determined the one-dimensional optimal system for the two-dimensional ideal gas equations. The nonrotating system it was found that is invariant under an eight-dimensional group of one-parameter point transformations. and there are twenty-three independent one-dimensional Lie algebras. One the other hand, when the Coriolis force is introduced, the dynamical admits seven Lie point symmetries and twenty one-dimensional Lie algebras.
For all the independent Lie algebras we determined all the invariant functions which correspond to all the independent similarity solutions.
In a future work we plan to classify all the independent one-dimensional Lie algebras which lead to analytic forms for the similarity solutions.
**Acknowledgements**
The author wants to thank Professor Kevin Duffy for gracious hospitality and Professor PGL Leach for a fruitful discussion on the subject.
Tables {#app1}
======
In this appendix we present the tables \[tabl2\], \[tabl2a\], [tabl2b]{}, \[tabl4\], \[tabl4a\] and \[tabl4b\] which are refereed in the main article.
--------------------------------------------------------------------------------- -------------------------------------------------- ------------------------------------------------- ------------------------------------------------- -------------------------------------------------- ------------------------------------------------- --------------------------- ---------------------------------- ---------------------------------------------------
$Ad\left( e^{\left( \varepsilon \mathbf{X}_{i}\right) }\right) \mathbf{X}_{j} $ $\mathbf{X}_{1}$ $\mathbf{X}_{2}$ $\mathbf{X}_{3}$ $\mathbf{X}_{4}$ $\mathbf{X}_{5}$ $\mathbf{X}_{6}$ $\mathbf{X}_{7}$ $\mathbf{X}_{8}$
$\mathbf{X}_{1}$ $X_{1}$ $X_{2}$ $X_{3}$ $X_{4}-\varepsilon X_{2}$ $X_{5}-\varepsilon X_{3}$ $X_{6}$ $X_{7}+\varepsilon \left( \gamma $X_{8}$
-1\right) X_{1}$
$\mathbf{X}_{2}$ $X_{1}$ $X_{2}$ $X_{3}$ $X_{4}$ $X_{5}$ $X_{6}+\varepsilon X_{3}$ $X_{7}$ $X_{8}-\varepsilon \left( \gamma
-1\right) X_{2}$
$\mathbf{X}_{3}$ $X_{1}$ $X_{2}$ $X_{3}$ $X_{4}$ $X_{5}$ $X_{6}-\varepsilon X_{2}$ $X_{7}$ $X_{8}-\varepsilon \left( \gamma
-1\right) X_{3}$
$\mathbf{X}_{4}$ $X_{1}+\varepsilon X_{2}$ $X_{2}$ $X_{3}$ $X_{4}$ $X_{5}$ $X_{6}+\varepsilon X_{5}$ $X_{7}-\varepsilon \left( \gamma $X_{8}-\varepsilon \left( \gamma -1\right) X_{4}$
-1\right) X_{4}$
$\mathbf{X}_{5}$ $X_{1}+\varepsilon X_{3}$ $X_{2}$ $X_{3}$ $X_{4}$ $X_{5}$ $X_{6}-\varepsilon X_{4}$ $X_{7}-\varepsilon \left( \gamma $X_{8}-\varepsilon \left( \gamma -1\right) X_{5}$
-1\right) X_{5}$
$\mathbf{X}_{6}$ $X_{1}$ $X_{2}\cos \varepsilon -X_{3}\sin \varepsilon $ $X_{2}\sin \varepsilon +X_{3}\cos \varepsilon $ $X_{4}\cos \varepsilon $X_{4}\sin \varepsilon +X_{5}\cos \varepsilon $ $X_{6}$ $X_{7}$ $X_{8}$
-X_{5}\sin \varepsilon $
$\mathbf{X}_{7}$ $e^{-\left( \gamma -1\right) \varepsilon }X_{1}$ $X_{2}$ $X_{3}$ $e^{-\left( \gamma -1\right) \varepsilon }X_{4}$ $e^{-\left( $X_{6}$ $X_{7}$ $X_{8}$
\gamma -1\right) \varepsilon }X_{5}$
$\mathbf{X}_{8}$ $X_{1}$ $e^{\left( \gamma -1\right) \varepsilon }X_{3}$ $e^{\left( \gamma -1\right) \varepsilon }X_{4}$ $e^{-\left( \gamma $e^{-\left( \gamma -1\right) \varepsilon $X_{6}$ $X_{7}$ $X_{8}$
-1\right) \varepsilon }X_{4}$ }X_{5}$
--------------------------------------------------------------------------------- -------------------------------------------------- ------------------------------------------------- ------------------------------------------------- -------------------------------------------------- ------------------------------------------------- --------------------------- ---------------------------------- ---------------------------------------------------
: Adjoint representation of the admitted Lie point symmetries for the nonrotating 2D Shallow water
\[tabl2\]
---------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
**Symmetry** ** Invariants**
$\mathbf{X}_{1}$ $x,~y,~h\left( x,y\right) ,~u\left( x,y\right) ,~v\left(
x,y\right) $
$\mathbf{X}_{2}$ $t,~y,~h\left( t,y\right) ,~u\left( t,y\right) ,~v\left(
t,y\right) $
$\mathbf{X}_{3}$ $t,~x,~h\left( t,x\right) ,~u\left( t,x\right) ,~v\left(
t,x\right) $
$\mathbf{X}_{4}$ $t,~y,~h\left( t,y\right) ,~\frac{x}{t}+U\left(
t,y\right) ,~v\left( t,y\right) $
$\mathbf{X}_{5}$ $t,~x,~h\left( t,x\right) ,~u\left( t,x\right) ,~\frac{y}{t}+V\left( t,x\right) $
$\mathbf{X}_{6}$ $t,~x^{2}+y^{2},~h\left( t,x^{2}+y^{2}\right) ,~\frac{xU\left( t,x^{2}+y^{2}\right) +yV\left( t,x^{2}+y^{2}\right) }{\sqrt{x^{2}+y^{2}}}~,~\frac{yU\left( t,x^{2}+y^{2}\right) -xV\left(
t,x^{2}+y^{2}\right) }{\sqrt{x^{2}+y^{2}}}$
$\mathbf{X}_{7}$ $\frac{x}{t},~\frac{y}{t},~h\left( \frac{x}{t},\frac{y}{t}\right) ,~u\left( \frac{x}{t},\frac{y}{t}\right) ,~v\left( \frac{x}{t},\frac{y}{t}\right) $
$\mathbf{X}_{8}$ $H\left( x,y\right) t^{\frac{2}{1-\gamma }},~U\left(
x,y\right) t^{-1}~,~V\left( x,y\right) t^{-1}$
$\xi _{\left( 12\right) }$ $x-\alpha t,~y,~h\left( x-\alpha t,y\right)
,~u\left( x-\alpha t,y\right) ,~v\left( x-\alpha t,y\right) $
$\xi _{\left( 13\right) }$ $x,~y-\alpha t,~h\left( x,y-\alpha t\right)
,~u\left( x,y-\alpha t\right) ,~v\left( x,y-\alpha t\right) $
$\xi _{\left( 14\right) }$ $x-\frac{\alpha }{2}t^{2},~y,~h\left( x-\frac{\alpha }{2}t^{2},y\right) ,~u\left( x-\frac{\alpha }{2}t^{2},y\right)
,~v\left( x-\frac{\alpha }{2}t^{2},y\right) $
$\xi _{\left( 15\right) }$ $x,~y-\frac{\alpha }{2}t^{2},~h\left( x,y-\frac{\alpha }{2}t^{2}\right) ,~u\left( x,y-\frac{\alpha }{2}t^{2}\right)
,~v\left( x,y-\frac{\alpha }{2}t^{2}\right) ~$
---------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Lie invariants for the optimal system of the nonrotating system
\[tabl2a\]
[cc]{} **Symmetry** & ** Invariants**\
$\xi _{\left( 16\right) }$ & $~~\begin{tabular}{l}
$t, e\^[-t]{}( x\^[2]{}+y\^[2]{}) , u( t,e\^[-t]{}x\^[2]{}+y\^[2]{}) ( t) +v( t,e\^[-t]{}x\^[2]{}+y\^[2]{}) ( t) $ \\
$, h( e\^[-t]{}x\^[2]{}+y\^[2]{},) , u( t,e\^[-t]{}x\^[2]{}+y\^[2]{}) ( t) -v( t,e\^[-t]{}x\^[2]{}+y\^[2]{}) ( t) $\end{tabular}$\
$\xi _{\left( 23\right) }$ & $t,~x-\alpha y,~h\left( t,x-\alpha y\right)
,~u\left( t,x-\alpha y\right) ,v\left( t,x-\alpha y\right) $\
$\xi _{\left( 34\right) }$ & $t,~y-\frac{x}{\alpha t},~h\left( t,y-\frac{x}{\alpha t}\right) ,~u\left( t,y-\frac{x}{\alpha t}\right) ,~v\left( t,y-\frac{x}{\alpha t}\right) $\
$\xi _{\left( 25\right) }$ & $t,~y-\alpha tx,~h\left( t,y-\alpha tx\right)
,~u\left( t,y-\alpha tx\right) ,~v\left( t,y-\alpha tx\right) $\
$\xi _{\left( 45\right) }$ & $t,~y-\alpha x,~h\left( t,y-\alpha x\right)
,~\alpha \frac{x}{t}+U\left( t,y-\alpha x\right) ,~\alpha \frac{x}{t}+V\left( t,y-\alpha x\right) $\
$\xi _{\left( 123\right) }$ & $t-\alpha x,~t-\beta y,~h\left( t-\alpha
x,t-\beta y\right) ,~u\left( t-\alpha x,t-\beta y\right) ,v\left( t-\alpha
x,t-\beta y\right) $\
$\xi _{\left( 145\right) }$ & $x-\frac{\alpha }{2}t^{2},~y-\frac{\beta }{2}t^{2},~h\left( x-\frac{\alpha }{2}t^{2},y-\frac{\beta }{2}t^{2}\right)
,~\alpha t+U\left( x-\frac{\alpha }{2}t^{2},~y-\frac{\beta }{2}t^{2}\right)
,~\beta t+V\left( x-\frac{\alpha }{2}t^{2},~y-\frac{\beta }{2}t^{2}\right) $\
$\xi _{\left( 125\right) }$ & $x-\alpha t,~y-\frac{\beta }{2}t^{2},~h\left( x-\alpha t,y-\frac{\beta }{2}t^{2}\right) ,~u\left( x-\alpha
t,y-\frac{\beta }{2}t^{2}\right) ,~\beta t+V\left( x-\alpha t,y-\frac{\beta
}{2}t^{2}\right) $\
$\xi _{\left( 134\right) }$ & $x-\frac{\beta }{2}t^{2},~y-\alpha t,~h\left(
x-\frac{\beta }{2}t^{2},y-\alpha t\right) ,~\beta t+U\left( x-\frac{\beta }{2}t^{2},y-\alpha t\right) ,~V\left( x-\frac{\beta }{2}t^{2},y-\alpha t\right)
$\
$\xi _{\left( 67\right) }$ &
-----------------------------------------------------------------------------------------------------------------------------------------------------
$\frac{\ln t}{\alpha },w=~\frac{t^{-\frac{\alpha +\sqrt{\alpha \left( \alpha
-4\right) -4}}{2\alpha }}}{2\sqrt{\alpha \left( \alpha -4\right) -4}}\left(
x-\left( \alpha +\sqrt{\alpha \left( \alpha -4\right) -4}\right) y\right)
,~z=~\frac{t^{-\frac{\alpha +\sqrt{\alpha \left( \alpha -4\right) -4}}{2\alpha }}}{2\sqrt{\alpha \left( \alpha -4\right) -4}}\left( x+\left( \alpha
+\sqrt{\alpha \left( \alpha -4\right) -4}\right) y\right) $
$~h\left( w,z\right) ,~U\left( w,z\right) \sin \left( \frac{\ln t}{\alpha }\right) +V\left( w,z\right) \sin \left( \frac{\ln t}{\alpha }\right)
~,~U\left( w,z\right) \cos \left( \frac{\ln t}{\alpha }\right) -~V\left(
w,z\right) \sin \left( \frac{\ln t}{\alpha }\right) $
-----------------------------------------------------------------------------------------------------------------------------------------------------
: Lie invariants for the optimal system of the nonrotating system
$~$\
$\xi _{\left( 68\right) }$ & $t,~x^{2}+y^{2}~,~x^{-\frac{2}{\gamma -1}}h\left( t,x^{2}+y^{2}\right) ~,~\frac{U\left( t,x^{2}+y^{2}\right) \cos
\left( \frac{\ln x}{\alpha }\right) +V\left( t,x^{2}+y^{2}\right) \sin
\left( \frac{\ln x}{\alpha }\right) }{x}~,~~\frac{U\left(
t,x^{2}+y^{2}\right) \sin \left( \frac{\ln x}{\alpha }\right) -V\left(
t,x^{2}+y^{2}\right) \cos \left( \frac{\ln x}{\alpha }\right) }{x}$\
$\xi _{\left( 78\right) }$ & $w=xt^{-\frac{\left( \gamma -1\right) }{\alpha
\left( \gamma -1\right) -2}},~z=yt^{-\frac{\left( \gamma -1\right) }{\alpha
\left( \gamma -1\right) -2}},~t^{-\frac{2\alpha }{\alpha \left( \gamma
-1\right) -2}}h\left( w,z\right) ,t^{-\frac{\left( \gamma -1\right) \alpha }{\alpha \left( \gamma -1\right) -2}}u\left( w,z\right) ,~t^{-\frac{\left(
\gamma -1\right) \alpha }{\alpha \left( \gamma -1\right) -2}}v\left(
w,z\right) $\
$\xi _{\left( 678\right) }$ &
---------------------------------------------------------------------------
$t~,~t^{-1-\beta }\left( x^{2}+y^{2}\right) ~,t^{-\beta }\left( U\left(
t,x^{2}+y^{2}\right) \sin \left( \alpha t\right) +V\left(
t,x^{2}+y^{2}\right) \cos \left( \alpha t\right) \right) $
$~t^{-\frac{2\beta }{\gamma -1}}H\left( t,x^{2}+y^{2}\right) ~,~t^{-\beta
}\left( U\left( t,x^{2}+y^{2}\right) \cos \left( \alpha t\right) -V\left(
t,x^{2}+y^{2}\right) \sin \left( \alpha t\right) \right) $
---------------------------------------------------------------------------
: Lie invariants for the optimal system of the nonrotating system
$~$\
\[tabl2b\]
--------------------------------------------------------------------------------- ---------------------------- ------------------------------------------------- ------------------------------------------------- --------------------------- --------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------------
$Ad\left( e^{\left( \varepsilon \mathbf{Y}_{i}\right) }\right) \mathbf{Y}_{j} $ $\mathbf{Y}_{1}$ $\mathbf{Y}_{2}$ $\mathbf{Y}_{3}$ $\mathbf{Y}_{4}$ $\mathbf{Y}_{5}$ $\mathbf{Y}_{6}$ $\mathbf{Y}_{7}$
$\mathbf{Y}_{1}$ $Y_{1}$ $Y_{2}$ $Y_{3}$ $Y_{4}$ $Y_{5}\cos \left( $Y_{5}\sin $Y_{7} $
f\varepsilon \right) -Y_{6}\sin \left( f\varepsilon \right) $ \left( f\varepsilon \right) +Y_{6}\cos \left( f\varepsilon \right) $
$\mathbf{Y}_{2}$ $Y_{1}$ $Y_{2}$ $Y_{3}$ $Y_{4}+\varepsilon Y_{3}$ $Y_{5}$ $Y_{6}$ $Y_{7}-\varepsilon \left( \gamma -1\right) Y_{2}$
$\mathbf{Y}_{3}$ $Y_{1}$ $Y_{2}$ $Y_{3}$ $Y_{4}-\varepsilon Y_{2}$ $Y_{5}$ $Y_{6}$ $Y_{7}-\varepsilon \left( \gamma -1\right) Y_{3}$
$\mathbf{Y}_{4}$ $Y_{1}$ $Y_{2}\cos \varepsilon -Y_{3}\sin \varepsilon $ $Y_{2}\sin \varepsilon +Y_{3}\cos \varepsilon $ $Y_{4}$ $Y_{5}\cos $Y_{6}\cos \varepsilon -Y_{5}\sin $Y_{7}$
\varepsilon +Y_{6}\sin \varepsilon $ \varepsilon $
$\mathbf{Y}_{5}$ $Y_{1}+f\varepsilon Y_{6}$ $Y_{2}$ $Y_{3}$ $Y_{4}-\varepsilon Y_{6}$ $Y_{5}$ $Y_{6}$ $Y_{7}-\varepsilon \left(
\gamma -1\right) Y_{5}$
$\mathbf{Y}_{6}$ $Y_{1}-f\varepsilon Y_{5}$ $Y_{2}$ $Y_{3}$ $Y_{4}+\varepsilon Y_{5}$ $Y_{5}$ $Y_{6}$ $Y_{7}-\varepsilon \left(
\gamma -1\right) Y_{6}$
$\mathbf{Y}_{7}$ $Y_{1}$ $e^{\left( \gamma -1\right) \varepsilon }Y_{2}$ $e^{\left( \gamma -1\right) \varepsilon }Y_{3}$ $Y_{4}$ $e^{\left( $e^{\left( \gamma -1\right) $Y_{7}$
\gamma -1\right) \varepsilon }Y_{5}$ \varepsilon }Y_{6}$
--------------------------------------------------------------------------------- ---------------------------- ------------------------------------------------- ------------------------------------------------- --------------------------- --------------------------------------------------------------- ---------------------------------------------------------------------- ---------------------------------------------------
: Adjoint representation of the admitted Lie point symmetries for the rotating 2D Shallow water
\[tabl4\]
----------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
**Symmetry** ** Invariants**
$\mathbf{Y}_{1}$ $x,~y,~h\left( x,y\right) ,~u\left( x,y\right) ,~v\left(
x,y\right) $
$\mathbf{Y}_{2}$ $t,~y,~h\left( t,y\right) ,~u\left( t,y\right) ,~v\left(
t,y\right) $
$\mathbf{Y}_{3}$ $t,~x,~h\left( t,x\right) ,~u\left( t,x\right) ,~v\left(
t,x\right) $
$\mathbf{Y}_{4}$ $t,~x^{2}+y^{2},~h\left( t,x^{2}+y^{2}\right) ,~\frac{xU\left( t,x^{2}+y^{2}\right) +yV\left( t,x^{2}+y^{2}\right) }{\sqrt{x^{2}+y^{2}}}~,~\frac{yU\left( t,x^{2}+y^{2}\right) -xV\left(
t,x^{2}+y^{2}\right) }{\sqrt{x^{2}+y^{2}}}$
$\mathbf{Y}_{5}$ $t,~x\cot \left( ft\right) -y~,~h\left( t,x\cot \left(
ft\right) -y\right) ,~fx\cot \left( ft\right) +U\left( t,x\cot \left(
ft\right) -y\right) ,~-fx+V\left( t,x\cot \left( ft\right) -y\right) $
$\mathbf{Y}_{6}$ $t,~x\tan \left( ft\right) +y,~h\left( t,x\tan \left(
ft\right) +y\right) ,~-fx\tan \left( ft\right) +U\left( t,x\tan \left(
ft\right) +y\right) ,~-fx+V\left( t,x\tan \left( ft\right) +y\right) $
$\mathbf{Y}_{7}$ $\frac{x}{t},~\frac{y}{t},~h\left( \frac{x}{t},\frac{y}{t}\right) ,~u\left( \frac{x}{t},\frac{y}{t}\right) ,~v\left( \frac{x}{t},\frac{y}{t}\right) $
$\chi _{\left( 12\right) }$ $x-\alpha t,~y,~h\left( x-\alpha t,y\right)
,~u\left( x-\alpha t,y\right) ,~v\left( x-\alpha t,y\right) $
$\chi _{\left( 13\right) }$ $x,~y-\alpha t,~h\left( x,y-\alpha t\right)
,~u\left( x,y-\alpha t\right) ,~v\left( x,y-\alpha t\right) $
$\chi _{\left( 14\right) }$ $~~\begin{tabular}{l}
$t, e\^[-t]{}( x\^[2]{}+y\^[2]{}) , u( t,e\^[-t]{}x\^[2]{}+y\^[2]{}) ( t) +v( t,e\^[-t]{}x\^[2]{}+y\^[2]{}) ( t) $ \\
$, h( e\^[-t]{}x\^[2]{}+y\^[2]{},) , u( t,e\^[-t]{}x\^[2]{}+y\^[2]{}) ( t) -v( t,e\^[-t]{}x\^[2]{}+y\^[2]{}) ( t) $\end{tabular}$
----------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Lie invariants for the optimal system of the rotating system
\[tabl4a\]
[cc]{} **Symmetry** & ** Invariants**\
$\chi _{\left( 15\right) }$ &
--------------------------------------------------------------------------------------------------------------------------
$x+\frac{\alpha }{f}\cos \left( ft\right) ,~y-\frac{\alpha }{f}\sin \left(
ft\right) ~,~h\left( \,x+\frac{\alpha }{f}\cos \left( ft\right) ,y-\frac{\alpha }{f}\sin \left( ft\right) \right) ,~$
$\alpha \sin \left( ft\right) +U\left( \,x+\frac{\alpha }{f}\cos \left(
ft\right) ,y-\frac{\alpha }{f}\sin \left( ft\right) \right) ,~\alpha \cos
\left( ft\right) +V\left( \,x+\frac{\alpha }{f}\cos \left( ft\right) ,y-\frac{\alpha }{f}\sin \left( ft\right) \right) $
--------------------------------------------------------------------------------------------------------------------------
: Lie invariants for the optimal system of the rotating system
\
$\chi _{\left( 16\right) }$ &
------------------------------------------------------------------------------------------------------------------------
$x-\frac{\alpha }{f}\sin \left( ft\right) ,~y-\frac{\alpha }{f}\cos \left(
ft\right) ~,~h\left( x-\frac{\alpha }{f}\sin \left( ft\right) ,y-\frac{\alpha }{f}\cos \left( ft\right) \right) ,~$
$\alpha \cos \left( ft\right) +U\left( x-\frac{\alpha }{f}\sin \left(
ft\right) ,y-\frac{\alpha }{f}\cos \left( ft\right) \right) ,~-\alpha \sin
\left( ft\right) +V\left( x-\frac{\alpha }{f}\sin \left( ft\right) ,y-\frac{\alpha }{f}\cos \left( ft\right) \right) $
------------------------------------------------------------------------------------------------------------------------
: Lie invariants for the optimal system of the rotating system
\
$\chi _{\left( 17\right) }$ & $xe^{-\alpha t},~ye^{-\alpha t},~e^{\frac{2\alpha }{\gamma -1}t}h\left( xe^{-\alpha t},ye^{-\alpha t}\right)
,~e^{\alpha t}u\left( xe^{-\alpha t},ye^{-\alpha t}\right) ,~e^{\alpha
t}v\left( xe^{-\alpha t},ye^{-\alpha t}\right) $\
$\chi _{\left( 23\right) }$ & $t,~x-\alpha y,~h\left( t,x-\alpha y\right)
,~u\left( t,x-\alpha y\right) ,~v\left( t,x-\alpha y\right) $\
$\chi _{\left( 45\right) }$ & $t,~w=\left( x^{2}+y^{2}-2x~\cos \left(
ft\right) +2y\sin \left( ft\right) \right) ,~\frac{U\left( t,w\right) +f\sin
\left( ft\right) }{V\left( t,w\right) +f\cos \left( ft\right) },~\frac{U\left( t,w\right) ^{2}+V\left( t,w\right) ^{2}}{2}+f\left( U\left(
t,w\right) \sin \left( ft\right) +V\left( t,w\right) \cos \left( ft\right)
\right) $\
$\chi _{\left( 46\right) }$ & $t,~w=\left( x^{2}+y^{2}-2x~\sin \left(
ft\right) -2y\cos \left( ft\right) \right) ,~\frac{U\left( t,w\right) -f\cos
\left( ft\right) }{V\left( t,w\right) +f\sin \left( ft\right) },~\frac{U\left( t,w\right) ^{2}+V\left( t,w\right) ^{2}}{2}+f\left( V\left(
t,w\right) \sin \left( ft\right) -U\left( t,w\right) \cos \left( ft\right)
\right) $\
$\chi _{\left( 56\right) }$ & $t,~z=y-\frac{x\left( \cos \left( ft\right)
-\alpha \sin \left( ft\right) \right) }{\sin \left( ft\right) +\alpha \cos
\left( ft\right) },~h\left( t,z\right) ,~f\frac{x\left( \cos \left(
ft\right) -\alpha \sin \left( ft\right) \right) }{\sin \left( ft\right)
+\alpha \cos \left( ft\right) }+U\left( t,z\right) ,~-x+V\left( t,z\right) $\
$\chi _{\left( 47\right) }$ & $t,~x^{2}+y^{2}~,~x^{-\frac{2}{\gamma -1}}h\left( t,x^{2}+y^{2}\right) ~,~\frac{U\left( t,x^{2}+y^{2}\right) \cos
\left( \frac{\ln x}{\alpha }\right) +V\left( t,x^{2}+y^{2}\right) \sin
\left( \frac{\ln x}{\alpha }\right) }{x}~,~~\frac{U\left(
t,x^{2}+y^{2}\right) \sin \left( \frac{\ln x}{\alpha }\right) -V\left(
t,x^{2}+y^{2}\right) \cos \left( \frac{\ln x}{\alpha }\right) }{x}$\
$\chi _{\left( 123\right) }$ & $t-\alpha x,~t-\beta y,~h\left( t-\alpha
x,t-\beta y\right) ,~u\left( t-\alpha x,t-\beta y\right) ,v\left( t-\alpha
x,t-\beta y\right) $\
$\chi _{\left( 147\right) }$ & $\,\begin{tabular}{l}
$z=e\^[-t( -1) ]{}( xt-yt) , w=e\^[-t( -1) ]{}( yt+xt) ,$ \\
$e\^[-t( -1) ]{}h( z,w) , e\^[-t( -1) ]{}( U( z,w) t-V( z,w) t) , e\^[-t( -1) ]{}( U( z,w) t+V( z,w) t) $\end{tabular}~$\
\[tabl4b\]
[99]{} P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, (1993)
G.W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, (1989)
N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CRS Press LLC, Florida (2000)
L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, New York, (1982)
S Jamal, P.G.L. Leach and A. Paliathanasis, Quaestiones Mathematicae 42, 125 (2019)
G.M. Webb, J. Phys A: Math. Gen. 23, 3885 (1990)
M. Tsamparlis, A. Paliathanasis and L. Karpathopoulos, J. Phys. A: Math. Theor. 45, 275201 (2012)
H. Azad and M.T. Mustafa, Commun. Nonlinear. Sci. Numer. Simul. 15, 1132 (2010)
M. Tsamparlis, A. Paliathanasis and A. Qadir, Int. J. Geom. Methods Mod. Phys. 12, 155003 (2005)
S.V. Meleshko and V.P. Shapeev, J. Nonl. Math. Phys. 18, 195 (2011)
A. Halder, A. Paliathanasis and P.G.L. Leach, Symmetry 10, 744 (2018)
K. S. Chou and C. Z. Qu, Acta Applicandae Mathematicae, 83, 257 (2004).
F. Galas and E. W. Richter, Phys. D, (50) 297 (1991).
S. V. Coggeshalla and J.Meyer-ter-Vehn, J.Math. Phys. 33, 3585 (1992).
X. Hu, Y. Li and Y. Chen, J. Math. Phys. 56, 053504 (2015)
G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, Cambridge University Press, Cambridge (2006)
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publ. (1948)
J. Kevorkian, Partial Differential Equations: Analytical Solutions Techniques, Chapman and Hall, New York (1990)
V. Caleffi, A. Valiani and A. Zanni, J. Hydraulic Research 41, 167 (2003)
R.A.D. Akkermans, L.P.J. Kamp, H.J.H. Clercx and G.J.F. van Heijst, Phys. Rev. E 82, 026314 (2010)
D.H. Kim, Y.S. Cho and Y.K. Yi, Ocean Engineering 34, 1164 (2007)
A.A. Chesnokov, Eur. J. Appl. Math. 20, 461 (2009)
X. Xin, L. Zhang, Y. Xia and H. Liu, Appl. Math. Lett. 94, 112 (2019)
S. Szatmari and A. Bihlo, Comm. Nonl. Sci. Num. Sim. 19, 530 (2014)
A.A. Chesnokov, J. Appl. Mech. Techn. Phys. 49, 737 (2008)
J.-G. Liu, Z.-F. Zeng, Y. He and G.-P. Ai, Int. J. Nonl. Sci. Num. Sim. 16, 114 (2013)
M. Pandey, Int. J. Nonl. Sci. Num. Sim. 16, 93 (2015)
A. Paliathanasis, to appear in Zeitschrift fur Naturforschung A DOI:10.1515/zna-2019-0063 \[arXiv:1906.00689\]
[^1]: Email: anpaliat@phys.uoa.gr
|
---
abstract: 'Let ${\mathcal{A}}= \{x_1, \dotsc, x_n\}$ be a subspace arrangement with a geometric lattice such that $\operatorname{codim}(x) \geq 2$ for every $x \in{\mathcal{A}}$. Using rational homotopy theory, we prove that the complement $M({\mathcal{A}})$ is rationally elliptic if and only if the sum $x_1^\perp + \dotso + x_n^\perp$ is a direct sum. The homotopy type of $M({\mathcal{A}})$ is also given : it is a product of odd dimensional spheres. Finally, some other equivalent conditions are given, such as Poincaré duality. Those results give a complete description of arrangements (with geometric lattice and with the codimension condition on the subspaces) such that $M({\mathcal{A}})$ is rationally elliptic, and show that most arrangements have an hyperbolic complement.'
address: |
UCL, Departement de mathematique\
Chemin du Cyclotron, 2\
B-1348 Louvain-la-neuve\
Belgium
author:
- Gery Debongnie
title: Rational homotopy type of subspace arrangements with a geometric lattice
---
[^1]
Introduction
============
Let $l$ be an integer. A subspace arrangement ${\mathcal{A}}$ is a finite set of affine subspaces in ${\mathbb{C}}^l$. We say that ${\mathcal{A}}= \{x_1, \dotsc, x_n\}$ is central if all the affine subspaces $x_i$ are vector subspaces.
To every arrangement, we associate the set of non empty intersections of elements of ${\mathcal{A}}$. This set $L({\mathcal{A}})$ is partially ordered by $x \leq y \iff y \subseteq x$. Let $x, y \in L({\mathcal{A}})$. If ${\mathcal{A}}$ is central, we can define two operations on $L({\mathcal{A}})$ : the meet $x \wedge y = \cap\{z \in L({\mathcal{A}}) {\mid}x \cup y \subset z\}$ and the join $x \vee y = x \cap y$. With these two operations, $L({\mathcal{A}})$ is a lattice.
For every $x \in L({\mathcal{A}})$, there exists a longest maximal chain ${\mathbb{C}}^l < x_1 < \dots < x_r = x$. We say that the rank of $x$, $\operatorname{rk}(x)$, is $r$. The lattice $L({\mathcal{A}})$ is called geometric if, for every $x, y \in L({\mathcal{A}})$, we have : $\operatorname{rk}(x) + \operatorname{rk}(y) \geq \operatorname{rk}(x \wedge y) + \operatorname{rk}(x \vee y)$. The complement of a subspace arrangement ${\mathcal{A}}$ is the topological space $$M({\mathcal{A}}) = {\mathbb{C}}^l \setminus \bigcup {\mathcal{A}}.$$ In 2002, S. Yuzvinsky described a rational model for $M({\mathcal{A}})$ in [@yu02]. Later on, in [@yu05], S. Yuzvinsky and E. Feichtner proved that if the lattice $L({\mathcal{A}})$ is geometric, then $M({\mathcal{A}})$ is formal and they give a simpler differential graded algebra model for $M({\mathcal{A}})$.
In section \[sec:rht\], we state some basic properties of rational homotopy theory. The rational model defined by Yuzvinsky for $M({\mathcal{A}})$ is recalled in section \[sec:rmsa\]. Arrangements with Poincaré duality are studied in section \[sec:pd\]. Finally, the main results are contained in section \[sec:proof\].
Briefly, the theorem \[theo:a\] shows that, under some conditions, the following statements are equivalent : the subspace arrangement ${\mathcal{A}}$ has a rationally elliptic complement $M({\mathcal{A}})$, $\operatorname{codim}\cap_{x \in {\mathcal{A}}} x = \sum \operatorname{codim}x$, $M({\mathcal{A}})$ has Poincaré duality, $M({\mathcal{A}})$ has the homotopy type of a product of odd dimensional spheres. The theorem \[theo:b\] gives a geometric interpretation : these statements are equivalent to the fact that $x_1^\perp + \dotso + x_n^\perp$ is a direct sum. So, every arrangement with a geometric lattice and rationally elliptic complement is obtained by taking a direct sum $y_1 \oplus \dotso \oplus y_n$ of vector subspaces in ${\mathbb{C}}^l$ and then taking their orthogonal complement : ${\mathcal{A}}= \{y_1^\perp, \dotsc, y_n^\perp\}$.
It shows that most arrangements have an hyperbolic complement. In that case, which is easy to check with the condition $\operatorname{codim}\cap_{x\in {\mathcal{A}}} x \neq \sum \operatorname{codim}(x)$, the sequence $\sum_{i \leq p} \operatorname{rk}\pi_i(M{\mathcal{A}}))$ has an exponential growth and for any integer $N$, there are infinitely many $q$ with $\operatorname{rk}\pi_q(M({\mathcal{A}})) \geq N$.
I would like to thank the referee for his/her work. In particular, the comments about the geometric interpretation were very helpful.
Rational homotopy theory {#sec:rht}
========================
For the basic facts on rational homotopy, we will refer to the classical references (see [@su77] or [@fe00]).
Let $V$ be a graded vector space. The free commutative algebra on $V$, $\Lambda V$, is by definition the tensor product of the symmetric algebra on $V^{\text{even}}$ by the exterior algebra on $V^{\text{odd}}$. A minimal model is a differential graded algebra of the form $(\Lambda V, d)$ where $d(V) \subset \Lambda^{\geq 2} V$, and such that there is a basis of $V$, $(x_a)_{a \in A}$, indexed by a well-ordered set with the property that $d(x_a) \in \Lambda(x_b)_{b < a}$.
Each 1-connected space $X$ with finite Betti numbers admits a minimal model $(\Lambda V, d)$ that is unique up to isomorphism and that contains all the rational homotopy type of $X$. In particular, $\dim V^n = \dim \pi_n(X) \otimes {\mathbb{Q}}$.
The space $X$ is called *formal* if there is a quasi-isomorphism $(\Lambda V, d) \to (H^\star(X, {\mathbb{Q}}), 0)$.
In the case of subspace arrangements, it is known that if the lattice $L({\mathcal{A}})$ is geometric then the space $M({\mathcal{A}})$ is formal (see [@yu05]).
The dichotomy theorem in rational homotopy theory states that finite 1-connected CW-complexes are either elliptic or hyperbolic, with the following properties : if $X$ is elliptic, then $\pi_n(X) =0$ for $n$ large enough and $H^\star(X;{\mathbb{Q}})$ satisfies Poincaré duality. If $X$ is hyperbolic, then the sequence $\dim \pi_n(X)$ has an exponential growth.
We will use the following theorem (see [@fe82] for details)
\[theo:fh\] If a space $X$ is elliptic and formal, then its minimal model has the form $(\Lambda V, d) = (\Lambda V_0 \oplus V_1, d)$ with $V_0 = V_0^{\text{odd}}\oplus V_0^{\text{even}}$, $\dim V_0^{\text{even}} = \dim V_1$, $V_1 = V_1^{\text{odd}}$, $dV_0 = 0$ and $dV_1 \subset \Lambda V_0$. Moreover, the injection $(\Lambda V_0^{\text{odd}}, 0) \to (\Lambda V, d)$ induces an injective map in cohomology.
Finally, to use rational homotopy theory, we need spaces that are 1-connected. The following lemmas will show that the space $M({\mathcal{A}})$ is 1-connected if the subspaces have all a codimension $\geq 2$.
\[lemm:ext\] Let ${\mathcal{A}}= \{x_1, \dotsc, x_q\}$ be a central arrangement in ${\mathbb{C}}^l$ such that $\operatorname{codim}x_i \geq 2$. Let $y_i = x_i \cap S^{2l-1}$ and $f{\colon}S^1 \to S^{2l-1} \setminus \cup_{i=1}^q y_i$ be a smooth map. Then $f$ extends to a map $\bar{f} {\colon}D^2 \to S^{2l-1} \setminus \cup_{i=1}^q y_i$ : $$\xymatrix@C=4mm@R=3mm{S^1 \ar[rr]^{f} \ar@{^{(}->}[dr] && S^{2l-1} \setminus \cup_{i=1}^q y_i \\ & D^2 \ar[ur]_{\bar{f}}}$$
The proof is done by induction on $q$. The case $q=1$ is a direct consequence of corollary 15.7 in [@br93]. Let’s assume the result true until $q-1$. Let $f{\colon}S^1 \to S^{2l-1} \setminus \cup_{i=1}^q y_i$. By induction, we know that there exists a $\tilde{f}{\colon}D^2 \to S^{2l-1} \setminus \cup_{i=1}^{q-1} y_i$ such that $\tilde{f}|_{S^1} = f$. Let $r >0$ such that $r < \operatorname{dist}(\tilde{f}(D^2), \cup_{i=1}^{q-1} y_i)$ and $$T = \{z \in S^{2l-1} {\mid}\operatorname{dist}(z, \cup_{i=1}^{q-1} y_i) < r\}.$$ The way we constructed $T$ implies that $\operatorname{im}\tilde{f} \subset S^{2l-1} \setminus T$, which is a $(2l-1)$-dimensional manifold with boundary. In it, $y_q \setminus T$ is a compact submanifold of dimension $< 2l-4$ (because it is of codimension $\geq 2$ in ${\mathbb{C}}^l$). The corollary 15.6 in [@br93] gives the existence of a smooth map $\bar{f} {\colon}D^2 \to S^{2l-1} \setminus T$ such that : $\bar{f}|_{S^1} = \tilde{f}|_{S^1} = f$ and $\bar{f}(D^2)$ is transverse to $y_q \setminus T$. But $\dim \bar{f}(D^2) + \dim(y_q \setminus T) \leq 2 + 2l-4 < 2l-1$. So, transversality can only happen if $\bar{f}(D^2) \cap (y_q \setminus T) = \emptyset$. Therefore, the application $\bar{f}(D^2)$ is such that $\operatorname{im}\bar{f}(D^2) \subset (S^{2l-1} \setminus T) \setminus (y_q \setminus T) \subset S^{2l-1} \setminus \cup_{i=1}^q y_i$.
\[lemm:muc\] Let ${\mathcal{A}}$ be a subspace arrangement such that for each $x \in {\mathcal{A}}$, $\operatorname{codim}x \geq 2$. Then the space $M({\mathcal{A}})$ is 1-connected.
Let $f{\colon}S^1 \to M({\mathcal{A}})$ be a map. Since the $x_i$ are vector spaces, we can define the homotopy $h_t = (1-t)f + t\frac{f}{||f||}$. We can assume that the map $h_1 {\colon}S^1 \to S^{2l-1} \setminus(\cup_{x \in {\mathcal{A}}} (x \cap S^{2l-1})$ is smooth. So, lemma \[lemm:ext\] can be applied and shows that $f \simeq h_1 \simeq \star$. It implies that $\pi_1(M({\mathcal{A}})) = 0$, so $M({\mathcal{A}})$ is 1-connected.
Rational model of subspace arrangements {#sec:rmsa}
=======================================
Let ${\mathcal{A}}$ be a central arrangement of subspaces in ${\mathbb{C}}^l$. Yuzvinsky defined the relative atomic differential graded algebra $D_{\mathcal{A}}= (D, d)$ associated with an arrangement as follows (see [@yu05]) : choose a linear order on ${\mathcal{A}}$. The chain complex $(D, d)$ is generated by all subsets $\sigma \subseteq {\mathcal{A}}$. For $\sigma = \{x_1, \dotsc, x_n\}$, we define the differential by $$d\sigma = \sum_{j:\vee(\sigma \setminus \{x_j\}) = \vee \sigma} (-1)^j(\sigma\setminus\{x_j\})$$ where the indexing of the elements in $\sigma$ follows the linear order imposed on ${\mathcal{A}}$. With $\deg(\sigma) = 2 \operatorname{codim}\vee \sigma - |\sigma|$, $(D,d)$ is a cochain complex. Finally, we need a multiplication on $(D,d)$. For $\sigma, \tau \subseteq {\mathcal{A}}$, $$\sigma \cdot \tau = \left\{\begin{aligned} (-1)^{\operatorname{sgn}\epsilon(\sigma, \tau)} \sigma \cup \tau &\text{ if } \operatorname{codim}\vee \sigma + \operatorname{codim}\vee \tau = \operatorname{codim}\vee(\sigma \cup \tau) \\ 0 & \text{ otherwise} \end{aligned}\right.$$ where $\epsilon(\sigma, \tau)$ is the permutation that, applied to $\sigma \cup \tau$ with the induced linear order, places elements of $\tau$ after elements of $\sigma$, both in the induced linear order.
A subset $\sigma \subseteq {\mathcal{A}}$ is said to be independant if $\operatorname{rk}( \vee \sigma) = |\sigma|$. When ${\mathcal{A}}$ is an arrangement with a geometric lattice, we have the following property : $H^\star(M({\mathcal{A}}))$ is generated by the classes $[\sigma]$, with $\sigma$ independant ([@yu05]).
Poincaré duality {#sec:pd}
================
Having Poincaré duality is a strong statement for a subspace arrangement with geometric lattice. That condition alone determines the minimal model of the complement.
For ${\mathcal{A}}= \{x_1, \dotsc, x_n\}$ a subspace arrangement with $L({\mathcal{A}})$ geometric, let $M_r$ be the greatest element in $L({\mathcal{A}})$ and ${\mathbb{C}}^l < M_1 < \dotso < M_{r-1} < M_r$ be any maximal chain in $L({\mathcal{A}})$. In particular, $\operatorname{rk}(M_i)=i$ for $1\leq i \leq r$. Let $X_i = \{x \in {\mathcal{A}}{\mid}x < M_i \}$. We can construct a chain complex $C^i_\star$ : $C^i_p$ is the module generated by all the linear combinations of the $\sigma \subset {\mathcal{A}}$ such that $\vee \sigma = M_i$ and $|\sigma| = p$. With the differential defined in the Yuzvinsky model of $M({\mathcal{A}})$ and $\deg(\sigma) = |\sigma|$, $C^i_\star$ is clearly a chain complex.
\[lemm:pda\] Let ${\mathcal{A}}= \{x_1, \dotsc, x_n\}$ be a subspace arrangement with geometric lattice. If $M({\mathcal{A}})$ has Poincaré duality and $1 \leq k \leq r$, then $\dim H_k(C^k_\star) = 1$.
In this proof, $X_0$ is the empty set. Let $E^i_p$ ($1 \leq i \leq r$) be the submodule of $C^i_p$ generated by
- all the $\sigma \subset {\mathcal{A}}$ such that $|\sigma| = p$, $\vee \sigma = M_i$ and $\sigma$ contains at least 2 elements of $X_i \setminus X_{i-1}$,
- all the elements $\{x_{i_1}, x_{i_2}, \dotsc, x_{i_{p-1}}, y_1\} - \{x_{i_1}, \dotsc, x_{i_{p-1}}, y_2\}$ with $x_{i_j} \in X_{i-1}$ and $y_1, y_2 \in X_{i} \setminus X_{i-1}$.
It is easy to check that $E^i_\star$ is a subcomplex of $C^i_\star$. We have the following short exact sequences : $$0 \to E^i_\star \to C^i_\star \to C^i_\star/ E^i_\star \to 0 .$$ Since ${\mathcal{A}}$ has a geometric lattice, the cohomology is generated by the classes $\sigma \in {\mathcal{A}}$ such that $\sigma$ is independant. So, the (reduced) homology of $C^i_\star$ is $0$ in every degree except possibly the $i^\text{th}$ degree. We have : $H_\star(C^i_\star) = H_{i}(C^i_\star)$. For $1 \leq i \leq r$, we know that $\operatorname{rk}(M_i) = i$. It means that $E^{i+1}_{i}$ is an empty set and $H_{i}(E^{i+1}_\star) = 0$. Hence, the long exact sequence in homology associated with the short exact sequence above implies that the map $H_{i+1}(C^{i+1}_\star) \to H_{i+1}(C^{i+1}_\star/E^{i+1}_\star)$ is surjective.
Since $X_{i+1} \setminus X_{i}$ is non empty (because $\vee X_{i+1} = M_{i+1}$ and $\vee X_i = M_i)$, we can fix some $y \in X_{i+1} \setminus X_{i}$. This $y$ define maps $\varphi_p {\colon}C^i_p \to C^{i+1}_{p+1}/E^{i+1}_{p+1}$ sending $\{x_1, \dotsc, x_p\}$ to $[\{x_1, \dotsc, x_p, y\}]$. Since the lattice is geometric, if $\vee \{x_1, \dotsc, \hat{x_j}, \dots, x_p\} < M_i$, then $\vee \{x_1, \dotsc, \hat{x_j}, \dotsc, x_p, y\} < M_{i+1}$. Therefore, the maps $\varphi_p$ commute with the differentials and define an isomorphism of chain complex $(C^i_\star)_p \to (C^{i+1}_\star / E^{i+1}_\star)_{p+1}$. Hence, $H_p(C^i_\star) = H_{p+1}(C^{i+1}_\star/ E^{i+1}_\star)$. But, we proved that the map $H_{i+1}(C^{i+1}_\star) \to H_{i+1}(C^{i+1}_\star/ E^{i+1}_\star)$ is a surjection. So$$\dim H_{i}(C^i_\star) = \dim H_{i+1}(C^{i+1}_\star/ E^{i+1}_\star) \leq \dim H_{i+1}(C^{i+1}_\star).$$ Since the space $M({\mathcal{A}})$ has Poincaré duality, there is a unique cohomology class in the highest degree in $H^\star(D_{\mathcal{A}})$. That cohomology class is represented by an independant $\sigma \in {\mathcal{A}}$ such that $|\sigma| = r$. If $\dim H_r(C_\star^r) \geq 2$, then there is another class $[\tau]$ and by Poincaré duality, there is an element $[\rho]$ in $H^\star(D_{\mathcal{A}})$ such that $[\sigma] = [\rho] [\tau]$, but this is impossible by the multiplication law because $\operatorname{codim}\vee \sigma= \operatorname{codim}\vee \tau$. Therefore, $\dim H_r(C_\star^r) = 1$. From the following sequence of inequalities $$1 = \dim H_1(C^1_\star) \leq \dim H_2(C^2_\star) \leq \dots \leq \dim H_r(C^r_\star) = 1.$$ we deduce that $\dim H_k(C^k_\star) = 1$ for all $1 \leq k \leq r$.
\[lemm:pdb\] Let ${\mathcal{A}}= \{x_1, \dotsc, x_n\}$ be a subspace arrangement such that $L({\mathcal{A}})$ is geometric and $M({\mathcal{A}})$ has Poincaré duality. Let $M \in L({\mathcal{A}})$ with $\operatorname{rk}(M) = i$ and let $X(M) = \{x \in {\mathcal{A}}{\mid}x \leq M \}$, then $\# X(M) = i$.
We prove the result by induction on $\operatorname{rk}M$. It is clear for $i = 1$. Now, let us suppose that it is true for all $N \in L({\mathcal{A}})$ with $\operatorname{rk}N \leq i-1$ and let $M \in L({\mathcal{A}})$ with $\operatorname{rk}M = i$. Denote by $M_1 < M_2 < \dotso < M_i = M < M_{i+1} < \dotso < M_r$ a maximal sequence in $L({\mathcal{A}})$, and write $$X(M_{i-1}) = \{x_1, \dotsc, x_{i-1}\} \quad \text{and} \quad
X(M) = \{x_1, \dotsc, x_{i-1}, x_{i-1+1}, \dotsc, x_{i-1+l}\}.$$ We consider the chain complex $C_\star^i$ defined in lemma 5 for that maximal chain. Remark first that if $\{ x_{n_1}, \dotsc, x_{n_{i+1}}\} \subset X(M)$ with $\vee x_{n_i} = M$, then for each $k$, $\vee \{ x_{n_1}, \dotsc, \hat{x}_{n_k}, \dotsc, x_{n_{i+1}}\} = M$, because otherwise $\vee \{ x_{n_1}, \dotsc, \hat{x}_{n_k}, \dotsc, x_{n_{i+1}}\}$ is an element $N$ in $L({\mathcal{A}})$ with $\operatorname{rk}N < i$. So, there are $i$ subspaces $y_j$ with $y_j < N$, in contradiction with our induction hypothesis. Therefore, if $\{x_{n_1}, \dotsc, x_{n_{i+1}}\} \subset C_{i+1}^i$, then $$d\{x_{n_1}, \dotsc, x_{n_{i+1}}\} = \sum_{j=1}^{i+1} (-1)^j \{x_{n_1}, \dotsc, \hat{x}_{n_j}, \dotsc, x_{n_{i+1}}\}.$$ In the complex $C_\star^i$, every cycle of degree $i$ is equivalent to a sum $\sum \alpha_j \{x_1, x_{j_2}, \dotsc, x_{j_i}\}$. Indeed, if $1 \not\in \{j_1, \dotsc, j_i\}$, then $$\{x_{j_1}, \dotsc, x_{j_i}\} = -d\{x_1, x_{j_1}, \dotsc, x_{j_i}\} + \sum_{k=2}^i (-1)^k\{x_1, \dotsc, \hat{x}_{j_k}, \dotsc, x_{j_i}\}.$$ Now, no cycle of the form $\sum_j \alpha_j\{x_1, x_{j_2}, \dotsc, x_{j_i}\}$ is a boundary. Suppose this is the case, we have : $$\sum\nolimits_j \alpha_j\{x_1, x_{j_2}, \dotsc, x_{j_i}\} = d\left[ \sum\nolimits_m \beta_m \{x_1, x_{m_1}, \dotsc, x_{m_i}\} + \sum\nolimits_n \gamma_n \{x_{n_1}, \dotsc, x_{n_{i+1}}\}\right]$$ with $1 \not\in \{x_1, \dotsc, n_{i+1}\}$. Developing the differential, we get $$0 = -\sum\nolimits_m \beta_m\{x_{m_1}, \dotsc, x_{m_i}\} + \sum\nolimits_n \gamma_n \left(\sum_{k=1}^{i+1} (-1)^k \{x_{n_1}, \dotsc, \hat{x}_{n_k}, \dotsc, x_{n_{i+1}}\} \right).$$ We deduce that $$\begin{gathered}
d\left(\sum\nolimits_n \gamma_n\{x_1, x_{n_1}, \dotsc, x_{n_{i+1}} \}\right) \\
= -\sum_n \gamma_n \{x_{n_1}, \dotsc, x_{n_{i+1}} \} - \sum_n \gamma_n\left( \sum_{k=1}^{i+1} (-1)^k \{x_1, x_{n_1}, \dotsc, \hat{x}_{n_k}, \dotsc, x_{n_{i+1}}\}\right) \\
= - \sum\nolimits_n \gamma_n \{x_{n_1}, \dotsc, x_{n_{i+1}} \} -\sum\nolimits_m \beta_m\{x_1, x_{m_1}, \dotsc, x_{m_i}\}.\end{gathered}$$ Since $d^2=0$, this gives $\sum_j \alpha_j\{x_1, x_{j_2}, \dotsc, x_{j_i} \} = 0$.
We deduce from the above calculation that $l=1$, i.e. $X(M) = \{x_1, \dotsc, x_i\}$, because otherwise, the cycles $\{x_1, \dotsc, x_i\}$ and $\{x_1, \dotsc, x_{i-1}, x_{i+1}\}$ would be linearly independant in homology, in contradiction with lemma \[lemm:pda\].
\[prop:pd\] Let ${\mathcal{A}}= \{x_1, \dotsc, x_n\}$ be a subspace arrangement. If $L({\mathcal{A}})$ is geometric and $M({\mathcal{A}})$ has Poincaré duality, then the minimal model of $M({\mathcal{A}})$ is the algebra $(\Lambda(y_1, \dotsc, y_n), 0)$ where $\deg y_i = 2 \operatorname{codim}x_i -1$.
It is a consequence from lemma \[lemm:pdb\]. This lemma shows that every subset $\sigma \subset {\mathcal{A}}$ are independant. Therefore, any product $\{x_{i_1}\} \cdot \dotso \cdot \{x_{i_l}\} \neq 0$, and all the products are different in cohomology.
Main results {#sec:proof}
============
Now, everything is in place to prove the main results. The first theorem uses rational homotopy theory and gives some equivalent conditions to the fact that $M({\mathcal{A}})$ is rationally elliptic. With some linear algebra, the second theorem shows that the condition (3) has a geometric interpretation in term of the orthogonal subspaces $x_i^\perp$.
\[theo:a\] Let ${\mathcal{A}}$ be a subspace arrangement with a geometric lattice such that every $x \in {\mathcal{A}}$ has $\operatorname{codim}(x) \geq 2$. Then the following conditions are equivalent :
1. $M({\mathcal{A}})$ is rationally elliptic,
2. $M({\mathcal{A}})$ has the rational homotopy type of a product of odd dimensional spheres,
3. $\operatorname{codim}\cap_{x \in {\mathcal{A}}} x = \sum \operatorname{codim}x$,
4. $M({\mathcal{A}})$ has the homotopy type of a product of odd dimensional spheres,
5. $M({\mathcal{A}})$ has Poincaré duality.
*(1) implies (2).* Since $L({\mathcal{A}})$ is geometric, we know (see [@yu05]) that $M({\mathcal{A}})$ is a formal space. If $M({\mathcal{A}})$ is elliptic, we can apply theorem \[theo:fh\]. By definition of the differential, every $x \in {\mathcal{A}}$, $\{x\}$ is a generator in cohomology for the rational model described in section \[sec:rmsa\]. The degree of $[\{x\}]$ is $2 \operatorname{codim}x - 1$. Therefore, the $(\{x\})_{x \in {\mathcal{A}}}$ form a linearly independant sequence in $V_0^\text{odd}$. By theorem \[theo:fh\], we have an injective map $$\rho{\colon}\Lambda_{x \in {\mathcal{A}}} [\{x\}] \to H^\star(\Lambda V).$$ In particular, for each sequence $x_1, \dotsc, x_n$ in ${\mathcal{A}}$, with $
x_i \neq x_j$, we have $$\{x_1 \} \cdot \{x_2\} \cdot \dotsc \cdot \{x_n\} \neq 0$$ because their product is non zero in cohomology. Therefore, we have the following equality $\prod_{i=1}^n \{x_i\} = \pm \{x_1, x_2, \dotsc, x_n\}$ and $[\{x_1, \dotsc, x_n\}] \neq 0$ (in cohomology).
The map $\rho$ is surjective because, for each independant set $\{x_1, \dotsc, x_n\}$ (which generates $H^\star(M({\mathcal{A}}))$), we have $[\{x_1, \dotsc, x_n\}] = \pm \prod_{i=1}^n [\{x_i\}]$, which is in the image of $\rho$. It implies that the map $\rho$ is an isomorphism. By lemma \[lemm:muc\], $M({\mathcal{A}})$ is 1-connected. Therefore, $M({\mathcal{A}})$ has the rational homotopy type of a product of odd dimensional spheres.
*(2) implies (3).* We showed that the product $\prod_{x \in {\mathcal{A}}} \{x\} \neq 0$. By definition of the product, it implies that $\operatorname{codim}\cap_{x \in {\mathcal{A}}} x = \sum_{x \in {\mathcal{A}}} \operatorname{codim}x$.
*(3) implies (4).* Let ${\mathcal{A}}= \{x_1, \dotsc, x_n\}$ be a subspace arrangement in ${\mathbb{C}}^l$ such that $\operatorname{codim}\cap x_i = \sum \operatorname{codim}x_i$. The quotient map $p {\colon}{\mathbb{C}}^n \to {\mathbb{C}}^n/(\cap x_i)$ induces a homotopy equivalence $({\mathbb{C}}^n \setminus \cup x_i) \to (({\mathbb{C}}^n / \cap x_i) \setminus \cup (x_i/\cap x_i))$. Hence we can assume that $\cap x_i = 0$.
Let’s write $x_i = \ker(H_i {\colon}{\mathbb{C}}^n \to {\mathbb{C}}^{n_i})$. The map $$(H_1, H_2, \dotsc, H_n) {\colon}{\mathbb{C}}^n \to \prod {\mathbb{C}}^{n_i}$$ is an isomorphism, which induces an homotopy equivalence $${\mathbb{C}}^n \setminus \cup x_i \to \prod_{i=1}^n ({\mathbb{C}}^{n_i} \setminus \{0\}).$$ But the injective map $\prod(S^{2n_i-1}) \to \prod({\mathbb{C}}^{n_i} \setminus \{0\})$ is an homotopy equivalence. Therefore $M({\mathcal{A}}) = {\mathbb{C}}^n \setminus \cup x_i$ has the homotopy type of a product of odd dimensional spheres.
*(4) implies (1).* Obvious.
*(1) implies (5).* Obvious.
*(5) implies (2).* Direct consequence from proposition \[prop:pd\].
\[theo:b\] Let ${\mathcal{A}}= \{x_1, \dotsc, x_n\}$ be a subspace arrangement. Then the following conditions are equivalent :
1. $\operatorname{codim}\cap_{i=1}^n x_i = \sum_{i=1}^n \operatorname{codim}x_i$,
2. the sum $x_1^\perp + \dotso + x_n^\perp$ is a direct sum.
First, let’s prove by induction on $k$, $2 \leq k \leq n$, that : $$\left(\cap_{j=1}^k x_j\right)^\perp = \sum\nolimits_{i=1}^k x_i^\perp.$$ For $k=2$, it gives $(x_1 \cap x_2)^\perp = x_1^\perp + x_2^\perp$, which is a well-known fact. Now, let’s suppose that the formula is true until $k-1$. We have : $$\left(\cap_{j=1}^k x_j\right)^\perp = (\cap_{j=1}^{k-1} x_j \cap x_k)^\perp = (\cap_{j=1}^{k-1} x_j )^\perp + x_k^\perp.$$ Using the induction hypothesis concludes the proof. Now, we can prove the theorem : $$\begin{gathered}
x_1^\perp + \dotso + x_n^\perp \text{ is a direct sum} \iff \sum\nolimits_{i=1}^n \dim x_i^\perp = \dim\left(\sum\nolimits_{i=1}^n x_i^\perp\right) \\
\iff \sum\nolimits_{i=1}^n \operatorname{codim}x_i = \dim \left( \cap_{i=1}^n x_i \right)^\perp \iff \sum\nolimits_{i=1}^n \operatorname{codim}x_i = \operatorname{codim}\cap_{i=1}^n x_i. \qedhere\end{gathered}$$
[xx]{} <span style="font-variant:small-caps;">G. Bredon</span>, *Topology and Geometry*, Graduate texts in Mathematics, Springer Verlag, 1993. <span style="font-variant:small-caps;">Y. Félix, S. Halperin</span>, *Formal spaces with finite-dimensional rational homotopy*, Trans. Amer. Math. Soc. 270 (1982), no. 2, 575–588. <span style="font-variant:small-caps;">Y. Félix, S. Halperin, J.-C. Thomas</span>, *Rational Homotopy Theory*, Springer-Verlag, 2000. <span style="font-variant:small-caps;">D. Sullivan</span>, *Infinitesimal computations in topology*, Publ. IHES 47 (1977), 267–331. <span style="font-variant:small-caps;">S. Yuzvinsky</span>, *Small rational model of subspace complement*, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1921–1945. <span style="font-variant:small-caps;">E. Feichtner, S. Yuzvinsky</span>, *Formality of the complements of subspace arrangements with geometric lattices*, <arXiv:math.AT/0504321>, 2005.
[^1]: The author is an “Aspirant” of the “Fonds National pour la Recherche Scientifique” (FNRS), Belgium.
|
---
author:
- 'Damien F. Frost , David A. Howey [^1]'
bibliography:
- 'references-frost.bib'
title: High Speed Peltier Calorimeter for the Calibration of High Bandwidth Power Measurement Equipment
---
Calorimeter, Peltier, thermoelectric cooler, high bandwidth, power measurement.
ACKNOWLEDGMENT {#acknowledgment .unnumbered}
==============
The authors acknowledge the financial support provided by Newtons4th Ltd. (N4L), the Natural Sciences and Engineering Research Council of Canada (NSERC) and Jesus College Oxford. This work also benefited from equipment funded by the John Fell Oxford University Press (OUP) Research Fund.
[Damien F. Frost]{} received the B.A.Sc. and M.A.Sc. degrees from the University of Toronto, Canada in 2007 and 2009, respectively. Before starting his D.Phil (PhD) degree in the Energy and Power Group, Department of Engineering Science, University of Oxford, Oxford, U.K. in 2013, he worked in the solar industry as a co-founder and power electronics designer at ARDA Power. He is currently working towards his D.Phil at Oxford, where his research is focused on the application of power electronics and control theory to battery management systems.
[David A. Howey]{} (M10) received the B.A. and M.Eng. degrees from Cambridge University, Cambridge, U.K., in 2002 and the Ph.D. degree from Imperial College London, London, U.K., in 2010. [He is currently an Associate Professor with the Energy and Power Group, Department of Engineering Science, University of Oxford, Oxford, U.K. His research is focused primarily on energy storage systems, including projects on model-based battery management, degradation, thermal management, and energy management for grid storage.]{}
[^1]: D.A. Howey and D.F. Frost are based in the Energy and Power Group, Department of Engineering Science of the University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom. Emails: [{damien.frost, david.howey} at eng.ox.ac.uk]{}.
|
---
abstract: 'In this paper, we analyze the predictability of the ocean currents using deep learning. More specifically, we apply the Long Short Term Memory (LSTM) deep learning network to a data set collected by the National Oceanic and Atmospheric Administration (NOAA) in Massachusetts Bay between November 2002-February 2003. We show that the current speed in two horizontal directions, namely u and v, can be predicted using the LSTM. We discuss the effect of training data set on the prediction error and on the spectral properties of predictions. Depending on the temporal or the spatial resolution of the data, the prediction times and distances can vary, and in some cases, they can be very beneficial for the prediction of the ocean current parameters. Our results can find many important applications including but are not limited to predicting the tidal energy variation, controlling the current induced vibrations of marine structures and estimation of the wave blocking point by the chaotic oceanic current and circulation.'
author:
- Cihan Bayindir
title: Predicting the Ocean Currents using Deep Learning
---
\[sec:level1\]Introduction
==========================
Ocean currents and circulations are one of the very important phenomena observed in the ocean hydrodynamics and they have attracted a lot of attention throughout the historical development of ocean science [@Richardson; @Peterson]. These currents and circulations can be generated by many different mechanisms. These mechanisms include but are not limited to winds [@Munk; @Sverdrup47], tides [@Parker], waves [@Stommel], changes in underwater topography [@Tomczak] and buoyancy fluxes [@Stommel; @Wust; @Sverdrup42; @Siedler]. There are different ways of classifying the currents. According to their forcing mechanisms, they may be classified as wind driven or thermohaline. One other possible classification is according to the depth of occurrence, currents may be classified as the surface or the subsurface currents. Majority of the currents occurring on the ocean surface are wind driven, however thermohaline currents are driven by the heat and salt concentration differences. Sinking of denser saltier water at colder parts of the ocean drives subsurface thermohaline currents. Yet another possible classification is to classify the currents according to their occurrence locations, i.e. the currents flowing some distance away the shore can be classified as an offshore current and the ones closer to shore can be classified as the inshore current. From a time series analysis point of view, a more appropriate classification is to classify them as periodic or mean currents. The currents which have changing speed and directions cyclically at regular intervals can be classified as periodic currents. However mean circulations in the ocean have mean parts which experience no or relatively little changes in time and space.
Ocean currents and circulations have very important effects in the marine environment. They are one of the driving mechanisms of the mass and heat transfer on earth. They can heat up or cool down the oceans and can transport biological agents, nutrients and chemicals. They may lead to many engineering problems for the offshore engineering structures and marine travel, such as excessive vibrations, increase of transit times and accidents risks in strategic waterways, such as the Bosphorus [@Oguz; @Jarosz]. Additionally, they may lead to formation of rogue waves due to their wave blocking effect [@Bayindir; @BayPRE1; @BayPRE2; @BayPLA]. They are the main driving mechanism of the tidal energy converters [@Magagna; @Uihlein; @Lynn].
Due to the reasons, which some are summarized above, the prediction of ocean currents and circulation in the marine environment is a vital problem for the safety of the marine operations and structures. Various attempts are made for the prediction of the different ocean processes including the ocean currents and circulations. Various wave models and time series methods are utilized for the prediction of the ocean wave height and energy in [@Roulston; @Reikard; @Baysci]. The predictability of the near surface winds using Kalman filtering technique is discussed in [@Malmberg]. A transfer function based method was proposed in [@Ho] for the wave height forecast. The usage of artificial neural networks are discussed for the prediction of waves is studied in [@Londhe]. A genetic programming approach is used for the real time wave forecasting in [@Gaur]. More recently, the prediction of wave conditions using a machine learning approach is proposed in [@James_machinewave]. Again a machine learning approach is proposed in [@Jiang_ml_thermocline] for the prediction of thermocline parameters. A statistical machine learning approach is utilized in [@Hollinger] for the prediction of ocean processes. Spatio-temporal prediction of the ocean currents using Gaussian processes is discussed in [@Sarkar]. Another approach was to apply the deep learning to ocean data inference and subgrid parameterization [@Bolton].
In this study, we discuss and analyze the predictability of the ocean currents using a deep learning approach [@MacKay]. To be more specific, we apply the LSTM deep learning network to a ocean current data set collected by NOAA in Massachusetts Bay between November 2002-February 2003. We discuss the performance of the LSTM for the prediction of the ocean current speed data and discuss the root-mean-square (rms) error and spectral properties of predictions. We also investigate the effect of the length of the training data set on predictions. We discuss our findings and comment on our results.
\[sec:level2\]Methodology
=========================
Review of the Long Short Term Memory
------------------------------------
One of the most commonly used deep learning networks is the LSTM network, which was introduced in [@Hochreiter]. One of the possible uses of the LSTM is the prediction of the various time series data [@Hochreiter; @Greff]. In order to predict the values of the data at the future time steps, a sequence-to-sequence regression LSTM network can be trained. For the LSTM network, training sequences with one time step shifted values are used as the responses [@Hochreiter], thus the LSTM network learns to predict the values of the next time step at every time step of the training sequence [@Hochreiter]. The LSTM layer architecture is shown in Fig. \[figLSTM\_layer\].
![LSTM layer architecture [@Hochreiter].[]{data-label="figLSTM_layer"}](LSTMarcht){width="3.7in"}
In the LSTM layer architecture, the flow of a time series having D features through an LSTM layer is illustrated. In this architecture, h shows the output and c shows the cell state [@Hochreiter]. The learned information at the previous time steps is contained in the cell state. The information is either added or removed from the cell state at each time step of the input sequence. The gates shown in Fig. \[figLSTM\_gates\], controls these updates.
![LSTM gates [@Hochreiter].[]{data-label="figLSTM_gates"}](LSTMgates){width="3.7in"}
The letters i, f, g, o in this figure denote the input, forget gates, cell candidate and the output gate, respectively [@Hochreiter]. At time step t, the cell state is computed by $$c_t = f_t \otimes c_{t-1} + i_t \otimes g_{t}
\label{eq01}$$ In here, $\otimes$ is the Hadamard product (element wise multiplication) [@Hochreiter]. At time t, the hidden state is computed using $$h_t = o_t \otimes \sigma_c(c_t)
\label{eq02}$$ where $\sigma_c$ is the state activation function. Among different possibilities, the tanh function is used for the state activation function throughout this study. The functions are shown in Fig. \[figLSTM\_components\]
![LSTM components and formula [@Hochreiter].[]{data-label="figLSTM_components"}](LSTMcomponents){width="2.2in"}
In here W shows the input weight, R shows the recurrent weight and b shows the bias [@Hochreiter]. $\sigma_g$ shows the gate activation function. A sigmoid function, $\sigma(x)=(1+e^{-x})^{-1}$ is used as the gate activation function [@Hochreiter]. In this approach, the training data is standardized to have zero mean and unit variance for a better fit. Then, the LSTM layer is specified to have 200 hidden units. The number of training epochs is selected as 250 and a gradient threshold value of 1 is used to avoid the divergence of the gradients. The predictions are started with an initial learning rate of 0.005, and after 200 epochs, it is dropped to 0.004. Then the predicted time series is unstandardized using the parameters discussed earlier. Additionally, in LSTM network approach one can also update the network state with observed values instead of predictions by utilizing the time steps between predictions which generally results in better prediction performance. All of the LSTM network based predictions in this work are performed using the parameters discussed above. Our paper focuses on discussing the usage of the deep learning for the prediction of ocean currents and circulations. Thus, the reader is referred to [@Hochreiter; @Greff] for a more comprehensive discussion and the details of the LSTM network.
Properties and Review of the Ocean Current Data
===============================================
In this study, we use an experimental data set to test the performance of the LSTM deep learning network for the ocean current and circulation predictions. The data set is recorded by NOAA in North Atlantic in the Massachusetts Bay at the location $42.3773N, 70.7819W$. The experimental data was recorded using a VMCM type moored current meter between the dates $2002/10/24$ and $2003/02/12$. The seafloor depth at this specific location was $33.7m$ and the current meter was deployed at a depth of $23.5m$. The horizontal current velocities, namely u and v are measured at every 3 minutes 44 seconds during this experiment. The original data set and further information can be seen at NOAA’s website . The time series of the 2D horizontal current velocities, u and v are depicted in Fig. \[fig1\].
![Ocean current speed time series a) u (m/s) b) v (m/s). []{data-label="fig1"}](fig1.eps){width="4.7in"}
\[sec:level3\]Results and Discussion
====================================
In this section, we assess the performance of the LSTM deep learning network for the prediction of the ocean current speed data depicted in Fig. \[fig1\] in the preceding section.
![Observed and predicted time series of the first component of the current velocity (u) obtained using the initial 95 % as the training sequence.[]{data-label="fig2"}](fig2.eps){width="4.3in"}
![a) Comparisons between the observed and predicted time series of the first component of the current velocity (u) obtained using the initial 95 % as the training sequence, b) the rms error of predictions.[]{data-label="fig3"}](fig3.eps){width="4.3in"}
In Fig. \[fig2\], the initial part of the first component of the original current speed data (u) between the dates October 24-October 28, 2002 is shown. In this prediction, we use the initial 95 % of this part of the current speed (u) data set as the training sequence. The training data set (in blue) and the predicted time series (in red) are depicted in Fig. \[fig2\]. We compare the predicted and observed time series and depict the root-mean-square (rms) error in Fig. \[fig3\]. As one can realize from these figures, the LSTM deep learning network performs quite well for the prediction of the first component of ocean current speed, u. The predictions are better for the first few time steps. As depicted in Fig. \[fig3\], the absolute value of the rms error remains less than $0.05m/s$ for the data having an absolute peak value around $0.15m/s$. It is useful to note that, the sampling interval of this data set is 3 minutes 44 seconds. An early prediction time on the order of few time steps can be very beneficial for many practices in marine science such as for tidal energy harvesting and vibration control, just to name a few. Depending on the temporal resolution of the data and with the advance of algorithms, this prediction times can vary and may be substantially improved.
![Observed and predicted time series of the first component of the current velocity (u) obtained using the initial 95 % as the training sequence and using the updates.[]{data-label="fig4"}](fig4.eps){width="4.3in"}
![a) Comparisons between the observed and predicted time series of the first component of the current velocity (u) obtained using the initial 95 % as the training sequence and using the updates, b) the rms error of predictions.[]{data-label="fig5"}](fig5.eps){width="4.3in"}
Next, we turn our attention to discuss the effects of using the observed values instead of predicted ones which are used for updating the LSTM network. We plot the first component of the predicted ocean current speed (u) time series and the rms error for this case in Fig. \[fig4\] and Fig. \[fig5\]. The same data set depicted Fig. \[fig2\] is used and its initial 95 % is used as the training sequence, as before. Checking Fig. \[fig4\] and Fig. \[fig5\], it is possible to conclude that the LSTM prediction with updates leads to better predictions and rms error significantly reduces. For the same data set, the peak absolute value of the rms error becomes less than $0.02m/s$ for this case.
![Observed and predicted time series of the second component of the current velocity (v) obtained using the initial 95 % as the training sequence.[]{data-label="fig6"}](fig6.eps){width="4.3in"}
![a) Comparisons between the observed and predicted time series of the second component of the current velocity (v) obtained using the initial 95 % as the training sequence, b) the rms error of predictions.[]{data-label="fig7"}](fig7.eps){width="4.3in"}
Focusing on the second component of the horizontal current velocity, namely v, we plot the predicted v time series and the rms error LSTM in Fig. \[fig6\] and Fig. \[fig7\]. The training sequence is selected as the initial 95 % of this part of the v time series. Checking these figures it is possible to realize that LSTM performs worse compared to the predictions of u. This is due to the difference in the behavior of time series. While u has a periodic steady state behavior, v exhibits a tendency of increase and does not have a steady state behavior. This structure of the data set makes the LSTM network predictions poorer. The absolute value of the rms error is on the order of $0.05m/s$ for the data having an absolute peak value around $0.10m/s$.
In order to discuss the effects of updating the LSTM network with observed values, we repeat the same simulations for the case with updates and depict the related results in Fig. \[fig8\] and Fig. \[fig9\]. Comparing Fig. \[fig6\] and Fig. \[fig7\] with Fig. \[fig8\] and Fig. \[fig9\] respectively, it is possible to argue that using the observed values to update the LSTM network significantly improves the prediction performance. The peak absolute value of the rms error becomes less than $0.04m/s$ for this case. It is also useful to note that using the observed values as updates limits the prediction time to one time step. The value of time step is 3 min 44 seconds for the data set used, however, depending on the size of the sampling interval the prediction time can vary and can still be very beneficial. Additionally, it is possible to upsample or downsample the data and adaptive time stepping can be utilized to enhance the prediction time scales.
![Observed and predicted time series of the second component of the current velocity (v) obtained using the initial 95 % as the training sequence and using the updates.[]{data-label="fig8"}](fig8.eps){width="4.3in"}
![a) Comparisons between the observed and predicted time series of the second component of the current velocity (v) obtained using the initial 95 % as the training sequence and using the updates, b) the rms error of predictions.[]{data-label="fig9"}](fig9.eps){width="4.3in"}
![a) Comparisons between the observed and predicted time series of the first component of the current velocity (u) obtained using the initial 70 % as the training sequence, b) the rms error of predictions.[]{data-label="fig10"}](fig10.eps){width="4.3in"}
![Comparison of the Fourier spectra of the observed and predicted time series of the first component of the current velocity (u) obtained using the initial 70 % as the training sequence.[]{data-label="fig11"}](fig11.eps){width="4.3in"}
![a) Comparisons between the observed and predicted time series of the first component of the current velocity (u) obtained using the initial 70 % as the training sequence and using the updates, b) the rms error of predictions.[]{data-label="fig12"}](fig12.eps){width="4.3in"}
![Comparison of the Fourier spectra of the observed and predicted time series of the first component of the current velocity (u) obtained using the initial 70 % as the training sequence and using the updates.[]{data-label="fig13"}](fig13.eps){width="4.3in"}
Next, we investigate the effects of the size of the LSTM training data set and the spectral properties of predictions obtained by the LSTM network. With this motivation, we use the initial part of time series between the dates October 24-November 19, 2002. We select the initial % 70 of this part of the time series data as the training data set and predict the remaining part using the LSTM network. Results for the first component of the current speed data (u) with no updates are depicted in Fig. \[fig10\] and its spectrum is depicted in Fig. \[fig11\]. The results for u with updates are depicted in Fig. \[fig12\] and its spectrum is depicted in Fig. \[fig13\].
Checking Figs. \[fig10\]-\[fig13\], it is possible to argue that LSTM with no updates produces quite sufficient results for the prediction of the ocean current speed. When the observed values are used as updates, the results significantly improve and predictions becomes excellent for many marine operations and marine engineering purposes. The depicted Fourier spectra in Fig. \[fig12\] and Fig. \[fig13\] clearly show that the predicted time series with LSTM network has a higher peak frequency and its bandwidth is significantly limited compared to the observations, when no observations are used as updates. Again, when the observations are used as updates, the Fourier spectrum of the predicted time series by the LSTM network is in excellent agreement with the spectrum of the observations. Comparing Fig. \[fig1\] and Fig. \[fig10\], it is also possible to state that when the training sequence is longer the prediction performance of the LSTM network significantly improves, as expected.
![a) Comparisons between the observed and predicted time series of the second component of the current velocity (v) obtained using the initial 70 % as the training sequence, b) the rms error of predictions.[]{data-label="fig14"}](fig14.eps){width="4.3in"}
![a) Comparisons between the observed and predicted time series of the second component of the current velocity (v) obtained using the initial 70 % as the training sequence and using the updates, b) the rms error of predictions.[]{data-label="fig15"}](fig15.eps){width="4.3in"}
![Comparison of the Fourier spectra of the observed and predicted time series of the second component of the current velocity (v) obtained using the initial 70 % as the training sequence and using the updates.[]{data-label="fig16"}](fig16.eps){width="4.3in"}
Lastly, we turn our attention again to the to the second component of the current speed time series, namely v. Again, we use the initial part of v time series between the dates October 24-November 19, 2002 and as before we select the initial % 70 of this part of the data set as our training sequence. The predicted v time series with no updates is depicted in Fig. \[fig14\] and the predicted v time series with updates is depicted in Fig. \[fig15\]. Checking these figures, it is possible to argue that the prediction with no updates is poor and can not represent the data after few time steps. This mismatch is a result of using a shorter training sequence as well as the unsteady nature of the v time series. However, when the observed values are used as updates in the LSTM network, the predicted time series and its spectrum become excellent for many possible engineering uses as depicted in Figs. \[fig15\] and \[fig16\].
\[sec:level1\]Conclusion and Future Work
========================================
In this paper, we have discussed the predictability of the ocean current speed time series by deep learning. Specifically, we have investigated the applicability of the LSTM deep learning network to the ocean current speed time series and discussed its prediction performance. The experimental data set used in our study was collected by NOAA in Massachusetts Bay between the dates November 2002-February 2003. Implementing the LSTM network based deep learning approach on this data set, we have analyzed the rms error and the Fourier spectra of the predicted time series. We have showed that, depending on the steady state behavior of the predicted time series, the LSTM network with no updates can be very successful in the prediction of the ocean current speed data, and may allow for accurate predictions at least a few time steps in advance. We showed that the LSTM network predictions with no updates have a higher peak frequency compared to the observed spectra. However, when the observed values are used as updates in the LSTM network, the prediction accuracy significantly develops. We have also showed that longer training sequences result in better prediction accuracy of the ocean current time series. Depending on the sampling interval of the data, the prediction times can significantly vary. Thus, LSTM network based deep learning approach proposed in this paper can be used to predict the ocean current time series data with very beneficial prediction times. Our results can be generalized for the prediction of other oceanic and atmospheric time series data including but are not limited to Gulf Stream, North Equatorial, Agulhas currents and East Wind drift. Possible usage areas include but are not limited to predicting the tidal energy variation, controlling the current induced vibrations of marine structures and estimation of the wave blocking point by the chaotic oceanic current and circulation.
[00]{}
P. Richardson, Benjamin Franklin and Timothy Folger’s first printed chart of the Gulf Stream, Science, 207, 643, (1980).
R. G. Peterson and L. Stramma and G. Kortum, Early concepts and charts of ocean circulation, Prog. Oceanogr., 37, 1, (1996).
W.H. Munk, On the wind driven ocean circulation, J. Meteor., 7, 79 (1950).
H. U. Sverdrup, Wind Driven Currents in a Baroclinic Ocean; With Application to the Equatorial Currents of the Eastern Pacific, Proc. Natl. Acad. Sci. USA, 33, 318, (1947).
B. B. Parker, Tidal Hydrodynamics, John Wiley and Sons, New York, (1991).
H. Stommel, A survey of ocean current theory, Deep Sea Res., 4, 149, (1957).
M. Tomczak and J.S. Godfrey, Regional Oceanography: an Introduction, Pergamon Press, Oxford, 1994.
G. Wust, Schichtung und Zirkulation des Atlantisches Ozeans. Die Stratosphare, Wissenschaftliche Ergebnisse der Deutschen Atlantischen Expedition auf dem Forschungs-und Vermessungsschiff, Meteor. 1925–1927, 6, 109 (1935).
H. U. Sverdrup and M.W. Johnson and R.H. Fleming, The Oceans: Their Physics, Chemistry and General Biology, Prentice Hall, New Jersey, 1942.
G. Siedler and J. Church and J. Gould, Ocean Circulation and Climate: Observing and Modelling the Global Ocean, Academic Press, New York, 2001.
T. O[ğ]{}uz and E. [Ö]{}zsoy and M. A. Latif and H. I. Sur and [Ü]{}. [Ü]{}nlüata, Modeling of hydraulically controlled exchange flow in the Bosphorus Strait, J. Phys. Oceanogr., 20, 945, (1990).
E. Jarosz and W. J. Teague and J. W. Book and [Ş]{}. Be[ş]{}iktepe, On flow variability in the Bosphorus Strait, J. Geophys. Res., 116, C08038, (2011).
C. Bay[i]{}nd[i]{}r, Shapes and statistics of the rogue waves generated by chaotic ocean current, Proc. of the 26th ISOPE, (2016).
C. Bay[i]{}nd[i]{}r, Rogue waves of the Kundu-Eckhaus equation in a chaotic wavefield, Phys. Rev. E., 93, 032201, (2016).
C. Bay[i]{}nd[i]{}r, Rogue wave spectra of the Kundu-Eckhaus equation, Phys. Rev. E., 93, 062215, (2016).
C. Bay[i]{}nd[i]{}r, Early detection of rogue waves by the wavelet transforms, Phys. Lett. A, 380, 156, (2016).
D. Magagna and A. Uihlein, Ocean energy development in Europe: current status and future perspectives, Int. J. Mar. Energy, 11, 84, (2015).
A. Uihlein and D. Magagna, Wave and tidal current energy – A review of the current state of research beyond technology, Renew. Sust. Energ. Rev., 58, 1070, (2016).
P.A. Lynn, Electricity from wave and tide: an introduction to marine energy, John Wiley and Sons, Chichester, UK, 2013.
M.S. Roulston and J. Ellepola and J. von Hardenberg and L.A. Smith, Forecasting wave height probabilities with numerical weather prediction models, Ocean. Eng., 32, 1841, (2005).
G. Reikard and P. Pinson and J-R. Bidlot, Forecasting ocean wave energy: the ECMWF wave model and time series methods, Ocean Eng., 38, 1089, (2011).
C. Bay[i]{}nd[i]{}r, Compressive spectral method for simulation of the nonlinear gravity waves, Scien. Rep., 22100 (2016).
A. Malmberg and U. Holst and J. Holst, Forecasting near-surface ocean winds with Kalman filter techniques, Ocean Eng., 32, 273, (2005).
P.C. Ho and J. Z. Yim, Wave height forecasting by the transfer function model, Ocean Eng., 33, 1230, (2006).
S.N. Londhe and V. Panchang, One-day wave forecasts based on artificial neural networks, J. Atmos. Ocean. Technol., 23, 1593, (2006).
S. Gaur and M.C. Deo, Real-time wave forecasting using genetic programming, Ocean. Eng., 35, 1166, (2008).
S. C. James and Y. Zhang and F. O’Donncha, A machine learning framework to forecast wave conditions, Coast. Eng., 137 , 1, (2018).
Y. Jiang and Y. Gou and T. Zhang and K. Wang and C. Hu, A Machine Learning Approach to Argo Data Analysis in a Thermocline, Sensors, 17, 2225, (2017).
G. Hollinger and A. Pereira and V. Ortenzi and G. Sukhatme, Towards improved prediction of ocean processes using statistical machine learning, Proc. Robotics: Science and Systems Workshop on Robotics for Environmental Monitoring (RSS), Sydney, Australia, (2012).
D. Sarkar and M. A. Osborne and T. A. Adcock, Spatio‐temporal prediction of tidal currents using Gaussian processes, J. Geophy. Res.: Oceans, (2019). doi:10.1029/2018jc014471.
T. Bolton and L. Zanna, Applications of Deep Learning to Ocean Data Inference and Subgrid Parameterization, J. Adv. Model. Earth Sys., 11, 376, (2019).
D. J. C. MacKay, Information theory, inference and learning algorithms, Cambridge University Press, Cambridge, UK , 2003.
S. Hochreiter and J. Schmidhuber, Long Short-Term Memory, Neur. Comp., 9, 1735, (1997).
K. Greff and R. K. Srivastava and J. Koutnik and B. R. Steunebrink and J. Schmidhuber, LSTM: A search space odyssey, IEEE Trans. on Neur. Netw. and Learn. Sys., 28, 2222, (2015).
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---
abstract: 'We explore the diversity of warped metric function in five-dimensional gravity including a scalar field and a $3$-brane. We point out that the form of the function is determined by a parameter introduced here. For a particular value of the parameter, the warped metric function is smooth without having a singularity, and we show that the bulk cosmological constant have a upper bound and must be positive and that the lower bound of five-dimensional fundamental scale is controlled by both the brane tension and four-dimensional effective Planck scale. The general warp factor obtained here may relate to models inspired by SUGRA or M-theory.'
---
Ł ł ł ¶ ø
DPNU-01-24\
hep-th/0109040
[ **Five-Dimensional Warped Geometry with a Bulk Scalar Field**]{}
Masato ITO [^1]
[ *Department of Physics, Nagoya University, Nagoya, JAPAN 464-8602* ]{}
Introduction
============
Randall and Sundrum proposed the setup of $3$-branes embedded in the five-dimensional theory with warped metric and discussed an alternative explanation of the hierarchy problem in the form of a fine-tuning between the negative bulk cosmological constant and brane tension [@Randall:1999ee; @Randall:1999vf]. The warped metric function in the model is an exponential scaling of the metric along the fifth dimension compactified on $S^{1}/Z_{2}$ orbifold. Originally, the introduction of warped metric have been made by Rabakov and Shaposhnikov who discussed the cosmological constant problem in six-dimensional theory with warped metric containing a singularity [@Rubakov:1983bz]. Recently, the extensions of the Randall-Sundrum scenario are widely made [@Collins:2001ni; @Collins:2001ed; @Ito:2001fd], in particular, several brane world scenarios with bulk scalar field are investigated [@Collins:2001ni; @Kachru:2000xs; @Kachru:2000hf; @Csaki:2000wz; @Binetruy:2000wn; @Carroll:2001zy]. Moreover, it is expected that this setup may connect with the AdS/CFT correspondence or string theory [@Kachru:2000xs; @Kachru:2000hf; @Nojiri:2000eb; @Anchordoqui:2001qc; @Anchordoqui:2000du; @Gherghetta:2001iv].
In this paper, we consider an alternative extension of warped metric function in the framework of five-dimensional gravity with a scalar field $\P$. In ref. [@Gherghetta:2001iv], it was shown that Randall-Sundrum brane-worlds arise form extremal D-brane configuration. Moreover, from a field theory point of view these brane-worlds consist of a warped geometry with a bulk scalar field, and the dual theories turn out to be non-conformal. To study the connection between RS model and SUGRA theories, it is important to investigate various types of warp factor in the setup with a bulk scalar fields. The metric taken here corresponds to the most general metric appearing to SUGRA theories. We show that the form of the warped metric is controlled by a parameter introduced here and investigate the behavior of the metric for the general value as well as particular value of the parameter.
This paper is organized as follows. In section 2, we describe the setup and derive the warped metric function and the jump conditions due to the existence of a brane. Moreover, we compute the four dimensional effective Planck scale by integrating out a fifth dimension. In section 3, a conclusion is described.
The Setup
=========
The physics of this model is governed by the following action $$\begin{aligned}
S&=&\int d^{5}x\;\sqrt{-G}
\left\{
\frac{1}{2\k^{2}_{5}}\R
-\frac{1}{2}\left(\nabla\phi\right)^{2}-\L
\right\}
+\int d^{4}x\;\sqrt{-g}\;
\left\{\;-f(\P)\;\right\}\,,
\label{eq1}
\end{aligned}$$ where $\cal R$ and $\L$ is the curvature and the cosmological constant in the bulk, respectively. Here $1/\kappa^{2}_{5}=M^{3}_{\ast}$ where $M_{\ast}$ is the fundamental scale of five-dimensional theory, and $G$ is the five-dimensional metric and $g$ is the induced four-dimensional metric on the brane which is located at $y=0$, where $y$ is the coordinate of fifth dimension. In Eq.(\[eq1\]), the second term denotes the brane tension with dependence of a scalar field. We take the following metric $$\begin{aligned}
ds^{2}&=&e^{2A(y)}g_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2B(y)}dy^{2}
\nonumber\\
&\equiv&G_{MN}dx^{M}dx^{N}\label{eq2}\,,
\end{aligned}$$ where $M,N=0,\cdots,3, 5$ and $g_{\mu\nu}=diag(-,+,+,+)$. We shall use the notation $\{x^{\mu}\}$ with $\mu=0,\cdots,3$ for the coordinates on the four-dimensional spacetime, and $x^{5}=y$ for the fifth coordinate on an extra dimension. Note that the appropriate change of $y$-coordinate can lead to the metric of the original Randall-Sundrum model, however, the original metric is not inspired by SUGRA theories. Therefore we can take most general metric Eq.(\[eq2\]) appearing in SUGRA theories.
Using the metric, Einstein equations are given by $$\begin{aligned}
\R_{MN}-\frac{1}{2}G_{MN}\R&=&
\k^{2}_{5}\left[\;\p_{M}\phi\p_{N}\phi
-G_{MN}\left\{
\frac{1}{2}\left(\nabla\phi\right)^{2} +\L
\right\}\;\right.\nonumber\\
&&\left.\hspace{1cm}
-\frac{\sqrt{-g}}{\sqrt{-G}}g_{\mu\nu}\d^{\mu}_{M}\d^{\nu}_{N}f(\P)\d(y)
\;\right]\,.
\label{eq3}
\end{aligned}$$ While, the equation of motion with respect to a scalar field $\P$ with $y$-dependence is given by $$\begin{aligned}
\p_{M}\left(\sqrt{-G}G^{MN}\p_{N}\P\right)=
\sqrt{-g}\;\frac{\p f}{\p\phi}\d(y)
\label{eq4}\,.
\end{aligned}$$ With the metric ansatz, these equations can be written as $$\begin{aligned}
&&\App + 2\left(\Ap\right)^{2}-\Ap\Bp=
-\frac{\k^{2}_{5}}{3}\left\{
\frac{1}{2}(\Pp)^{2}+e^{2B}\L\right\}
-\frac{\k^{2}_{5}}{3}e^{B}f(\P)\d (y)
\label{eq5}\,,\\
&&\left(\Ap\right)^{2}=
\frac{1}{12}\k^{2}_{5}\left(\Pp\right)^{2}-\frac{\k^{2}_{5}}{6}e^{2B}\L
\label{eq6}\,,\\
&&
4\Ap\Pp-\Bp\Pp+\Ppp
=e^{B}\frac{\p f}{\p\P}\d(y)\,,\label{eq7}
\end{aligned}$$ where the prime represents the derivative with respect to the $y$.
We take the simple ansatz: $$\begin{aligned}
B=\alpha A \,,\label{eq8}
\end{aligned}$$ where $\alpha$ is a parameter at this stage. Below, we solve the equation of motion in the bulk and study the warped metric function for arbitrary $\alpha$. Integrating out the equation in the bulk of Eq.(\[eq7\]), we have $$\begin{aligned}
\Pp=c\; e^{(\alpha-4)A}\,.\label{eq9}
\end{aligned}$$ Hence $c$ is the integration constant and it has mass dimension $[5/2]$. Substituting the above equation into Eq.(\[eq6\]), the equation is expressed as $$\begin{aligned}
\Ap=\e\frac{\sqrt{3}}{6}\k_{5}\left|c\right|
e^{(\alpha-4)A}\sqrt{1-\frac{2\L}{c^{2}}e^{8A}}\,,
\label{eq10}
\end{aligned}$$ where $\e=\pm$, and the selection of the sign $\e$ determines the branch of the square root. Note that this solution make sense when the argument of the square root in Eq.(\[eq10\]) is positive.
For $\alpha\neq 4,12$, this equation can be obtained by using hypergeometric function as follows $$\begin{aligned}
e^{(4-\alpha)A}\;
{}_{2}F_{1}\left(\;\frac{1}{2}\,,\frac{4-\alpha}{8}\,;
\frac{12-\alpha}{8}\,;\,\frac{2\L}{c^{2}}e^{8A}\;\right)
=\e\frac{\sqrt{3}}{6}\k_{5}\left|c\right|(4-\alpha)
\left(y+d\right)\,,
\label{eq11}
\end{aligned}$$ where $d$ is the integration constant and $e^{A(0)}$ can be normalized to be unity. Note that the above equation is solution to be consistent with Eq.(\[eq5\]), consequently. In the case of $\alpha=4,12$, we cannot use the integral representation of hypergeometric function due to vanishing of second or third argument. Later, we describe the case of $\alpha=4$. For $4<\alpha<12$, since the second argument in the hypergeometric function is negative, it becomes positive by using the transformation formulas of the function. In the case of $\alpha>12$, both the second and the third argument become negative. The simplest warped metric of $\alpha=0$ have been already investigated in Ref. [@Collins:2001ni; @Collins:2001ed]. Note that the value of $\alpha$ determines the form of the metric function $A(y)$. Furthermore, the solution of Eq.(\[eq11\]) is well-defined on one side of $y=-d$ due to ${}_{2}F_{1}\geq 0$ ( depending on the sign of $\e(4-\alpha)$ ) [@Kachru:2000xs; @Kachru:2000hf].
The existence of a brane at $y=0$ leads to the jump conditions with respect to the first derivative of $A$ and $\P$. From Eqs.(\[eq5\]) and (\[eq7\]), the jump conditions are $$\begin{aligned}
\Ap(0+)-\Ap(0-)&=&-\frac{1}{3}\k^{2}_{5}e^{\alpha A(0)}f(\P(0))\,,
\nonumber\\
\Pp(0+)-\Pp(0-)&=&e^{\alpha A(0)}\frac{\p f}{\p \P}(\P(0))\,.
\label{eq12}
\end{aligned}$$ We denote the sign and the integration constants for the positive region $(y>0)$ by $\e_{+}\,,c_{+}\,,d_{+}$ and those for the negative region $(y<0)$ by $\e_{-}\,,c_{-}\,,d_{-}$. Imposing the fact with $A(0)=0$, the jump conditions yield $$\begin{aligned}
\e_{+}\frac{1}{\td_{+}}-\e_{-}\frac{1}{\td_{-}}
&=&\frac{\p f}{\p\P}(\P(0))\,,\nonumber\\
\e_{+}\sqrt{\frac{1}{\td^{2}_{+}}-2\L}-
\e_{-}\sqrt{\frac{1}{\td^{2}_{-}}-2\L}&=&
-\frac{2}{\sqrt{3}}\k_{5}f(\P(0))\,,\label{eq13}
\end{aligned}$$ where $$\begin{aligned}
\frac{1}{\td_{\pm}}=\frac{6}{\sqrt{3}\k_{5}(4-\alpha)}\;
{}_{2}F_{1}\left(\;\frac{1}{2}\,,\frac{4-\alpha}{8}\,;
\frac{12-\alpha}{8}\,;\,\frac{2\L}{c^{2}_{\pm}}\;\right)
\frac{1}{d_{\pm}}\,.\label{eq14}
\end{aligned}$$ Furthermore, if the fifth dimension is assumed to have certain symmetry, the sign $\e_{\pm}$ and the integration constants $c_{\pm},d_{\pm}$ are related each other.
If this setup arises from string theory, it is natural that a scalar field $\P$ in the bulk is regarded as dilaton field. Namely, in the framework of perturbative string theory, $f(\P)$ corresponds to the tree-level dilaton coupling with dependence of exponential type $( f(\P)\propto e^{a\P})$. Thus, the jump conditions of Eq.(\[eq13\]) gives the information on the general form of dilation coupling [@Kachru:2000xs]. As for this point, we discuss elsewhere.
For $\alpha=4$, we can directly solve the differential equation in Eq.(\[eq10\]), the warped function with $Z_{2}$ symmetry is given by $$\begin{aligned}
e^{8A_{\pm}(y)}=
\frac{c^{2}}{2\L}\left\{
1-\tanh^{2}\frac{2}{\sqrt{3}}\k_{5}|c|(y\pm d)
\right\}\,,\label{eq15}
\end{aligned}$$ where plus ( minus ) corresponds to the function for $y>0$ ( $y<0$ ) region. Note that this warped metric function is smooth without having a singularity. The normalization condition $A(0)=0$ gives the constraint on the integration constants $$\begin{aligned}
\tanh^{2}\frac{2}{\sqrt{3}}\k_{5}|c|d=1-\frac{2\L}{c^{2}}\,,\label{eq16}
\end{aligned}$$ therefore, the allowed range of the bulk cosmological constant is $$\begin{aligned}
0<\L\leq \frac{c^{2}}{2}\,,\label{eq17}
\end{aligned}$$ where we assume $\L\neq 0$. From Eq.(\[eq12\]), the jump condition of $\Ap$ yields the relation between the bulk cosmological constant and the brane tension $$\begin{aligned}
V=\frac{\sqrt{3(c^{2}-2\L})}{\k_{5}}\,,\label{eq18}
\end{aligned}$$ where $f(\P(0))=V$.
It assumes that the fifth dimension is noncompact. Integrating out $y$ in the action Eq.(\[eq1\]), the resulting four dimensional effective Planck scale $M_{\rm Pl}$ is finite $$\begin{aligned}
M^{2}_{\rm Pl}
=\frac{1}{\k^{2}_{5}}\int^{\infty}_{-\infty}\;dy\;
\left. e^{(2+\alpha) A}\right|_{\alpha=4}=
\frac{\sqrt{3}}{2\k^{3}_{5}|c|}\;
{}_{2}F_{1}\left(\;\frac{3}{4}\,,\frac{1}{2}\,;\frac{7}{4}\,;
\frac{2\L}{c^{2}}\;\right)\,.\label{eq19}
\end{aligned}$$ Taking account for the range of $\L$ in Eq.(\[eq17\]), since $1<{}_{2}F_{1}(\frac{3}{4},\frac{1}{2};\frac{7}{4};z)\leq
\frac{\Gamma(\frac{1}{2})\Gamma(\frac{7}{4})}{\Gamma(\frac{5}{4})}
\sim 1.79721$ for $0<z\leq 1$, we obtain $$\begin{aligned}
M^{2}_{\rm Pl}\lsim \frac{M^{9/2}_{\ast}}{|c|}\,,\label{eq20}
\end{aligned}$$ where we used $1/\k^{2}_{5}=M^{3}_{\ast}$. From Eqs.(\[eq18\]) and (\[eq20\]), the integration constant can be eliminated, and the bulk cosmological constant is expressed as $$\begin{aligned}
2\L\lsim \frac{M^{9}_{\ast}}{M^{4}_{\rm Pl}}-\frac{V^{2}}{3M^{3}_{\ast}}
\label{eq21}\,,
\end{aligned}$$ imposing the fact that $\L$ is positive, we have $$\begin{aligned}
M_{\ast}>V^{\frac{1}{6}}M^{\frac{1}{3}}_{\rm Pl}
\label{eq22}\,.
\end{aligned}$$ Thus, the lower bound of the five dimensional fundamental scale is controlled by both the brane tension and four dimensional effective Planck scale. For instance, if the value of brane tension $V$ is TeV scale, we then obtain $$\begin{aligned}
M_{\ast}> 10^{8}\;{\rm GeV}\,.
\end{aligned}$$ Consequently, the lower bound of $M_{\ast}$ decreases as the value of the brane tension decreases. Especially, in the case of vanishing brane tension ( $V=0$ ), if we take $M_{\ast}\sim 1\;{\rm TeV}$, the inequality Eq.(\[eq21\]) becomes $$\begin{aligned}
\L\lsim \left(\;1\;{\rm eV}\;\right)^{5}\,,\label{eq23}
\end{aligned}$$ therefore, Eq.(\[eq18\]) leads that $c^2\lsim 2({\rm eV})^{5}$.
As for the value of $\alpha=12$, solving Eq.(\[eq10\]), since the exact form of the warped metric function is complicated, we are going to investigate it in next paper [@Ito:2001gk]. Furthermore, whether the effective four-dimensional Planck scale is finite depends on the value of $\alpha$ [@Ito:2001gk].
Conclusion
==========
Under the conjecture that Randall-Sundrum model arises from the underlying theories (string or M-theory), we showed that warp factors has various types in the setup with a bulk scalar field. Namely, we explore the diversity of the five-dimensional warped geometry, where the warped metric function with $y$-dependence of four-dimensional spacetime metric part and the one of fifth dimension metric part are tied through a parameter $\alpha$. The metric taken here corresponds to the metric appearing in SUGRA theories to be low energy effective theories of D-brane configurations. For arbitrary $\alpha$, the form of the warped metric function is implicitly described in terms of $y$ by using a hypergeometric function. Taking a specific value of $\alpha=4$, the warped metric function without having a singularity can be explicitly obtained, and we pointed out that the resulting four-dimensional effective Planck scale is finite and that the lower bound of five dimensional fundamental scale is determined by the brane tension. Furthermore, there exist the allowed bound on the bulk cosmological constant. Although we concretely investigate a simple case of $\alpha=4$ as mentioned above, it is necessary to explore the behavior of the warped metric function for various $\alpha$ [@Ito:2001gk]. However, it is unknown whether the value of $\alpha$ corresponds to concrete types of D-brane configuration. As for this point, we are going to describe elsewhere in future. As preliminary study of connection between RS model and SUGRA theories, the general results obtained here are important.
[99]{}
L. Randall and R. Sundrum, “A large mass hierarchy from a small extra dimension,” Phys. Rev. Lett. [**83**]{}, 3370 (1999), hep-ph/9905221.
L. Randall and R. Sundrum, “An alternative to compactification,” Phys. Rev. Lett. [**83**]{}, 4690 (1999), hep-th/9906064.
V. A. Rubakov and M. E. Shaposhnikov, “Extra Space-Time Dimensions: Towards A Solution To The Cosmological Constant Problem,” Phys. Lett. B [**125**]{}, 139 (1983).
H. Collins and B. Holdom, “The Randall-Sundrum scenario with an extra warped dimension,” Phys. Rev. D [**64**]{}, 064003 (2001), hep-ph/0103103.
H. Collins and B. Holdom, “The cosmological constant and warped extra dimensions,” Phys. Rev. D [**63**]{}, 084020 (2001), hep-th/0009127.
M. Ito, “Warped geometry in higher dimensions with an orbifold extra dimension,” hep-th/0105186.
S. Kachru, M. Schulz and E. Silverstein, “Bounds on curved domain walls in 5d gravity,” Phys. Rev. D [**62**]{}, 085003 (2000), hep-th/0002121.
S. Kachru, M. Schulz and E. Silverstein, “Self-tuning flat domain walls in 5d gravity and string theory,” Phys. Rev. D [**62**]{}, 045021 (2000), hep-th/0001206.
C. Csaki, J. Erlich, C. Grojean and T. Hollowood, “General properties of the self-tuning domain wall approach to the cosmological constant problem,” Nucl. Phys. B [**584**]{}, 359 (2000), hep-th/0004133.
P. Binetruy, J. M. Cline and C. Grojean, “Dynamical instability of brane solutions with a self-tuning cosmological constant,” Phys. Lett. B [**489**]{}, 403 (2000), hep-th/0007029.
S. M. Carroll and L. Mersini, “Can we live in a self-tuning universe?,”, hep-th/0105007.
S. Nojiri, S. D. Odintsov and S. Zerbini, “Quantum (in)stability of dilatonic AdS backgrounds and holographic renormalization group with gravity,” Phys. Rev. D [**62**]{}, 064006 (2000), hep-th/0001192.
L. Anchordoqui, J. Edelstein, C. Nunez, S. Perez Bergliaffa, M. Schvellinger,M. Trobo and F. Zyserman, “Brane worlds, string cosmology, and AdS/CFT,”, hep-th/0106127.
L. Anchordoqui, C. Nunez and K. Olsen, “Quantum cosmology and AdS/CFT,” JHEP [**0010**]{}, 050 (2000), hep-th/0007064.
T. Gherghetta and Y. Oz, “Supergravity, non-conformal field theories and brane-worlds,”, hep-th/0106255.
M. Ito, “Various types of five dimensional warp factor and effective Planck scale,”,hep-th/0109140, to be published in PLB.
[^1]: E-mail address: mito@eken.phys.nagoya-u.ac.jp
|
[**HP Lyr – possibly the hottest RV Tau type object**]{}\
[**Dariusz Graczyk$\,^{1,4}\!\!$, Maciej Miko[ł]{}ajewski$\,^1\!$, Laurits Leedj[ä]{}rv$\,^2\!$, Sylwia M. Frackowiak$\,^1\!$, Jȩdrzej P. Osiwa[ł]{}a$\,^1\!$, Alar Puss$^{2,3}\!\!$, and Toma Tomov$\,^1\!$**]{}\
[$^{1}$Centre for Astronomy, Nicolaus Copernicus University, Gagarina 11, 87-100 Toru[ń]{}, Poland\
e-mail: (DG) weganin@astri.uni.torun.pl, (MM) mamiko@astri.uni.torun.pl\
$^{2}$Tartu Observatory, 61602 T[õ]{}ravere, Estonia, e-mail: leed@aai.ee\
$^{3}$Department of Physics, Tartu University, 4 T[ä]{}he Street, 51010 Tartu, Estonia\
$^{4}$Institute of Astronomy, Lubuska 2, 65-265 Zielona G[ó]{}ra, Poland]{}\
ABSTRACT\
[We report Johnson’s $UBVRI$ photometric and optical spectroscopic observations of a long period variable HP Lyr which up to now has been considered an eclipsing binary with a period of 140 days. The spectral type changes continually from A2-3 at maxima to A7-F2 at minima. We propose that the brightness changes are caused by the pulsation of the star with two periods: $P_1=69.35$, and $P_2=2\times P_1=138.7$ days. These periods decreased by more than 1% between 1960 and 1980. The spectral luminosity class determination gives an A type supergiant Iab. HP Lyr is also the optical counterpart of the infrared source IRAS 19199+3950. Relatively high galactic latitude ($b=+11^{o}\!\!.7$) and high radial velocity ($-$113 km/s) indicate that HP Lyr is an evolved, most likely post-AGB star. All presented features argue that this star is an RV Tau type object.\
[**Key words:**]{} binaries: eclipsing – variable: RV Tau stars – individual: HP Lyr]{}\
Introduction
============
The star HP Lyr is a poorly known long-period variable discovered by Morgenroth (1935). The photographic photometry showed the light curve of a semiregular type with the amplitude of brightness variations $\Delta m_{pg}=0\fm 5$ and the period of 70.4 days (Sandig 1939). The brightness at maximum was 105 in $m_{pg}$ and no secular changes of it were reported. Wenzel (1960) determined the spectral type of the variable as A6 and proposed that HP Lyr be an eclipsing binary consisting of a pair of A6 stars orbiting in a circular orbit with the period of $\sim$ 140 days. He classified the light curve as a $\beta$ Lyrae type with both minima of similar depth.
Observations
============
HP Lyr was one of the targets in our project of $U\!BV\!RI$ photometric monitoring of long period binaries in Piwnice Observatory of Nicolaus Copernicus University in Toru[ń]{} (Poland). The photometry was carried out during 1998 and 1999 using a one-channel photometer on the 60-cm Cassegrain telescope equipped with EMI9558B photomultiplier. As a comparison star, we chose a very close to HP Lyr A0V star HD 182592 ($V$=8.01, $U-B$=0.06, $B-V$=0.07, $V-R$=0.07, $R-I$=0.01). The effective wavelengths of our instrumental $ri$ bands were significantly shorter than those of Johnson’s system: 6390 Å$\,$ and 7420 Å, respectively. Nevertheless, for a single star we can use the formal transformation, found by observations of Johnson’s standards: $(R-I)=1.40(r-i)$, and $R=r-0.37(R-I)$. Our original data are presented on Fig. \[LiCu\]. The mean observational errors were 0.04, 0.03, 0.01, 0.01, 0.03 in particular $UBVri$ bands, respectively.
-------------------------------------------------------------------------------------- ------------ ------- ------- ------------------ --------------- ------- ------------------- ------------------ ------- -------
Date JD Phase Exp. $\lambda\lambda$ Disp. Spec. $T_{eff}$ Rad. vel. N[^1] O[^2]
(2450000+) (min) (nm) (Å$\!/\!$pix) class (K) (km/s)
26/27.08.00 1783.375 0.693 30 375-480 2.0 A4 8500 -81 $\!\pm\!$ 8 3 T
05/06.09.00 1793.375 0.765 40 “ & ” A5 8200 -112 $\!\pm\!$ 15 3 T
09/10.09.00 1797.374 0.794 30 “ & ” A5 8200 -102 $\!\pm\!$ 10 3 T
17/18.09.00 1805.377 0.851 60 625-665 0.8 – -122 $\!\pm\!$ 2 14 T
19/20.10.00 1837.258 0.081 60 “ & ” – -135 $\!\pm\!$ 4 11 T
" & 1837.348& 0.082 & 50 & 375-480 & 2.0& A7& 7700&-103 $\!\pm\!$ 12 & 3&T\
27/28.11.00 & 1876.189& 0.362 & 60 & 625-665 & 0.8& &–& -123 $\!\pm\!$ 3 &13&T\
02/03.05.01 & 2032.433& 0.489 & 20 & 350-550 & 2.0& F2& 7000& -88 $\!\pm\!$ 8 & 5&P\
08/09.05.01 & 2038.388& 0.529 & 60 & 625-665 & 0.8& &–& -106 $\!\pm\!$ 5 &15&T\
24/25.05.01 & 2054.538& 0.648 & 10 & 350-550 & 2.0& A3& 8800&-135 $\!\pm\!$ 9 & 5&P\
-------------------------------------------------------------------------------------- ------------ ------- ------- ------------------ --------------- ------- ------------------- ------------------ ------- -------
: Journal of the spectroscopic observations of HP Lyr[]{data-label="Jour"}
We have obtained some CCD spectrograms of this star in Tartu Observatory in Estonia and two spectra in Piwnice Observatory. The observations were carried out using the Cassegrain grating spectrographs attached to 1.5m (Tartu) and 0.9m (Piwnice) telescopes. Table \[Jour\] presents the journal of all our spectroscopic observations and results of the radial velocity measurments and the spectral class classification. The spectra were reduced under the IRAF and the MIDAS packages.
Period searching
================
We have performed frequency analysis of our photometric observations using the Lomb-Scargle periodograms (Lomb 1976, Scargle 1982). In each band we detected a strong peak at the frequency of about $f_1=0.0144$ (69.5 days) and the second one at a frequency $f_2=0.0072$ (139 days), whereas two sidelobes around $f_1$ signal reflect the shape of the spectral window (Fig. \[LoSca\]). These two detected signals $f_1$ and $f_2$ are in the 2:1 resonance. The mean period derived from $f_1$ frequency obtained in all filters and assuming a 2:1 resonance is $P=139.4 \pm 0.7$ days, i.e. about 1% less than the previous estimations (Wenzel 1960, Kreiner et al. 2001).
Looking for the ephemeris of HP Lyr we have carried out the timing analysis of observed minima. The Wenzel’s (1960) original ephemeris: $${\rm Min\, I} = {\rm JD}\,\, 2\,426\,910 + 140\fd 75\, E. \label{Eqn1}$$ was based on 66 independent moments of 56 photographically observed minima between 1931 and 1959. His O-C’s residua are shown in Fig.\[OC1\] for $E<72$. During 1960-1980 there were no observations mentioned in the literature. Since 1981 several moments of minima, estimated visually, were observed by Tristram Brellstaff (JD 2444817, 2444893, 2445171, 2445240, 2445510, 2445587, 2446217, 2447807) and published in a number of BAA VSS circulars (Markham and Pickard 2001). The next set of minima dates was collected by Kreiner et al. (2001). These data contain photoelectric and visual observations done by W. Braune et al. (JD 2445236.5, 2445309.2) and J. Heubscher et al. (JD 2445516, 2445586, 2445656.5, 2449464.0, 244953.0, 2450998.2, 2451062.6) and were published in several issues of B.A.V.Mitt. We have also added two moments of minima from our observations: JD 2425062.0 and JD 2451341.0 (Fig. \[LiCu\]).
All data after 1980 ($E>120$) show significant deviation from Wenzel’s ephemeris (Fig. \[OC1\]). We found a satisfactory, cubic solution connecting all minima which is presented in Fig. \[OC1\]: $${\rm Min\, I} = {\rm JD}\,\, 2\,426\,907 + 140\fd 74\, E
- 0\fd 0043\, E^2 - 4\fd 9\!\cdot\! 10^{-5}\, E^3 \label{Eph3}$$ However the second possibility – the linear equation for the later data – gives a slightly better fit assuming the existence of an abrupt period decrease: $${\rm Min\, I} = {\rm JD}\,\, 2\,444\,893 + 138\fd 66\, (E-128). \label{Eqn2}$$ The numbers $E$ are the same as in Wenzel’s ephemeris (Eq. \[Eqn1\]). Epoch $E=128$ corresponds to the first observed “primary” minimum after the 1960-1980 gap. If the abrupt period change is real, it should most probably occur in 1974 at the 112th cycle according to Wenzel’s ephemeris. Observations should soon distinguish between the cubic and the linear ephemeris.
Eclipsing or pulsating star?
============================
Our photometry shows that initially the amplitude of the light variations was about 05 in $V$ band, slightly smaller in red $ri$ filters and was raising in blue filters up to 10 ($U$ band). But, after about JD 2451150, all amplitudes decreased by a factor of two, whereas the mean brightness remained unchanged (Fig. \[LiCu\]). This is rather not typical behaviour for an eclipsing binary and we decided to show the mean light curves using $V-R$ and $R-I$ transformed Johnson colours (Fig. \[NLiC\]). Our $V$ light curve in Fig. \[NLiC\] resembles the $\beta$ Lyr type curve with a slightly deeper “primary” minimum at phase 0.5. However, we have a gap in observations around phase 1.0. Also, the colour index curves seem to be deeper at phase 0.5. This fact is confirmed by our spectral observations which give earlier spectral type and higher temperature around phase 1.0 than around phase 0.5 (Fig. \[radtem\]). Wenzel (1960) reported that both minima are of similar depth but the inspection of his photographic light curve shows that the minimum numbered by him as 1 (phase 0.0) may be slightly deeper. This alternation between the depth of minima is very difficult to understand in a binary system. It is real if the cubic solution (Eq. \[Eph3\]) is a valid model of both sets of the data in Fig. \[OC1\]. On the other hand, if the binary consists of two similar stars, then any mass loss or exchange of matter should lead to an increase (not a decrease) of the orbital period.
\
In order to rule out definitively the binary hypothesis we have calculated synthetic light curves using WD code (Wilson & Devinney 1971). In general, only the ellipsoidal variations in a binary system can produce reddening of both minima. Especially in early type stars with radiative envelopes, a significant gravitational reddening effect should be expected following von Zeipel’s (1924) theorem. We have tested two possible models with extreme von Zeipel’s effect: 1) an overcontact binary with two similar and evolved A6 components and 2) a semidetached binary consisting of an A6 star filling its Roche-lobe and massive, compact, optically invisible companion. Both ellipsoidal models can roughly reproduce the V band light curve but failed to reproduce the colour variations – Fig. \[NLiC\]. No binary model can explain the observed reddening with the amplitude about $\Delta (B-V)\approx 0.3$ during both minima.
Changes of colours create complex loops on the $U-B$, $B-V$ diagram (Fig. \[col\]). However, they show a general trend to align with the supergiant sequence from A2 at maximum to F0 at minimum with constant $E(B-V)=0.42$. Nearly the same spectral changes from A3 at maximum to F2 at minimum (Fig. \[BluC\]), were obtained from our blue spectra (Table \[Jour\]) using MK criteria by Morgan et al. (1978). The temperature changes derived from Straizys’ (1982) calibration strictly follow the light and colour variations (Fig. \[radtem\]). The most probable explanation of this behaviour is pulsations of a single star. Although our radial velocity data are insufficient for interpretation in terms of pulsations, they can preclude the binary model. It is not possible to join the four radial velocity points obtained from metallic lines (Fig. \[radtem\]) by one sinusoidal line with the condition of crossing the barycentric velocity close to spectroscopic conjunctions (“eclipses” at phases 0.0 and 0.5).
Evolutionary status
===================
The supergiant luminosity class Ib-Iab was derived by comparison with the spectra of the MK standards – Fig. \[Lumclas\]. However, relatively high galactic latitude ($b=+11^{o}\!\!.7$) and high radial velocity ($-$113 km/s) indicates that HP Lyr is most likely an evolved star of Intermediate Population II. The massive young object of Population I stars should be expected close to the Galactic plane. The spatial distribution of the interstellar extinction at the same galactic longitude ($l=70\fg - 75\fg$), but lower latitude ($b=5\fg - 6\fg$) shows practically constant extinction $A_v = 1.2-1.6$ above a distance of about 1.5 kpc (Neckel & Klare 1980, Miko[ł]{}ajewska & Miko[ł]{}ajewski 1980). This value of $A_v$ corresponds well to $E(B-V)$ for HP Lyr and gives a limit for the absolute magnitude $M_v < -$15.
A comparison between the spectra in the $H_{\alpha}$ region 6420 - 6600 Åobtained during two descending branches of the light curve is presented in Fig. \[Hacom\]. There are numerous metallic absorption lines used for radial velocity measurements (Table \[Jour\]). On the blue spectra, only three Balmer lines (and additionally CaII K and FeII 5169 Å$\,$ at Piwnice) were measured. The mean heliocentric radial velocity measured from the Balmer lines is $-104\pm 5$ km/s and from metallic lines $-122\pm 5$ km/s. It seems that all extremaly sharp Balmer absorptions are affected by the P Cyg emission components related to shock expanding in the atmosphere. Such weak emmission is clearly visible in the $H_\alpha$ profile in Fig. \[Hacom\].
HP Lyr was positionally associated with IRAS source 19199+3950 by Friedemann et al. (1996), but they rejected this identification because of its early spectral type. HP Lyr has the very good positional coincidence $3^"$ with an IRAS source (it is much better then most identified IRAS sources in their catalog), whereas the weak red star which Friedemann et al. suggested as a possible optical counterpart lies 8 times further (i.e. at the distance $24^"$). We found HP Lyr in maps of the Two Micron All Sky Survey[^3] with $J=8.98$, $H=8.44$ and $K=7.73$. Within a radius of 60$^{"}$ around HP Lyr there is no other $JHK$ source. We conclude that HP Lyr is the only countepart of the IRAS infrared source. Such infrared excess is typical for evolved Population II stars.
HP Lyr as the RV Tau variable
=============================
The $\beta$ Lyr type shape of the light curve and the other observed photometric and spectroscopic properties of HP Lyr show many important similarities with RV Tau group of variables. The RV Tau stars are luminous, pulsating variables located in the brightest part of the Population II instability strip and overlap in the Hertzsprung-Russell diagram with the W Vir type II Cepheids (Wahlgren 1992). Typical members have spectral type between F and K, luminosity class Ia-II, periods of their light variations in the range from about 30 to 150 days, large spectral and colour changes from maxima to minima. Alternating deep and shallow minima are caused by two dominant frequencies in their power spectra in 2:1 resonance with the ratio of their amplitudes close or smaller than unity. There is general agreement that RV Tau stars are pulsating low-mass post-AGB stars in the transition into the planetary nebulae phase (Jura 1986, Giridhar et al. 2000).
Most photometric peculiarities of HP Lyr are also typical for RV Tau stars. The photometric study of Pollard et al. (1996) revealed several objects such as AR Pup which during the period of observations did not show any distinction between primary and secondary minima (just like HP Lyr). Since the mean brightness of HP Lyr remains unchanged, it belongs to the RVa subgroup of RV Tau stars. Possible alternation of the “primary” and the “secondary” minima observed in HP Lyr is also typical for RV Tau stars (e.g. R Sct). The asymmetric, “bowed” shape of the colour curves are typical for these variables.
Another spectacular feature observed in HP Lyr photometric behaviour was the period change(s). The pulsation period of the RV Tau stars often varies with a complex way in a long timescale. Their O$-$C diagrams show many period instabilities and abrupt changes when the period increses or decreases from 0.001$P$ to 0.01$P$ (Erleksova 1971, Percy et al. 1991). The timescale over which the period changes can be from 20 to over 100 cycles.
The observed spectroscopic features of HP Lyr correspond quite well to common spectroscopic characteristics of RV Tau stars (Pollard et al. 1997 and references therein). For example, from many $H_\alpha$ profiles of some RV Tau stars, IW Car profiles are most similar to those of HP Lyr. IW Car itself was reported to be a binary system with spectral classification of A4 Ib-II: + F7/8 (Houk 1987), but the radial velocity measurements contradict this hypothesis. Most of the RV Tau type stars have more prominent P Cyg profiles in the $H_\alpha$ lines. It can be explained by their lower temperature and weaker photospheric components of the Balmer absorptions. One of our spectra of HP Lyr taken at a minimum (2/3 May) shows a weak CN I blend at 3883. Thus, in the spectroscopic subclassification scheme proposed by Preston et al. (1963), HP Lyr should belong to the the spectroscopic group “B” as an extremely hot pulsating RV Tau star (mean $T_{eff}\sim 7700$ K). The hottest RV Tau star known previously, IW Car, has the temperature $T_{eff}=6700$ (Giridhar et al. 1994).
Many RV Tau stars also exhibit infrared excesses (Raveendran 1989) from, most probably, dusty circumstellar shells. The presence of infrared excess in the case of RV Tau stars may be a sign of their old evolutionary stage as AGB stars. On the R(12/25) – R(25/60) diagram HP Lyr lies in the area populated by RVB stars (Fig. \[Iras\]). Also, the colours: $J-H = 0.54$ and $H-[25] = H+2.5\log(F_{25\mu m}) = 9.88$ of HP Lyr place it inside the well defined area populated by RV Tau stars at the near- far-infrared colour-colour diagram (Fujii et al. 2001)
Alcock et al. (1998) derived a single P-L relation for blue RV Tau stars in the Large Magellanic Cloud (Eq. 6 in their paper). If we apply this relation for the halfperiod of 70 days, we estimate the absolute magnitude of HP Lyr: $M_v \sim -4.5$. This value implies the distance to the star $\sim$ 5 kpc and the distance from the galactic plane $z\sim 1$ kpc. The high absolute magnitude of HP Lyr implied by Alcock et al.’s P-L relation may not be correct. However, the low amplitude of brightness changes in HP Lyr ($\Delta V = 0.5$ mag) is in a very good agreement with the slope of the relation $V$ amplitude-period in Fig. 7(b) of Alcock et al.(1998)
Conclusions
===========
The results of the photometric and spectroscopic survey of the long period variable HP Lyr have been reported. The star has behaviour typical for RV Tau stars but is apparently hotter than any other known RV Tau object. It means that this star makes a substantial extension to the Type II Cepheids/RV Tau instability strip to higher temperatures. Thus HP Lyr can be very important from an evolutionary point of view. Further photometric monitoring is very interesting to follow the period changes.
[**Acknowledgements.**]{} We are thankful to J.L. Janowski for making part of the photometric observations and to T. Eenmäe for taking one spectrum of HP Lyr. We thank P. Moskalik for very fruitful comments in the beginning of this work. We thank J.M. Kreiner and T. Markham and R. Pickard for kindly making their times of HP Lyr minima data available. We are grateful to U. Maciejewska and B. Roukema for English corrections in the text. This study was supported by Polish KBN Grant No. 5 P03D 003 20 and Estonian Science Foundation grants No. 3166 and 5006.\
REFERENCES\
[Alcock, C. et al. (MACHO collaboration)]{}, [1998]{}, [AJ]{}, [115]{}, [1921]{}\
[Erleksova, G.E.]{}, [1971]{}, [Peremennye Zvezdy]{}, [18]{}, [53]{}\
[Friedemann, C., Guertler, G., Loewe, M.]{}, [1996]{}, [A&AS]{}, [117]{}, [205]{}\
[Fujii, T., Nakada, Y., Parthasarathy, M.]{}, [2001]{}, [in “Post-AGB objects of stellar evolution”, Ed. R. Szczerba and S.K. G[ó]{}rny, ASSL 265, Kluwer]{},[ ]{}[111]{}\
[Giridhar, S., Lambert D., Gonzales G.]{}, [2000]{}, [ApJ]{}, [531]{}, [521]{}\
[Giridhar, S., Rao, K.N., Lambert, D.]{}, [1994]{}, [ApJ]{}, [437]{}, [476]{}\
[Houk, N.]{}, [1978]{}, [Michigan Spectral Catalogue of Two-dimensional Spectral Types for the HD Stars, University of Michigan, Ann Arbor]{}, [ Vol. 1]{}[ ]{}\
[Jura, M.]{}, [1986]{}, [ApJ]{}, [309]{}, [732]{}\
[Kreiner, J.M., Kim, C., Nha, I.]{}, [2001]{}, ["O$-$C Diagrams of Eclipsing dawnictwo Naukowe AP, Krak[ó]{}w]{}[ ]{}[ ]{}\
[Lomb, N.R.]{}, [1976]{}, [Ap&SS]{}, [39]{}, [447]{}\
[Markham, T., Pickard, R.]{}, [2001]{}, [private communication]{}[ ]{}[ ]{}\
[Miko[ł]{}ajewska J., Miko[ł]{}ajewski M.]{}, [1980]{}, [AcA]{}, [30]{}, [347]{}\
[Morgan, W.M., Abt, H.A., Tapscott, J.W.]{}, [1978]{}, [“Revised MK spectral stars earlier than the sun”, Yerkes Observatory]{}, [ ]{}, [ ]{}\
[Morgenroth, O.]{}, [1935]{}, [Astron. Nachr.]{}, [255]{}, [425]{}\
[Neckel, Th., Klare, G.]{}, [1980]{}, [A&AS]{}, [42]{}, [251]{}\
[Payne-Gaposchkin, C., Brenton, V.K., Gaposchkin, S.]{}, [1943]{},\
[Percy, J.R., Sasselov, D.D., Alfred, A., Scott, G.]{}, [1991]{}, [ApJ]{}, [375]{}, [691]{}\
[Pollard, K.R., Cottrell, P.L., Kilmartin, P.M., Gilmore, A.C.]{}, [1996]{}, ,\
[Pollard, K.R., Cottrell, P.L., Lawson, W.A., Albrow, M.D., Tobin, W.]{}, [1997]{}, , [286]{}, [1]{}\
[Preston, G.W., Krzemi[ń]{}ski, W., Smak, J., Williams, J.A.]{}, [1963]{}, [ApJ]{}, [137]{}, [401]{}\
[Raveendran, A.V.]{}, [1989]{}, [MNRAS]{}, [238]{}, [945]{}\
[Sandig, H.-U.]{}, [1939]{}, [Astron. Nachr.]{}, [276]{}, [177]{}\
[Scargle, J.D.]{}, [1982]{}, [ApJ]{}, [263]{}, [835]{}\
[Straizys, V.]{}, [1977]{}, [“Multicolour Stellar Photometry”, Mokslas, Vilnius]{}[ ]{}[ ]{}\
[Straizys, V.]{}, [1982]{}, [“Metal-deficient Stars”, Mokslas, Vilnius]{}[ ]{}[ ]{}\
[Van der Veen, W.E.C.J., Habing, H.J.]{}, [1988]{}, [A&A]{}, [194]{}, [125]{}\
[von Zeipel, H.]{}, [1924]{}, [MNRAS]{}, [84]{}, [665]{}\
[Wahlgren, G.M.]{}, [1992]{}, [AJ]{}, [104]{}, [1174]{}\
[Wenzel, W.]{}, [1960]{}, [Mitt. Ver. Sterne]{}, [499-500]{}[ ]{}\
[Wilson, R.E., Devinney, E.J.]{}, [1971]{}, [ApJ]{}, [166]{}, [605]{}\
[^1]: The number of lines
[^2]: T – Tartu Observatory, P – Piwnice Observatory
[^3]:
|
---
abstract: 'We study Fermi liquid properties of a weakly interacting 2D gas of single-component fermionic polar molecules with dipole moments $d$ oriented perpendicularly to the plane of their translational motion. This geometry allows the minimization of inelastic losses due to chemical reactions for reactive molecules and, at the same time, provides a possibility of a clear description of many-body (beyond mean field) effects. The long-range character of the dipole-dipole repulsive interaction between the molecules, which scales as $1/r^3$ at large distances $r$, makes the problem drastically different from the well-known problem of the two-species Fermi gas with repulsive contact interspecies interaction. We solve the low-energy scattering problem and develop a many-body perturbation theory beyond the mean field. The theory relies on the presence of a small parameter $k_Fr_*$, where $k_F$ is the Fermi momentum, and $r_*=md^2/\hbar^2$ is the dipole-dipole length, with $m$ being the molecule mass. We obtain thermodynamic quantities as a series of expansion up to the second order in $k_Fr_*$ and argue that many-body corrections to the ground-state energy can be identified in experiments with ultracold molecules, like it has been recently done for ultracold fermionic atoms. Moreover, we show that only many-body effects provide the existence of zero sound and calculate the sound velocity.'
author:
- 'Zhen-Kai Lu$^{1,2,3}$ and G. V. Shlyapnikov$^{2,4}$'
title: Novel Fermi Liquid of 2D Polar Molecules
---
Introduction
============
The recent breakthrough in creating ultracold diatomic polar molecules in the ground ro-vibrational state [@Ni; @Deiglmayr2008; @Inouye; @Nagerl] and cooling them towards quantum degeneracy [@Ni] has opened fascinating prospects for the observation of novel quantum phases [@Baranov2008; @Lahaye2009; @Pupillo2008; @Wang2006; @Buchler2007; @Taylor; @Cooper2009; @Pikovski2010; @Lutchyn; @Potter; @Barbara; @Sun; @Miyakawa; @Gora; @Ronen; @Parish; @Babadi; @Baranov]. A serious problem in this direction is related to ultracold chemical reactions, such as KRb+KRb$\Rightarrow$K$_2$+Rb$_2$ observed in the JILA experiments with KRb molecules [@Jin; @Jin2], which places severe limitations on the achievable density in three-dimensional samples. In order to suppress chemical reactions and perform evaporative cooling, it has been proposed to induce a strong dipole-dipole repulsion between the molecules by confining them to a (quasi)two-dimensional (2D) geometry and orienting their dipole moments (by a strong electric field) perpendicularly to the plane of the 2D translational motion [@Bohn1; @Baranov1]. The suppression of chemical reactions by nearly two orders of magnitude in the quasi2D geometry has been demonstrated in the recent JILA experiment [@Ye]. At the same time, not all polar molecules of alkali atoms, on which experimental efforts are presently focused, may undergo these chemical reactions [@Jeremy]. In particular, they are energetically unfavorable for RbCs bosonic molecules obtained in Innsbruck [@Nagerl], or for NaK and KCs molecules which are now being actively studied by several experimental groups (see, e.g. [@MITZ]). It is thus expected that future experimental studies of many-body physics will deal with non-reactive polar molecules or with molecules strongly confined to the 2D regime.
Therefore, the 2D system of fermionic polar molecules attracts a great deal of interest, in particular when they are in the same internal state. Various aspects have been discussed regarding this system in literature, in particular the emergence and beyond mean field description of the topological $p_x+ip_y$ phase for microwave-dressed polar molecules [@Cooper2009; @Gora], interlayer superfluids in bilayer and multilayer systems [@Pikovski2010; @Ronen; @Potter; @Zinner], the emergence of density-wave phases for tilted dipoles [@Sun; @Miyakawa; @Parish; @Babadi; @Baranov]. The case of superfluid pairing for tilted dipoles in the quasi2D geometry beyond the simple BCS approach has been discussed in Ref. [@Baranov]. The Fermi liquid behavior of this system has been addressed by using the Fourier transform of the dipole-dipole interaction potential [@Das1; @Das2; @Miyakawa; @Taylor; @Pu; @Das3; @Baranov] and then employing various types of mean field approaches, such as the Hartree-Fock approximation [@Miyakawa] or variational approaches [@Taylor; @Pu]. It should be noted, however, that the short-range physics can become important for the interaction between such polar molecules, since in combination with the long-range behavior it introduces a peculiar momentum dependence of the scattering amplitude [@Gora].
On the other hand, there is a subtle question of many-body (beyond mean field) effects in the Fermi liquid behavior of 2D polar molecules, and it can be examined in ultracold molecule experiments. For the case of atomic fermions, a milestone in this direction is the recent result at ENS, where the experiment demonstrated the many-body correction to the ground state energy of a short-range interacting two-species fermionic dilute gas [@Salomon1; @Salomon2]. This correction was originally calculated by Huang, Lee, and Yang [@Huang; @Lee] by using a rather tedious procedure. Later, it was found by Abrikosov and Khalatnikov [@Abr] in an elegant way based on the Landau Fermi liquid theory [@Landau].
In this paper, we study a weakly interacting 2D gas of fermionic polar molecules which are all in the same internal state. It is assumed that each molecule has an average dipole moment $d$ which is perpendicular to the plane of the translational motion, so that the molecule-molecule interaction at large separations $r$ is $$\label{VO}
U(r)=\frac{d^2}{r^3}=\frac{\hbar^2 r_*}{mr^3},$$ where $r_*=md^2/\hbar^2$ is the characteristic dipole-dipole distance, and $m$ is the molecule mass. The value of $d$ depends on the external electric field. At ultralow temperatures that are much smaller than the Fermi energy, characteristic momenta of particles are of the order of the Fermi momentum $k_F$, and the criterion of the weakly interacting regime is: $$\label{weakint}
k_Fr_*\ll 1.$$
As a consequence, the Fermi liquid properties of this system, such as the ground state energy, compressibility, effective mass, can be written as a series of expansion in the small parameter $k_Fr_*$. We obtain explicit expressions of these quantities up to the second order in $k_Fr_*$, which requires us to reveal the role of the short-range physics in the scattering properties and develop a theory beyond the mean field. Our analysis shows that only many-body (beyond mean field) effects provide the existence of undamped zero sound in the collisionless regime.
The paper is organized as follows. In Section II we analyze the low-energy 2D scattering of the polar molecules due to the dipole-dipole interaction. We obtain the scattering amplitude for all scattering channels with odd orbital angular momenta. The leading part of the amplitude comes from the so-called anomalous scattering, that is the scattering related to the interaction between particles at distances of the order of their de Broglie wavelength. This part of the amplitude corresponds to the first Born approximation and, due to the long-range $1/r^3$ character of the dipole-dipole interaction, it is proportional to the relative momentum $k$ of colliding particles for any orbital angular momentum $l$. We then take into account the second Born correction, which gives a contribution proportional to $k^2$. For the $p$-wave scattering channel it is necessary to include the short-range contribution, which together with the second Born correction leads to the term behaving as $k^2\ln{k}$. In Section III, after reviewing the Landau Fermi liquid theory for 2D systems, we specify two-body (mean field) and many-body (beyond mean field) contributions to the ground state energy for 2D fermionic polar molecules in the weakly interacting regime. We then calculate the interaction function of quasiparticles on the Fermi surface and, following the idea of Abrikosov-Khalatnikov [@Abr], obtain the compressibility, ground state energy, and effective mass of quasiparticles. In Section IV we calculate the zero sound velocity and stress that the many-body contribution to the interaction function of quasiparticles is necessary for finding the undamped zero sound. We conclude in Section V, emphasizing that the 2D gas of fermionic polar molecules represents a novel Fermi liquid, which is promising for revealing many-body effects. Moreover, we show that with present facilities it is feasible to obtain this system in both collisionless and hydrodynamic regimes.
Low-energy scattering of fermionic polar molecules in 2D
========================================================
General relations
-----------------
We first discuss low-energy two-body scattering of identical fermionic polar molecules undergoing the 2D translational motion and interacting with each other at large separations via the potential $U(r)$ (\[VO\]). The term low-energy means that their momenta satisfy the inequality $kr_*\ll 1$. In order to develop many-body theory for a weakly interacting gas of such molecules, we need to know the off-shell scattering amplitude defined as $$\label{offshellgen}
f({\mathbf{k}}^{\prime},{\mathbf{k}})=\int \exp(-i{\mathbf{k}}^{\prime}{\mathbf{r}})U(r)\tilde{\psi}_{{\mathbf{k}}}({\mathbf{r}})d^{2}{\mathbf{r}},$$ where $\tilde{\psi}_{{\mathbf{k}}}({\mathbf{r}})$ is the true wavefunction of the relative motion with momentum ${\mathbf{k}}$. It is governed by the Schrödinger equation $$\label{Schrgen}
\left(-\frac{\hbar^{2}}{m}\Delta+U(r)\right)\tilde{\psi}_{{\mathbf{k}}}({\mathbf{r}})=\frac{\hbar^{2}k^2}{m}\tilde{\psi}_{{\mathbf{k}}}({\mathbf{r}}).$$ For $|{\bf k}'|=|{\bf k}|$ we have the on-shell amplitude which enters an asymptotic expression for $\psi_{{\mathbf{k}}}({\mathbf{r}})$ at $r\rightarrow\infty$ [@Lan2; @Gora]: $$\label{psiassgen}
\tilde{\psi}_{{\mathbf{k}}}({\mathbf{r}})=\exp(i{\bf kr})-\frac{m}{\hbar^2}\sqrt{\frac{i}{8\pi kr}}f(k,\varphi)\exp(ikr),$$ with $\varphi$ being the scattering angle, i.e. the angle between the vectors ${\bf k}'$ and ${\bf k}$.
The wavefunction $\tilde\psi_{{\bf k}}({\bf r})$ can be represented as a sum of partial waves $\tilde\psi_l(k,r)$ corresponding to the motion with a given value of the orbital angular momentum $l$: $$\label{psil}
\tilde\psi_{{\bf k}}({\bf r})=\sum_{l=-\infty}^{\infty}\tilde\psi_l(k,r)i^{l}\exp(il\varphi).$$ Using the relation $$\label{expl}
\exp(i{\bf kr})=\sum_{l=-\infty}^{\infty} i^{l} J_{l}(kr)\exp[il(\varphi_{k}-\varphi_{r})],$$ where $J_l$ is the Bessel function, and $\varphi_k$ and $\varphi_r$ are the angles of the vectors ${\bf k}$ and ${\bf r}$ with respect to the quantization axis. Eqs. (\[psil\]) and (\[expl\]) allow one to express the scattering amplitude as a sum of partial-wave contributions: $$\label{offshellexp}
f({\mathbf{k}}^{\prime},{\mathbf{k}})=\sum_{l=-\infty}^{\infty}\exp(il\varphi)f_{l}(k',k),$$ with the off-shell $l$-wave amplitude given by $$\label{offshelll}
f_l(k',k)=\int_0^{\infty}J_l(k'r)U(r)\tilde\psi_l(k,r)2\pi rdr.$$ Similar relations can be written for the on-shell scattering amplitude: $$\begin{aligned}
&&f(k,\varphi)=\sum_{l=-\infty}^{\infty}\exp(il\varphi)f_{l}(k), \label{fsum} \\
&&f_l(k)=\int_{0}^{\infty}J_{l}(k'r)U(r)\tilde{\psi}_{l}(k,r)2\pi rdr. \label{flgen} \end{aligned}$$
The asymptotic form of the wavefunction of the $l$-wave relative motion at $r\rightarrow\infty$ may be represented as $$\label{psideltal}
\tilde\psi_l(k,r)\propto\frac{\cos(kr-\pi/4+\delta_l(k))}{\sqrt{kr}},$$ where $\delta_l(k)$ is the scattering phase shift. This is obvious because in the absence of scattering the $l$-wave part of the plane wave $\exp(i{\bf kr})$ at $r\rightarrow\infty$ is $(kr)^{-1/2}\cos(kr-\pi/4)$. Comparing Eq. (\[psideltal\]) with the $l$-wave part of Eq. (\[psiassgen\]) we obtain a relation between the partial on-shell amplitude and the phase shift: $$\label{fldeltal}
f_{l}(k)=-\frac{4\hbar^{2}}{m}\frac{\tan \delta_l(k)}{1-i \tan \delta_l(k)}.$$ Note that away from resonances the scattering phase shift is small in the low-momentum limit $kr_*\ll 1$.
For the solution of the scattering problem it is more convenient to normalize the wavefunction of the radial relative motion with orbital angular momentum $l$ in such a way that it is real and for $r\rightarrow\infty$ one has: $$\begin{aligned}
\label{nor}
\psi_l(k,r)&&=\left[J_l(kr)-\tan \delta_l(k)N_l(kr)\right] \nonumber \\
&&\propto \cos(kr-l\pi/2-\pi/4+\delta_l(k)),\end{aligned}$$ where $N_l$ is the Neumann function. One checks straightforwardly that $$\tilde\psi_l(k,r)=\frac{\psi_l(k,r)}{1-i\tan\delta_l(k)}.$$ Using this relation the off-shell scattering amplitude (\[offshelll\]) can be represented as $$\label{barfloff}
f_l(k',k)=\frac{{\bar f}_l(k',k)}{1-i\tan\delta_l(k)},$$ where ${\bar f}_l(k',k)$ is real and follows from Eq. (\[offshelll\]) with $\tilde\psi_l(k,r)$ replaced by $\psi_l(k,r)$. Setting $k'=k$ we then obtain the related on-shell scattering amplitude: $$\label{barflon}
{\bar f}_l(k,k)\equiv{\bar f}_l(k)=-\frac{4\hbar^2}{m}\tan \delta_l (k).$$
Low-energy $p$-wave scattering
------------------------------
As we will see, the slow $1/r^3$ decay of the potential $U(r)$ at sufficiently large distances makes the scattering drastically different from that of short-range interacting atoms. For identical fermionic polar molecules, only the scattering with odd orbital angular momenta $l$ is possible. For finding the amplitude of the $p$-wave scattering in the ultracold limit, $kr_*\ll 1$, we employ the method developed in Ref. [@Gora] and used there for the scattering potential containing an attractive $1/r^3$ dipole-dipole tail. We divide the range of distances into two parts: $r<r_0$ and $r>r_0$, where $r_0$ is in the interval $r_*\ll r_0\ll k^{-1}$. In region I where $r<r_0$, the $p$-wave relative motion of two particles is governed by the Schrödinger equation with zero kinetic energy: $$\label{Schrp1}
-\frac{\hbar^2}{m}\left(\frac{d^2\psi _I}{d r^2}+\frac{1}{r}\frac{d \psi_I}{d r}-\frac{\psi_I}{r^2}\right)+U(r)\psi_I=0.$$ At distances where the potential $U(r)$ already acquires the form (\[VO\]), the solution of Eq. (\[Schrp1\]) can be expressed in terms of growing and decaying Bessel functions: $$\label{short}
\psi_I(r)\propto \left[AI_2\left(2\sqrt{\frac{r_*}{r}}\right)+K_2\left(2\sqrt{\frac{r_*}{r}}\right)\right].$$ The constant $A$ is determined by the behavior of $U(r)$ at short distances where Eq. (\[VO\]) is no longer valid. If the interaction potential $U(r)$ has the form (\[VO\]) up to very short distances, then $A=0$, so that for $r\rightarrow 0$ equation (\[short\]) gives an exponentially decaying wavefunction.
It should be noted here that for the quasi2D regime obtained by a tight confinement of the translational motion in one direction, we can encounter the situation where $r_*\lesssim l_0$, with $l_0$ being the confinement length. However, we may always select $r_0\gg l_0$ if the condition $kl_0\ll 1$ is satisfied. Therefore, our results for the 2D $p$-wave scattering obtained below in this section remain applicable for the quasi2D regime. The character of the relative motion of particles at distances $r\lesssim l_0$ is only contained in the value of the coefficient $A$, and the extra requirement is the inequality $kl_0\ll 1$.
At large distances, $r>r_0$, the relative motion is practically free and the potential $U(r)$ can be considered as perturbation. To zero order, the relative wavefunction is given by $$\label{psiII0}
\psi_{II}^{(0)}(r)=J_1(kr)-\tan \delta_I(k)N_1(kr),$$ where the phase shift $\delta_I(k)$ is due to the interaction between particles in region I. Equalizing the logarithmic derivatives of $\psi_I(r)$ and $\psi_{II}^{(0)}$ at $r=r_0$ we obtain: $$\label{deltaI}
\!\!\tan \delta_I\!=\!-\frac{\pi k^2r_0r_*}{8}\left[1\!+\!\frac{r_*}{r_0}\left(2C\!-\!\frac{1}{2}\!-\!2A\!+\!\ln\frac{r_*}{r_0}\right)\right],\!\!$$ with $C=0.5772$ being the Euler constant.
We now include perturbatively the contribution to the $p$-wave scattering phase shift from distance $r>r_0$. In this region, to first order in $U(r)$, the relative wavefunction is given by $$\label{sec}
\!\psi^{(1)}_{II}(r)=\psi^{(0)}_{II}(r)\!-\!\int_{r_0}^{\infty}G(r,r')U(r')\psi^{(0)}_{II}(r')2\pi r' dr'\!,$$ where the Green function for the free $p$-wave motion obeys the radial equation: $$\begin{aligned}
-\frac{\hbar^2}{m}\left(\frac{d^2}{dr^2}+\frac{1}{r}\frac{d}{dr}-\frac{1}{r^2}+k^2 \right)G(r,r')=\frac{\delta(r-r')}{2\pi r}. \nonumber\end{aligned}$$ For the normalization of the relative wavefunction chosen in Eq. (\[nor\]), we have: $$G(r,r')=-\frac{m}{4\hbar^2}
\begin{cases}
\psi^{(0)}_{II}(r')N_1(kr), &r>r'\\
\\
\psi^{(0)}_{II}(r)N_1(kr'). &r<r'
\end{cases}$$ Substituting this Green function into Eq. (\[sec\]) and taking the limit $r\rightarrow \infty$, for the first order contribution to the phase shift we have: $$\label{f1leadingint}
\!\!\tan \delta_1^{(1)}(k)\!=\!\tan\delta_I(k)\!-\!\frac{m}{4\hbar^2}\int_{r_0}^{\infty}\!\![\psi^{(0)}_{II}(r)]^2U(r)2\pi rdr.$$ Using Eqs. (\[psiII0\]) and (\[deltaI\]) we then obtain: $$\!\!\!\tan\delta_1^{(1)}(k)\!=\!-\frac{2kr_*}{3}\!-\!\frac{\pi k^2 r_*^2}{8}\left(\!\!-2A\!+\!2C\!+\!\ln\frac{r_*}{r_0}\!-\!\frac{3}{2}\!\right).\!$$
To second order in $U(r)$, we have the relative wavefunction: $$\begin{aligned}
\label{secondwave}
\psi^{(2)}_{II}(r)&&=\psi^{(1)}_{II}(r)+\int_{r_0}^{\infty} G(r,r')U(r')2\pi r' dr' \nonumber\\
&&\times\int_{r_0}^{\infty} G(r',r'')U(r'')\psi^{(0)}_{II}(r'')2\pi r'' dr''.\end{aligned}$$ Taking the limit $r\rightarrow\infty$ in this equation we see that including the second order contribution, the scattering phase shift becomes: $$\begin{aligned}
\tan\delta_1(k)&&=\tan\delta^{(1)}(k)-\frac{m^2}{8\hbar^4}\int_{r_0}^{\infty}\psi^{(0)}_{II}(r)^{2}U(r)2\pi r dr\nonumber \\
&&\times\int_{r}^{\infty}N_1(kr')U(r')\psi^{(0)}_{II}(r')2\pi r' dr'.\end{aligned}$$ As we are not interested in terms that are proportional to $k^3$ or higher powers of $k$, we may omit the term $\tan\delta_I(k)N_1(kr)$ in the expression for $\psi^{(0)}_{II}(r)$. Then the integration over $dr'$ leads to: $$\begin{aligned}
&&\tan\delta_1(k)=\tan\delta_1^{(1)}(k)-\frac{(\pi kr_*)^2}{2}\int_{kr_0}^{\infty} \frac{J^2_1(x)}{x^2}dx\nonumber \\
&&\times \Big[\frac{2}{3}x\left(N_0(x)J_2(x)-N_1(x)J_1(x)\right) \nonumber\\
&& \;\;\;\;-\frac{1}{2}N_0(x)J_1(x)+\frac{1}{6}N_1(x)J_2(x)-\frac{1}{\pi x}\Big].\end{aligned}$$ For the first four terms in the square brackets, we may put the lower limit of integration equal to zero and use the following relations: $$\begin{aligned}
&\int_{0}^{\infty} J^3_1(x)N_1(x)\frac{dx}{x}=-\frac{1}{4\pi},\\
&\int_{0}^{\infty} J^2_1(x)J_2(x)N_0(x)\frac{dx}{x}=\frac{1}{8\pi},\\
&\int_{0}^{\infty} J^3_1(x)N_0(x)\frac{dx}{x^2}=\frac{1}{16\pi},\\
&\int_{0}^{\infty} J^2_1(x)J_2(x)N_1(x)\frac{dx}{x^2}=-\frac{1}{16\pi}.\end{aligned}$$ For the last term in the square brackets we have: $$\int_{kr_0}^{\infty} J^{2}_1(x)\frac{dx}{x^3}\approx\frac{1}{16}-\frac{C}{4}+\frac{\ln 2}{4}-\frac{1}{4}\ln kr_0.$$ We then obtain: $$\begin{aligned}
\tan\delta_1(k)&=\tan\delta_1^{(1)}(k)-\frac{\pi(kr_*)^2}{8}\left[\frac{7}{12}+C-\ln 2+\ln kr_0\right] \nonumber\\
&=-\frac{2kr_*}{3}-\frac{\pi k^2 r^2_*}{8}\ln\xi kr_*, \label{deltap}\end{aligned}$$ where: $$\label{xi}
\xi=\exp\left(3C-\ln 2 -\frac{11}{12}-2A\right).$$
Using Eqs. (\[barflon\]) and (\[deltap\]) we represent the on-shell $p$-wave scattering amplitude ${\bar f}_1(k)$ in the form: $${\bar f}_1(k)={\bar f}^{(1)}_1(k)+{\bar f}^{(2)}_1(k),$$ with $$\label{f1leading}
{\bar f}^{(1)}_1(k)=\frac{8\hbar^2}{3m}kr_*$$ and $$\label{f12}
{\bar f}^{(2)}_1(k)=\frac{\pi\hbar^2}{2m}(kr_*)^2\ln\xi kr_*.$$ The leading term is ${\bar f}^{(1)}_1(k)\propto k$. It appears to first order in $U(r)$ and comes from the scattering at distances $r\sim 1/k$. This term can be called “anomalous scattering” term (see [@Lan2]). The term $f^{(2)}_1(k)\propto k^2\ln\xi kr_*$ comes from both large distances $\sim 1/k$ and short distances. Note that the behavior of the wavefunction at short distances where $U(r)$ is no longer given by Eq. (\[VO\]), is contained in Eq. (\[deltap\]) only through the coefficient $\xi$ under logarithm.
Scattering with $|l|>1$
-----------------------
The presence of strong anomalous $p$-wave scattering, i.e. the scattering from interparticle distances $\sim 1/k$, originates from the slow $1/r^3$ decay of the potential $U(r)$ at large $r$. The strong anomalous scattering is also present for partial waves with higher $l$. In this section we follow the same method as in the case of the $p$-wave scattering and calculate the amplitude of the $l$-wave scattering with $|l|>1$. For simplicity we consider positive $l$, having in mind that the scattering amplitude and phase shift depend only on $|l|$.
To zero order in $U(r)$, the wavefunction of the $l$-wave relative motion at large distances $r>r_0$ is written as: $$\label{large}
\psi^{(0)}_{l(II)}(k,r)=\left[J_l(kr)-\tan \delta_{l(I)}(k)N_l(kr)\right],$$ where $\delta_{l(I)}(k)$ is the $l$-wave scattering phase shift coming from the interaction at distances $r<r_0$. We then match $\psi^{(0)}_{l(II)}(k,r)$ at $r=r_0$ with the short-distance wavefunction $\psi_{l(I)}(r)$ which follows from the Schrödinger equation for the $l$-wave relative motion in the potential $U(r)$ at $k=0$. This immediately gives a relation: $$\label{arg}
\tan \delta_{l(I)}(k)=\frac{k J'_l(kr_0)-w_lJ_l(kr_0)}{k N'_l(kr_0)-w_lN_l(kr_0)},$$ where the momentum-independent quantity $w_l$ is the logarithmic derivative of $\psi_{l(I)}(r)$ at $r=r_0$. Since we have the inequality $kr_0\ll 1$, the arguments of the Bessel functions in Eq. (\[arg\]) are small and they reduce to $J_l(x)\sim x^l \; ,\; N_l(x)\sim x^{-l}$. This leads to $\tan \delta_{l(I)}(k) \sim (kr_0)^{2l}$. Thus, the phase shift coming from the interaction at short distances is of the order of $(kr_0)^{2l}$. As we confine ourselves to second order in $k$, we may put $\tan \delta_{l(I)}(k)=0$ for the scattering with $|l|>1$.
Then, like for the $p$-wave scattering, we calculate the contribution to the phase shift from distances $r>r_0$ by considering the potential $U(r)$ as perturbation. To first and second order in $U(r)$, at $r>r_0$ we have similar expressions as Eq. (\[f1leadingint\]), (\[secondwave\]) for the relative wavefunction of the $l$-wave motion. Following the same method as in the case of the $p$-wave scattering and retaining only the terms up to $k^2$, for the first order phase shift we have: $$\begin{aligned}
\label{flleadingint}
&&\hspace{-17mm}\tan \delta_l^{(1)}(k)=-\frac{m}{4\hbar^2}\int_{r_0}^{\infty}[\psi^{(0)}_{l(II)}(r)]^2U(r)2\pi rdr \nonumber\\
&&\simeq -\frac{\pi kr_*}{2}\int_{kr_0}^{\infty}J_l^2(x)\frac{1}{x^2} dx=-\frac{2 kr_*}{4 l^2-1}.\end{aligned}$$ The second order phase shift is: $$\begin{aligned}
\label{ex}
&&\tan\delta^{(2)}_l(k)=-\frac{m^2}{8\hbar^4}\int_{r_0}^{\infty}\psi^{(0)}_{l(II)}(r)^{2}U(r)2\pi r dr \nonumber \\
&&\;\;\times\int_{r}^{\infty}N_l(kr')U(r')\psi^{(0)}_{l(II)}(r')2\pi r' dr' \nonumber\\
&&\simeq -\frac{(\pi k r_*)^2}{2}\int_{kr_0}^{\infty}\frac{J^2_l(x)}{x^2} dx \int_{x}^{\infty}\frac{N_l(y)J_l(y)}{y^2} dy,\end{aligned}$$ and we may put the lower limit of integration equal to zero. For the integral over $dy$, we obtain : $$\begin{aligned}
&\int_{x}^{\infty}\frac{N_l(y)J_l(y)}{y^2} dy \nonumber \\
&=\frac{1}{2l(2l-1)}J_{l}(x)N_{l-1}(x)+\frac{1}{2l(2l+1)}J_{l+1}(x)N_{l}(x) \nonumber\\
&\!\!+\frac{2x}{4l^2-1}\big[N_{l-1}(x)J_{l+1}(x)\!-\!J_l(x)N_l(x)\big]-\frac{1}{\pi lx}.\end{aligned}$$ Then, using the relations: $$\int_{0}^{\infty} \frac{J_{l}^{2}(x)}{x^3} dx=\frac{1}{4l (l^2-1)},$$ $$\int_{0}^{\infty}\frac{J_{l}^{2}(x)}{x} N_{l-1}(x) J_{l+1}(x) dx =\frac{1}{4l (l+1)\pi},$$ $$\int_{0}^{\infty}\frac{J_{l}^{3}(x)}{x} N_{l}(x) dx =-\frac{1}{4l^2\pi},$$ $$\int_{0}^{\infty}\frac{J_{l}^{2}(x)}{x^2} J_{l}(x)N_{l-1}(x) dx =\frac{1}{8l^2(l+1)\pi},$$ $$\int_{0}^{\infty}\frac{J_{l}^{2}(x)}{x^2} J_{l+1}(x)N_{l}(x) dx =-\frac{1}{8l^2(l+1)\pi},$$ we find the following result for the second order phase shift: $$\begin{aligned}
\tan\delta^{(2)}_l(k)=\frac{3\pi(kr_*)^2}{8}\frac{1}{l(l^2-1)(4l^2-1)}.\end{aligned}$$ So, the total phase shift is given by $$\begin{aligned}
\tan \delta_l(k)&=\tan \delta_l^{(1)}(k)+\tan\delta_l^{(2)}(k)\nonumber\\
&=-\frac{2 kr_*}{4l^2-1}+\frac{3\pi(kr_*)^2}{8l(l^2-1)(4l^2-1)}.\end{aligned}$$ Then, according to Eq. (\[barflon\]) the on-shell scattering amplitude ${\bar f}_l(k)$ is $${\bar f}_l(k)={\bar f}^{(1)}_l(k)+{\bar f}^{(2)}_l(k),$$ where $$\label{flleading}
{\bar f}^{(1)}_l(k)=\frac{8\hbar^2 kr_*}{m}\frac{1}{4 l^2-1},$$ $$\label{fl2}
{\bar f}^{(2)}_l(k)=-\frac{3\pi\hbar^2}{2m}(kr_*)^2\frac{1}{|l|(l^2-1)(4l^2-1)}.$$ Note that Eqs. (\[flleading\]) and (\[fl2\]) do not contain short-range contributions as those are proportional to $k^{2|l|}$ and can be omitted for $|l|>1$.
First order Born approximation and the leading part of the scattering amplitude
-------------------------------------------------------------------------------
As we already said above, in the low-momentum limit for both $|l|=1$ and $|l|>1$ the leading part of the on-shell scattering amplitude ${\bar f}_l(k)$ is ${\bar f}_l^{(1)}(k)$ and it is contained in the first order contribution from distances $r>r_0$. For $|l|>1$ it is given by Eq. (\[flleading\]) and follows from Eq. (\[flleadingint\]) with $\psi^{(0)}_{l(II)}=J_l(kr)$. In the case of $|l|=1$ this leading part is given by Eq. (\[f1leading\]) and follows from the integral term of Eq. (\[f1leadingint\]) in which one keeps only $J_1(kr)$ in the expression for $\psi^{(0)}_{II}(r)$. This means that ${\bar f}_l^{(1)}(k)$ actually follows from the first order Born approximation.
The off-shell scattering amplitude can also be represented as ${\bar f}_l(k',k)={\bar f}_l^{(1)}(k',k)+{\bar f}_l^{(2)}(k',k)$, and the leading contribution ${\bar f}_l^{(1)}(k',k)$ follows from the first Born approximation. It is given by Eq. (\[offshelll\]) in which one should replace $\tilde\psi_l(k,r)$ by $J_l(kr)$: $$\label{leadingoffshelll}
{\bar f}_l^{(1)}(k',k)=\int_{0}^{\infty} J_{l}(kr) J_{l}(kr')U(r) 2\pi r dr.$$ Note that it is not important that we put zero for the lower limit of the integration, since this can only give a correction which behaves as $k^2$ or a higher power of $k$. Then, putting $U(r)=\hbar^2r_*/mr^3$ in Eq. (\[leadingoffshelll\]), we obtain: $$\begin{aligned}
\label{off}
{\bar f}^{(1)}_l(k',k)= &\frac{\pi \hbar^2}{m}\frac{\Gamma(l-1/2)}{\sqrt{\pi}}\frac{k^l r_*}{(k')^{l-1}} \nonumber \\
& \times F\left(-\frac{1}{2},-\frac{1}{2}+l,1+l,\frac{k^2}{k'^2}\right),\end{aligned}$$ where $F$ is the hypergeometric function. The result of Eq. (\[off\]) corresponds to $k<k'$, and for $k>k'$ one should interchange $k$ and $k'$.
For identical fermions the full scattering amplitude contains only partial amplitudes with odd $l$. Since the scattered waves with relative momenta ${\bf k}'$ and $-{\bf k}'$ correspond to interchanging the identical fermions, the scattering amplitude can be written as (see, e.g. [@Lan2]): $$\label{ffermion}
\tilde f({\bf k}',{\bf k})=f({\bf k}',{\bf k})-f(-{\bf k}',{\bf k}).$$ Then, according to equation (\[fsum\]) one can write: $$\label{ffermionsum}
\tilde f({\bf k}',{\bf k})=2\sum_{l\,odd}f_l(k',k)\exp(il\varphi).$$
In the first Born approximation there is no difference between $f_l(k',k)$ and ${\bar f}_l(k',k)$ because $\tan\delta_l(k)$ in the denominator of Eq. (\[barfloff\]) is proportional to $k$ and can be disregarded. Therefore, one may use ${\bar f}_l^{(1)}(k',k)$ of Eq.(\[off\]) for $f_l(k',k)$ in Eq. (\[ffermionsum\]). One can represent $\tilde f({\bf k}',{\bf k})$ in a different form recalling that in the first Born approximation we have: $$\label{fBorn}
f({\bf k}',{\bf k})=\int U(r)\exp[i({\bf k}-{\bf k}'){\bf r}]d^2r.$$ Performing the integration in this equation, with $U(r)$ given by Eq. (\[VO\]), and using Eq. (\[ffermion\]) we obtain: $$\label{tildefBorn}
\tilde f({\bf k}',{\bf k})=\frac{2\pi\hbar^2r_*}{m}\{|{\bf k}+{\bf k}'|-|{\bf k}-{\bf k}'|\}.$$ Equation (\[tildefBorn\]) is also obtained by a direct summation over odd $l$ in Eq. (\[ffermionsum\]), with $f_l(k',k)$ following from Eq. (\[off\]).
Thermodynamics of a weakly interacting 2D gas of fermionic polar molecules at $T=0$
===================================================================================
General relations of Fermi liquid theory
----------------------------------------
Identical fermionic polar molecules undergoing a two-dimensional translational motion and repulsively interacting with each other via the potential (\[VO\]) represent a 2D Fermi liquid. General relations of the Landau Fermi liquid theory remain similar to those in 3D (see, e.g. [@Landau]). The number of “dressed" particles, or quasiparticles, is the same as the total number of particles $N$, and the (quasi)particle Fermi momentum is $$\label{kF}
k_F=\sqrt{\frac{4\pi N}{S}},$$ where $S$ is the surface area. At $T=0$ the momentum distribution of free quasiparticles is the step function $$\label{nstep}
n({\bf k})=\theta(k_F-k),$$ i.e. $n({\bf k})=1$ for $k<k_F$ and zero otherwise.The chemical potential is equal to the boundary energy at the Fermi circle, $\mu=\epsilon_F\equiv\epsilon(k_F)$.
The quasiparticle energy $\epsilon({\bf k})$ is a variational derivative of the total energy with respect to the distribution function $n({\bf k})$. Due to the interaction between quasiparticles, the deviation $\delta n$ of this distribution from the step function (\[nstep\]) results in the change of the quasiparticle energy: $$\label{1}
\delta\epsilon ({\mathbf{k}})=\int F({\mathbf{k}},{\mathbf{k}}') \delta n({\mathbf{k}}') \frac{d^2k'}{(2\pi)^2}.$$ The interaction function of quasiparticles $F({\mathbf{k}},{\mathbf{k}}')$ is thus the second variational derivative of the total energy with regard to $n({\bf k})$. The quantity $\delta n({\bf k})$ is significantly different from zero only near the Fermi surface, so that one may put ${\bf k}=k_F{\bf n}$ and ${\bf k}'=k_F{\bf n}'$ in the arguments of $F$ in Eq. (\[1\]), where ${\bf n}$ and ${\bf n}'$ are unit vectors in the directions of ${\bf k}$ and ${\bf k}'$. The quasiparticle energy near the Fermi surface can be written as: $$\label{2}
\epsilon({\mathbf{k}})=\epsilon_{F}+\hbar v_{F}(k-k_{F})+\int F({\mathbf{k}},{\mathbf{k}}')\delta n({\mathbf{k}}')\frac{d^2k'}{(2\pi)^2}.$$ The quantity $v_F=\partial\epsilon({\bf k})/\hbar\partial k|_{k=k_F}$ is the Fermi velocity, and the effective mass of a quasiparticle is defined as $m^*=\hbar k_F/v_F$. It can be obtained from the relation (see [@Landau]): $$\begin{aligned}
\label{eff}
\frac{1}{m}=\frac{1}{m^{*}}+\frac{1}{(2\pi\hbar)^{2}}\int_0^{2\pi} F(\theta) \cos\theta d \theta,\end{aligned}$$ where $\theta$ is the angle between the vectors ${\bf n}$ and ${\bf n}'$, and $F(\theta)=F(k_F{\bf n},k_F{\bf n}')$.
The compressibility $\kappa$ at $T=0$ is given by [@Landau]: $$\label{kappa-1}
\kappa^{-1}=\frac{N^2}{S}\frac{\partial \mu}{\partial N}.$$ The chemical potential is $\mu=\epsilon_{F}$, and the variation of $\mu$ due to a change in the number of particles can be expressed as $$\label{mu}
\delta \mu=\int F(k_F{\bf n},{\mathbf{k}}')\delta n({\mathbf{k'}})\frac{d^2k'}{(2\pi)^2}+\frac{\partial \epsilon_{F}}{\partial k_{F}} \delta k_{F}.$$ The quantity $\delta n({\mathbf{k'}})$ is appreciably different from zero only when ${\mathbf{k'}}$ is near the Fermi surface, so that we can replace the interaction function $F$ by its value on the Fermi surface. Then the first term of Eq. (\[mu\]) becomes $$\begin{aligned}
\int F(\theta) \frac{d\theta}{2\pi}\int\delta n({\mathbf{k'}})\frac{d^2k'}{(2\pi)^2}=\frac{\delta N}{2\pi S}\int F(\theta) d \theta. \nonumber\end{aligned}$$ The second term of Eq. (\[mu\]) reduces to $$\begin{aligned}
\frac{\partial \epsilon_{F}}{\partial k_{F}} \delta k_{F}=\frac{\hbar^2 k_{F}}{m^{*}}\delta k_{F}=\frac{2\pi\hbar^{2}}{m^{*}}\frac{\delta N}{S}.\end{aligned}$$ We thus have (see [@Landau]): $$\begin{aligned}
\label{u1}
\frac{\partial \mu}{\partial N}&=\frac{1}{2\pi S}\int_0^{2\pi} F(\theta) d\theta +\frac{2\pi \hbar^{2}}{m^{*}S} \nonumber\\
&=\frac{2\pi\hbar^{2}}{mS}+\frac{1}{2\pi S}\int_0^{2\pi} (1-\cos\theta) F(\theta) d\theta. \end{aligned}$$ Equation (\[u1\]) shows that the knowledge of the interaction function of quasiparticles on the Fermi surface, $F(\theta)$, allows one to calculate $\partial\mu/\partial N$ and, hence, the chemical potential $\mu=\partial E/\partial N$ and the ground state energy $E$. This elegant way of finding the ground state energy has been proposed by Abrikosov and Khalatnikov [@Abr]. It was implemented in Ref. [@Abr] for a two-component 3D Fermi gas with a weak repulsive contact (short-range) interspecies interaction.
We develop a theory beyond the mean field for calculating the interaction function of quasiparticles for a single-component 2D gas of fermionic polar molecules in the weakly interacting regime. We obtain the ground state energy as a series of expansion in the small parameter $k_Fr_*$ and confine ourselves to the second order. In this sense our work represents a sort of Lee-Huang-Yang [@Huang; @Lee] and Abrikosov-Khalatnikov [@Abr] calculation for this dipolar system. As we will see, the long-range character of the dipole-dipole interaction makes the result quite different from that in the case of short-range interactions.
Two-body and many-body contributions to the ground state energy
---------------------------------------------------------------
We first write down the expression for the kinetic energy and specify two-body (mean field) and many-body (beyond mean field) contributions to the interaction energy. The Hamiltonian of the system reads: $$\label{ha}
\!\!{\oldhat{\cal H}}\!=\!\sum_{{\mathbf{k}}}\frac{\hbar^2k^2}{2m}{\oldhat{a}}_{{\mathbf{k}}}^{\dag}{\oldhat{a}}_{{\mathbf{k}}}\!+\!\frac{1}{2S}\!\!\!\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}},{\mathbf{q}}}\!\!\!U({\mathbf{q}}) {\oldhat{a}}_{{\mathbf{k_{1}}}+{\mathbf{q}}}^{\dag}{\oldhat{a}}_{{\mathbf{k_{2}}}-{\mathbf{q}}}^{\dag}{\oldhat{a}}_{{\mathbf{k_{2}}}}{\oldhat{a}}_{{\mathbf{k_{1}}}},\!\!$$ where ${\oldhat{a}}^{\dagger}_{{\bf k}}$ and ${\oldhat{ a}}_{{\bf k}}$ are creation and annihilation operators of fermionic polar molecules, and $U({\mathbf{q}})$ is the Fourier transform of the interaction potential $U(r)$: $$\label{Uq}
U({\mathbf{q}})=\int d^2{\mathbf{r}} U(r)e^{-i{\mathbf{q}}\cdot{\mathbf{r}}},$$ The first term of Eq. (\[ha\]) represents the kinetic energy and it gives the main contribution to the total energy $E$ of the system. This term has only diagonal matrix elements, and using the momentum distribution (\[nstep\]) at $T=0$ we have: $$\begin{aligned}
\label{Ekin}
\frac{E_{kin}}{S}=\int_0^{k_F}\frac{\hbar^2k^2}{2m}\frac{2\pi kdk}{(2\pi)^2}=\frac{\hbar^2k_F^4}{16m}.\end{aligned}$$
The interaction between the fermionic molecules is described by the second term in Eq. (\[ha\]) and compared to the kinetic energy it provides a correction to the total energy $E$. The first order correction is given by the diagonal matrix element of the interaction term of the Hamiltonian: $$\begin{aligned}
\label{E1}
E^{(1)}&=\frac{1}{2S}\sum_{{\bf{k_1}},{\bf{k_2} },{\bf q}}U({\mathbf{q}})\langle {\oldhat{a}}_{{\mathbf{k_{1}}}+{\mathbf{q}}}^{\dag}{\oldhat{a}}_{{\mathbf{k_{2}}}-{\mathbf{q}}}^{\dag}{\oldhat{a}}_{{\mathbf{k_{2}}}}{\oldhat{a}}_{{\mathbf{k_{1}}}}\rangle \nonumber\\
&=\frac{1}{2S}\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}}}\left[U(0)-U({\mathbf{k_{2}}}-{\mathbf{k_{1}}})\right]n_{{\mathbf{k_{1}}}}n_{{\mathbf{k_{2}}}}.\end{aligned}$$ The second order correction to the energy of the state $\ket{j}$ of a non-interacting system can be expressed as: $$\begin{aligned}
E_{j}^{(2)}=\sum_{m\neq j} \frac{V_{jm}V_{mj}}{E_{j}-E_{m}},\end{aligned}$$ where the summation is over eigenstates $\ket{m}$ of the non-interacting system, and $V_{jm}$ is the non-diagonal matrix element. In our case the symbol $j$ corresponds to the ground state and the symbol $m$ to excited states. The non-diagonal matrix element is $$\!\!V_{jm}\!=\!\frac{1}{2S}\left<\!m\left|\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}},{\mathbf{q}}} U({\mathbf{q}}) {\oldhat{a}}_{{\mathbf{k_{1}}}+{\mathbf{q}}}^{\dag}{\oldhat{a}}_{{\mathbf{k_{2}}}-{\mathbf{q}}}^{\dag}{\oldhat{a}}_{{\mathbf{k_{2}}}}{\oldhat{a}}_{{\mathbf{k_{1}}}}\right|j\!\right>. \!$$ This matrix element corresponds to the scattering of two particles from the initial state ${\mathbf{k_{1}}}$, ${\mathbf{k_{2}}}$ to an intermediate state ${\mathbf{k'_{1}}}$, ${\mathbf{k'_{2}}}$, and the matrix element $V_{mj}$ describes the reversed process in which the two particles return from the intermediate to initial state. Taking into account the momentum conservation law ${\bf k}_1+{\bf k}_2={\bf k}'_1+{\bf k}'_2$ the quantity $V_{jm}V_{mj}=|V_{jm}|^2$ is given by $$\begin{aligned}
|V_{jm}|^2&=\frac{1}{(2S)^2}n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}(1-n_{\mathbf{k_{1}'}})(1-n_{\mathbf{k_{2}'}})\nonumber\\
&\times\left|U({\mathbf{k'_{1}}}-{\mathbf{k_{1}}})-U({\mathbf{k'_{2}}}-{\mathbf{k_{1}}})\right|^2,\end{aligned}$$ and the second order correction to the ground state energy takes the form: $$\begin{aligned}
\label{E2}
E^{(2)}=&\frac{1}{(2S)^2}\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}},{\mathbf{k_{1}'}}} \Bigg[\left| U({\bf k}'_1-{\bf k}_1)-U({\bf k}'_2-{\bf k}_1)\right|^{2} \nonumber\\
&\times \frac{n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}(1-n_{\mathbf{k_{1}'}})(1-n_{\mathbf{k_{2}'}})}{\hbar^2({\mathbf{k^{2}_{1}}}+{\mathbf{k^{2}_{2}}}-{\mathbf{k'^{2}_{1}}}-{\mathbf{k'^{2}_{2}}})/2m}\Bigg].\end{aligned}$$
From Eq. (\[E2\]) we see that the second order correction diverges because of the term proportional to $n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}$, which is divergent at large $k'_1$. This artificial divergence is eliminated by expressing the energy correction in terms of a real physical quantity, the scattering amplitude. The relation between the Fourier component of the interaction potential and the off-shell scattering amplitude is given by [@Lan2]: $$\label{renorm}
f({\mathbf{k'}},{\mathbf{k}})=U({\mathbf{k}}'-{\mathbf{k}})+\frac{1}{S}\sum_{{\mathbf{k''}}}\frac{U({\mathbf{k}}'-{\mathbf{k''}})f({\mathbf{k''}},{\mathbf{k}})}{(E_{{\mathbf{k}}}-E_{{\mathbf{k''}}}-i0)},$$ where $E_{\mathbf{k}}=\hbar^2{\bf k}^2/m$ and $E_{\mathbf{k''}}=\hbar^2{\bf k}''^2/m$ are relative collision energies. Obviously, we have: $E_{{\mathbf{k}}}-E_{{\mathbf{k''}}}=\hbar^2({\mathbf{k_1}}^2+{\mathbf{k_2}}^2-{\mathbf{k''_1}}^2-{\mathbf{k''_2}}^2)/2m $, with ${\bf k}_1$, ${\bf k}_2$ (${\bf k}''_1$, ${\bf k}''_2$) being the momenta of colliding particles in the initial (intermediate) state, as the relative momenta are given by ${\mathbf{k}}=({\mathbf{k}}_1-{\mathbf{k}}_2)/2$, ${\mathbf{k''}}=({\mathbf{k''_1}}-{\mathbf{k''_2}})/2$. We thus can write: $$\label{ren}
\!\!\!U({\mathbf{k}}'\!-{\mathbf{k}})\!=\!f({\mathbf{k}}'\!,\!{\mathbf{k}})\!-\!\frac{2m}{\hbar^2 S}\sum_{{\mathbf{k_{1}''}}}\frac{U({\mathbf{k}}'\!-\!{\mathbf{k}}'')f({\mathbf{k''}}\!,\!{\mathbf{k}})}{{\mathbf{k^{2}_{1}}}\!+\!{\mathbf{k^{2}_{2}}}\!-\!{\mathbf{k''^{2}_{1}}}\!-\!{\mathbf{k''^{2}_{2}}}\!-\!i0}.\!\!$$
Then, putting ${\bf k}'={\bf k}$ we have $$\begin{aligned}
U(0)=f({\bf k},{\bf k})-\frac{2m}{\hbar^2S}\sum_{{\bf k}''_1}\frac{U({\bf k}-{\bf k}'')f({\bf k}'',{\bf k})}{{\mathbf{k^{2}_{1}}}+{\mathbf{k^{2}_{2}}}-{\mathbf{k''^{2}_{1}}}-{\mathbf{k''^{2}_{2}}}-i0}, \nonumber\end{aligned}$$ and setting ${\bf k}'=-{\bf k}$ we obtain $$\begin{aligned}
\!\!U({\bf k}_2\!-\!{\bf k}_1)=f(-{\bf k},{\bf k})\!-\!\frac{2m}{\hbar^2S}\sum_{{\bf k}''_1}\frac{U(-{\bf k}\!-\!{\bf k}'')f({\bf k}'',{\bf k})}{{\mathbf{k^{2}_{1}}}\!+\!{\mathbf{k^{2}_{2}}}\!-\!{\mathbf{k''^{2}_{1}}}\!-\!{\mathbf{k''^{2}_{2}}}\!-\!i0}, \nonumber\end{aligned}$$ Using these relations the first order correction (\[E1\]) takes the form: $$\begin{aligned}
\label{1str}
&E^{(1)}=\frac{1}{2S}\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}}}\left[f({\mathbf{k}},{\mathbf{k}})-f({\mathbf{-k}},{\mathbf{k}})\right]n_{{\mathbf{k_{1}}}}n_{{\mathbf{k_{2}}}} \nonumber \\
&\!\!-\frac{1}{2S^2}\!\!\!\!\!\sum_{{\mathbf{k_{1}}}, {\mathbf{k_{2}}},{\mathbf{k'_{1}}}}\!\!\frac{[U({\mathbf{k}}\!-\!{\mathbf{k'}})\!-\!U({\!-\bf k}\!-\!{\bf k}')]f({\mathbf{k'}},{\mathbf{k}})}{\hbar^2({\mathbf{k^{2}_{1}}}\!+\!{\mathbf{k^{2}_{2}}}\!-\!{\mathbf{k'^{2}_{1}}}\!-\!{\mathbf{k'^{2}_{2}}}\!-\!i0)/2m}n_{{\bf k}_1}n_{{\bf k}_2}. \!\!\end{aligned}$$
The quantity $[U({\bf k}-{\bf k}')-U(-{\bf k}-{\bf k}')]$ in the second term of Eq. (\[1str\]), being expanded in circular harmonics $\exp(il\varphi)$ contains terms with odd $l$. Therefore, partial amplitudes with even $l$ in the expansion of the multiple $f({\bf k}',{\bf k})$ vanish after the integration over $d^2k'$. Hence, this amplitude can be replaced by $[f({\bf k}',{\bf k})-f({\bf k}',-{\bf k})]/2$. As we are interested only in the terms that behave themselves as $\sim k$ or $\sim k^2$, the amplitudes in the second term of Eq. (\[1str\]) are the ones that follow from the first Born approximation and are proportional to $k$. Therefore, we may put $[U({\bf k}-{\bf k}')-U(-{\bf k}-{\bf k}')]=[f({\bf k},{\bf k}')-f(-{\bf k},{\bf k}')]$ and $f({\bf k}',{\bf k})=f^*({\bf k},{\bf k}')$. Then the first order correction takes the form: $$\begin{aligned}
\label{1str1}
&E^{(1)}=\frac{1}{2S}\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}}}\left[f({\mathbf{k}},{\mathbf{k}})-f({\mathbf{-k}},{\mathbf{k}})\right]n_{{\mathbf{k_{1}}}}n_{{\mathbf{k_{2}}}}-\frac{1}{(2S)^2}\nonumber \\
&\times\!\!\sum_{{\mathbf{k_{1}}}, {\mathbf{k_{2}}},{\mathbf{k'_{1}}}}\frac{|f({\mathbf{k'}},{\mathbf{k}})\!-\!f({\bf k}',-{\bf k})|^2}{\hbar^2({\mathbf{k^{2}_{1}}}\!+\!{\mathbf{k^{2}_{2}}}\!-\!{\mathbf{k'^{2}_{1}}}\!-\!{\mathbf{k'^{2}_{2}}}\!-\!i0)/2m}n_{{\bf k}_1}n_{{\bf k}_2}.\end{aligned}$$
Using the expansion of the full scattering amplitude in terms of partial amplitudes as given by Eq. (\[ffermionsum\]) we represent the first order correction as $$\begin{aligned}
\label{E1l}
&E^{(1)}=\frac{1}{S}\sum_{{\bf k}_1,{\bf k}_2}\sum_{l\,odd}f_l(k)n_{{\bf k}_1}n_{{\bf k}_2}-\frac{1}{S^2}\sum_{{\bf k}_1,{\bf k}_2}\sum_{l\,odd}\nonumber\\
&\!\times\!\!\int\! \frac{d^2k'}{(2\pi)^2}\frac{f_l^2(k)}{\hbar^2({\mathbf{k^{2}_{1}}}\!+\!{\mathbf{k^{2}_{2}}}\!-\!{\mathbf{k'^{2}_{1}}}\!-\!{\mathbf{k'^{2}_{2}}}\!-\!i0)/2m}n_{{\bf k}_1}n_{{\bf k}_2}.\!\!\end{aligned}$$ The contribution of the pole in the integration over $d^2k'$ in the second term of Eq. (\[E1l\]) gives $imf_l^2(k)/4\hbar^2$ for each term in the sum over ${\bf k}_1$, ${\bf k}_2$, and $l$, and we may use here the amplitude ${\bar f}^{(1)}_l(k)$. In the first term of Eq. (\[E1l\]) we should use $f_l(k)=f_l^{(1)}(k)+f_l^{(2)}(k)$. However, we may replace $f_l^{(2)}$ by ${\bar f}_l^{(2)}$ because the account of $\tan\delta(k)$ in the denominator of Eq. (\[barfloff\]) leads to $k^3$ terms and terms containing higher powers of $k$. For the amplitude $f_l^{(1)}(k)$, we use the expression: $$f_l^{(1)}(k)={\bar f}^{(1)}_l+i\tan\delta(k){\bar f}^{(1)}_l={\bar f}^{(1)}_l-im[{\bar f}^{(1)}_l]^2/4\hbar^2,$$ which assumes a small scattering phase shift. The second term of this expression, being substituted into the first line of Eq. (\[E1l\]), exactly cancels the contribution of the pole in the second term of (\[E1l\]). Thus, we may use the amplitude ${\bar f}_l$ in the first term of equation (\[E1l\]) and take the principal value of the integral in the second term. The resulting expression for the first order correction reads: $$\begin{aligned}
\label{1str2}
E^{(1)}&=\frac{1}{S}\sum_{{\bf k}_1,{\bf k}_2}{\bar f}({\bf k})n_{{\bf k}_1}n_{{\bf k}_2}-\frac{1}{(2S)^2}\nonumber\\
&\times\sum_{{\mathbf{k_{1}}}, {\mathbf{k_{2}}},{\mathbf{k'_{1}}}} \frac{2m|f({\mathbf{k}},{\mathbf{k'}})-f({-\bf k},{\bf k}')|^2}{\hbar^2({\mathbf{k^{2}_{1}}}+{\mathbf{k^{2}_{2}}}-{\mathbf{k'^{2}_{1}}}-{\mathbf{k'^{2}_{2}}})}n_{{\bf k}_1}n_{{\bf k}_2},\end{aligned}$$ where ${\bar f}({\bf k})=\sum_{l\,\,odd}{\bar f}_l(k)$.
The second order correction (\[E2\]) can also be expressed in terms of the scattering amplitude by using Eq.(\[renorm\]). Replacing $U({\bf k}_1-{\bf k}'_1)=U({\bf k}-{\bf k}')$ and $U({\bf k}'_2-{\bf k}_1)=U(-{\bf k}-{\bf k}')$ by $f({\bf k}',{\bf k})$ and $f(-{\bf k},{\bf k}')$, respectively, we have: $$\begin{aligned}
\label{2ndr}
E^{(2)}=\frac{1}{(2S)^2}&\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}},{\mathbf{k_{1}'}}}\Big[\frac{ \left| f({\mathbf{k'}},{\mathbf{k}})- f({\mathbf{k'}},-{\mathbf{k}})\right|^{2}}{\hbar^2({\mathbf{k^{2}_{1}}}+{\mathbf{k^{2}_{2}}}-{\mathbf{k'^{2}_{1}}}-{\mathbf{k'^{2}_{2}}})/2m}\nonumber \\
&\times n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}(1-n_{\mathbf{k_{1}'}})(1-n_{\mathbf{k_{2}'}})\Big].\end{aligned}$$
Note that the divergent term proportional to $n_{{\bf k}_1} n_{{\bf k}_2}$ in Eq. (\[2ndr\]) and the (divergent) second term of Eq. (\[1str2\]) exactly cancel each other, and the sum of the first and second order corrections can be represented as $E^{(1)}+E^{(2)}=\tilde E^{(1)}+\tilde E^{(2)},$ where $$\label{tildeE1}
\tilde E^{(1)}=\frac{1}{S}\sum_{{\bf k}_1,{\bf k}_2}{\bar f}({\bf k})n_{{\bf k}_1}n_{{\bf k}_2},$$ and $$\begin{aligned}
\label{tilde2}
\tilde E^{(2)}=\frac{1}{(2S)^2}&\sum_{{\mathbf{k_{1}}}, {\mathbf{k_{2}}},{\mathbf{k'_{1}}}}\Big\{\frac{|f({\mathbf{k'}},{\mathbf{k}})-f({\bf k'},-{\bf k})|^2}{\hbar^2({\mathbf{k^{2}_{1}}}+{\mathbf{k^{2}_{2}}}-{\mathbf{k'^{2}_{1}}}-{\mathbf{k'^{2}_{2}}})/2m} \nonumber\\
&\times n_{{\bf k}_1}n_{{\bf k}_2}[(1-n_{{\bf k}'_1})(1-n_{{\bf k}'_2})-1]\Big\}.\end{aligned}$$ The term $\tilde E^{(1)}$ originates from the two-body contributions to the interaction energy and can be quoted as the mean field term. The term $\tilde E^{(2)}$ is the many-body contribution, which is beyond mean field.
It is worth noting that the term proportional to the product of four occupation numbers vanishes because its numerator is symmetrical and the denominator is antisymmetrical with respect to an interchange of ${\bf k}_1,{\bf k}_2$ and ${\bf k}'_1,{\bf k}'_2$. The terms containing a product of three occupation numbers, $n_{{\bf k}_1}n_{{\bf k}_2}n_{{\bf k}'_1}$ and $n_{{\bf k}_1}n_{{\bf k}_2}n_{{\bf k}'_2}$ are equal to each other because the denominator is symmetrical with respect to an interchange of ${\bf k}'_1$ and ${\bf k}'_2$. We thus reduce Eq. (\[tilde2\]) to $$\label{tildeE2}
\!\!\!\!\tilde E^{(2)}\!\!=\!\!-\frac{1}{2S^2}\!\!\!\!\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}},{\mathbf{k'_{1}}}}\!\!\!\frac{2m|f({\mathbf{k'}},{\mathbf{k}})\!-\!f({\bf k'},\!-{\bf k})|^2}{\hbar^2({\mathbf{k^{2}_{1}}}\!+\!{\mathbf{k^{2}_{2}}}\!-\!{\mathbf{k'^{2}_{1}}}\!\!-\!{\mathbf{k'^{2}_{2}}})}n_{{\bf k}_1}\!n_{{\bf k}_2}\!n_{{\bf k}'_1}. \!\!\!\!$$ Equations (\[tildeE1\]) and (\[tildeE2\]) allow a direct calculation of the ground state energy. With respect to the mean field term $\tilde E^{(1)}$ this is done in Appendix \[App1\]. However, a direct calculation of the many-body correction $\tilde E^{(2)}$ is even a more tedious task than in the case of two-component fermions with a contact interaction. We therefore turn to the Abrikosov-Khalatnikov idea of calculating the ground state energy (and other thermodynamic quantities) through the interaction function of quasiparticles on the Fermi surface.
Interaction function of quasiparticles
--------------------------------------
The interaction function of quasiparticles $F({\bf k},{\bf k}')$ is the second variational derivative of the total energy with respect to the distribution $n_{{\bf k}}$. The kinetic energy of our system is linear in $n_{{\bf k}}$ (see Eq. (\[ha\])), and the second variational derivative is related to the variation of the interaction energy $\tilde E$. We have [@Landau]: $$\label{tildeF}
\delta \tilde E=\frac{1}{2S}\sum_{{\bf k},{\bf k}'}F({\bf k},{\bf k}')\delta n_{{\mathbf{k}}} \delta n_{{\mathbf{k'}}},$$ where $\tilde E=\tilde E^{(1)}+\tilde E^{(2)}$, and the quantities $\tilde E^{(1)}$ and $\tilde E^{(2)}$ are given by equations (\[tildeE1\]) and (\[tildeE2\]). On the Fermi surface we should put $|{\bf k}|=|{\bf k}'|=k_F$, so that the interaction function will depend only on the angle $\theta$ between ${\bf k}$ and ${\mathbf{k'}}$. Hereinafter it will be denoted as $\tilde F(\theta)$.
The contribution $\tilde F^{(1)}(\theta)=2S\delta\tilde E^{(1)}/\delta n_{{\bf k}}\delta n_{{\bf k}'}$ is given by $$\label{tildef1}
\tilde F^{(1)}(\theta)=2f\left(\frac{|{\bf k}-{\bf k}'|}{2}\right)=2\sum_{l\,odd}{\bar f}_l\left(k_F|\sin{\frac{\theta}{2}}|\right),$$ where ${\bar f}_l={\bar f}_l^{(1)}+{\bar f}_l^{(2)}$, and the amplitudes ${\bar f}_l^{(1)}$ and ${\bar f}_l^{(2)}$ follow from Eqs. (\[f1leading\]) and (\[f12\]) at $|l|=1$, and from Eqs. (\[flleading\]), (\[fl2\]) at $|l|>1$. We thus may write equation (\[flleading\]), $${\bar f}^{(1)}_{l}(k)=\frac{8\hbar^2}{m}\frac{1}{4 l^2-1} kr_*,$$ for any odd $l$, and $$\begin{aligned}
{\bar f}^{(2)}_l(k)=\frac{\pi\hbar^2}{2m}(kr_*)^2\times
\begin{cases}
\ln(\xi kr_{*}); & \text{$|l|=1$}\\
-\frac{3}{|l|(l^2-1)(4l^2-1)}; & \text{$|l|>1$}
\end{cases} \nonumber
\end{aligned}$$ with $\xi$ from Eq. (\[xi\]). Making a summation over all odd $l$ we obtain: $${\bar f}^{(1)}(k)=\sum_{l\,odd} f^{(1)}_{l}(k)=\frac{2\pi\hbar^2}{m}kr_*, \label{barf1} \\$$ $$\!\!\!{\bar f}^{(2)}(k)\!=\!\!\!\sum_{l\,odd} f^{(2)}_{l}(k)\!\!=\!\!\frac{\pi\hbar^2}{m}(kr_{*}\!)^2\!\!\left[\ln(\xi kr_{*}\!)\!-\!\frac{25}{12}\!+\!3\!\ln 2\!\right]\!\!.\!\!\!\label{barf2}$$ Putting $k=k_F|\sin(\theta/2)|$ and substituting the results of equations (\[barf1\]) and (\[barf2\]) into Eq. (\[tildef1\]) we find: $$\begin{aligned}
\tilde F^{(1)}(\theta)&=\frac{4\pi\hbar^2 k_Fr_{*}}{m}|\sin\frac{\theta}{2}| +\frac{2\hbar^2}{m}(k_Fr_*)^2 \nonumber \\
&\times\pi\sin^2\frac{\theta}{2}\left[\ln|\xi r_{*}k_F\sin\frac{\theta}{2}|-\frac{25}{12}+3\ln 2\right]. \label{tildeF1}\end{aligned}$$
The many-body correction (\[tildeE2\]) we represent as $\tilde E^{(2)}=\tilde E_1^{(2)}+\tilde E_2^{(2)}$, where $$\begin{aligned}
\!\!\!&\tilde E_1^{(2)}\!\!\!=\!\!-\frac{8(\pi\hbar r_*)^2}{mS^2}\!\!\!\!\!\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}},{\mathbf{k_{1}'}}}\!\!\!\frac{|{\mathbf{k'_1}}\!-\!{\mathbf{k_1}}|^2}{{\mathbf{k^{2}_{1}}}\!+\!{\mathbf{k^{2}_{2}}}\!-\!{\mathbf{k'^{2}_{1}}}\!-\!{\mathbf{k'^{2}_{2}}}}n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}n_{\mathbf{k_{1}'}}\!, \!\!\!\label{tildeE12} \\
\!\!\!&\tilde E_2^{(2)}\!\!\!=\!\frac{8(\pi\hbar r_*)^2}{mS^2}\!\!\!\!\!\sum_{{\mathbf{k_{1}}},{\mathbf{k_{2}}},{\mathbf{k_{1}'}}}\!\!\!\frac{|{\mathbf{k_1}}\!-\!{\mathbf{k'_1}}|\!\cdot\!|{\mathbf{k_2}}\!-\!{\mathbf{k'_1}}|}{{\mathbf{k^{2}_{1}}}\!+\!{\mathbf{k^{2}_{2}}}\!-{\mathbf{k'^{2}_{1}}}\!-\!{\mathbf{k'^{2}_{2}}}}n_{\mathbf{k_{1}}}n_{\mathbf{k_{2}}}n_{\mathbf{k_{1}'}}\!, \!\!\label{tildeE22}\end{aligned}$$ and we used Eqs (\[ffermion\]) and (\[tildefBorn\]) for the scattering amplitudes. The contribution to the interaction function from $\tilde E_1^{(2)}$ is calculated in Appendix \[App2\] and it reads: $$\label{tildeF12}
\!\!\!\tilde F_1^{(2)}\!(\theta)\!\!=\!\!\frac{2\hbar^2(k_Fr_*\!)^2}{m}\!\left[\!3\pi\!+\!2\pi\sin^2\frac{\theta}{2}\left(\!\frac{4}{3}\!-\!\ln\!|\tan \frac{\theta}{2}|\right)\right]\!.\!\!$$ The contribution from $\tilde E_2^{(2)}$ is calculated in Appendix \[F22\]. It is given by
$$\begin{aligned}
\tilde{F}^{(2)}_2(\theta)=&\frac{2\hbar^2k^2_Fr^2_*}{m}\Big\{-\sin^2\frac{\theta}{2}\left(\pi\ln2+\frac{\pi}{2}-\pi\ln|\sin\frac{\theta}{2}|+4\ln|\cos\frac{\theta}{2}|-4\ln(1+|\sin\frac{\theta}{2}|)+\mathcal{G}(\theta)+
\frac{4\arcsin|\sin\frac{\theta}{2}|-2\pi}{|\cos\frac{\theta}{2}|} \right) \nonumber\\
&-\frac{1}{|\cos\frac{\theta}{2}|}\left(\pi-2\arcsin|\sin\frac{\theta}{2}|+|\sin\theta|\right)-4\left[\cos^2\frac{\theta}{2}\ln\frac{1+|\sin(\theta/2)|}{1-|\sin(\theta/2)|}+2|\sin\frac{\theta}{2}|\right]\Big\}, \label{tildeF22}\end{aligned}$$
where $$\label{G}
\mathcal{G}(\theta)=\int_{0}^{\pi} 2\sin^2\varphi \ln\left(\sin\varphi+\sqrt{\sin^2\frac{\theta}{2}+\cos^2\frac{\theta}{2}\sin^2\varphi}\right) d\varphi,$$ so that $$\label{dGdtheta}
\frac{d\mathcal{G}(\theta)}{d\theta}=\frac{\pi}{2}\cot\frac{\theta}{2}-\frac{|\sin(\theta/2)|}{\sin(\theta/2)}\frac{1}{\cos(\theta/2)}+\frac{\arcsin|\cos(\theta/2)|}{|\cos(\theta/2)|}\left(\tan\frac{\theta}{2}-\cot\frac{\theta}{2}\right).$$
We thus have $\tilde F(\theta)=\tilde{F}^{(1)}(\theta)+\tilde{F}^{(2)}_1(\theta)+\tilde{F}^{(2)}_2(\theta)$, where $\tilde F^{(1)}$, $\tilde F^{(2)}_1$, $\tilde F^{(2)}_2$ follow from Eqs. (\[tildeF1\]), (\[tildeF12\]), and (\[tildeF22\]). This allows us to proceed with the calculation of thermodynamic quantities.
Compressibility, ground state energy, and effective mass
--------------------------------------------------------
We first calculate the compressibility at $T=0$. On the basis of Eq. (\[u1\]) we obtain:
$$\begin{aligned}
\label{dmudN}
&\frac{\partial\mu}{\partial N}=\frac{2\pi\hbar^{2}}{mS}+\frac{1}{2\pi S}\int (1-\cos\theta) \left[\tilde{F}^{(1)}(\theta)+\tilde{F}_1^{(2)}(\theta)+\tilde{F}_2^{(2)}(\theta)\right] d\theta \nonumber\\
&=\frac{2\pi\hbar^2}{mS}+\frac{32\hbar^2}{3 mS}k_Fr_*+\frac{3\pi\hbar^2}{2mS}(k_Fr_*)^2\left(\ln[4\xi k_Fr_*]-\frac{3}{2}\right)+\frac{6\pi\hbar^2}{mS}(k_Fr_*)^2-\frac{\hbar^2}{\pi mS}(k_Fr_*)^2(30-8G+21\zeta(3)),\end{aligned}$$
where $G=0.915966$ is the Catalan constant, and $\zeta(3)=1.20206$ is the Riemann zeta function. Calculating coefficients and recalling that $k_F=\sqrt{4\pi N/S}$ we represent the inverse compressibility following from Eq. (\[kappa-1\]) in a compact form: $$\label{kappa*}
\!\!\kappa^{-1}\!=\!\frac{\hbar^2k_F^2}{2m}\frac{N}{S}\!\left(\!1\!+\!\frac{16}{3\pi}k_Fr_*\!+\!\frac{3}{4}(k_Fr_*)^2\ln(\zeta_1k_Fr_*)\right)\!,\!\!\!\!$$ where we obtain the coefficient $\zeta_1=2.16\exp(-2A)$ by using Eq. (\[xi\]) for the coefficient $\xi$ which depends on the short-range behavior through the constant $A$ (see Eq. (\[short\])). For the chemical potential and ground state energy we obtain:
$$\begin{aligned}
\mu&=&\frac{2\pi\hbar^2N}{mS}+\frac{64\hbar^2N}{9mS}k_Fr_*+\frac{3\pi\hbar^2N}{4mS}(k_Fr_*)^2\left(\ln[4\xi k_Fr_*]-\frac{7}{4}\right)+\frac{3\pi\hbar^2N}{mS}(k_Fr_*)^2-\frac{\hbar^2N}{2\pi mS}(k_Fr_*)^2(30-8G+21\zeta(3)) \nonumber \\
&=&\frac{\hbar^2k_F^2}{2m}\left(1+\frac{32}{9\pi}k_Fr_*+\frac{3}{8}(k_Fr_*)^2\ln(\zeta_2k_Fr_*)\right). \label{mug}\end{aligned}$$
$$\begin{aligned}
\frac{E}{N}&=&\frac{\pi\hbar^2N}{mS}+\frac{128\hbar^2N}{45mS}k_Fr_*+\frac{\pi\hbar^2N}{4mS}(k_Fr_*)^2\left(\ln[4\xi k_Fr_*]-\frac{23}{12}\right)+\frac{\pi\hbar^2N}{mS}-\frac{\hbar^2N}{6\pi mS}(30-8G+21\zeta(3)) \nonumber \\
&=&\frac{\hbar^2k_F^2}{4m}\left(1+\frac{128}{45\pi}k_Fr_*+\frac{1}{4}(k_Fr_*)^2\ln(\zeta_3k_Fr_*)\right), \label{Eg}\end{aligned}$$
with numerical coefficients $\zeta_2=1.68\exp(-2A)$ and $\zeta_3=1.43\exp(-2A)$. Note that the first term in the second line of Eq. (\[dmudN\]) and the first terms in the first lines of Eqs. (\[mug\]) and (\[Eg\]) represent the contributions of the kinetic energy, the second and third terms correspond to the contributions of the mean field part of the interaction energy, and the last two terms are the contributions of the many-body effects.
The effective mass is calculated in a similar way by using Eq. (\[eff\]): $$\begin{aligned}
\!\!\!\!\!\!&\frac{1}{m^*}\!=\!\frac{1}{m}\!-\!\frac{1}{(2\pi\hbar)^{2}}\int_{0}^{2\pi}\!(F^{(1)}(\theta)\!+\!F^{(2)}_1(\theta)\!+\!F^{(2)}_2(\theta))\cos\theta d\theta\!\! \nonumber \\
\!\!\!\!\!\!&\!=\!\!\frac{1}{m}\!\!\left[\!1\!\!+\!\frac{4k_Fr_{\!*}}{3\pi}\!+\!\frac{(k_Fr_{\!*})^{\!2}}{4}\!\!\left(\!\ln{\![4k_Fr_{\!*}\xi]}\!\!-\!\frac{8}{3}\!+\!\frac{48G\!\!-\!\!20\!-\!\!14\zeta(3)}{\pi^2}\!\right)\!\right]\!\! \nonumber \\
\!\!\!\!\!\!&\!=\frac{1}{m}\left[1+\frac{4}{3\pi}k_Fr_*+\frac{1}{4}(k_Fr_*)^2\ln(\zeta_4k_Fr_*)\right], \label{mg*}\end{aligned}$$ where the numerical coefficient $\zeta_4=0.65\exp(-2A)$. Note that if the potential $U(r)$ has the dipole-dipole form (\[VO\]) up to very short distances, we have to put $A=0$ in the expressions for the coefficients $\zeta_1,\,\zeta_2,\,\zeta_3,\,\zeta_4$. Considering the quasi2D regime, this will be the case for $r_*$ greatly exceeding the length of the sample in the tightly confined direction, $l_0$. Then, as one can see from equations (\[kappa\*\]), (\[mug\]), (\[Eg\]), and (\[mg\*\]), the terms proportional to $(k_Fr_*)^2$ are always negative in the considered limit $k_Fr_*\ll 1$. These terms may become significant for $k_Fr_*>0.3$.
Zero sound
==========
In the collisionless regime of the Fermi liquid at very low temperatures, where the frequency of variations of the momentum distribution function greatly exceeds the relaxation rate of quasiparticles, one has zero sound waves. For these waves, variations $\delta n({\bf q},{\bf r},t)$ of the momentum distribution are related to deformations of the Fermi surface, which remains a sharp boundary between filled and empty quasiparticle states. At $T\rightarrow 0$ the equilibrium distribution $n_{\bf q}$ is the step function (\[nstep\]), so that $\partial n_{\bf q}/\partial{\bf q}=-{\bf n}\delta(q-k_F)=-\hbar{\bf v}\delta(\epsilon_q-\epsilon_F)$, where ${\bf v}=v_F{\bf n}$, with ${\bf n}$ being a unit vector in the direction of ${\bf q}$. Then, searching for the variations $\delta n$ in the form: $$\delta n({\bf q},{\bf r},t)=\delta(\epsilon_q-\epsilon_F)\nu({\bf n})\exp{i({\bf kr}-\omega t)}$$ and using Eq. (\[1\]), from the kinetic equation in the collisionless regime: $$\begin{aligned}
\frac{\partial \delta n}{\partial t}+{\mathbf{v}}\cdot\frac{\partial \delta n}{\partial {\mathbf{r}}} -\frac{\partial n_{\bf q}}{\partial {\mathbf{q}}}\cdot\frac{\partial\delta\epsilon_q}{\hbar\partial {\mathbf{r}}}=0, \nonumber\end{aligned}$$ one obtains an integral equation for the function $\nu({\bf n})$ representing displacements of the Fermi surface in the direction of ${\bf n}$ [@Landau]: $$\begin{aligned}
(\omega-v_F{\mathbf{n}}\cdot{\mathbf{k}})\nu({\mathbf{n}})=\frac{k_F}{(2\pi)^2\hbar}{\mathbf{n}}\cdot{\mathbf{k}}\int F(k_F{\mathbf{n}},k_F{\mathbf{n'}})\nu ({\mathbf{n'}})d{\bf n}'. \nonumber\end{aligned}$$ Introducing the velocity of zero sound $u_0=\omega/k$ and dividing both sides of this equation by $v_Fk$ we have: $$\label{inteq1}
(s-\cos\theta)\nu(\theta)=\frac{m^*\cos\theta}{(2\pi\hbar)^2}\int_0^{2\pi}\tilde F(\theta-\theta')\nu(\theta')d\theta',$$ where $s=u_0/v_F$, and $\theta,\,\theta'$ are the angles between ${\bf k}$ and ${\bf n},\,{\bf n}'$, so that $\theta-\theta'$ is the angle between ${\bf n}$ and ${\bf n}'$. The dependence of the interaction function of quasiparticles $\tilde F=\tilde F^{(1)}+\tilde F^{(2)}_1+\tilde F^{(2)}_2$ on $(\theta-\theta')$ follows from Eqs. (\[tildeF1\]), (\[tildeF12\]), and (\[tildeF22\]) in which one has to replace $\theta$ by $(\theta-\theta')$.
The solution of equation (\[inteq1\]) gives the function $\nu(\theta)$ and the velocity of zero sound $u_0$, and in principle one may obtain several types of solutions. It is important to emphasize that undamped zero sound requires the condition $s>1$, i.e. the sound velocity should exceed the Fermi velocity [@Landau]. We will discuss this issue below.
For solving Eq. (\[inteq1\]) we represent the interaction function $\tilde F$ as a sum of the part proportional to $k_Fr_*$ and the part proportional to $(k_Fr_*)^2$. As follows from Eqs. (\[tildeF1\]), (\[tildeF12\]), and (\[tildeF22\]), we have: $$\label{kandkk}
\!\!\!\!\tilde F(\theta\!-\!\theta')\!=\!\frac{4\pi\hbar^2}{m}k_Fr_*\!\left|\sin\frac{\theta\!-\!\theta'}{2}\right|+\frac{2\hbar^2}{m}(k_Fr_*\!)^2\Phi(\theta\!-\!\theta'),\!\!\!\!$$ where the function $\Phi(\theta-\theta')$ is given by the sum of three terms. The first one is the term in the second line of Eq. (\[tildeF1\]), the second term is the expression in the square brackets in Eq. (\[tildeF12\]), the third term is the one in curly brackets in Eq. (\[tildeF22\]), and we should replace $\theta$ by $(\theta-\theta')$ in all these terms. It is important that the function $\Phi(\theta-\theta')$ does not have singularities and $\Phi(0)=\Phi(\pm 2\pi)=2\pi$. Using Eq. (\[kandkk\]) the integral equation (\[inteq1\]) is reduced to the form: $$\begin{aligned}
(s-\cos\theta)\nu(\theta)&=\beta\cos\theta\int_0^{2\pi}\nu(\theta')\left|\sin\frac{\theta-\theta'}{2}\right|d\theta' \nonumber \\
&+\frac{\beta^2m}{2m^*}\cos\theta\int_0^{2\pi}\nu(\theta')\Phi(\theta-\theta')d\theta',\label{inteq2}\end{aligned}$$ where $\beta=(m^*/\pi m)k_Fr_*\ll 1$.
We now represent the function $\nu(\theta)$ as $$\label{nucos}
\nu(\theta)=\sum_{p=0}^{\infty}C_p\cos p\theta.$$ Then, integrating over $d\theta'$ in Eq. (\[inteq2\]), multiplying both sides of this equation by $\cos{j\theta}$ and integrating over $d\theta$, we obtain a system of linear equations for the coefficients $C_j$. We write this system for the coefficients $\eta_j=C_j(1-\beta/(j^2-1/4))$, so that $C_j=\eta_j(1+\beta U_j)$, where $U_j=(j^2-1/4-\beta)^{-1}$. The system reads: $$\begin{aligned}
&\!\!\!\!(s-1)(1+\beta U_0)\eta_0+[\eta_0-\frac{1}{2}\eta_1]+\beta U_0\eta_0=\frac{\beta^2}{2}{\bar \Phi}_0;\! \label{systemf0}\\
&\!\!\!\!(s\!-\!1)(1\!+\!\beta U_1)\eta_1\!+\![\eta_1-\eta_0-\frac{1}{2}\eta_2]+\beta U_1\eta_1\!\!=\!\frac{\beta^2}{2}{\bar \Phi}_1;\label{systemf1} \\
&\!\!\!\!(\!s\!\!-\!\!1\!)(\!1\!\!+\!\!\beta U_j\!)\eta_j\!\!+\!\![\eta_j\!\!-\!\!\frac{1}{2}\!(\!\eta_{j\!-\!1}\!\!+\!\eta_{j\!+\!1)\!}\!]\!\!+\!\!\beta U_j\eta_j\!\!=\!\!\frac{\beta^2}{2}\!{\bar \Phi}_j\!;\,j\!\!\geq\!\!2,\!\!\!\!\!\!\!\!\!\!\!\label{systemfj}\end{aligned}$$ where $$\label{Phij}
\!\!{\bar \Phi}_j\!=\!\frac{\tilde C_j}{\pi}\!\!\int_0^{2\pi}\!\!\!\!\!\!\!\!\cos\theta\,\cos j\theta d\theta\!\!\int_0^{2\pi}\!\!\sum_{p=0}^{\infty}C_p\cos p\theta'\Phi(\theta-\theta')d\theta'\!,\!\!\!\!\!\!\!$$ with $\tilde C_j=1$ for $j\geq 1$ and $\tilde C_0=1/2$, and we put $m^*=m$ in the terms proportional to $\beta^2$.
In the weakly interacting regime the velocity of zero sound is close to the Fermi velocity and, hence, we have $(s-1)\ll 1$ (see, e.g. [@Landau]). Since $\beta\ll 1$, we first find coefficients $\eta_j$ omitting the terms proportional to $\beta$ and $\beta^2$ in Eqs. (\[systemf0\])-(\[systemfj\]). For $j\gg 1$ equation (\[systemfj\]) then becomes: $$(s-1)\eta_j-\frac{1}{2}\frac{d^2\eta_j}{dj^2}=0,$$ and searching for $s>1$ we may write $$\label{fjbig}
\eta_j\simeq\exp\{-\sqrt{2(s-1)}j\};\,\,\,j\gg 1.$$ If $j\ll 1/\sqrt{s-1}$, then we may also omit the terms proportional to $(s-1)$ in the system of linear equations for $\eta_j$ (\[systemf0\])-(\[systemfj\]). The system then takes the form: $$\begin{aligned}
&\eta_0-\frac{1}{2}\eta_1=0; \nonumber \\
&\eta_1-\eta_0-\frac{1}{2}\eta_2=0; \nonumber \\
&\eta_j-\frac{1}{2}[\eta_{j-1}+\eta_{j+1}]=0;\,\,\,\,j\geq 2. \nonumber\end{aligned}$$ Without loss of generality we may put $\eta_0=1/2$. This immediately gives $\eta_j=1$ for $j\geq 1$, which is consistent with Eq. (\[fjbig\]) at $j\ll 1/\sqrt{s-1}$. We thus have the zero order solution: $$\label{fzo}
\begin{cases}
\eta_0=1/2; \\
\eta_j=1;\,\,\,1\leq j\ll 1/\sqrt{s-1}.
\end{cases}$$
In order to find the coefficients $\eta_j$ taking into account the terms linear in $\beta$, we consider $j$ such that $\beta U_j\sim\beta/j^2\gg (s-1)$, i.e. $j\ll\sqrt{\beta/(s-1)}$. Then we may omit the terms proportional to $(s-1)$ in equations (\[systemf0\])-(\[systemfj\]). Omitting also the terms proportional to $\beta^2$ this system of equations becomes: $$\begin{aligned}
&\eta_0-\frac{1}{2}\eta_1+\beta U_0\eta_0=0; \label{system0} \\
&\eta_1-\eta_0-\frac{1}{2}\eta_2+\beta U_1\eta_1=0; \label{system1} \\
&\eta_j-\frac{1}{2}[\eta_{j-1}+\eta_{j+1}]+\beta U_j\eta_j=0;\,\,j\geq 2. \label{systemj}\end{aligned}$$ Putting again $\eta_0=1/2$ the solution of these equations reads: $$\begin{aligned}
&\eta_1=1+\beta U_0; \nonumber \\
&\eta_j=1+\beta jU_0+2\beta\sum_{p=1}^{j-1}(j-p)U_p;\,\,\,j\geq 2. \nonumber\end{aligned}$$ Confining ourselves to terms linear in $\beta$ we put $U_p=1/(p^2-1/4)$ and, hence, $U_0=-4$. Then, using the relation $$\sum_{p=1}^{j-1}\frac{1}{p^2-1/4}=\frac{4(j-1)}{2j-1},$$ which is valid for $j\geq 2$, we obtain: $$\begin{aligned}
\label{fbeta}
\begin{cases}
&\eta_0=\frac{1}{2}; \\
\\
&\eta_1=1-4\beta; \\
\\
&\eta_j=1-2\beta\left\{\frac{2j}{2j-1}+\sum_{p=1}^{j-1}\frac{p}{p^2-1/4}\right\};\,\,\,j\geq 2.
\end{cases}\end{aligned}$$ For $j\gtrsim\sqrt{\beta/(s-1)}$ we should include the terms proportional to $(s-1)$ in Eq. (\[systemj\]). This leads to the solution in the form of the decaying Bessel function: $\eta_j\simeq\sqrt{2(s-1)/\pi}K_{\sqrt{1/4+\beta}}(\sqrt{s-1}j)$, which for small $\beta$ is practically equivalent to Eq. (\[fjbig\]).
We now make a summation of equations (\[systemf0\])-(\[systemfj\]) from $j=0$ to $j=j_*\ll1/\sqrt{s-1}$. The summation of the second terms of these equations gives $\sqrt{(s-1)/2}$, whereas the contribution of the terms proportional to $(s-1)$ is much smaller and will be omitted. The sums $\sum_{j=0}^{j_*}U_j\eta_j$ and $\sum_{j=0}^{j_*}{\bar \Phi}_j$ converge at $j\ll 1/\sqrt{s-1}$, and the upper limit of summation in these terms can be formally replaced by infinity. We thus obtain a relation: $$\label{sum1}
\sqrt{\frac{s-1}{2}}+\sum_{j=0}^{\infty}\beta\eta_jU_j-\frac{\beta^2}{2}\sum_{j=0}^{\infty}{\bar \Phi}_j=0.$$ Confining ourselves to contributions up to $\beta^2$, in the second term on the left hand side of Eq. (\[sum1\]) we use coefficients $\eta_j$ given by Eqs. (\[fbeta\]), and write $U_j=1/(j^2-1/4)+\beta/(j^2-1/4)^2$. In the expressions for ${\bar \Phi}_j$ we use $C_0=1/2$ and $C_p=1$ for $p\geq 1$. We then have: $$\!\!\sum_{j=0}^{\infty}\beta\eta_jU_j\!=\!-2\beta\!+\!\sum_{j=1}^{\infty}\frac{\beta}{\!j^2\!-\!1/4}\!+\!\beta^2\{\!8\!+\!S_1\!-\!2S_2\!-2S_3\!\}. \!\!\!\!\label{sum2}$$ The contribution linear in $\beta$ vanishes because $\sum_{j=1}^{\infty}1/(j^2-1/4)=2$. The quantities $S_1,\,S_2$, and $S_3$ are given by $$\begin{aligned}
&\!\!\!S_1=\sum_{j=1}^{\infty}\frac{1}{(j^2-1/4)^2}=\pi^2-8; \nonumber \\
&\!\!\!S_2\!=\!\sum_{j=2}^{\infty}\frac{1}{j^2\!-\!1/4}\!\sum_{p=1}^{j-1}\frac{p}{p^2\!-\!1/4}\!=\!\sum_{j=1}\frac{j}{(j^2\!-\!1/4)(j\!+\!1/2)}; \nonumber \\
&\!\!\!S_3=\sum_{j=1}^{\infty}\frac{j}{(j-1/2)(j^2-1/4)}, \nonumber\end{aligned}$$ so that $$S_2+S_3=\sum_{j=1}^{\infty}\frac{2j^2}{(j^2-1/4)^2}=\frac{\pi^2}{2}.$$ We thus see that the contribution quadratic in $\beta$ also vanishes because the term in the curly brackets in Eq. (\[sum2\]) is exactly equal to zero. Hence, we have $\sum_{j=0}^{\infty}\beta\eta_jU_j=0$ up to terms proportional to $\beta^2$.
The sum in the third term on the left hand side of Eq. (\[sum1\]), after putting $C_0=1/2$ and $C_p=1$ for $p\geq 1$ in the relations for ${\bar \Phi}_j$, reduces to $$\begin{aligned}
\label{sum3}
\sum_{j=0}^{\infty}{\bar \Phi}_j&=\frac{1}{4\pi}\!\sum_{j=-\infty}^{\infty}\!\int_0^{2\pi}\!\!\!\!\cos\theta\,\cos{j\theta}d\theta \nonumber \\
&\times\sum_{p=-\infty}^{\infty}\!\int_0^{2\pi}\!\!\!\!\cos{p\theta'}\Phi(\theta\!-\!\theta')d\theta'.\!\!\!\!\end{aligned}$$ For $\theta$ in the interval $0\leq\theta\leq 2\pi$ we have a relation: $$\sum_{j=-\infty}^{\infty}\cos{j\theta}=\pi[\delta(\theta)+\delta(\theta-2\pi)],$$ which transforms Eq. (\[sum3\]) to $$\label{sum4}
\sum_{j=0}^{\infty}{\bar \Phi}_j=\frac{\pi}{4}[2\Phi(0)+\Phi(2\pi)+\Phi(-2\pi)]=2\pi^2,$$ and equation (\[sum1\]) becomes: $$\begin{aligned}
\sqrt{\frac{s-1}{2}}-\beta^2\pi^2=0. \nonumber\end{aligned}$$ This gives $s=1+2(\beta\pi)^4$, and recalling that $\beta=k_Fr_*/\pi$ (we put $m^*=m$) we obtain for the velocity of zero sound: $$\label{u0}
u_0=v_F[1+2(k_Fr_*)^4].$$
Note that in contrast to the 3D two-species Fermi gas with a weak repulsive contact interaction (scattering length $a$), where the correction $(u_0-v_F)$ exponentially depends on $k_Fa$, for our 2D dipolar gas we obtained a power law dependence. This is a consequence of dimensionality of the system.
It is important that confining ourselves to only the leading part of the interaction function $\tilde F$, which is proportional to $k_Fr_*$ and is given by the first term of Eq. (\[tildeF1\]), we do not obtain undamped zero sound ($s>1$) [@comment]. This corresponds to omitting the terms $\beta^2{\bar \Phi}_j/2$ in equations (\[systemf0\])-(\[systemfj\]) and is consistent with numerical calculations [@Baranov]. Only the many-body corrections to the interaction function of quasiparticles, given by equations (\[tildeF12\]) and (\[tildeF22\]), provide non-zero positive values of $\Phi(0)$ and $\Phi(\pm 2\pi)$, thus leading to a positive value of $(u_0-v_F)$. One then sees that many-body effects are crucial for the propagation of zero sound.
In principle, we could obtain the result of Eq. (\[u0\]) in a simpler way, similar to that used for the two-species Fermi gas with a weak repulsive interaction (see, e.g. [@Landau]). Representing the function $\nu(\theta)$ as $\nu(\theta)=\cos{\theta}\tilde\nu(\theta)/(s-\cos{\theta})$ we transform Eq. (\[inteq1\]) to the form: $$\label{inteq3}
\tilde\nu(\theta)=\frac{m^*}{(2\pi\hbar)^2}\int_0^{2\pi}\frac{\tilde F(\theta-\theta')\tilde\nu(\theta')\cos{\theta'}}{s-\cos{\theta'}}d\theta'.$$ Since $s$ is close to unity, it looks reasonable to assume that the main contribution to the integral in Eq. (\[inteq3\]) comes from $\theta'$ close to zero and to $2\pi$. Using the fact that $\tilde F(\theta)=\tilde F(2\pi -\theta)$ we then obtain: $$\label{tildenu}
\tilde\nu(\theta)=\frac{m^*\tilde F(\theta)\tilde\nu(0)}{4\pi\hbar^2}\sqrt{\frac{2}{s-1}}.$$ We now take the limit $\theta\rightarrow 0$ and substitute $\tilde F(0)=(4\pi\hbar^2/m)(k_Fr_*)^2$ as follows froms Eqs. (\[tildeF1\]), (\[tildeF12\]), and (\[tildeF22\]). Putting $m^*=m$ we then obtain $s=1+2(k_Fr_*)^4$ and arrive at Eq. (\[u0\]).
Note, however, that for very small $\theta$ or $\theta$ very close to $2\pi$ the dependence $\tilde F(\theta)$ is very steep. For $\theta\rightarrow 0$ the leading part of the interaction function, which is linear in $k_Fr_*$, vanishes, and only the quadratic part contributes to $\tilde F(0)$. Therefore, strictly speaking the employed procedure of calculating the integral in Eq. (\[inteq3\]) is questionable for very small $\theta$. This prompted us to make the analysis based on representing $\nu(\theta)$ in the form (\[nucos\]) and on solving the system of linear equations (\[systemf0\])-(\[systemfj\]).
Equation (\[inteq3\]) is useful for understanding why undamped zero sound requires the condition $s>1$ so that $u_0>v_F$. For $s<1$ there is a pole in the integrand of Eq. (\[inteq3\]), which introduces an imaginary part of the integral. As a result, the zero sound frequency $\omega$ will also have an imaginary part at real momenta $k$, which means the presence of damping (see, e.g. [@Landau]).
We could also consider an odd function $\nu(\theta)$, namely such that $\nu(2\pi-\theta)=-\nu(\theta)$ and $\nu(0)=\nu(2\pi)=0$. In this case, however, we do not obtain an undamped zero sound.
Concluding remarks
==================
We have shown that (single-component) fermionic polar molecules in two dimensions constitute a novel Fermi liquid, where many-body effects play an important role. For dipoles oriented perpendicularly to the plane of translational motion, the many-body effects provide significant corrections to thermodynamic functions. Revealing these effects is one of the interesting goals of up-coming experimental studies. The investigation of the full thermodynamics of 2D polar molecules, including many-body effects, can rely on the in-situ imaging technique as it has been done for two-component atomic Fermi gases [@Salomon1; @Salomon2]. This method can also be extended to 2D systems for studying thermodynamic quantities [@Zwierlein; @Dalibard2]. Direct imaging of a 3D pancake-shaped dipolar molecular system has been recently demonstrated at JILA [@Ye2]. For 2D polar molecules discussed in our paper, according to equations (\[kappa\*\])-(\[Eg\]), the contribution of many-body corrections proportional to $(k_Fr_*)^2$ can be on the level of $10\%$ or $20\%$ for $k_Fr_*$ close to $0.5$. Thus, finding many-body effects in their thermodynamic properties looks feasible.
It is even more important that the many-body effects are responsible for the propagation of zero sound waves in the collisionless regime of the 2D Fermi liquid of polar molecules with dipoles perpendicular to the plane of translational motion. This is shown in Section IV of our paper, whereas mean-field calculations do not find undamped zero sound [@Baranov]. Both collisionless and hydrodynamic regimes are achievable in on-going experiments. This is seen from the dimensional estimate of the relaxation rate of quasiparticles. At temperatures $T\ll \epsilon_F$ the relaxation of a non-equilibrium distribution of quasiparticles occurs due to binary collisions of quasiparticles with energies in a narrow interval near the Fermi surface. The width of this interval is $\sim T$ and, hence, the relaxation rate contains a small factor $(T/\epsilon_F)^2$ (see, e.g. [@Landau]). Then, using the Fermi Golden rule we may write the inverse relaxation time as $\tau^{-1}\sim (g_{eff}^2/\hbar)(m/\hbar^2)n(T/\epsilon)^2$, where $n$ is the 2D particle density, the quantity $\sim m/\hbar^2$ represents the density of states on the Fermi surface, and the quantity $g_{eff}$ is the effective interaction strength. Confining ourselves to the leading part of this quantity, from Eqs. (\[1str2\]) and (\[barf1\]) we have $g_{eff}\sim \hbar^2k_Fr_*/m$. We thus obtain: $$\label{taurel}
\frac{1}{\tau}\sim\frac{\hbar n}{m}(k_Fr_*)^2\left(\frac{T}{\epsilon_F}\right)^2.$$ Note that as $\epsilon_F\approx\hbar^2k_F^2/2m\approx 2\pi\hbar^2n/m$, for considered temperatures $T\ll \epsilon_F$ the relaxation time $\tau$ is density independent. Excitations with frequencies $\omega\ll 1/\tau$ are in the hydrodynamic regime, where on the length scale smaller than the excitation wavelength and on the time scale smaller than $1/\omega$ the system reaches a local equilibrium. On the other hand, excitations with frequencies $\omega\gg 1/\tau$ are in the collisionless regime. Assuming $T\sim 10$nK, for KRb molecules characterized by the dipole moment $d\simeq 0.25$ D in the electric field of $5$kV/cm as obtaind in the JILA experiments, we find $\tau$ on the level of tens of milliseconds. The required condition $T\ll\epsilon_F$ is satisfied for $\epsilon_F\gtrsim 70$ nK, which corresponds to $n\gtrsim 2\cdot 10^8$ cm$^{-2}$. In such conditions excitations with frequencies of the order of a few Hertz or lower will be in the hydrodynamic regime, and excitations with larger frequencies in the collisionless regime.
The velocity of zero sound is practically equal to the Fermi velocity $v_F=\hbar k_F/m^*$. This is clearly seen from Eq. (\[u0\]) omitting a small correction proportional to $(k_Fr_*)^4$. Then, using Eq. (\[mg\*\]) for the effective mass and retaining only corrections up to the first order in $k_Fr_*$, we have: $$\label{u0Simple}
u_0\simeq\frac{\hbar k_F}{m}\left(1+\frac{4}{3\pi}k_Fr_*\right).$$ In the hydrodynamic regime the sound velocity is: $$\label{usimple}
u=\sqrt{\frac{N}{m}\frac{\partial\mu}{\partial N}}\simeq\frac{\hbar k_F}{m}\left(1+\frac{8}{3\pi}k_Fr_*\right),$$ where we used Eq. (\[dmudN\]) for $\partial\mu/\partial N$ and retained corrections up to the first order in $k_Fr_*$. The hydrodynamic velocity $u$ is slightly larger than the velocity of zero sound $u_0$, and the difference is proportional to the interaction strength. This is in sharp contrast with the 3D two-component Fermi gas, where $u_0\approx v_F>u\approx v_F/\sqrt{3}$.
We thus see that it is not easy to distinguish between the hydrodynamic and collisionless regimes from the measurement of the sound velocity. A promising way to do so can be the observation of damping of driven excitations, which in the hydrodynamic regime is expected to be slower. Another way is to achieve the values of $k_Fr_*$ approaching unity and still discriminate between $u_0$ and $u$ in the measurement of the sound velocity. For example, in the case of dipoles perpendicular to the plane of their translational motion the two velocities are different from each other by about $20\%$ at $k_Fr_*\simeq 0.5$. These values of $k_Fr_*$ are possible if the 2D gas of dipoles still satisfies the Pomeranchuk criteria of stability. These criteria require that the energy of the ground state corresponding to the occupation of all quasiparticle states inside the Fermi sphere, remains the minimum energy under an arbitrarily small deformation of the Fermi sphere. The generalization of the Pomeranchuk stability criteria to the case of the 2D single-component Fermi liquid with dipoles perpendicular to the plane of their translational motion reads: $$\label{Pom}
1+\frac{m^*}{(2\pi\hbar)^2}\int_0^{2\pi}\tilde F(\theta)\cos{j\theta}\,d\theta>0,$$ and this inequality should be satisfied for any integer $j$. As has been found in Ref. [@Baranov], the Pomeranchuk stability criteria (\[Pom\]) are satisfied for $k_Fr_*$ approaching unity from below if the interaction function of quasiparticles contains only the first term of Eq. (\[tildeF1\]), which is the leading mean field term. We have checked that the situation with the Pomeranchuk stability does not change when we include the full expression for the interaction function, $\tilde F(\theta)=\tilde F^{(1)}(\theta)+\tilde F^{(2)}_1(\theta)+\tilde F^{(2)}_2(\theta)$, following from Eqs. (\[tildeF1\]), (\[tildeF12\]), and (\[tildeF22\]). Thus, achieving $k_Fr_*$ approaching unity looks feasible. For KRb molecules with the (oriented) dipole moment of $0.25$ D the value $k_Fr_*\approx 0.5$ requires densities $n\approx 2\cdot 10^8$ cm$^{-2}$.
Finally, we would like to emphasize once more that our results are applicable equally well for the quasi2D regime, where the dipole-dipole length $r_*$ is of the order of or smaller than the confinement length $l_0=(\hbar/m\omega_0)^{1/2}$, with $\omega_0$ being the frequency of the tight confinement. The behavior at distances $r\lesssim l_0$ is contained in the coefficient $A$ defined in Eq. (\[short\]). Therefore, the results for the velocity of zero sound which is independent of $A$, are universal in the sense that they remain unchanged when going from $r_*\gg l_0$ to $r_*\lesssim l_0$. The only requirement is the inequality $k_Fl_0\ll 1$. It is, however, instructive to examine the ratio $r_*/l_0$ that can be obtained in experiments with ultracold polar molecules. Already in the JILA experiments using the tight confinement of KRb molecules with frequency $\omega_0\approx 30$ kHz and achieving the average dipole moment $d\simeq 0.25$ D in electric fields of $5$ kV/cm, we have $r_*\simeq 100$ nm and $l_0\simeq 50$ nm so that $r_*/l_0\simeq 2$. A decrease of the confinement frequency to $5$ kHz and a simultaneous decrease of the dipole moment by a factor of 2 leads to $r_*/l_0\sim 0.2$. On the other hand, for $d$ close to $0.5$ (which is feasible to obtain for other molecules) one can make the ratio $r_*/l_0$ close to $10$ at the same confinement length.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to M.A. Baranov and S.I. Matveenko for fruitful discussions. We acknowledge support from EPSRC Grant No. EP/F032773/1, from the IFRAF Institute, and from the Dutch Foundation FOM. This research has been supported in part by the National Science Foundation under Grant No. NSF PHYS05-51164. LPTMS is a mixed research unit No. 8626 of CNRS and Université Paris Sud.
Direct calculation of the first order contribution to the interaction energy {#App1}
============================================================================
For directly calculating the first order (mean field) contribution to the interaction energy $\tilde E^{(1)}$ (\[tildeE1\]), we represent it as $\tilde E^{(1)}=\tilde E^{(1)}_1+\tilde E^{(1)}_2$ where $$\begin{aligned}
&\tilde E^{(1)}_1=\int {\bar f}^{(1)}\left(\frac{|{\bf k}_1-{\bf k}_2|}{2}\right)n_{{\bf k}_1}n_{{\bf k}_2}\frac{d^2k_1d^2k_2}{(2\pi)^4},\label{E11} \\
&\tilde E^{(1)}_2=\int {\bar f}^{(2)}\left(\frac{|{\bf k}_1-{\bf k}_2|}{2}\right)n_{{\bf k}_1}n_{{\bf k}_2}\frac{d^2k_1d^2k_2}{(2\pi)^4},\label{E12} \end{aligned}$$ and the amplitudes ${\bar f}^{(1)}$ and ${\bar f}^{(2)}$ are given by Eqs. (\[barf1\]) and (\[barf2\]), respectively. In the calculation of the integrals for $\tilde E^{(1)}_1$ and $\tilde E^{(1)}_2$ we turn to the variables ${\bf x}=({\bf k}_1-{\bf k}_2)/2k_F$ and ${\bf y}=({\bf k}_1+{\bf k}_2)/2k_F$, so that $d^2k_1d^2k_2=8\pi k_F^4d^2xd^2yd\varphi$, where $\varphi$ is the angle between the vectors ${\bf x}$ and ${\bf y}$, and the integration over $d\varphi$ should be performed from $0$ to $2\pi$. The distribution functions $n_{{\bf k}_1}$ and $n_{{\bf k}_2}$ are the step functions (\[nstep\]). The integration over $dk_1$ and $dk_2$ from $0$ to $k_F$ corresponds to the integration over $dy$ from $0$ to $y_0(x,\varphi)=-x|\cos\varphi|+\sqrt{1-x^2\sin2\varphi}$ and over $dx$ from $0$ to $1$. Using Eq. (\[barf1\]) we reduce Eq. (\[E11\]) to $$\label{intermE11}
\tilde E^{(1)}_1=\frac{S\hbar^2k_F^4}{\pi^2m}k_Fr_*I_1,$$ where $$\begin{aligned}
&I_1=\int_0^{2\pi}d\varphi\int_0^1x^2dx\int_0^{y_0(x,\varphi)}ydy=\frac{1}{2}\int_0^{2\pi}d\varphi\int_0^1x^2 dx\nonumber\\
&\times[1-2|\cos\varphi|\sqrt{1-x^2\sin^2\varphi}+x^2(\cos^2\varphi-\sin^2\varphi)].\nonumber\end{aligned}$$ The last term of the second line vanishes, and the integration of the first two terms over $d\varphi$ and $dx$ gives: $$I_1=\int_0^1x^2\left(\pi-2x\sqrt{1-x^2}-2\arcsin{x}\right)=\frac{8}{45}.$$ Then Eq. (\[intermE11\]) yields: $$\label{E11final}
\tilde E^{(1)}_1=\frac{8S}{45\pi^2}\frac{\hbar^2k_F^4}{m}k_Fr_*=\frac{N^2}{S}\frac{128}{45}\frac{\hbar^2k_F^2}{m}k_Fr_*,$$ which exactly coincides with the second term of the first line of Eq. (\[Eg\]).
Using Eq. (\[barf2\]) the contribution $\tilde E^{(1)}_2$ takes the form: $$\label{E12interm}
\!\!\!\!\tilde E^{(\!1\!)}_2\!\!\!=\!\frac{S\hbar^2\!k_F^4}{2\pi^2m}(k_Fr_*\!)^2\!\left\{\!\left[\ln(\xi k_Fr_*\!)\!-\!\frac{25}{12}\!+\!3\ln{2}\right]\!I_2\!+\!I_3\!\right\}\!\!,\!\!\!\!\!\!\!\!$$ where the integrals $I_2$ and $I_3$ are given by $$\begin{aligned}
I_2&=\int_0^{2\pi}d\varphi\int_0^1x^3dx\int_0^{y_0(x,\varphi)}ydy =\frac{1}{2}\int_0^{2\pi}d\varphi\int_0^1x^3 dx\nonumber\\
&\times\big[1-2|\cos\varphi|x\sqrt{1-x^2\sin^2\varphi}+x^2(\cos^2\varphi-\sin^2\varphi)\big] \nonumber \\
&=\frac{1}{2}\int_0^1x^3[2\pi-4x\sqrt{1-x^2}-4\arcsin{x}]dx=\frac{\pi}{32},\nonumber\end{aligned}$$ and $$\begin{aligned}
&I_3=\int_0^{2\pi}d\varphi\int_0^1x^3\ln{x}dx\int_0^{y_0(x,\varphi)}ydy=\frac{1}{2}\int_0^{2\pi}d\varphi\int_0^1 dx\nonumber\\
&\times x^3\ln{x}\big[1-2|\cos\varphi|x\sqrt{1-x^2\sin^2\varphi}+x^2(\cos^2\varphi-\sin^2\varphi)\big]\nonumber \\
&=\frac{1}{2}\int_0^1x^3\ln{x}[2\pi-4x\sqrt{1-x^2}-4\arcsin{x}]dx\nonumber\\
&=\frac{\pi}{32}\left(\frac{1}{6}-\ln{2}\right).\nonumber\end{aligned}$$
Substituting the calculated $I_2$ and $I_3$ into Eq. (\[E12interm\]) we obtain: $$\begin{aligned}
&\tilde E^{(1)}_2=\frac{S\hbar^2k_F^4}{64\pi m}(k_Fr_*)^2\left[\ln(4\xi k_Fr_*)-\frac{23}{12}\right] \nonumber \\
&=\frac{N^2}{S}\frac{\pi\hbar^2}{4m}(k_Fr_*)^2\left[\ln(4\xi k_Fr_*)-\frac{23}{12}\right]. \label{E12final}\end{aligned}$$ This exactly reproduces the third term of the first line of Eq. (\[Eg\]).
Calculation of the interaction function $\tilde F_{1}^{(2)}$ {#App2}
============================================================
The interaction function $\tilde F_{1}^{(2)}$ is the second variational derivative of the many-body contribution to the interaction energy, $\tilde E^{(2)}_1$ (\[tildeE12\]), with respect to the momentum distribution function. It can be expressed as $$\label{F12initial}
\tilde F_{1}^{(2)}({\mathbf{k}},{\mathbf{k'}})=-\frac{2\hbar^2}{m}(k_Fr_*)^2 (\tilde I_1+\tilde I_2+\tilde I_3),$$ where $$\begin{aligned}
\!\!&\tilde I_1=2\!\int_{|{\mathbf{k_1}}|<k_F}\!\frac{d^2k_1}{k_F^2}\frac{|{\mathbf{k}}-{\mathbf{k_1}}|^2}{{\mathbf{k^2}}\!+\!{\mathbf{k'^2}}\!-\!{\mathbf{k^2_1}}\!-\!{\mathbf{k^2_2}}}\delta_{{\mathbf{k}}\!+\!{\mathbf{k'}}\!-\!{\mathbf{k_1}}\!-\!{\mathbf{k_2}}},
\label{tildeI1} \\
\!\!&\tilde I_2=2\!\int_{|{\mathbf{k_1}}|<k_F}\!\frac{d^2k_1}{k_F^2}\frac{|{\mathbf{k}}-{\mathbf{k'}}|^2}{{\mathbf{k^2}}\!+\!{\mathbf{k^2_1}}\!-\!{\mathbf{k'^2}}\!-\!{\mathbf{k^2_2}}}\delta_{{\mathbf{k}}\!+\!{\mathbf{k_1}}\!-\!{\mathbf{k'}}\!-\!{\mathbf{k_2}}},
\label{tildeI2} \\
\!\!&\tilde I_3=2\!\int_{|{\mathbf{k_1}}|<k_F}\!\frac{d^2k_1}{k_F^2}\frac{|{\mathbf{k_1}}-{\mathbf{k'}}|^2}{\!{\mathbf{k^2_1}}\!+\!{\mathbf{k^2}}\!-\!{\mathbf{k'^2}}\!-\!{\mathbf{k^2_2}}}\delta_{{\mathbf{k_1}}\!+\!{\mathbf{k}}\!-\!{\mathbf{k'}}\!-\!{\mathbf{k_2}}},
\label{tildeI3}\end{aligned}$$ and the presence of the Kronecker symbols $\delta_{\bf q}$ reflects the momentum conservation law. On the Fermi surface we put $|{\bf k}|=|{\bf k}'|=k_F$ and denote the angle between ${\bf k}$ and ${\bf k}'$ as $\theta$. Due to the symmetry property: $F({\mathbf{k}},{\mathbf{k'}})=F({\mathbf{k'}},{\mathbf{k}})$ we have $F(\theta)=F(2\pi-\theta)$ and may consider $\theta$ in the interval from $0$ to $\pi$.
In order to calculate the integral $\tilde I_1$, we use the quantities ${\mathbf{s}}=({\mathbf{k}}+{\mathbf{k'}})/2k_F$ and ${\mathbf{m}}=({\mathbf{k}}-{\mathbf{k'}})/2k_F$ and turn to the variable ${\mathbf{x}}=({\mathbf{k_1}}-{\mathbf{k_2}})/2k_F=(2{\mathbf{k_1}}-{\mathbf{s}})/2k_F$. For given vectors ${\mathbf{k}}$ and ${\mathbf{k'}}$, the vectors ${\mathbf{s}}$ and ${\mathbf{m}}$ are fixed and $|{\mathbf{s}}|=\cos(\theta/2)$, $|{\mathbf{m}}|=\sin(\theta/2)$. The integral can then be rewritten as: $$\tilde I_1=\int \frac{m^2+x^2}{m^2-x^2} d^2x.$$ The integration region is shown in Fig.\[fig:f1\], where the distance between the points $O_1$ and $O_2$ is ${\bf R}_{O_1O_2}={\mathbf{s}}$. The distance between the points $O_1$ and $N$ is ${\bf R}_{O_1N}={\mathbf{k_1}}/2k_F$, and ${\bf R}_{NO_2}={\mathbf{k_2}}/2k_F$, so that ${\bf R}_{ON}={\mathbf{x}}/2$. The quantity $|{\bf x}|$ changes from $0$ to $l_1(\varphi)$ where $$l_1^2(\varphi)+\cos^2\frac{\theta}{2}-2l_1(\varphi)\cos\frac{\theta}{2}\cos\varphi=1,$$ and $l_1(\varphi)\cdot l_1(\varphi+\pi)=\sin^2(\theta/2)$, with $\varphi$ being an angle between ${\bf m}$ and ${\bf x}$. In the polar coordinates the integral $\tilde I_1$ takes the form: $$\tilde I_1=\int_0^{2\pi}d\varphi\int_0^{l_1(\varphi)}\left(-1+2\sin^2\frac{\theta}{2}\frac{1}{\sin^2(\theta/2)-x^2}\right)xdx,$$ and after a straightforward integration we obtain: $$\label{tildeI1final}
\tilde I_1=\pi\left(2\sin^2\frac{\theta}{2}\ln|\tan\frac{\theta}{2}|-1\right).$$
![(color online). Left: The integration area for $\tilde I_1$ (in blue). The distance between the points $O_1$ and $P$ is ${\bf R}_{O_1P}={\mathbf{k}}/2k_F$, ${\bf R}_{PO_2}={\mathbf{k'}}/2k_F,$ and ${\bf R}_{O_1N}={\mathbf{k_1}}/2k_F$. Right: The integration area for $\tilde I_2$ and $\tilde I_3$ (in red). The distance between the points $O_2$ and $N$ is ${\bf R}_{O_2N}={\mathbf{k_1}}/2k_F$, ${\bf R}_{O_1P}={\bf k}/2k_F$, and ${\bf R}_{O_2P}={\bf k}'/2k_F$. \[fig:f1\]](1.eps "fig:"){width="0.5\columnwidth"}![(color online). Left: The integration area for $\tilde I_1$ (in blue). The distance between the points $O_1$ and $P$ is ${\bf R}_{O_1P}={\mathbf{k}}/2k_F$, ${\bf R}_{PO_2}={\mathbf{k'}}/2k_F,$ and ${\bf R}_{O_1N}={\mathbf{k_1}}/2k_F$. Right: The integration area for $\tilde I_2$ and $\tilde I_3$ (in red). The distance between the points $O_2$ and $N$ is ${\bf R}_{O_2N}={\mathbf{k_1}}/2k_F$, ${\bf R}_{O_1P}={\bf k}/2k_F$, and ${\bf R}_{O_2P}={\bf k}'/2k_F$. \[fig:f1\]](2.eps "fig:"){width="0.5\columnwidth"}
In the integral $\tilde I_2$, using the variable ${\mathbf{y}}=({\mathbf{k_1}}+{\mathbf{k_2}})/2k_F$ we observe that it changes from $0$ to $l_2(\tilde\varphi)$ where $$l_2^2(\tilde\varphi)+\sin^2\frac{\theta}{2}-2l_2(\tilde\varphi)\sin\frac{\theta}{2}\cos(\tilde\varphi)=1$$ and $l_2(\tilde\varphi)-l_2(\tilde\varphi+\pi)=2\sin\frac{\theta}{2}\cos\tilde\varphi$, with $\tilde\varphi$ being an angle between ${\bf y}$ and ${\bf m}$. We then have: $$\begin{aligned}
\tilde I_2&=-2\int \frac{m^2}{{\mathbf{m}}\cdot{\mathbf{y}}}d^2y=-2\sin\frac{\theta}{2}\int_{0}^{2\pi}d\tilde\varphi\int_0^{l_2(\tilde\varphi)} \frac{dy}{\cos\tilde\varphi}\nonumber \\
&=-4\pi\sin^2\frac{\theta}{2}. \label{tildeI2final}\end{aligned}$$
For the integral $\tilde I_3$ we have: $$\begin{aligned}
&\tilde I_3=-\frac{1}{2}\int \frac{s^2+y^2-2{\mathbf{s}}\cdot{\mathbf{y}}}{{\mathbf{m}}\cdot{\mathbf{y}}} \nonumber\\
&=-\frac{1}{2\sin\frac{\theta}{2}}\int_{0}^{2\pi}\!\!\frac{d\tilde\varphi}{\cos\tilde\varphi}\!\int_{0}^{l_2(\tilde\varphi)}\!\!\!dy\left[y^2+\cos^2\frac{\theta}{2}-2y\cos\frac{\theta}{2}\sin\tilde\varphi\right] \nonumber\\
&=-2\pi \left(\cos^2\frac{\theta}{2}+\frac{1}{3}\sin^2\frac{\theta}{2}\right), \label{tildeI3final}\end{aligned}$$ where we used the relation $l_2(\tilde\varphi)=l_2(-\tilde\varphi)$.
Using integrals $\tilde I_1$ (\[tildeI1final\]), $\tilde I_2$ (\[tildeI2final\]), and $\tilde I_3$ (\[tildeI3final\]) in Eq. (\[F12initial\]), we obtain equation (\[tildeF12\]): $$\!\!\!\tilde F_{1}^{(2)}(\theta)\!\!
=\!\!\frac{2\hbar^2r^2_*k^2_F}{m}\!\!\left[\! 3\pi\! +\!2\pi\sin^2\frac{\theta}{2}\left(\frac{4}{3} \!-\!\ln|\tan \frac{\theta}{2}|\right)\!\right].$$ .
Calculation of the interaction function $\tilde F_{2}^{(2)}$ {#F22}
============================================================
The interaction function $\tilde F_{2}^{(2)}$ is the second variational derivative of the many-body contribution to the interaction energy, $\tilde E^{(2)}_2$ (\[tildeE22\]), with respect to the momentum distribution. It reads: $$\label{F22initial}
\tilde F_{2}^{(2)}({\mathbf{k}},{\mathbf{k'}})=\frac{2\hbar^2}{m}(k_Fr_*)^2(I'_1+I'_2),$$ where $$\begin{aligned}
\!\!&I'_1=2\!\int_{|{\mathbf{k_1}}|<k_F}\!\frac{d^2k_1}{k_F^2}\frac{|{\mathbf{k}}\!-\!{\mathbf{k_1}}|\cdot|{\mathbf{k'}}\!-\!{\mathbf{k_1}}|}{{\mathbf{k^2}}\!+\!{\mathbf{k'^2}}\!-\!{\mathbf{k^2_1}}\!-\!{\mathbf{k^2_2}}}\delta_{{\mathbf{k}}\!+\!{\mathbf{k'}}\!-\!{\mathbf{k_1}}\!-\!{\mathbf{k_2}}}, \label{I1prime} \\
\!\!&I'_2=4\!\int_{|{\mathbf{k_1}}|<k_F}\!\frac{d^2k_1}{k_F^2}\frac{|{\mathbf{k}}\!-\!{\mathbf{k'}}|\cdot|{\mathbf{k_1}}\!-\!{\mathbf{k'}}|}{{\mathbf{k^2}}\!+\!{\mathbf{k^2_1}}\!-\!{\mathbf{k'^2}}\!-\!{\mathbf{k^2_2}}}\delta_{{\mathbf{k}}\!+\!{\mathbf{k_1}}\!-\!{\mathbf{k'}}\!-\!{\mathbf{k_2}}} \label{I2prime}\end{aligned}$$ The integration area for $I'_1$ is shown in Fig. \[fig:f2\], where the distance between the points $O_1$ and $P$ is ${\bf R}_{O_1P}={\bf k}/2k_F$, ${\bf R}_{PO_2}={\bf k}'/2k_F$, ${\bf R}_{O_1N}={\bf k}_1/2k_F$, and ${\bf R}_{ON}={\bf x}/2$. We thus have ${\bf R}_{NP}=({\bf k}-{\bf k}_1)/2k_F$ and ${\bf R}_{NP'}=({\bf k}'-{\bf k}_1)/2k_F$. In the region of integration we should have $|{\bf R}_{O_1N}|=k_1/2k_F\leq 1/2$. This leads to $$\begin{aligned}
I'_1&=4\int \frac{|{\bf R}_{NP}|\cdot|{\bf R}_{NP'}|}{m^2-x^2}d^2n=-\int_0^{2\pi} d\varphi\int_{0}^{l_3(\varphi)}xdx \nonumber \\
&\times\frac{\sqrt{[x^2+\sin^2(\theta/2)]^2-4x^2\sin^2(\theta/2)\cos^2\varphi}}{x^2-\sin^2(\theta/2)},\end{aligned}$$ where $\varphi$ is the angle between ${\bf x}$ and ${\bf m}$ (see Fig. \[fig:f2\]), and the quantity $l_3(\varphi)$ obeys the equation $$l_3^2(\varphi)-2\cos\frac{\theta}{2}\sin\varphi \cdot l_3(\varphi)+\cos^2\frac{\theta}{2}=1.$$ Turning to the variable $z=r^2-\sin^2(\theta/2)$ the integral $I'_1$ is reduced to $$\begin{aligned}
\label{I1primeinterm}
I'_1=-\frac{1}{2}\int_0^{2\pi} d\varphi \int_{-\sin^2(\theta/2)}^{l_3^2(\varphi)-\sin^2(\theta/2)} \frac{\sqrt{R}}{z} dz,\end{aligned}$$ with $$R=z^2+4z\sin^2\frac{\theta}{2}\sin^2\varphi+4\sin^4\frac{\theta}{2}\sin^2\varphi.$$
![(color online). Left: The integration area for $I'_1$ (in blue): ${\bf R}_{O_1P}={\mathbf{k}}/2k_F$, ${\bf R}_{PO_2}={\mathbf{k'}}/2k_F$, ${\bf R}_{O_1N}={\mathbf{k_1}}/2k_F$, and $\varphi$ is the angle between the vectors ${\bf R}_{OP}$ and ${\bf R}_{ON}$, which is the same as the angle between ${\bf m}$ and ${\bf x}$. Right: The integration area for $ I'_2$ (in red): ${\bf R}_{O_1P}={\mathbf{k}}/2k_F$, ${\bf R}_{O_2P}={\mathbf{k'}}/2k_F$, ${\bf R}_{O_2N}={\mathbf{k_1}}/2k_F$, $\alpha$ is the angle between ${\bf R}_{PM}$ and ${\bf R}_{PN}$, and $\phi$ is the angle between ${\bf R}_{PN}$ and ${\bf R}_{OO_2}$. \[fig:f2\]](3.eps "fig:"){width="0.5\columnwidth"}![(color online). Left: The integration area for $I'_1$ (in blue): ${\bf R}_{O_1P}={\mathbf{k}}/2k_F$, ${\bf R}_{PO_2}={\mathbf{k'}}/2k_F$, ${\bf R}_{O_1N}={\mathbf{k_1}}/2k_F$, and $\varphi$ is the angle between the vectors ${\bf R}_{OP}$ and ${\bf R}_{ON}$, which is the same as the angle between ${\bf m}$ and ${\bf x}$. Right: The integration area for $ I'_2$ (in red): ${\bf R}_{O_1P}={\mathbf{k}}/2k_F$, ${\bf R}_{O_2P}={\mathbf{k'}}/2k_F$, ${\bf R}_{O_2N}={\mathbf{k_1}}/2k_F$, $\alpha$ is the angle between ${\bf R}_{PM}$ and ${\bf R}_{PN}$, and $\phi$ is the angle between ${\bf R}_{PN}$ and ${\bf R}_{OO_2}$. \[fig:f2\]](4.eps "fig:"){width="0.5\columnwidth"}
It is easy to see that: $$\begin{aligned}
&I_r=\int_{-\sin^2(\theta/2)}^{l_3^2(\varphi)-\sin^2(\theta/2)}\frac{\sqrt{R}}{z}dz \nonumber \\
&=\Big\{\sqrt{R}-\sqrt{a}\ln\left(2a+bz+2\sqrt{aR}\right)\nonumber \\
&+\frac{b}{2}\ln\left(2\sqrt{R}+2z+b\right)\Big\}\Big|^{l_3^2(\varphi)-\sin^2(\theta/2)}_{-\sin^2(\theta/2)} \nonumber \\
&+\sqrt{a} \cdot P\int_{-\sin^2(\theta/2)}^{l_3^2(\varphi)-\sin^2(\theta/2)}\frac{dz}{z}=I_{r\uparrow}-I_{r\downarrow},\end{aligned}$$ where $a=4\sin^4(\theta/2)\sin^2\varphi$, $b=4\sin^2(\theta/2)\sin^2\varphi$, and the symbol $P$ stands for the principal value of the integral. The quantities $I_{r\uparrow}$ and $I_{r\downarrow}$ denote the values of the integral at the upper and lower bounds, respectively (in the last line we have to take the principal value of the integral and, hence, if the upper bound of the integral is positive we have to replace the lower bound with $\sin^2(\theta/2)$). Then $I_{r\uparrow}$ and $I_{r\downarrow}$ are given by:
$$\begin{aligned}
&I_{r\uparrow}=2|\sin\varphi|\cdot l(\varphi)-2\sin^2\frac{\theta}{2}|\sin\varphi|\cdot\left[\ln\left(8\sin^2\frac{\theta}{2}\sin^2\varphi\right)+\ln\left(\sin^2\frac{\theta}{2}+\cos\frac{\theta}{2}\sin\varphi \cdot l(\varphi)+l(\varphi)\right)\right] \nonumber\\
&+2\sin^2\frac{\theta}{2}|\sin\varphi|\cdot\ln|2\cos\frac{\theta}{2}\sin\varphi \cdot l(\varphi)|+2\sin^2\frac{\theta}{2}\sin^2\varphi\left[\ln4+\ln\left(|\sin\varphi|\cdot l(\varphi)+\cos\frac{\theta}{2}\sin\varphi\cdot l(\varphi)+\sin^2\frac{\theta}{2}\sin^2\varphi\right)\right], \nonumber \\
&I_{r\downarrow}=\sin^2\frac{\theta}{2}-2\sin^2\frac{\theta}{2}|\sin\varphi|\cdot\left[\ln\left(4\sin^4\frac{\theta}{2}\right)+\ln\left(\sin^2\varphi+|\sin\varphi|\right)\right]+2\sin^2\frac{\theta}{2}|\sin\varphi|\cdot\ln\left(\sin^2\frac{\theta}{2}\right)\nonumber\\
&+2\sin^2\frac{\theta}{2}\sin^2\varphi\cdot\ln\left(4\sin^2\frac{\theta}{2}\sin^2\varphi\right). \nonumber\end{aligned}$$
The integral $I'_1$ can be expressed as: $$I'_1=-\frac{1}{2}\int_{0}^{2\pi}[I_{r\uparrow}-I_{r\downarrow}]d\varphi,$$ and for performing the calculations we notice that $l_3(\varphi)\cdot l_3(\varphi+\pi)=\sin^2\frac{\theta}{2}$, $l_3(\varphi)-l_3(\varphi+\pi)=2\cos\frac{\theta}{2}\sin\varphi$, and $l_3(\varphi)+l_3(\varphi+\pi)=2\sqrt{\cos^2\frac{\theta}{2}\sin^2\varphi+\sin^2\frac{\theta}{2}}$. We then obtain:
$$\begin{aligned}
\label{I11}
I'_{1}=&-\sin^2\frac{\theta}{2}\left(\pi\ln2+\pi/2-\pi\ln\sin\frac{\theta}{2}+4\ln|\cos\frac{\theta}{2}|-4\ln(1+\sin\frac{\theta}{2})+\mathcal{G}(\theta)-\frac{2\pi}{|\cos\frac{\theta}{2}|}-\frac{4\arcsin(\sin\frac{\theta}{2})}{|\cos\frac{\theta}{2}|} \right) \nonumber\\
&-\frac{k^2_F}{|\cos\frac{\theta}{2}|}\left(\pi-2\arcsin(\sin\frac{\theta}{2})+|\sin\theta|\right),\end{aligned}$$
with $$\mathcal{G}(\theta)=\int_{0}^{\pi} 2\sin^2\varphi \ln\left(\sin\varphi+\sqrt{\sin^2\frac{\theta}{2}+\cos^2\frac{\theta}{2}\sin^2\varphi}\right) d\varphi.$$
The integration area for $I'_2$ is shown in Fig. \[fig:f2\], and we get: $$I'_2=-4\int \frac{|{\mathbf{m}}|\cdot|{\bf R}_{PN}|}{{\mathbf{m}}\cdot{\mathbf{y}}}d^2y=-8\int \frac{d^2\rho}{\cos \phi},$$ where we denote ${\bf R}_{PN}=$ $\boldsymbol\rho$, and $\phi=\alpha-\theta/2$ is the angle between the vectors ${\mathbf{m}}$ and $\boldsymbol\rho$, with $\alpha$ being the angle between the vectors ${\bf R}_{PM}$ and ${\bf R}_{PN}$ (see Fig. \[fig:f2\]). We then have: $$\begin{aligned}
\label{I12}
I'_2&=-8\int_{0}^{\pi} d\alpha\int_{0}^{\sin\alpha}\frac{\rho d\rho}{\cos (\alpha-\theta/2)}\nonumber\\
&=-4\left[\cos^2\frac{\theta}{2}\ln\frac{1+\sin(\theta/2)}{1-\sin(\theta/2)}+2\sin\frac{\theta}{2}\right].\end{aligned}$$
Using $I'_1$ (\[I11\]) and $I'_2$ (\[I12\]) in Eq. (\[F22initial\]) we get equation (\[tildeF22\]) for the interaction function $\tilde F^{(2)}_2(\theta)$.
[10]{} K.-K. Ni, S.Ospelkaus, M.H.G. de Miranda, A.Pe’er, B.Neyenhuis, J.J.Zirbel, S.Kotochigova, P.S.Julienne, D.S.Jin and J.Ye, Science **322**, 231(2008). J. Deiglmayr, A. Grocola, M. Repp, K. M[ö]{}rtlbauer, C. Gl[ü]{}ck, J. Lange, O. Dulieu, R. Wester, and M. Weidem[ü]{}ller, Phys. Rev. Lett. [**101**]{}, 133004 (2008). K. Aikawa, D. Akamatsu, M. Hayashi, K. Oasa, J. Kobayashi, P. Naidon, T. Kishimoto, M. Ueda, and S. Inouye, Phys. Rev. Lett. [**105**]{}, 203001 (2010). M. Deatin, T. Takekoshi, R.Rameshan, L. Reichsollner, F. Felaino, R. Grimm, R. Vexiau, N. Bouloufa, O. Dulieu, and H.-C. Naegerl, arXiv:1106.0129 and an up-coming preprint. M.A. Baranov, Physics Reports [**464**]{}, 71 (2008). T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, Rep. Prog. Phys. [**72**]{}, 126401 (2009). , edited by R.V. Krems, B. Friedrich, and W.C. Stwalley, (CRC Press, Taylor & Francis Group 2009). D.-W. Wang, M.D. Lukin, and E. Demler, Phys. Rev. Lett. [**97**]{}, 180413 (2006). H.P. Büchler, E. Demler, M. Lukin, A. Micheli, N. Prokof’ev, G. Pupillo, and P. Zoller, Phys. Rev. Lett. [**98**]{}, 060404 (2007). G. M. Bruun and E. Taylor, Phys. Rev. Lett. **101**, 245301 (2008). N.R. Cooper and G.V. Shlyapnikov, Phys. Rev. Lett. [**103**]{}, 155302 (2009). A. Pikovski, M. Klawunn, G.V. Shlyapnikov, and L. Santos, Phys. Rev. Lett. [**105**]{}, 215302 (2010). R.M. Lutchyn, E. Rossi, and S. Das Sarma, Phys. Rev. A [**82**]{}, 061604 (2010). A.C. Potter, E. Berg, D.-W. Wang, B.I. Halpern, and E. Demler, Phys. Rev. Lett. [**105**]{}, 220406 (2010). B. Capagrosso-Snasone, C. Trefzger, M. Lewenstein, P. Zoller, and G. Pupillo, Phys. Rev. Lett. [**104**]{}, 125301 (2010). K. Sun, C. Wu, and S. Das Sarma, Phys. Rev. B [**82**]{}, 075105 (2010). Y. Yamaguchi, T. Sogo, T. Ito, and T. Miyakawa, Phys. Rev. A **82**, 013643 (2010). J. Levinsen, N. R. Cooper, and G. V. Shlyapnikov, Phys. Rev. A **84**, 013603 (2011) M.A. Baranov, A. Micheli, S. Ronen, and P. Zoller, Phys. Rev. A [**83**]{}, 043602 (2011). M.M. Parish and F.M. Marchetti, arXiv:1109.2464. M. Babadi and E. Demler, arXiv:1109.3755. L. M. Sieberer, M. A. Baranov, arXiv:1110.3679. S. Ospelkaus, K.-K. Ni, D. Wang, M.H.G. de Miranda, B. Neyenhuis, G. Quemener, P. S. Julienne, J. L. Bohn, D. S. Jin, and J. Ye, Science **327**, 853 (2010). K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M.H.G. de Miranda, J.L. Bohn, J. Ye, and D.S. Jin, Nature **464**, 1324 (2010). G. Quéméner and J. L. Bohn, Phys. Rev. A **81**, 060701(2010). A. Micheli, Z. Idziaszek, G. Pupillo, M. A. Baranov, P. Zoller, and P. S. Julienne, Phys. Rev. Lett. **105**, 073202(2010) M.H.G. de Miranda, A. Chotia, B. Neyenhuis, D. Wang, G. Quemener, S. Ospelkaus, J.L. Bohn, J. Ye, and D.S. Jin, Nature Phys. **7**, 502 (2011). P.S. Zuchowski and J.M. Hutson, Phys. Rev. A [**81**]{}, 060703(R) (2010). J.W. Park, C.-H. Wu, I. Santiago, T.G. Tiecke, P. Ahmadi, and M.W. Zwierlein, arXiv:1110.4552. N.T. Zinner, B. Wunsch, D. Pekker, and D.-W. Wang, arXiv:1009.2030. J. P. Kestner and S. Das Sarma, Phys. Rev. A **82**, 033608 (2010). C-K Chan, C Wu, W-C Lee and S. Das Sarma, Phys. Rev. A **81**, 023602 (2010). T.Miyakawa, T. Sogo, and H. Pu, Phys. Rev. A **77**, 061603 (2008). Q. Li, E. H. Hwang, and S. Das Sarma, Phys. Rev. B **82**, 235126 (2010). N. Navon, S. Nascimbene, F. Chevy and C. Salomon, Science **328**, 729 (2010). S. Nascimbéne, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon, Nature **463**, 1057 (2010). K. Huang and C.N. Yang, Phys. Rev. [**105**]{}, 767 (1957). T. D. Lee and C. N. Yang, Phys. Rev. **105**, 1119 (1957). A.A.Abrikosov and I.M.Khalatnikov, Sov. Phys. JETP **6**, 888 (1958) E.M.Lifshitz and L.P.Pitaevskii, [*Statistical Physics, Part 2*]{} (Pergamon Press, Oxford, 1980). L.D. Landau and E.M. Lifshitz, [*Quantum Mechanics, Non-Relativistic Theory*]{} (Butterworth-Heinemann, Oxford, 1999). In fact, we arrived at this conclusion confining ourselves to the contributions up to $\beta^2$. In stead of considering higher order contributions we checked numerically that the solution with $s>1$ is absent if we omit the $(k_Fr_*)^2$-terms in the interaction function of quasiparticles. M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, arXiv:1110.3309. T. Yefsah, R. Desbuquois, L. Chomaz, K. J. Gunter, and J. Dalibard, Phys. Rev. Lett. **107**, 130401 (2011). D. Wang, B. Neyenhuis, M. H. G. de Miranda, K.-K. Ni, S. Ospelkaus, D. S. Jin, and J. Ye, Phys. Rev. A **81**, 061404(R) (2010).
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abstract: 'Based on density functional calculations, we propose a possible orbital ordering in MnV$_2$O$_4$ which consists of orbital chains running along crystallographic $a$ and $b$ directions with orbitals rotated alternatively by about 45$^{\circ}$ within each chain. We show that the consideration of correlation effects as implemented in the local spin density approximation (LSDA)+U approach is crucial for a correct description of the space group symmetry. This implies that the correlation-driven orbital ordering has a strong influence on the structural transitions in this system. Inclusion of spin-orbit effects does not seem to influence the orbital ordering pattern. We further find that the proposed orbital arrangement favours a noncollinear magnetic ordering of V spins, as observed experimentally. Exchange couplings among V spins are also calculated and discussed.'
author:
- 'S. Sarkar$^{1}$, T. Maitra$^{2}$, Roser Valent[í]{}$^{3}$, T. Saha-Dasgupta$^{1}$'
title: 'Proposed Orbital Ordering in MnV$_2$O$_4$ from First-principles Calculations'
---
The spinel compounds with a chemical formula of AB$_2$X$_4$ where B sites are usually transition metal ions, form a frustrated pyrochlore lattice with corner-sharing tetrahedra. These compounds show a complex behavior including structural transitions from cubic to tetragonal symmetries which are often accompanied by an orbital order-disorder transition as well as complicated magnetic orderings at low temperatures[@radaelli].
The spinel [MnV$_2$O$_4\,\,$]{}has experienced a recent surge in activities due to new experimental observations in single crystals[@garlea] revealing a lower symmetry structure than previously suggested[@adachi]. This has important implications for the related orbital order at low temperatures which is still unclear. The presence of two magnetic ions in [MnV$_2$O$_4\,\,$]{}(Mn with spin 5/2 and V with spin 1) translates into more complex magnetic phase transitions in this system than in other vanadium spinel oxides such as ZnV$_2$O$_4$, MgV$_2$O$_4$ or CdV$_2$O$_4$ with nonmagnetic A-site ions. Recent experimental findings[@garlea; @suzuki] indicated that [MnV$_2$O$_4\,\,$]{}undergoes a phase transition from paramagnetic to a collinear ferrimagnetic phase at 56K where the Mn and V spin moments point in opposite directions. At T = 53K a second magnetic phase transition to noncollinear ferrimagnetism follows accompanied by a structural transition from cubic to tetragonal phase.
The cubic to tetragonal structural transition in [MnV$_2$O$_4\,\,$]{}is, similar to other vanadium spinels, associated with a compression of the VO$_6$ octahedron ($c_T/a_T=0.98$). The octahedral environment of V (VO$_6$) splits the $d$ states into lower t$_{2g}$ and higher e$_g$. Since V$^{+3}$ is in a 3$d^2$ configuration, the t$_{2g}$ orbitals are partially filled and possible orbital orderings may occur. Earlier experimental observations [@adachi] indicated the tetragonal space group to be $I4_1/amd$. However, recent precise measurements on a single crystal[@garlea; @suzuki] showed that the tetragonal space group is $I4_1/a$. Since the orbital order and, accordingly, the magnetic order are closely related to the underlying space group symmetry, it is very important to establish the space group symmetry unambiguously.
The $I4_1/a$ space group breaks the mirror and glide symmetries present in the $I4_1/amd$ space group, which implies that two of the four V-O bonds in the $ab$ plane are shorter whereas in $I4_1/amd$ symmetry all four V-O bond lengths are the same. Garlea [*et al.*]{}[@garlea] proposed a staggered A-type orbital ordering for this system based on their observations of the structural and magnetic phases at low temperature. A similar ordering was also proposed by Suzuki [*et al.*]{}[@suzuki]. Though the magnetic structure at low temperatures has been unambiguously established by the above mentioned experiments, there has not yet been any experiment such as X-ray resonant spectroscopy to directly probe the orbital order. Determination of exchange couplings using neutron scattering techniques by Chung [*et al.*]{}[@chung] is in apparent contradiction with the proposed staggered A-type orbital ordering. As pointed out by these authors, the proposed orbital order in Refs. lacks the consideration of trigonal distortion, which is found to be strongest in [MnV$_2$O$_4\,\,$]{}among all the vanadium spinels. The trigonal distortion has often shown to have significant effects on the orbital order [@maitra; @anisimov].
In this Letter we show, based on density functional theory (DFT) calculations, that the ground state tetragonal space group symmetry at low temperatures is $I4_1/a$ and strongly driven by correlation effects. We propose an orbital ordering consisting of orbital chains running along the axes $a$ and $b$ with orbitals rotated by about 45$^{\circ}$ within each chain. This ordering favors a noncollinear arrangement of spins, as observed experimentally, which is a convincing indication of its existence.
For our DFT calculations we considered a combination of three different methods, namely: (a) plane wave-based method (b) linear augmented plane wave (LAPW) method and (c) muffin-tin orbital (MTO) based N-th order MTO (NMTO) method. Results were cross-checked among the three schemes in terms of total energy differences, density of states and band structures. Since first principles calculations take into account all structural and chemical aspects appropriately, we expect to gain a better understanding of the nature of the structural phase transition and possible orbital ordering.
We first performed a structural optimization using the plane wave method as implemented in the Vienna Ab-initio Simulation Package (VASP)[@vasp] and considered exchange-correlation functionals within LSDA, generalized gradient approximation (GGA) and LSDA+U[@Anisimov_93] in order to investigate the relative stability between $I4_1/amd$ and $I4_1/a$ symmetries in [MnV$_2$O$_4\,\,$]{}. We used projector augmented wave (PAW) potentials[@paw] and the wavefunctions were expanded in the plane wave basis with a kinetic energy cut off of 450 eV. Reciprocal space integration was carried out with a k-mesh of 6$\times$6$\times$6.
---- ------------------- ------------------- ----------------------
LSDA GGA LSDA+U
(U=4.5 eV)
Mn 0.0 0.25 0.125 0.0 0.25 0.125 0.0 0.25 0.125
V 0.0 0.0 0.5 0.0 0.0 0.5 0.0 0.0 0.5
O 0.0 0.0243 0.7392 0.0 0.0236 0.7394 0.0059 0.0244 0.7383
---- ------------------- ------------------- ----------------------
: Energy-minimized structural parameters for [MnV$_2$O$_4\,\,$]{}. Lattice constants were kept at the experimental value[@adachi]. The LSDA+U optimized structural parameters show the O x-coordinate to be non-zero, signaling the change of space group symmetry to $I4_1/a$.
Optimization of the atomic positions[@note] within LSDA as well as GGA assuming ferrimagnetic spin arrangements between Mn and V atoms gave us a ground state structure of $I4_1/amd$ symmetry where the tetragonal distortion is found to be substantially reduced compared to the experimental estimate[@adachi]. In order to check the influence of electron-electron correlation on the structural optimization, which has been found to be important in previous reports[@u-effect], we have further optimized the atomic positions within the LSDA+U approach with different choices of U values[@uvalue] (U=0.5, 1, 2, 3, 4.5 and 6 eV) for both Mn and V. $J$ was chosen to be 1 eV for all calculations. Remarkably, we observe that with the consideration of U beyond 2eV, the $I4_1/a$ symmetry becomes the ground state structure (see Table I). This optimized structure shows a tetragonal distortion close to the experimentally reported one[@adachi]. These results indicate the importance of correlation effects for the description of the correct orbital ordering and the low temperature structure.
![(Color online) LSDA+U V-$d$ partial DOS for U=4.5 eV in the APW+lo basis. Only the DOS for the majority spin channel is shown (the minority spin channel is unoccupied).[]{data-label="dos"}](fig1_new){width="8cm"}
We analyzed the resulting orbital order with the full potential LAPW method as implemented in the Wien2k code [@wien2k]. The atomic sphere radii were chosen to be 2.01, 1.98 and 1.77 a.u. for Mn, V and O respectively. We chose the APW+lo as the basis set and the expansion in spherical harmonics for the radial wave functions was taken up to $l=10$. The charge densities and potentials were represented by spherical harmonics up to $l=6$. For Brillouin- zone (BZ) integrations we considered a 52 $ k$ points mesh in the irreducible wedge and the modified tetrahedron method was applied[@tetra]. The collinear ferrimagnetic spin arrangements between Mn and V was taken the same as for the structural optimization calculations. In all further calculations we considered the LSDA+U approximation[@double] and fixed the value of U at 4.5 eV which reproduces the experimentally observed orbital moment in vanadium, as will be discussed later.
In Fig. \[dos\] we show the electronic density of states (DOS) calculated within the LSDA+U approximation. In the partial DOS one observes the usual t$_{2g}$ (consisting of x$^{2}$-y$^{2}$, xz and yz orbitals defined in the crystallographic co-ordinate system)[@note1] and e$_g$ (consisting of xy, 3z$^{2}$) splitting of V $d$-orbitals due to the O octahedral crystal field. Inclusion of correlation effects in the V $d$-orbitals through the LSDA+U approach, splits the t$_{2g}$ states further and opens a gap of 1.1 eV. The degeneracy between all the three t$_{2g}$ orbitals is lifted in the low symmetry $I4_1/a$ group[@comment]. All t$_{2g}$ orbitals are partially occupied with higher x$^{2}$-y$^{2}$ and yz occupancy compared to xz. This becomes more evident in the band structure results. Fig. \[bands\] shows the t$_{2g}$ bandstructure in the majority spin channel, which is separated from occupied O-p dominated bands by a gap of 1.5 eV and from unoccupied e$_{g}$-like bands by a gap of 0.2 eV. The fatness of the bands indicate the projected band characters of x$^{2}$-y$^{2}$, xz and yz orbitals.
![LSDA+U bandstructure of [MnV$_2$O$_4\,\,$]{}(APW+lo basis) projected onto V- x$^{2}$-y$^{2}$, xz and yz character (from left to right) in the energy range \[-3 eV, 1 eV\]. The high symmetry path of the tetragonal Brillouin zone was considered.[]{data-label="bands"}](fig2_new){width="9cm"}
Significant mixing of orbitals happens due to the low symmetry of the $I4_1/a$ space group. In Fig. \[eldens\] we show the three-dimensional electron density of occupied V t$_{2g}$ orbitals on a real space grid. We identify a long range order pattern for the orbital distribution. Contrary to the proposed staggered A-type order[@garlea; @suzuki], we observe orbital chains along $a$ and $b$ directions (indicated by solid and dashed lines) with the orbitals within each chain rotated alternatively by about 45$^{\circ}$ (shown by the arrows).
![[ (Color online) Three dimensional electron density plot showing the orbital ordering. The black solid and dashed lines designate the orbital chains. The arrows superimposed on the electron density at each V site, mark the rotation sense of the orbitals as one moves to neighboring V sites within a given chain. The atoms at the alternate corners of the distorted cubes are occupied by V and O respectively. The isovalue was chosen as 0.1 e$^{-}$/$(\AA^{3})$.]{}[]{data-label="eldens"}](fig3_test){width="6.5cm"}
In order to assign the precise V orbital compositions we have performed NMTO-downfolding[@nmto] calculations to construct a V-t$_{2g}$-e$_g$ only low-energy Hamiltonian by integrating out degrees of freedom other than V-t$_{2g}$-e$_g$, starting with a full LSDA+U Hamiltonian. Diagonalization of the on-site energy block of this 5 x 5 Hamiltonian gives rise to eigenstates given by:
$$\begin{aligned}
\vert 1 \rangle =\phantom{-}.78 \vert x^2-y^2 \rangle - .59 \vert xz \rangle - .21\vert yz \rangle
+ .07 \vert xy \rangle + .02 \vert z^2 \rangle\\
\vert 2 \rangle =-.35 \vert x^{2}-y^{2} \rangle
- .15 \vert xz \rangle - .92 \vert yz
\rangle - .09 \vert xy \rangle - .07 \vert z^2 \rangle\\
\vert 3 \rangle =\phantom{-}.52 \vert x^{2}-y^{2} \rangle +
.79 \vert xz \rangle - .31 \vert yz \rangle
- .13 \vert xy \rangle + .02 \vert z^2 \rangle\\
\vert 4 \rangle =\phantom{-}.05 \vert x^{2}-y^{2} \rangle - .08 \vert xz \rangle +
.11 \vert yz \rangle
- .66 \vert xy \rangle - .74 \vert z^2 \rangle\\
\vert 5 \rangle =-.02 \vert x^{2}-y^{2} \rangle - .11 \vert xz \rangle +
.04 \vert yz \rangle - .73 \vert xy \rangle + .67 \vert z^2 \rangle\end{aligned}$$
with energies 0.81, 1.19, 1.47, 2.05, 2.28 eV respectively.
We observe that the lowest energy state has predominant x$^{2}$-y$^{2}$ character -which is expected due to the tetragonal distortion with the compression of VO$_6$ octahedron along the [*c*]{}-direction- with a significant mixing of xz character. The next higher energy state is dominated by yz character. Therefore, the second electron of $V^{3+}$ always occupies the orbital with predominant yz character in all V sites. The rotation of orbitals with respect to each other within the chain and between the chains (see Fig. \[eldens\]), can therefore be explained due to the staggered trigonal distortion that is present both within the $ab$-plane and along the [*c*]{}-direction. Despite an apparent [*antiferro-orbital*]{} ordering, we call the ordering [*ferro-orbital*]{} since it is in all sites the same orbital that is occupied by the second electron, and not an alternating occupation of xz and yz.
The spin-orbit effect has been observed to play a significant role in dictating the nature of orbital order[@tchernyshyov; @maitra] in ZnV$_2$O$_4$ and was proposed to be important for the magnetic and orbital physics of [MnV$_2$O$_4\,\,$]{}[@plumier]. We performed LSDA+U+SO calculations with the same U values as mentioned above, where the spin-orbit effects have been introduced as a second variation using the scalar relativistic approximation. Contrary to the case of ZnV$_2$O$_4$[@maitra], we do not observe any significant difference in charge density, from that of LSDA+U. The value of the orbital moment depends sensitively on U. The experimental V moment is best described for U=4.5 eV. At this U value we obtain an orbital moment of about 0.34 $\mu_B$ at V site which is antiparallel to the spin-moment (1.65$\mu_B$). The total magnetic moment of 1.31 $\mu_B$ is close to the measured value[@garlea] of 1.3 $\mu_B$. Also, the calculated magnetic moment at the Mn site is found to be 4.24 $\mu_B$ in good agreement with the experimental estimate [@garlea]. The orbital moment at the V site seems to develop an appreciable value only beyond a critical U value, U$_c$ (3.0 eV $<$ U$_c$ $\le$ 4.5 eV)[@orb], which may be interpreted as [*Coulomb enhanced spin-orbit effect*]{}[@oka].
We note that the perfect antiferro-orbital ordering as proposed by Refs. and would imply a quenching of orbital moment. The presence of a finite orbital moment can be associated with the breakdown of perfect antiferro-orbital ordering and may explain the domain alignment by magnetic field as observed by Ref. .
We have also computed the magnetic exchange couplings from first principles by considering LSDA+U total energy calculations with the PAW basis for different spin alignments of V atoms within the V tetrahedra. Mapping the total energies to a Heisenberg like model, we obtain exchange interactions along the orbital chains (J) of 11 meV and between the chains (J$^{'}$) of 2 meV. This implies $\alpha$ = J$^{'}$/J $\approx$ 0.2 compared to 0.3 found by Chung [*et al.*]{}[@chung]. Perfect antiferro-orbital ordering with xz and yz alternately occupied along the [*c*]{}-axis would however yield much smaller ratios of J$^{'}$/J, since the overlap between orthogonal yz and xz orbitals at neighboring sites would have been nearly zero. The moderately strong value of J$^{'}$, as obtained in the DFT calculation, originates from large mixing of different t$_{2g}$ orbitals influencing the overlap of the renormalized orbitals at neighboring sites.
Our calculations described so far assume the collinear arrangement of V spins, while experiment reports a transition from collinear to noncollinear spin arrangements coincident with the structural phase transition. In order to check whether our proposed orbital order sustains a noncollinear arrangement of V spins, we performed PAW calculations where we relaxed the V spin orientation keeping the Mn spins aligned parallel to the c axis[@garlea]. The relaxed spin structure shows the V spins to be canted with respect to the c axis by about 63$^{\circ}$, which is in very good agreement with the experimentally estimated canting of 65$^{\circ}$[@garlea]. The noncollinear spin arrangement was found to be slightly favoured over the collinear ferrimagnetic spin arrangement by an energy gain of 3 meV. Though this energy difference is almost within the accuracy limit of DFT, the good agreement between theory and experimental estimates is encouraging.
To conclude, we have carried out DFT-based first-principles calculations to investigate the nature of the orbital ordering in [MnV$_2$O$_4\,\,$]{}which is closely associated with the transition from a high temperature cubic structure to a low temperature tetragonal structure. Our geometry-optimized structures for [MnV$_2$O$_4\,\,$]{}show a strong influence of correlation effects in the choice of the [*correct*]{} low temperature structure. The obtained ground state structure, $I4_1/a$ looses the mirror and glide symmetry compared to the alternative proposed candidate $I4_1/amd$. The O in $I4_1/a$ are in 16f positions with nonzero x-coordinate, which makes the V-O bondlengths even in the ab-plane to be unequal. This lowering of symmetry necessarily breaks the degeneracy of the t$_{2g}$ states completely and also introduces mixing between different t$_{2g}$ states. The resulting eigenstates therefore turn out to be of [*mixed-character*]{} and [*nondegenerate*]{}, which get filled up by two V electrons. The occupied orbitals follow the site symmetry of vanadium which is 4-fold rotation times inversion to give rise to orbital chains with orbitals rotated with respect to each other both within and between the chains. Our DFT computed V-V magnetic coupling is found to be in agreement with the experimental findings[@chung]. These results provide an explanation of the controversy between [*antiferro-orbital ordering*]{} versus the strong exchange between orbital chains (J$^{'}$). We further showed that our proposed orbital ordering is capable of predicting correctly the noncollinear spin structure as observed experimentally[@garlea]. Further experiments like X-ray resonant spectroscopy would be helpful to probe directly our proposed orbital order.
[*Acknowledgements-*]{} We acknowledge useful discussions with J. Glinnemann and D. Khomskii. TSD thanks Swarnajayanti Grant and MPI, Stuttgart through partnergroup program. RV thanks the DFG for financial support through the SFB/TRR49 program. SS thanks CSIR for financial support.
[99]{} P. G Radaelli, New J. Phys. [**7**]{}, 53 (2005). V. O. Garlea [*et. al.*]{}, Phys. Rev. Lett. [**100**]{}, 066404 (2008). K. Adachi [*et. al.*]{}, Phys. Rev. Lett. [**95**]{}, 197202 (2005). T. Suzuki [*et. al.*]{}, Phys. Rev. Lett. [**98**]{}, 127203 (2007). J.-H. Chung [*et. al.*]{}, Phys. Rev. B [**77**]{} 054412 (2008). T. Maitra and R. Valentí, Phys. Rev. Lett. [**99**]{}, 126401, (2007). V.I. Anisimov [*et al.*]{}, Phys. Rev. Lett [**83**]{}, 364 (1999). G. Kresse and J. Hafner, Phys. Rev. B 47, RC558 (1993); Phys. Rev. B 48, 13115 (1993); Phys. Rev. B 49, 14251 (1994). V.I. Anisimov [*et al.*]{}, Phys. Rev. B [**48**]{}, 16929 (1993). P.E. Blöchl, Phys. Rev. [**50**]{}, 17953 (1994). Since Mn and V occupy the high symmetry 4a and 8d positions, optimization essentially involves optimization of the internal degrees of freedom associated with O. L. Pisani and R. Valentí, Phys. Rev. B [**71**]{}, 180409 (2005); C. J. Fennie and K. M. Rabe, Phys. Rev. B [**72**]{}, 214123 (2005). Mn and V are neighbors in the periodic table and it is not to be expected that their U values will be very different. P. Blaha, K. Schwartz, G. K. H. Madsen, D. Kvasnicka and J. Luitz; WIEN2K, An Augmented Plane Wave + Local Orbitals Program for calculating crystal properties (K. Schwarz, Techn. University Wien, Austria, 2001), ISBN 3-9501031-1-2. P. E. Blöchl [*et. al.*]{}, Phys. Rev B 49, 16223 (2004). In all calculations we considered the self interaction double counting correction (SIC)[@Anisimov_93]. We also performed calculations with the around mean field correction (AMF) in order to check the influence of double counting correction. Our conclusions remained unchanged. $I4_1/a$ is favored with respect to $I4_1/amd$ by 0.07 eV in the SIC calculation and 0.04 eV in the AMF calculation in APW+lo basis. The crystallographic $ab$-plane is rotated by 45$^{\circ}$ from the basal plane defined by the V-O bonds, thereby forming t$_{2g}$ states of x$^{2}$-y$^{2}$, xz and yz symmetry instead of more commonly used convention of xy, xz and yz. This is in contrast to the situation in $I4_1/amd$ symmetry where the degeneracy between xz and yz is maintained (see [*e.g.*]{} Fig.2 in Ref. ). O. K. Andersen and T. Saha-Dasgupta, Phys. Rev. B [**62**]{}, R16219 (2000). O. Tchernyshyov, Phys. Rev. Lett. [**93**]{}, 157206 (2004). R. Plumier and M. Sougi, Physica B [**155**]{}, 315 (1989). The V orbital moment computed for U values 3.0 eV, 4.5 eV and 6.0 eV were found to be 0.04 $\mu_B$, 0.34 $\mu_B$ and 0.38 $\mu_B$ respectively. Guo-Qiang Liu [*et. al.*]{}, Phys. Rev. Lett. [**101**]{}, 026408 (2008).
|
---
abstract: |
We investigate optimal geographical caching in heterogeneous cellular networks, where different types of base stations (BSs) have different cache capacities. The content library contains files with different popularities. The performance metric is the total hit probability.
The problem of optimally placing content in all BSs jointly is not convex in general. However, we show that when BSs are deployed according to homogeneous Poisson point processes (PPP), independently for each type, we can formulate the problem as a convex problem. We give the optimal solution to the joint problem for PPP deployment. For the general case, we provide a distributed local optimization algorithm (LOA) that finds the optimal placement policies for different types of BSs. We find the optimal placement policy of the small BSs (SBSs) depending on the placement policy of the macro BSs (MBSs). We show that storing the most popular content in the MBSs is almost optimal if the SBSs are [[ using an optimal]{}]{} placement policy. Also, for the SBSs no such heuristic can be used; the optimal placement is significantly better than storing the most popular content. Finally, we numerically verify that LOA gives the same hit probability as the joint optimal solution for the PPP model.
author:
- 'Berksan Serbetci, and Jasper Goseling [^1] [^2] [^3]'
title: Optimal Geographical Caching in Heterogeneous Cellular Networks
---
Introduction
============
In recent years, there is extreme growth in data traffic over cellular networks. The growth rate of the demand is expected to increase in the upcoming years [@cisco] such that current network infrastructures will not be able to support this data traffic [@femtocells]. In order to tackle this problem, an obvious approach is to increase the number of base stations. These base stations require a high-speed backhaul to make this system work properly and it is costly to connect every base station to the core network in real life. A solution to this problem is to reduce backhaul traffic by reserving some storage capacity at both macro base stations (MBSs) and small base stations (SBSs) and use these as caches [@femtod2d]. In this way, part of the data is stored at the wireless edge and the backhaul is used only to refresh this stored data. Data replacement will depend on the users’ demand distribution over time. Since this distribution is varying slowly, the stored data can be refreshed at off-peak times. In this way, caches containing popular content changing over time depending on the demand will serve as helpers to the overall system and decrease the backhaul traffic.
Recently, there has been growing interest in caching in cellular networks. In [@femtocaching] Shanmugam *et al.* focus on the content placement problem and analyze which files should be cached by which helpers for the given network topology and file popularity distribution by minimizing the expected total file delay. In [@approximation] Poularakis *et al.* provide an approximation algorithm for the problem of minimizing the user content requests routed to macrocell base stations with constrained cache storage and bandwidth capacities. In [@optimalgeographic] B[ł]{}aszczyszyn *et al.* revisit the optimal content placement in cellular caches by assuming a known distribution of the coverage number and provide the optimal probabilistic placement policy which guarantees maximal total hit probability. In [@fundamental] Maddah-Ali *et al.* developed an information-theoretic lower bound for the caching system for local and global caching gains. In [@distrcach] Ioannidis *et al.* propose a novel mechanism for determining the caching policy of each mobile user that maximize the system’s social welfare. In [@exploiting] Poularakis *et al.* consider the content storage problem of encoded versions of the content files. In [@cacheenabled] Bastug *et al.* couple the caching problem with the physical layer. In [@diststorage] Altman *et al.* compare the expected cost of obtaining the complete data under uncoded and coded data allocation strategies for caching problem. Cache placement with the help of stochastic geometry and optimizing the allocation of storage capacity among files in order to minimize the cache miss probability problem is presented by Avrachenkov *et al.* in [@optimizationofcaching]. A combined caching scheme where part of the available cache space is reserved for caching the most popular content in every small base station, while the remaining is used for cooperatively caching different partitions of the less popular content in different small base stations, as a means to increase local content diversity is proposed by Chen *et al.* in [@cooperativecaching]. In [@utility] Dehghan *et al.* associate with each content a utility, which is a function of the corresponding content hit probabilities and propose utility-driven caching, where they formulate an optimization problem to maximize the sum of utilities over all contents.
The *main contribution* of this paper is to find optimal placement strategies that maximize total hit probability in heterogeneous cellular networks. [[ Different from [@optimalgeographic; @optimizationofcaching] our focus is on heterogeneous cellular networks in which an operator wants to jointly optimize the cached content in macro base stations (MBSs) and small base stations (SBSs) with different storage capacities and different coverage radii.]{}]{} This problem is not convex in general conditions. We show that it is possible to reformulate the problem and make it convex when base stations are deployed according to homogeneous Poisson point processes (PPP). [[ To our knowledge, this is the first paper that provides an optimal solution to the optimization of heterogeneous (and multi-tier) caching devices. We show that optimal placement strategies of the base stations can be flexibly distributed over different types if the sum of the file placement probabilities times the density parameters of the base stations satisfy a certain capacity constraint.]{}]{} As the general problem is not convex, we provide a distributed local optimization algorithm (LOA) and optimize only one type of cache (e.g. SBS) using the information coming from other types of caches (e.g. MBS and other SBSs with different cache capacities) at each iteration step. We numerically verify that for PPP deployment scenario, LOA converges to the optimal hit probability that is found by solving the joint convex optimization problem after one round. We also illustrate with numerical examples how LOA performs for non-PPP deployment scenarios.
For several configurations we show that whether MBSs use the optimal deployment strategy or store “the most popular content", has no impact on the total hit probability after deploying the SBSs with optimal content placement policies. We show that it is crucial to optimize the content placement strategy of the SBSs in order to maximize the overall performance. We show that heuristic policies like storing the popular content that is not yet available in the MBSs result in significant performance penalties.
The placement strategies that are proposed in this paper are probabilistic in nature. Therefore, they provide a very low-complexity solution to content placement in large networks. In [@gibbsian] and [@sigmetrics] non-probabilistic strategies are proposed that take into account exact base station locations and the overlap in coverage regions. These strategies will result in higher hit probabilities, but come at significantly larger complexity.
The remainder of this paper is organized as follows. In Section \[model\] we start the paper with model and problem definition. In Section \[sec:jointoptimization\] we present the joint optimal placement strategy problem for the PPP model and give required tools to solve it. In Section \[sec:distributedoptimization\] we provide a distributed local optimization algorithm for the general non-convex joint optimization problem. In Section \[sec:performance\] we continue with performance evaluation of the optimal placement strategies for different probabilistic deployment scenarios. In Section \[discussion\] we conclude the paper with discussions.
Model and Problem Definition {#model}
============================
In this section we will present the general model and the problem formulation.
Throughout the paper we will be interested in different types of base stations, namely MBSs and SBSs with different cache capacities. We will give the most general formulation of the problem as it is possible to have MBSs and SBSs with different storage capacities in some network topologies.
We consider a heterogeneous cellular network with L-different types of base stations in the plane. These base stations are distributed according to a spatial point process [@baccelli1]. [[ Type$-\ell$, $\ell = 1,\dots, L$, base stations have coverage radius $r_\ell$]{}]{}. Let $\mathcal{N}_\ell$ denote the number of base stations of type$-\ell$ that are covering a user at an arbitrary location in the plane. Furthermore, let $p_{\ell}(n_\ell) := \mathbb{P}[\mathcal{N}_\ell = n_\ell]$ denote the probability of a user being covered by $n_\ell$ type$-\ell$ caches, and $p(\bm{n}) := \mathbb{P}[\bm{\mathcal{N}} = \bm{n}]$ [[ denote the joint probability of a user being covered by $n_\ell$ type$-\ell$ caches, $\forall \ell = 1,\dots,L$]{}]{}, where $\bm{\mathcal{N}}=(\mathcal{N}_1,\dots, \mathcal{N}_L)$ and $\bm{n} = \left(n_1, \dots, n_L\right).$ We also define $\bm{\mathcal{N}}^{(-\ell)} = (\mathcal{N}_i)_{i\neq\ell}$ and $\bm{n}^{(-\ell)} = (n_i)_{i\neq\ell}$ as the corresponding $(L-1)$-tuples excluding the $\ell$th component.
Caches store files from a content library $\mathcal{C} = \{c_1, c_2, \dots, c_J\}$, where an element $c_j$ is a file with normalized size $1$. The probability that file $c_j$ is requested is denoted as $a_j$. Without loss of generality, $a_1 \geq a_2 \geq \dots \geq a_J$.
Type$-\ell$ caches have cache memory size $K_\ell \geq 1$, $\ell = 1, \dots, L$. We use the probabilistic content placement policy of [@optimalgeographic]. We denote the probability that the content $c_j$ is stored at a type$-\ell$ cache as $$b_j^{(\ell)} := \mathbb{P}\left(c_j \text{ stored in type$-\ell$ cache}\right),$$ and the placement strategy $\bm{b}^{(\ell)} = (b_1^{(\ell)}, \dots, b_J^{(\ell)})$ as a J-tuple for any type$-\ell$ cache. The content is independently placed in the cache memories according to the same distribution for the same type of caches. The placement procedure is as follows. The memory of a type$-\ell$ cache is divided into $K_\ell$ continuous memory intervals of unit length. Then $b_j^{(\ell)}$ values fill the cache memory sequentially and continue filling the next slot if not enough space is available in the memory slot that it has started filling in as in the end completely covering the $K_\ell$ memory intervals. Then, for any type$-\ell$ cache, a random number from the interval $[0,1]$ is picked and the intersecting $K_\ell$ files are cached. An example is shown in Figure \[realization\].
Then the overall placement strategy for all types of caches can be denoted by $\bm{B} = \left[\bm{b}^{(1)}; \dots; \bm{b}^{(L)}\right]$ as a $L \times J$ matrix.
Next, we introduce our performance metric. The performance metric is the total miss probability ($1$-minus-hit probability) for the users located in the plane and is given by $$f\left(\mathbf{B}\right) = \sum_{j = 1}^J a_j \sum_{n_1=0}^{\infty} \dots \sum_{n_L=0}^{\infty} p(\bm{n}) \prod_{\ell = 1}^L (1 - b_j^{(\ell)})^{n_\ell}.
\label{missprob}$$
We define the optimization problem to find the optimal placement strategy minimizing the total miss probability as follows:
\[orgprb\] $$\begin{aligned}
&\min \text{ } f\left(\mathbf{B}\right)\nonumber\\
&\text{ }\mathbf{s.t.}\quad b_1^{(\ell)} + \dots + b_J^{(\ell)} = K_\ell, \quad b_j^{(\ell)} \in [0,1],\quad \forall j, \ell. \label{constraints}\end{aligned}$$
Deployment models and file popularities
=======================================
In this section we present specific cache deployment models and file popularity distributions that will be used in this paper.
Deployment models
-----------------
### Homogeneous PPP deployment model
In this model we assume that a user at an arbitrary location in the plane can connect to all type$-\ell$ caches that are within radius $r_\ell$. The caches follow a two-dimensional (2D) spatial homogeneous Poisson process with type$-\ell$ caches independently distributed in the plane with density $\lambda_\ell > 0$ where $\ell = 1, \dots, L$. Type$-\ell$ caches within radius $r_\ell$ follows a Poisson distribution with parameter $t_\ell = \lambda_\ell \pi r_\ell^2$. Then, we conclude that $$\begin{aligned}
p_{\ell}(n_\ell) &= P(\text{$n_\ell$ type$-\ell$ caches within radius $r_\ell$})\nonumber\\
&= \frac{t_\ell^{n_\ell}}{n_\ell!}e^{-t_\ell}. \label{poisdist}\end{aligned}$$
The user is covered by $n_\ell$ type$-\ell$ caches and distributions of the different types of caches are independent of each other. Therefore, the total coverage distribution probability mass function $p_{\bm{n}}$ will be the product of individual probability distributions $$\label{PPPcoverage}
p\left(\bm{n}\right) = p_{1}(n_1) p_{2}(n_2)\dots p_{L}(n_L).$$
### M-or-None deployment model
In this model once again we assume that a user at an arbitrary location in the plane can connect to [[ all type$-\ell$ caches that are within radius $r_\ell$]{}]{}. Type$-1$ caches represent macro base stations and follow a two-dimensional (2D) spatial homogeneous Poisson process with density $\lambda_1 > 0$. As a consequence, the number of type$-1$ caches within radius $r_1$ follows a Poisson distribution satisfying for $\ell = 1$.
We assume that if a user is covered by at least one macro base station (type$-1$ cache), then it will have $M$ helpers (other types of caches) in total. As a result, network operators serve users with providing them $M$ helpers as long as they are connected to at least one of the macro base stations. If a user is not covered by a type$-1$ cache, then it doesn’t receive any service from other caches either. Therefore, we have $$P\left(\bm{\mathcal{N}}^{(-1)} = \bm{n}^{(-1)} \vert \mathcal{N}_1 = 0\right) = \left\{
\begin{array}{rl}
0 & \text{if } \sum_{l=2}^L n_l \neq 0,\\
1 & \text{if } \sum_{l=2}^L n_l = 0,
\end{array} \right.$$ and, $$P\left(\bm{\mathcal{N}}^{(-1)} = \bm{n}^{(-1)} \vert \mathcal{N}_1 = n_1 \right) = \left\{
\begin{array}{rl}
0 & \text{if } \sum_{l=2}^L n_l \neq M,\\
1 & \text{if } \sum_{l=2}^L n_l = M,
\end{array} \right.$$ when $n_1 > 0$.
File Popularities
-----------------
In this section we will introduce file popularity distributions. Even though any popularity distribution can be used, our numerical results will be based on Zipf distribution. Specifically we will use standard and perturbed Zipf models.
### Zipf distribution
For this model, without loss of generality, $a_1 \geq a_2 \geq \dots \geq a_J$. The probability that a user will ask for content $c_j$ is then equal to $$a_j = \frac{j^{-\gamma}}{\sum_{j=1}^J j^{-\gamma}}, \label{zipfpars}$$ where $\gamma > 0$ is the Zipf parameter.
### Perturbed Zipf distribution
In practice, one might not have the exact file popularities available. Instead, only estimates might be available. Suppose that $a_j^{\text{pert}}$ values are the actual file popularity values and that $a_j$ values are estimates of these popularities. We propose a perturbed Zipf model for the actual popularity distribution. In this model the probability that a user will ask for content $c_j$ is equal to $$a_j^{\text{\text{pert}}} = \frac{\left(a_j + Z_j \right)^+}{\sum_{j=1}^J \left(a_j + Z_j \right)^+}, \label{zipfnoisy}$$ where $a_j$ follows a Zipf distribution with given $\gamma > 0$, $Z_j$ is the noise, where $Z_j$ is independent and identically distributed and drawn from a zero-mean normal distribution with variance $\sigma_j^2$. Note that the difference between the available popularity values $a_j$ and the actual file popularity values $a_j^{\text{pert}}$ increases as the variance $\sigma_j^2$ of the perturbation increases.
Joint Optimization for the Homogeneous Poisson Point Process (PPP) model {#sec:jointoptimization}
========================================================================
Finding the optimal placement strategy for all types of caches jointly is an interesting problem. However, this joint optimization problem presented in Problem \[orgprb\] is not convex in general conditions.
When each type of cache is deployed according to a homogeneous spatial Poisson process, it is possible to reformulate the joint optimization problem such that the problem becomes convex. We will present such a formulation in the next subsection and continue with the general structure afterwards.
Since each type of cache is deployed according to a homogeneous spatial Poisson process, we can see for any file $j$ being present in a type$-i$ cache as a thinned Poisson process [@stochasticgeometry]. Then, for any user in the plane, the probability of missing file $j$ is equal to the joint probability of the thinned Poisson processes. We will continue with formulation and the optimal solution of this problem.
Formulation of the problem
--------------------------
In this model, we assume that a user at an arbitrary location in the plane can connect to all type$-\ell$ caches that are within radius $r_\ell$. The caches follow a two-dimensional (2D) spatial homogeneous Poisson process with type$-\ell$ caches independently distributed in the plane with density $\lambda_\ell > 0$, where $\ell = 1, \dots, L$. The number of type$-\ell$ caches within radius $r_\ell$ follows a Poisson distribution with parameter $\lambda_\ell \pi r_\ell^2$. Then, from thinning the Poisson process, it follows that type$-\ell$ caches storing the file $c_j$ follows the Poisson distribution with parameter $b_j^{(\ell)}\lambda_\ell \pi r_\ell^2$. The performance metric is the total miss probability which is the probability that a user will miss the content that he requires in one of the caches that he is covered by.
With the thinned Poisson process argument, the total miss probability is given by $$f_{\text{joint}}\left(\bm{B}\right) = \sum_{j=1}^J a_j \exp\left\{-\sum_{\ell = 1}^L b_j^{(\ell)} \lambda_\ell \pi r_\ell^2\right\}.
\label{thinprob}$$
By using the thinning argument, the total probability of miss for type$-\ell$ cache is equal to the probability of being covered by $0$ type$-\ell$ caches storing the file. Therefore, we have $$f_{\ell}\left(\bm{b}^{(\ell)}\right) = \sum_{j=1}^J a_j \exp\left\{-b_j^{(\ell)} \lambda_\ell \pi r_\ell^2\right\}.$$ When different types of caches’ locations are all following homogeneous Poisson processes, the probability of missing a specific file is equal to the joint probability of being not covered by any cache storing that specific file over all types of caches. Hence, [[ the union of independent Poisson point processes is a Poisson point process with a density equal to the sum of the respective densities and the total miss probability is given by Eq. ]{}]{}.
We define the optimization problem to find the optimal placement strategy minimizing the total hit probability for all caches jointly as follows:
\[prb:PPPorgprb\] $$\begin{aligned}
&\min \text{ } f_{\text{joint}}\left(\bm{B}\right)\nonumber\\
&\text{ }\mathbf{s.t.}\quad b_1^{(\ell)} + \dots + b_J^{(\ell)} = K_\ell, \quad b_j^{(\ell)} \in [0,1],\quad \forall j, \ell. \label{constraints2}\end{aligned}$$
Solution of the optimization problem
------------------------------------
In this section, we will analyze the structure of the optimization problem.
\[convex\] Problem \[prb:PPPorgprb\] is a convex optimization problem.
[[ The exponential of an affine function is known to be convex. Our performance metric is a positively-weighted sum of the exponential of affine combinations, which is still convex.]{}]{}
We already showed that $f_{\text{joint}}\left(\bm{B}\right)$ is convex by Lemma \[convex\] and the constraint set is linear as given in . Thus, the Karush-Kuhn-Tucker (KKT) conditions provide necessary and sufficient conditions for optimality [@KKTref]. We define a new parameter $d_j$ as the sum of the intensities of all thinned Poisson processes for file $c_j$ as follows: $$d_j = \sum_{\ell = 1}^L b_j^{(\ell)} \lambda_\ell \pi r_\ell^2 \label{djdef},$$ and the following vector consisting of the sum of the intensities of all types of caches for all files: $\bm{D} = (d_1, \dots, d_J)$ as a J-tuple.
Then, the total miss probability is given by $$f_{\text{sum}}\left(\bm{D}\right) = \sum_{j=1}^J a_j \exp(-d_j),
\label{missprob}$$ and we have a optimization problem to find the optimal placement strategy minimizing the total hit probability for all caches when caches are following PPP as follows:
\[prb:PPPmodprb\] $$\begin{aligned}
&\min \text{ } f_{\text{sum}}\left(\bm{D}\right)\nonumber\\
&\text{ }\mathbf{s.t.}\quad \sum_{j = 1}^J d_j = \sum_{\ell = 1}^L K_\ell \lambda_\ell \pi r_\ell^2,\label{PPPmodconstraints1}\\
&\hspace{1cm}0 \leq d_j \leq \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2,\quad \forall j. \label{PPPmodconstraints2}\end{aligned}$$
Note that follows from the combination of the capacity constraint in and the definition of the parameter $d_j$ as presented in (and changing the summation order.). Similarly, directly follows from the boundary constraint in and the definition of the parameter $d_j$ as presented in .
Problem \[prb:PPPmodprb\] is a nonlinear resource allocation problem and has the same structure as the problem presented in [@nonlinearresource]. As such, although a solution algorithm to give the optimal solution is available, a closed-form expression for this class of problems is not available in general. One of the contributions of this paper is an explicit closed-form solution for $\bm{D}$. Also, we will demonstrate how to find the optimal placement strategies for all types of caches, [[*i.e.*]{}, ]{}how to find $\bm{B}$ from $\bm{D}$.
The Lagrangian function corresponding to Problem \[prb:PPPmodprb\] is $$\begin{aligned}
&L\left(\bm{d}, \nu, \bm{\eta}, \bm{\omega}\right) = \sum_{j=1}^J a_j \exp({-d_j})+\nu \left(\sum_{j=1}^J d_j -
{{\color{black} \sum_{\ell = 1}^ L}}K_\ell \lambda_\ell \pi r_\ell^2 \right)\nonumber\\
&- \sum_{j=1}^J \eta_j d_j +\sum_{j=1}^J \omega_j \left(d_j - \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2 \right), \nonumber\end{aligned}$$ where $\bm{d}$, $\bm{\eta}$, $\bm{\omega} \in \mathbb{R}_+^{J}$ and $\nu \in \mathbb{R}$.
Let $\bar{\bm{d}}$, $\bar{\bm{\eta}}$, $\bar{\bm{\omega}}$ and $\bar{\nu}$ be primal and dual optimal. The KKT conditions for Problem \[prb:PPPmodprb\] state that $$\begin{aligned}
\sum_{j=1}^J \bar{d}_j = \sum_{\ell = 1}^L K_\ell \lambda_\ell \pi r_\ell^2&, \label{kkt2ppp}\\
0 \leq \bar{d}_j \leq \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2&, \quad \forall j = 1,\dots, J \label{kkt1ppp}\\
\bar{\eta}_j \geq 0&,\quad \forall j = 1,\dots, J, \label{kkt3ppp}\\
\bar{\omega}_j \geq 0&,\quad \forall j = 1,\dots, J, \label{kkt4ppp}\\
\bar{\eta}_j \bar{d}_j = 0&,\quad \forall j = 1,\dots, J, \label{kkt5ppp}\\
\bar{\omega}_j \left(\bar{d}_j - \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2\right) = 0&, \quad \forall j = 1,\dots, J, \label{kkt6ppp}\\
-a_j \exp(-\bar{d}_j) + \bar{\nu} - \bar{\eta}_j + \bar{\omega}_j = 0&,\quad \forall j = 1,\dots, J \label{kkt7ppp}.\end{aligned}$$
\[thm:PPPoptsol\] The optimal placement strategy for Problem \[prb:PPPmodprb\] satisfies $$\begin{aligned}
\label{PPP:dfunc}
\bar{d}_j = \left\{
\begin{array}{rl}
\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2, & \text{if } j < s_1\\
\log{\frac{a_j}{\bar{\nu}}}, & \text{if } s_1 \leq j \leq s_2,\\
0, & \text{if } j > s_2,
\end{array} \right.\end{aligned}$$ where $$\begin{aligned}
\label{PPP:gfunc}
g_j(\nu) = \left\{
\begin{array}{rl}
\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2, & \text{if } \nu \leq a_j \exp\left(-\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2\right)\\
0, & \text{if } \nu \geq a_j,\\
\log{\frac{a_j}{\nu}}, & \text{otherwise},
\end{array} \right.\end{aligned}$$ and $g: \mathbb{R} \rightarrow \left[0, \sum_{\ell = 1}^L K_\ell \lambda_\ell \pi r_\ell^2\right]$, where $g(\nu) = \sum_{j=1}^J g_j(\nu)$, $$\begin{aligned}
s_1 = \min\biggl\{\,1 \leq k \leq J \mathrel{\Big|} g\left(a_k \exp\left(-\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2 \right)\right) \geq \sum_{\ell = 1}^L K_\ell \lambda_\ell \pi r_\ell^2 \biggr\},\label{PPPs1}\end{aligned}$$ $$s_2 = \max\biggl\{1 \leq k \leq J \mathrel{\Big|} g(a_k) \leq \sum_{\ell = 1}^L K_\ell \lambda_\ell \pi r_\ell^2 \biggr\}, \label{PPPs2}$$ and $$\bar{\nu} = \exp \left\{ \frac{\sum_{j = s_1}^{s_2} \log a_j -\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2\left(K_\ell - s_1 + 1\right)}{s_2 - s_1 + 1}\right\}.\label{PPPnubar}$$
From , and , we have $$\bar{\omega}_j = \frac{\bar{d}_j}{\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2} \left[a_j \exp\left(-\bar{d}_j\right) - \bar{\nu}\right], \label{omegaeqppp}$$ which, when insterted into , gives $$\frac{\bar{d}_j}{\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2} \left[a_j \exp\left(-\bar{d}_j\right) - \bar{\nu}\right] \left(\bar{d}_j - \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2\right) = 0 \label{starppp}.$$ From , we see that $0 < \bar{d}_j < \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2$ only if $$\bar{\nu} = a_j \exp\left(-\bar{d}_j\right).$$ Since we know that $0 \leq d_j \leq \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2$, this implies that $$\begin{aligned}
\bar{\nu} \in \left[a_j \exp\left(-\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2\right), a_j \right].\end{aligned}$$ If $\bar{\nu} \leq a_j \exp\left(-\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2\right)$, we have $\bar{\omega}_j > 0$. Thus, from , we have $\bar{d}_j = \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2$. Similarly, if $\bar{\nu} \geq a_j$, we have $\bar{\eta}_j > 0$. Hence, from , we have $\bar{d}_j = 0$.
Recalling $d_j$ from Eq. , when $\bar{d}_j = \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2$, it means that $\bar{b}_j^{(\ell)} = 1$, $\forall \ell = 1,\dots,L$. Similarly, when $\bar{d}_j = 0$, $\bar{b}_j^{(\ell)} = 0$, $\forall \ell = 1,\dots,L$. Then, it follows that there exist $s_1, s_2 \in [1, J]$ such that $\bar{b}^{(\ell)}_1 = \bar{b}^{(\ell)}_2 = \dots = \bar{b}^{(\ell)}_{s_1 -1} = 1$ and $\bar{b}^{(\ell)}_{s_2 + 1} = \bar{b}^{(\ell)}_{s_2 + 2} = \dots = \bar{b}^{(\ell)}_J = 0$, $\forall \ell = 1,\dots,L$. Then $s_1$ is given by $$s_1 = \min\left\{1 \leq k \leq J \mathrel{\Big|} \bar{\nu} > a_k \exp\left(-\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2 \right) \right\}. \label{eqs1earppp}$$ In order to satisfy the capacity constraint , above minimum is guaranteed to exist. Then $s_1$ is obtained by inserting function $g$ to and applying the capacity constraint . [[ Similarly, ]{}]{}$s_2$ is found by applying the same steps and given by $$s_2 = \max\left\{1 \leq k \leq J \mathrel{\Big|} \bar{\nu} <a_k \right\}. \label{eqs2earppp}$$ Using the same argument, in order to satisfy the capacity constraint the above maximum is guaranteed to exist. The proof is completed by solving for $\bar{\nu}$ in $g(\bar{\nu}) = \sum_{\ell = 1}^L K_\ell \lambda_\ell \pi r_\ell^2$.
\[thm:PPPstrategy\] The optimal placement strategy for Problem \[prb:PPPorgprb\] satisfies $$\begin{aligned}
\label{PPP:bfunc}
\bar{b}_j^{(\ell)} = \left\{
\begin{array}{rl}
1, \forall \ell, & \text{if } j < s_1\\
\phi_j(\ell), & \text{if } s_1 \leq j \leq s_2,\\
0, \forall \ell, & \text{if } j > s_2,
\end{array} \right.\end{aligned}$$ where $\phi_j(\ell)$ is any solution over $b^{(\ell)}_j\in [0,1]$ satisfying $$\sum_{\ell = 1}^{L} \bar{b}_j^{(\ell)} \lambda_\ell \pi r_\ell^2 = \log\frac{a_j}{\bar{\nu}}, \label{midconstr}$$ $$\bar{b}_{s_1}^{(\ell)} + \dots + \bar{b}_{s_2}^{(\ell)} = K_\ell - s_1 + 1, \label{thm3bconstrst}$$ for all $s_1\leq j\leq s_2$ and $1\leq\ell\leq L$ and where $s_1$, $s_2$ and $\bar{\nu}$ are given by from Eq. , and , respectively.
We will combine the solution given in Theorem \[thm:PPPoptsol\] with Eq. for the proof. In Theorem \[thm:PPPoptsol\] we give the solution for the joint problem in terms of $\bm{D}$. This solution is unique in $d_j$’s, and $\bar{d}_j$’s are given in Theorem \[thm:PPPoptsol\]. Even though the solution is unique in $d_j$’s, it is easy to observe that the solution is not unique in $b_j$’s from Eq. . To give an optimal placement strategy in terms of $b_j$’s, we will first show that the relation between the $d_j$’s and $b_j$’s correspond to a [[ balanced]{}]{} capacitated transportation problem [@combinatorialoptimization]. Finally, since it is known that the greedy solution is optimal for the balanced capacitated transportation problem, we can obtain $\bar{b}_j^{\ell}$’s greedily.
When $\bar{d}_j = \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2$, for any file index $j < s_1$, $\bar{b}_j^{(\ell)} = 1$, $\forall \ell = 1,\dots,L$, [[*i.e.*]{}, ]{}file $c_j$ is stored in all types of caches. Similarly, when $\bar{d}_j = 0$, it means that for any file index $j > s_2$, $\bar{b}_j^{(\ell)} = 0$, $\forall \ell = 1,\dots,L$, [[*i.e.*]{}, ]{}file [[ $c_j$ is not stored in any type of caches.]{}]{} This implies that, the remaining capacity that can be used for files $s_1,\dots,s_2$ in caches of type $\ell$ is $K_\ell - s_1 + 1$. These files should follow and and it remains to show that a solution to this system of equations exists.
For notational convenience let $f_j^{(\ell)}=\lambda_\ell\pi r_\ell^2 b_j^{(\ell)}$. We observe that and correspond to a capacitated transportation problem [@combinatorialoptimization] in the variables $f_j^{(\ell)}$, i.e. the $f_j^{(\ell)}$ need to satisfy $$\begin{aligned}
\sum_{\ell=1}^L f_j^{(\ell)} &= \log\frac{a_j}{\bar{\nu}}, \label{eq:transp1} \\
\sum_{j=s_1}^{s_2} f_j^{(\ell)} &= \lambda_\ell\pi r_\ell^2(K_\ell - s_1 + 1), \label{eq:transp2} \\ 0 \leq f_j^{(\ell)} &\leq \lambda_\ell\pi r_\ell^2. \label{eq:transp3} \end{aligned}$$ In fact, this is a balanced transportation problem, since, by we have $$\begin{aligned}
\sum_{j = s_1}^{s_2} \log\frac{a_j}{\bar{\nu}} &= \sum_{j = s_1}^{s_2} \log{a_j} - (s_2 - s_1 + 1)\log{\bar{\nu}} \notag \\
&= \sum_{j = s_1}^{s_2} \log{a_j} - \sum_{j = s_1}^{s_2} \log a_j +\sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2\left(K_\ell - s_1 + 1\right) \notag \\
&= \sum_{\ell = 1}^L \lambda_\ell \pi r_\ell^2\left(K_\ell - s_1 + 1\right). \label{eq:transp4} \end{aligned}$$ Moreover, by we have $$\log\frac{a_j}{\bar{\nu}} \leq \sum_{\ell=1}^L \lambda_\ell\pi r_\ell^2. \label{eq:transp5}$$ Due to , the $f_j^{(\ell)}$ can be found for each file satisfying and . Finally, it is readily verified that due to this can be done greedily by considering each file consecutively.
Local Optimization {#sec:distributedoptimization}
==================
Since the joint optimization problem presented in Problem \[orgprb\] is not convex for general deployment scenarios, [[*i.e.*]{}, ]{}if the cache deployments are not following homogeneous PPP, we will provide a heuristic algorithm that finds the optimal solution for a single type of cache assuming that all other types are storing files with fixed probabilities at any iteration step. The main aim is to see how the overall system performance behaves as the algorithm solves for all types of caches iteratively. The procedure is as follows. At each iteration step we find the optimal strategy for a specific type of cache assuming that the placement strategies for other types of caches are known and fixed. Then we continue with the same procedure for the next type and we continue iterating over different types. In the ensuing subsections, first we will formulate the local optimization problem and give the optimal solution for a single type of cache class. Then we continue with presenting our Local Optimization Algorithm by using the local optimization solution we have obtained.
Formulation and solution of the problem
---------------------------------------
In this section, we will formulate the local optimization problem for a single type of cache where all the other types of caches’ placement strategies are known and fixed and give the solution to the problem.
We start this section by defining our performance metric and the formulation of the optimization problem. The performance metric is the total miss probability which is the probability that a user will not find the content that she requires in any of the [*caches*]{} that she is covered by. We assume that the placement strategy for the probability distribution over $J$ files through all $L-1$ types is fixed and known and we will solve the optimization problem for only one type. In this section, without loss of generality, we consider the optimization of type$-1$. For notational convenience, superscript $^c$ in the notation $b_j^{(i)^c}$ indicates that the placement strategy for type$-i$ is known and constant. Then, the total miss probability is given by $$f_{\text{local}}^{(1)}\left(\bm{b}^{(1)}\right) = \sum_{j=1}^J a_j \sum_{n_1= 0}^{\infty}(1-b_j^{(1)})^{n_1}p_{1}(n_1) q_1(j, n_1),
\label{hitprob}$$ where $$\begin{aligned}
\label{qfunc}
&q_m(j, n_m) = P(\text{non type$-m$ caches miss the file $c_j$}) \nonumber\\
&= \sum_{\substack{n_k = 0\\k = 1,\dots,L\\k \neq m}}^{\infty} \prod_{\substack{l = 1\\l \neq m}}^L (1-b_j^{(l)^c})^{n_l} P(\bm{\mathcal{N}}^{(-m)} = \bm{n}^{(-m)} \vert \mathcal{N}_m = n_m).\end{aligned}$$
We define the optimization problem to find the optimal placement strategy minimizing the total miss probability for type$-1$ caches as follows:
\[modprb\] $$\begin{aligned}
&\min \text{ } f_{\text{local}}^{(1)}\left(\bm{b}^{(1)}\right)\nonumber\\
&\text{ }\mathbf{s.t.}\quad b_1^{(1)} + \dots + b_J^{(1)} = K_1, \quad b_j^{(1)} \in [0,1],\quad \forall j. \label{constraints3}\end{aligned}$$
Next, we will analyze the structure of the optimization problem.
\[convex2\] Problem \[modprb\] is a convex optimization problem.
The objective function is separable with respect to (w.r.t.) $b_1^{(1)}, \dots, b_J^{(1)}$. $f_{\text{local}}^{(1)}\left(\bm{b}^{(1)}\right)$ is convex in $b_j^{(1)}$, $\forall j$. Hence, it is a convex function of $\left(b_1^{(1)}, \dots, b_J^{(1)}\right)$ since it is a sum of convex subfunctions.
Since $f_{\text{local}}^{(1)}\left(\bm{b}^{(1)}\right)$ is convex by Lemma \[convex2\] and the constraint set is linear as given in , the Karush-Kuhn-Tucker (KKT) conditions provide necessary and sufficient conditions for optimality. The Lagrangian function corresponding to Problem \[modprb\] becomes $$\begin{aligned}
&L\left(\bm{b}^{(1)}, \nu, \bm{\lambda}, \bm{\omega}\right) = \sum_{j=1}^J a_j \sum_{n_1= 0}^{\infty}(1-b_j^{(1)})^{n_1} p_{1}(n_1) q_1(j, n_1) \nonumber\\
&+ \nu \left(\sum_{j=1}^J b_j^{(1)} - K_1\right) - \sum_{j=1}^J \lambda_j b_j^{(1)} + \sum_{j=1}^J \omega_j \left(b_j^{(1)} - 1\right), \nonumber\end{aligned}$$ where $\bm{b}^{(1)}$, $\bm{\lambda}$, $\bm{\omega} \in \mathbb{R}_+^J$ and $\nu \in \mathbb{R}$.
Let $\bar{\bm{b}}^{(1)}$, $\bar{\bm{\lambda}}$, $\bar{\bm{\omega}}$ and $\bar{\nu}$ be primal and dual optimal. The KKT conditions for Problem \[modprb\] state that $$\begin{aligned}
\sum_{j=1}^J \bar{b}^{(1)}_j &= K_1, \label{kkt2}\\
0 \leq \bar{b}^{(1)}_j &\leq 1, \quad \forall j = 1,\dots, J, \label{kkt1}\\
\bar{\lambda}_j &\geq 0, \quad \forall j = 1,\dots, J,\label{kkt3}\\
\bar{\omega}_j &\geq 0,\quad \forall j = 1,\dots, J,\label{kkt4}\\
\bar{\lambda}_j \bar{b}^{(1)}_j &= 0,\quad \forall j = 1,\dots, J,\label{kkt5}\\
\bar{\omega}_j \left(\bar{b}^{(1)}_j - 1\right) &= 0, \quad \forall j = 1,\dots, J,\label{kkt6}\\
a_j \sum_{n_1= 0}^{\infty}n_1(1-b_j^{(1)})^{n_1-1}p_{1}(n_1) &q_1(j, n_1) + \bar{\lambda}_j - \bar{\omega}_j = \bar{\nu}, \forall j = 1,\dots, J \label{kkt7}.\end{aligned}$$
\[optsol\] The optimal placement strategy for Problem \[modprb\] is $$\begin{aligned}
\label{bfunc}
\bar{b}^{(1)}_j = \left\{
\begin{array}{rl}
1, & \text{if } \bar{\nu} < a_j p_{1}(1) q_1(j, 1)\\
0, & \text{if } \bar{\nu} > a_j \sum_{n_1=0}^{\infty}n_1 p_{1}(n_1) q_1(j, n_1),\\
\phi(\bar{\nu}), & \text{otherwise},
\end{array} \right.\end{aligned}$$ where $\phi(\bar{\nu})$ is the solution over $b^{(1)}_j$ of $$a_j \sum_{n_1= 0}^{\infty}n_1(1-b_j^{(1)})^{n_1-1} p_{1}(n_1) q_1(j, n_1) = \bar{\nu}, \label{eqsb}$$ and $\bar{\nu}$ can be obtained as the unique solution to the additional constraint $$\bar{b}_1^{(1)} + \dots + \bar{b}_J^{(1)} = K_1. \label{eqK}$$
From , and , we have $$\bar{\omega}_j = \bar{b}_j^{(1)} \left[a_j \sum_{n_1=0}^{\infty} n_1 (1-b_j^{(1)})^{n_1 -1} p_{1}(n_1) q_1(j, n_1) - \bar{\nu}\right], \label{omegaeq}$$ which, when inserted into , gives $$\bar{b}_j^{(1)}(\bar{b}_j^{(1)} - 1)\left[a_j \sum_{n_1=0}^{\infty}n_1 (1-b_j^{(1)})^{n_1 -1} p_{1}(n_1) q_1(j,n_1)\right] = \bar{\nu} \label{star}.$$ From , we see that $0 < \bar{b}_j^{(1)} < 1$ only if $$\bar{\nu} = a_j \sum_{n_1=0}^{\infty}n_1 (1-b_j^{(1)})^{n_1 -1} p_{1}(n_1) q_1(j, n_1).$$ Since we know that $0 \leq b_j^{(i)} \leq 1$, this implies that $$\begin{aligned}
\bar{\nu} \in \left[a_j p_{1}(1) q_1(j, 1),\text{ } a_j \sum_{n_1=0}^{\infty} n_1 p_{1}(n_1) q_1(j,n_1) \right].\end{aligned}$$ If $\bar{\nu} < a_j p_{1}(1) q_1(j, 1)$, we have $$\bar{\omega}_j = \bar{\lambda}_j + a_j \sum_{n_1=0}^{\infty} n_1 (1-b_j^{(1)})^{n_1 -1} p_{1}(n_1) q_1(j, n_1) - \bar{\nu} > 0.$$ Thus, from , we have $\bar{b}_j^{(1)} = 1$. Similarly, if $\bar{\nu} > a_j \sum_{n_1=0}^{\infty} n_1 p_{1}(n_1) q_1(j, n_1)$, we have $$\bar{\lambda}_j = \bar{\omega}_j + \bar{\nu} -a_j \sum_{n_1=0}^{\infty}n_1 (1-b_j^{(1)})^{n_1 -1} p_{1}(n_1) q_1(j, n_1) > 0.$$ Hence, from , we have $\bar{b}_j^{(1)} = 0$.
Finally, since $\sum_{j=1}^J {b}_j^{(1)}$ is a decreasing function in $\nu$, solving $J$ equations of satisfying give the unique solution $\bar{\nu}$.
Local Optimization Algorithm (LOA) {#sec:loa}
----------------------------------
The basic idea of our algorithm is to repeatedly perform local optimization. Let ${\tilde{\mathbf{b}}}^{(\ell)}$ denote the optimal the placement strategy for type$-\ell$ caches given by Theorem \[optsol\].
As different types of caches share information with each other, the idea is to see if applying distributed optimization iteratively and updating the file placement strategies over different types of caches gives $\mathbf{\bar{b}}^{(\ell)}$ for all $\ell \in [1:L]$ yielding to the global optimum of Problem \[orgprb\]. To check this, we define the following algorithm.
For Local Optimization Algorithm (LOA), we update the caches following the sequence of the indices of the different types of caches. We assume that all types of caches are initially storing the most popular $K_\ell$ files depending on their cache capacities. The algorithm stops when $f^{(\ell)}(\mathbf{b}^{(\ell)})$ converged $\forall \ell \in \{1,\dots,L\}$, [[*i.e.*]{}, ]{}a full round over all types of caches $\{1,\dots,L\}$ does not give an improvement in hit probability. LOA is shown in Algorithm \[loa\].
[latex@errorgobble]{}
\[loa\]
initialize $\mathbf{b}^{(\ell)} = [\underbrace{1, \dots, 1}_{\text{$K_\ell$ many}}, 0, \dots, 0]$, $\forall \ell \in \{1,\dots,L\}$ set imp = 1
Next, we will present the placement strategies obtained by LOA for two deployment models we presented earlier.
LOA for PPP deployment model {#sec:PPPmodel}
----------------------------
We already showed that we can reformulate Problem \[orgprb\] and find an analytical solution to the joint optimization problem when all types of caches are deployed according to homogeneous PPP models in Section \[sec:jointoptimization\]. In this section, we will follow the local optimization approach and find the optimal solution for different types of caches at every iteration step. The idea behind this is to see if our LOA converges to the optimal solution given in Theorem \[thm:PPPstrategy\]. We will give the numerical results in Section \[sec:performance\].
Again, without loss of generality, we consider the local optimization of type$-1$. Since $P(\bm{\mathcal{N}}^{(-1)} = \bm{n}^{(-1)} \vert \mathcal{N}_1 = n_1) = p_{2}(n_2) \dots p_{L}(n_L)$ is independent of $n_1$, for the sake of simplicity we can define a new parameter: $q_1(j) =: q_1(j,n_1)$, $\forall n_1$.
[[ The proof of the next result follows along the same lines as the proof of Theorem \[optsol\] and is skipped due to space constraints.]{}]{}
\[PPPopt\] LOA solution for of Problem \[modprb\] for PPP model is given by $$\begin{aligned}
\label{bfuncPPPLOA}
\tilde{b}^{(1)}_j = \left\{
\begin{array}{rl}
1, & \text{if } j < s_1,\\
\frac{1}{t_1}\log\frac{a_j t_1 q_1(j)}{\bar{\nu}}, &\text{if } s_1 \leq j \leq s_2,\\
0, & \text{if } j > s_2.
\end{array} \right.\end{aligned}$$ where $$\begin{aligned}
\label{gfuncPPPLOA}
g_j(\nu) = \left\{
\begin{array}{rl}
&1, \quad \text{if } \nu < a_j p_{1}(1) q_1(j),\\
&0, \quad \text{if } \nu > a_j \sum_{n_1=0}^{\infty}n_1 p_{1}(n_1) q_1(j),\\
&\frac{1}{t_1}\log\frac{a_j t_1 q_1(j)}{\nu}, \quad \text{otherwise},
\end{array} \right.\end{aligned}$$ and $g: \mathbb{R} \rightarrow [0, K_1]$, where $g(\nu) = \sum_{j=1}^J g_j(\nu)$, $$s_1 = \min\left\{1 \leq \ell \leq J \vert g\left(a_{\ell} t_1 e^{-t_1}q_1(\ell)\right) \geq K_1\right\}, \label{eqs1}$$ $$s_2 = \max\left\{1 \leq \ell \leq J \middle\vert g\left(a_{\ell} t_1 q_1(\ell)\right) \leq K_1\right\}, \label{eqs2}$$ and $$\begin{aligned}
\bar{\nu} = \exp\Biggl\{\frac{\sum_{j = s_1}^{s_2} \log\left(a_j\right) - t_1\left(K_1 - s_1 + 1\right)}{s_2 - s_1 +1}+\log\left[t_1 q_1(j)\right]\Biggr\}. \label{LOAPPPnu}\end{aligned}$$
LOA for M-or-None deployment model
----------------------------------
Again, without loss of generality, we first consider the local optimization of type$-1$. For M-or-None deployment model we will first analyze the behavior of the function $q_1(j, n_1)$. Using , we have $$q_1(j,n_1) = \sum_{n_2=0}^{M} \sum_{n_3=0}^{M - n_2} \dots \sum_{n_L = 0}^{M - (n_2 + \dots + n_{L-1})} \prod_{l = 2}^L (1-b_j^{(l)^c})^{n_l},$$ which is not a function of $n_1$, but $M$. Then again for the sake of simplicity we define $q_1^M(j) =: q(j,n_1)$ for M-or-None deployment model. The rest of the analysis is the same as the one that is shown for PPP model.
\[MorNoneopt\] LOA solution of Problem \[modprb\] for type$-1$ caches for M-or-None model is given by $$\begin{aligned}
\label{bfuncMorNoneLOA}
\tilde{b}^{(1)}_j = \left\{
\begin{array}{rl}
1, & \text{if } j < s_1\\
\frac{1}{t_1}\log\frac{a_j t_1 q_1^M(j)}{\bar{\nu}}, &\text{if } s_1 \leq j \leq s_2,\\
0, & \text{if } j > s_2.
\end{array} \right.\end{aligned}$$ where $$\begin{aligned}
\label{gfuncMorNoneLOA}
g_j(\nu) = \left\{
\begin{array}{rl}
&1, \quad \text{if } \nu < a_j p_{1}(1) q_1^M(j),\\
&0, \quad \text{if } \nu > a_j \sum_{n_1=0}^{\infty}n_1 p_{1}(n_1) q_1^M(j),\\
&\frac{1}{t_1}\log\frac{a_j t_1 q_1^M(j)}{\nu}, \quad \text{otherwise},
\end{array} \right.\end{aligned}$$ and $g: \mathbb{R} \rightarrow [0, K_1]$, where $g(\nu) = \sum_{j=1}^J g_j(\nu)$, $$s_1 = \min\left\{1 \leq \ell \leq J \vert g\left(a_{\ell} t_1 e^{-t_1}q_1^M(\ell)\right) \geq K_1\right\}, \label{eqs1mon}$$ $$s_2 = \max\left\{1 \leq \ell \leq J \vert g\left(a_{\ell} t_1 q_1^M(\ell)\right) \leq K_1\right\}, \label{eqs2mon}$$ and $$\begin{aligned}
\bar{\nu} = \exp\Biggl\{\frac{\sum_{j = s_1}^{s_2} \log\left(a_j\right) - t_1\left(K_1 - s_1 + 1\right)}{s_2 - s_1 +1}+\log\left[t_1 q_1^M(j)\right]\Biggr\}. \label{LOAMorNonenu}\end{aligned}$$
Proof is the same as of PPP model with replacing $q_1(j)$ with $q_1^M(j)$. The only important thing here to note is that $q_1(j,n_1)$ is not a function of $n_1$ for both deployment models and constant for $M$. Thus, solution to can be further exploited with some manipulations.
For the helpers, the analysis is different. Again, without loss of generality, we will consider the local optimization of type$-2$ helpers. For M-or-None deployment model we will first analyze the behavior of the function $q_2(j, n_2)$, [[*i.e.*]{}, ]{}the probability that non type$-2$ caches are missing file $j$. First for the sake of simplicity, we define a new parameter for the probability of other helpers missing file $j$ as $$\zeta_2(j,n_2) = \sum_{n_3=0}^{M - n_2} \dots \sum_{n_L = 0}^{M - \sum_{k = 2}^{L-1} n_k} \prod_{l = 3}^L \left(1-b_j^{(l)^c}\right)^{n_l}.$$ Then, using , we have $$\begin{aligned}
q_2(j,n_2) &= \sum_{n_1 = 0}^\infty p_1(n_1) \left(1-b_j^{(1)^c}\right)^{n_1} \zeta_2(j,n_2)\\
&= e^{\left(1-b_j^{(1)^c}\right)} \zeta_2(j,n_2).\end{aligned}$$ $q_2(j,n_2) $ is now a function of $n_2$ and we can not manipulate Eq. further to get a closed form solution for the helpers for M-or-None deployment model.
[[ The proof of the next result follows along the same lines as the proofs of Theorems \[optsol\] and \[MorNoneopt\] and is skipped due to space constraints.]{}]{}
\[MorNoneopt2\] LOA solution of Problem \[modprb\] for type$-2$ caches for M-or-None model is given by $$\begin{aligned}
\label{bfuncMorNone2}
\tilde{b}^{(2)}_j = \left\{
\begin{array}{rl}
1, & \text{if } \bar{\nu} < a_j e^{\left(1-b_j^{(1)^c}\right)} \zeta_2(j,1)\\
0, & \text{if } \bar{\nu} > a_j \sum_{n_2=0}^{M}n_2 e^{\left(1-b_j^{(1)^c}\right)} \zeta_2(j,n_2),\\
\phi(\bar{\nu}), & \text{otherwise},
\end{array} \right.\end{aligned}$$ where $\phi(\bar{\nu})$ is the solution over $b^{(2)}_j$ of $$a_j \sum_{n_2 = 0}^{M}n_2(1-b_j^{(2)})^{n_2-1} e^{\left(1-b_j^{(1)^c}\right)} \zeta_2(j,n_2) = \bar{\nu},$$ and $\bar{\nu}$ can be obtained as the unique solution to the additional constraint $$\bar{b}_1^{(2)} + \dots + \bar{b}_J^{(2)} = K_2.$$
The solution for other helper types can be obtained by following the same procedure by replacing $\zeta_2(j,n_2)$ by $\zeta_\ell(j,n_\ell)$ for type$-\ell$ caches.
Performance Evaluation {#sec:performance}
======================
In this section, first we will present some heuristic placement strategies. Next we will specify different network coverage models and show the performances of the proposed algorithms .
Heuristics
----------
In this subsection we will introduce some heuristic placement strategies. The main aim of proposing these heuristics is to compare the hit probability performance of the system when the optimal strategy is used with the hit probability obtained when these heuristics are used. Later we will show by numerical results that the hit probability is increased remarkably by using the placement strategies that our proposed algorithms give compared to these heuristics.
### Heuristic 1 (H1) {#policyh1}
The first heuristic is to use is to store the first $K_i$ most popular files in type-$i$ caches, denoted by H1. For H1, $$\bm{{b}}^{(i)} = \left(\underbrace{1, 1, \dots, 1}_\text{$K_i$ many}, 0, \dots, 0\right).$$
### Heuristic 2 (H2) {#policyh2}
We will introduce an example to explain how H2 works. In some scenarios type-$1$ caches may store the first $K_1$ files with high probabilities. Then, it is wiser to come up with a smarter heuristic than H1 since the first $K_1$ files are already available for the users covered by type$-1$ caches. Hence, the second heuristic we propose suggests not to store the most popular first $K_1$ files in type-$2$ caches, and store the next $K_2$ popular files with probability $1$, and continue with the same procedure for type-$3$ caches and so on. The second heuristic is called Heuristic 2 (H2). For H2, $$\bm{{b}}^{(2)} = \left(\underbrace{0, 0, \dots, 0}_\text{$K_1$ many}, \underbrace{1, \dots, 1}_\text{$K_2$ many}, 0, \dots, 0\right),$$ and $$\bm{{b}}^{(3)} = \left(\underbrace{0, 0, \dots, 0}_\text{$K_1 + K_2$ many}, \underbrace{1, \dots, 1}_\text{$K_3$ many}, 0, \dots, 0\right),$$ and so on.
### Heuristic 3 (H3) {#policyh3}
We will introduce a smarter deployment heuristic here that also takes the deployment densities of the different types of caches into account. Suppose there are type-$1$ caches in the plane with density $\lambda_1$ and type-$2$ caches are to be deployed in the plane with density $\lambda_2$. Then, we store the first $K_2\lceil \frac{\lambda_2\ r_2^2}{\lambda_1 r_1^2}\rceil$ files with probability $\frac{1}{\lceil \frac{\lambda_2 r_2^2}{\lambda_1 r_1^2}\rceil}$. Namely, for H3, $$\bm{{b}}^{(2)} = \left(\underbrace{\frac{1}{\lceil \frac{\lambda_2 r_2^2}{\lambda_1 r_1^2}\rceil}, \frac{1}{\lceil \frac{\lambda_2 r_2^2}{\lambda_1 r_1^2}\rceil}, \dots, \frac{1}{\lceil \frac{\lambda_2 r_2^2}{\lambda_1 r_1^2}\rceil}}_\text{$K_2\lceil \frac{\lambda_2 r_2^2}{\lambda_1 r_1^2}\rceil$ many}, 0, \dots, 0\right).$$
Poisson Point Process (PPP) deployment model
--------------------------------------------
[[ In this subsection we will consider various scenarios for the case where different types of caches are all following homogeneous PPP. First, we will show the hit probability evolution for the case where helpers (SBSs) with different coverage radii (Femtocells have a typical coverage radius of $10$ m, picocells have $150$ m, and macrocells have $1-2$ km in rural areas [@mobnetguide].) and different cache capacities. We will illustrate how optimal placement policies behave for LOA, and compare LOA performance with the joint solution and the heuristics. Furthermore, we will show that LOA indeed gives the optimal solution by comparing it with the solution of the joint problem which has been proven to be optimal for homogeneous PPP. Also, we will provide numerical results for the case where the file popularities follow distributions with different Zipf parameters and where there is incomplete information on file popularities.]{}]{}
### Files with popularities following the Zipf distribution
First, we will present a scenario where the SBSs have different coverage radius, $r_2$. Consider the case of two types of caches in the plane. Type$-1$ caches represent MBSs and type$-2$ caches represent SBSs, with $K_1$ and $K_2$-slot cache memories, respectively. The content library size is $J = 1000$. We set $K_1 = 10$ (1
[[ Next, we will present a scenario where the SBSs have a fixed coverage radius and different deployment densities with different cache memories. We assume that our SBSs are picocells. We consider the same parameters in the previous setting, except we set $r_2 = 150 m$ and we will consider various values for the deployment densities for SBSs, namely for $\lambda_2$.]{}]{}
[[ In [[ Figure \[hitprobevo\]]{}]{} we see the hit probability evolution for the different deployment densities of the SBSs with different cache capacities. Let us consider the two curves where $K_2 = 10$ and $K_2 = 20$. It can be easily verified that the hit probability is equal for the cases when the SBSs have cache memory of $K_2 = 10$ at $\lambda_2/\lambda_1 = 2$ and of $K_2 = 20$ at $\lambda_2/\lambda_1 = 1$. Note that this holds for any ratio, and confirms the validity of the analytical results (for instance, the relation can be seen in Eq. .).]{}]{}
[[ Next, we will show how LOA works for the PPP model. Since we can already obtain the optimal solution for the joint problem for this model, we know what the optimal hit probability is. Therefore, we would like to give an insight on how LOA works and performs by comparing it with the joint solution. We consider the same parameters in the previous setting, except we set the content library size $J = 100$ in order to effectively show the difference between the optimal placement probabilities obtained via the joint solution and LOA.]{}]{}
![Resulting placement strategies obtained via the joint solution and LOA.[]{data-label="JointVsLoaPlacement"}](figures/JointVsLoaPlacement.pdf){width="0.6\columnwidth"}
In Figure \[JointVsLoaPlacement\] we see that the resulting placement strategies for MBSs and SBSs differ for the solution of the joint problem and LOA. For LOA solution, first we find the optimal solution for MBSs assuming that there are no SBSs in the plane. Solving this optimization problem gives the blue straight curve $\bm{\bar{b}}^{(1)}$ (LOA) in Figure \[JointVsLoaPlacement\]. Then we set $\bm{b}^{(1)^c} = \bm{\bar{b}}^{(1)}$ (LOA), take it as an input, add SBSs into the plane and find the optimal placement strategy $\bm{\bar{b}}^{(2)}$ (LOA) for SBSs. Solving this optimization problem gives the red dashed curve $\bm{\bar{b}}^{(2)}$ (LOA) in Figure \[JointVsLoaPlacement\]. The solution of the joint problem gives the blue dotted curve $\bm{\bar{b}}^{(1)}$ (Joint) and the red dashed dotted red curve $\bm{\bar{b}}^{(2)}$ (Joint), for the MBSs and SBSs, respectively. Computing the hit probabilities for both the joint solution and LOA gives $ f\left(\bm{\bar{b}} \text{ (Joint)}\right) = f\left(\bm{\bar{b}} \text{ (LOA)}\right) = 0.6125$.
Now let us pick a file index and verify the validity of Theorem \[thm:PPPstrategy\] numerically. Suppose we pick the file $c_1$. $\bar{b}_1^{(1)} \text{ (Joint)} = 0.1195$, $\bar{b}_1^{(1)} \text{ (LOA)} = 0.1216$, $\bar{b}_1^{(2)} \text{ (Joint)} = 0.0837$ and $\bar{b}_1^{(2)} \text{ (LOA)} = 0.0608$. It is easy to verify that holds both for the joint problem and LOA. In fact, holds for any file index and we conclude that LOA gives the optimal solution for this specific scenario even though LOA’s resulting placement probabilities are different than the optimal solution of the joint problem.
This validates our claim in Theorem \[thm:PPPstrategy\] numerically, namely $d_j$’s given in Theorem \[thm:PPPoptsol\] is unique, and $b_j$’s can be obtained greedily by using LOA, satisfying the unique solution given by Theorem \[thm:PPPstrategy\].
We would like to briefly give the intuition behind LOA. The algorithm starts with optimizing type$-1$ caches. This solution has already proven to be optimal when there are no other types of caches in the network. On the other hand, adding other types of caches can only help type$-1$ caches, and can not cause any harm to the optimal strategy that had already been obtained. The structure of Theorem \[thm:PPPstrategy\] shows that the contribution for the coverage per file per any type of cache comes with two important parameters: (a) the density of the caches and (b) the probability of storing the file. The cumulative contribution of different types of caches will then give the ultimate hit probability. Therefore, just as in Theorem \[PPPopt\], starting with the worst case strategy (no information coming from other types for type$-1$) and updating all types of caches sequentially by providing more information to the next type at each step is simply equivalent to the optimal strategy for the joint problem as provided in Theorem \[thm:PPPstrategy\]. As a conclusion, LOA is a distributed algorithm that gives the optimal placement strategy when caches are deployed according to PPP.
[[ Numerical results for different Zipf parameters are consistent with the presented results. ]{}]{}
Our next aim is to compare the optimal placement strategy with various heuristics. We use the same simulation parameters as in the above cases with the content library size $J = 100$.
![Hit probability evolution for various heuristics.[]{data-label="gamma1"}](figures/gamma1Range.pdf){width="0.6\columnwidth"}
In Figure \[gamma1\], we see the hit probability evolution for various heuristics. We observe that:
- LOA is optimal since it performs equally well as the joint solution.
- As $\lambda_2 \pi r_2^2 >> \lambda_1 \pi r_1^2$, using H3 for the deployment of SBSs gives an improvement in hit probability as long as MBSs are using the optimal placement strategy \[OPT\] -which is simply the first iteration step of LOA, [[*i.e.*]{}, ]{}finding the optimal solution for MBSs assuming that there are no SBSs in the plane.-. However, it requires an unrealistically dense deployment of the SBSs in order to reach the optimal performance.
- Applying first step of LOA to MBSs and using H1 and H2 for SBSs results in significant performance penalties.
- Using H1 for the MBSs and applying the second step of LOA to SBSs, [[*i.e.*]{}, ]{}taking H1 strategy for the MBSs fixed, and finding the optimal strategy for SBSs, performs significantly well and converges to optimal performance as $\lambda_2 \pi r_2^2 >> \lambda_1 \pi r_1^2$.
- Applying heuristics to both MBSs and SBSs results in significant performance penalties.
- Running LOA iteratively does not improve the hit probability, namely running one round over each type of caches gives the optimal solution.
### Incomplete information on file popularities
We use the same notation in the previous subsection. The content library size is $J = 100$. We set $K_1 = 1$ and $K_2 = 2$. We set $\gamma = 1$, $a_j$ takes the values from and $a_j^{\text{pert}}$ takes the values from by adjusting $\sigma_j^2$ such that the signal-to-noise ratio ($SNR$) between $a_j$ and $\sigma_j^2$ is set accordingly, [[*i.e.*]{}, ]{}the signal power is equal to $POW_{a_j} = 10\log_{10}\left(\vert a_j \vert ^2\right)$, and the noise power is equal to $POW_{\sigma_j^2} = POW_{a_j} - SNR$ in dB. We set $\lambda_1 = 0.5$ and $r_1 = r_2 = 1$.
\[PPPSmartPerturb\]
As proposed earlier, in real life most of the time the file popularities will not follow a smooth distribution as the Zipf distribution. Recalling from the model that $a_j^{\text{pert}}$ values are the actual file popularity values that can not be obtained in real-time. We have the approximated $a_j$ values available and difference between the available popularity values $a_j$ and the actual file popularity values $a_j^{\text{pert}}$ increases as $\sigma_j^2$ increases. In Figure \[PPPSmartPerturb\] we show the total hit probability evolution for LOA. Straight lines indicate the ideal maximum hit probability that could be reached if the optimal deployment strategy was found by using $a_j^{\text{pert}}$ values. Dashed lines show the hit probability when the system is optimized with the already available $a_j$ values. It is [[ not surprising]{}]{} that the difference between the ideal and actual hit probability decreases as $\sigma_j^2$ decreases. We see a similar behavior under the \[H1/OPT\] strategy in [[ our simulations.]{}]{}
M-or-None deployment model
--------------------------
In this subsection we will present the performance evaluation of the placement strategies for the files following the Zipf distribution for M-or-None deployment model.
### Files with popularities following the Zipf distribution
We use the same notation in the previous subsection. The content library size is $J = 100$. We set $K_1 = 1$, and $K_2 = 5$. We set $\gamma = 1$ and taking $a_j$ according to . Also we set $\lambda_1 = 1.8324 \times 10^{-5}$ and $r_1 = 700 m$.
For the first step of LOA, the optimal placement strategy for MBSs is $\bm{\bar{b}}^{(1)} = \left(0.1220, 0.0973,\right.\\\left.0.0829, 0.0727, \dots \right)$, and the resulting hit probability is $ f\left(\bm{\bar{b}}^{(1)}\right) = 0.5875$. Next, we solve the problem for Type-2 caches.
From Figure \[cachingpolicyMorNoneSmart\], we see that the probability of storing less popular files increases as $M$ increases. As in the PPP case, repeatedly updating $\bm{\bar{b}}^{(1)}$ and $\bm{\bar{b}}^{(2)}$, does not improve the hit probability. We could not come up with an analytical solution to the joint problem of this deployment model since it is not convex, however from the result we obtained from the PPP model, it is very likely that LOA algorithm performs quite well for M-or-None deployment model as well.
For the \[H1/OPT\] scenario, we have $\bm{\bar{b}}^{(1)} = \left(1, 0, \dots, 0\right)$ and the resulting hit probability is $ f\left(\bm{\bar{b}}^{(1)}\right) = 0.2040$. Then we set $\bm{b}^{(1)^c} = \bm{\bar{b}}^{(1)}$, take it as a fixed input, and find the optimal placement strategy $\bm{\bar{b}}^{(2)}$ for SBSs.
![The total hit probability evolution for different $M$ values (M-or-None).[]{data-label="hitprobMorNone"}](figures/Figure93.pdf){width="0.6\columnwidth"}
From Figure \[cachingpolicyMorNoneMostpop\], as $c_1$ is stored in MBSs with probability $1$ and SBSs are present in the system only when $n_1 > 0$, $c_1$ is never stored at SBSs. We see that probability of storing $c_2$ and $c_3$ decreases and probability of storing other files increases as $M$ increases.
From Figure \[hitprobMorNone\], we see that the hit probabilities under LOA and \[H1/OPT\] become identical and increase as $M$ increases, [[*i.e.*]{}, ]{}we can use a heuristic placement policy for MBSs as long as we compensate the penalty by optimizing SBSs. The hit probability remains constant for until some point as $M$ increases due to the nature of the M-or-None model. Until the point where the hit probability starts increasing, only type-1 caches are present in the system (note that $\lambda_1 \pi r_1^2 \approx 28.2$). For heuristic SBS deployment policies, \[OPT/H1\], \[H1/H1\] and \[H1/H2\] policies achieve significantly lower hit probability than the optimal policy. We use a small variant of H3 here (since we do not have the density ratios for H3, we used the ratio of $\frac{M - \lambda_1 \pi r_1^2}{\lambda_1 \pi r_1^2}$.) \[OPT/H3\] and \[H1/H3\] achieve a higher probability compared to other heuristics, however optimizing the SBSs still gives a much higher hit probability.
[[ We see a similar behavior in M-or-None deployment model for the incomplete information on file popularities as in PPP case, and we skip the illustration due to space constraints.]{}]{}
Discussion and Conclusion {#discussion}
=========================
In this paper we have shown that whether MBSs use the optimal deployment strategy or store “the most popular content", has very limited impact on the total hit probability if the SBSs are using the optimal deployment strategy when the deployment densities of the SBSs are much larger than the MBSs’. Namely, when MBSs do not use the optimal placement strategy, it is possible to compensate this performance penalty, [[*i.e.*]{}, ]{}it is important to optimize the content placement strategy of the SBSs and the total hit probability is increased significantly when the SBSs use the optimal deployment strategy.
[[ For the PPP model, we have defined a new parameter for the probability of storing a file times the deployment density for an individual cache type, and show that the solution is unique for the sum of these new parameters over all cache types. We have shown that the relation between the newly presented parameter and the optimal placement probabilities follow a capacitated transportation problem and we show that the optimal placement probabilities can be obtained greedily. Consecutively, one has the flexibility of choosing the optimal placement strategies of the different types of caches as long as some certain capacity constraint is satisfied.]{}]{}
It is shown that heuristic policies for SBSs like storing the popular content that is not yet available in the MBS results in significant performance penalties. We have also proposed [[ a heuristic]{}]{} that takes deployment densities of different types of caches into account. We have shown that even though this heuristic gives a better hit probability performance compared to other heuristics, using optimal placement strategy still gives a better hit probability.
To conclude, using the optimal deployment strategy for the SBSs (typically SBSs have the higher deployment density) is crucial and it ensures the overall network to have the greatest possible total hit probability independent of the deployment policy of MBSs. We have shown that solving the individual problem to find optimal placement strategy for different types of base stations iteratively, namely repeatedly updating the placement strategies of the different types, does not improve the hit probability. Finally, we have shown numerically that LOA gives the same hit probability as the optimal placement strategy of the joint optimization problem of the PPP model by running a single cycle over different types.
[99]{} B. Serbetci, and J. Goseling, “On Optimal Geographical Caching In Heterogeneous Cellular Networks", *IEEE Wireless Communications and Networking Conference (WCNC)*, San Francisco, CA, USA, March 2017.
“Cisco VNI Global Mobile Data Traffic Forecast, 2015-2020", San Jose, CA, USA, Feb. 2016.
J. Andrews, H. Claussen, M. Dohler, S. Rangan, and M. Reed, “Femtocells: Past, Present and Future", *IEEE Journal on Selected Areas in Communications*, vol. 30, no. 3, pp. 497–508, April 2012.
N. Golrezaei, A. F. Molisch, A. G. Dimakis, and G. Caire, “Femtocaching and Device-to-Device Collaboration: A New Architecture for Wireless Video Distribution", *IEEE Communications Magazine*, vol. 51, no. 4, pp. 142–149, April 2013.
K. Shanmugam, N. Golrezaei, A. G. Dimakis, A. F. Molisch, and G. Caire, “FemtoCaching: Wireless Content Delivery Through Distributed Caching Helpers", *IEEE Transactions on Information Theory*, vol. 59, no. 12, pp. 8402–8413, December 2013.
K. Poularakis, G. Iosifidis, and L. Tassiulas, “Approximation Algorithms for Mobile Data Caching in Small Cell Networks", *IEEE Transactions on Communications*, vol. 62, no. 10, pp. 3665–3677, October 2014.
B. B[ł]{}aszczyszyn, and A. Giovanidis, “Optimal Geographic Caching In Cellular Networks", *IEEE International Conference on Communications (ICC) 2015*, pp. 3358–3363, London, UK, June 2015.
M. A. Maddah-Ali, and U. Niesen “Fundamental Limits of Caching", *IEEE Transactions on Information Theory*, vol. 60, pp. 2856–2867, May 2014.
S. Ioannidis, L. Massoulie, and A. Chaintreau, “Distributed Caching over Heterogeneous Mobile Networks", *SIGMETRICS 2010*, pp. 311–322, NY, USA, June 2010.
K. Poularakis, and L. Tassiulas, “Exploiting User Mobility for Wireless Content Delivery", *IEEE International Symposium on Information Theory Proceedings (ISIT) 2013*, pp. 1017–1021, Istanbul, Turkey, 2013.
E. Baştuğ, M. Bennis, and M. Debbah, “Cache-enabled Small Cell Networks: Modeling and Tradeoffs", *11th International Symposium on Wireless Communications Systems*, pp. 649–653, 2014.
E. Altman, K. Avrachenkov, and J. Goseling, “Distributed Storage in the Plane", *Networking Conference, IFIP 2014*, pp. 1–9, Trondheim, Norway, June 2014.
K. Avrachenkov, X. Bai, and J. Goseling, “Optimization of Caching Devices with Geometric Constraints", *Performance Evaluation*, vol. 113, pp. 68–82, 2017.
Z. Chen, J. Lee, T. Q. S. Quek, and M. Kountouris, “Cooperative Caching and Transmission Design in Cluster-Centric Small Cell Networks", *IEEE Transactions on Wireless Communications*, vol. 16, no. 5, pp. 3401–3415, 2017.
M. Dehghan, L. Massoulie, D. Towsley, D. Menasche, and Y. C. Tay, “A Utility Optimization Approach to Network Cache Design", *INFOCOM 2016*, San Francisco, CA, USA, April 2016.
A. Chattopadhyay, and B. B[ł]{}aszczyszyn, “Gibbsian on-line distributed content caching strategy for cellular networks", *arXiv: 1610.02318*, 2016.
K. Avrachenkov, J. Goseling, and B. Serbetci, “A Low-Complexity Approach to Distributed Cooperative Caching with Geographic Constraints", *Proceedings of the ACM on Measurement and Analysis of Computing Systems (POMACS)*, vol.1, no.1, pp. 1–25, June 2017.
F. Baccelli, and B. Blaszczyszyn, *[Stochastic Geometry and Wireless Networks, Volume I - Theory]{}*, ser. Foundations and Trends in Networking Vol. 3: No 3–4, pp. 249–449.1em plus 0.5em minus 0.4emNoW Publishers, 2009, vol. 1.
D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf, *[Stochastic geometry and its applications, Volume II]{}*, Wiley Chichester, 1995.
H. W. Kuhn, and A. W. Tucker, “Nonlinear programming", *Proc. of 2nd Berkeley Symposium*, pp. 481–492, 1950.
K. M. Bretthauer, and B. Shetty, “The Nonlinear Resource Allocation Problem", *Operations Research*, vol. 43, no. 4, pp. 670–683, 1995.
A. Schrijver, *[Combinatorial Optimization: Polyhedra and Efficiency]{}*, Springer Science & Business Media, 2002.
OpenMobileNetwork, .
[ Mobile Network Guide,]{} .
[^1]: A part of this work [@wcncpaper] has been presented at the IEEE Wireless Communications and Networking Conference 2017, held at San Francisco, CA, USA.
[^2]: B. Serbetci was with the Stochastic Operations Research, University of Twente, Enschede, 7522 NB, The Netherlands. He is now with the Communication Systems Department, EURECOM, 06410, Biot, France (e-mail: berksan.serbetci@eurecom.fr).
[^3]: J. Goseling is with the Stochastic Operations Research, University of Twente, Enschede, 7522 NB, The Netherlands. (e-mail: j.goseling@utwente.nl).
|
---
author:
- |
The IceCube Collaboration[^1]\
[*<http://icecube.wisc.edu/collaboration/authors/icrc19_icecube>*]{}\
E-mail:
bibliography:
- 'references.bib'
title: Science Case of a Scintillator and Radio Surface Array at IceCube
---
Introduction {#sec:info}
============
Understanding the origin of cosmic rays is the primary science goal of the IceCube neutrino observatory at the South Pole. In particular, astrophysical neutrinos measured by the $1\,$km$^3$ large in-ice array [@IceCube:2018cha] can reveal sources of cosmic rays. IceCube’s astrophysical neutrinos seem to be dominated by extragalactic sources [@Albert:2018vxw], and one extragalactic source has already been identified in a multi-messenger observation of neutrinos and gamma-rays [@IceCube:2018dnn]. IceTop is the $1\,$km$^2$ surface array of IceCube above its in-ice detector [@IceCube:2012nn; @Aartsen:2016nxy] and is comprised of 162 ice-Cherenkov detectors. Icetop measures cosmic-ray air showers in the energy range of about $250\,$TeV to $1\,$EeV [@Aartsen:2013wda], where the most energetic Galactic cosmic rays are presumed. Since both, the most energetic Galactic and extragalactic sources are unknown, IceTop and the in-ice detector complement each other in the quest for identifying the unknown cosmic-ray sources. Moreover, cosmic-ray measurements directly contribute to the neutrino science of IceCube by calibration of the in-ice detector with muons, the possibility to veto air showers, and by improving our understanding of the atmospheric background in the neutrino measurements. Furthermore, IceTop addresses further science goals such as the search for PeV photons [@Aartsen:2012gka], and the study of solar cosmic rays at low energies [@Abbasi:2008vr].
IceTop has collected data for more than 10 years. Hence, many analyses are not limited by statistics, but by systematic uncertainties. In particular, the snow accumulation on the IceTop tanks increases the detection threshold for air showers and the systematic uncertainty because current models for the snow attenuation lack accuracy. Therefore, an enhancement of the surface detector by scintillation detectors was originally planned to mitigate the effect of the snow [@ScintillatorsICRC:2017]. Recently, the geometry of the array and the station layout have been optimized to enable a significantly broader science case, and to reduce the deployment effort without deteriorating its performance. By adding radio antennas to the same array, the accuracy and sky coverage for high-energy cosmic rays will be enhanced significantly beyond the original performance of IceTop. Therefore, radio antennas were added to the existing prototype station at the South Pole [@IceScint_ICRC2019; @IceRad_ICRC2019]. In 2020, this prototype station will be refurbished to the new layout (see next section), and the installation of the complete array is planned for the subsequent years.[^2] While scintillators are an established technique for air-shower arrays, the radio technique has matured in the last years and has reached similar measurement accuracy as established optical techniques [@SchroederReview2016; @HuegeReview2016]. By this, the scintillator-radio upgrade of IceTop will enable new science goals as well as progress in the current goals of IceCube, such as key contributions to the science of the highest energy Galactic cosmic rays [@Astro2020_GCR_WhitePaper].
![Conceptual layout of the scintillator-radio hybrid array (left) comprised of 32 stations (right). Each station consists of 8 scintillation panels arranged in pairs, one pair at the center of the station where the local data-acquisition is located in an elevated field hub, and three pairs at $72\,$m distance from the station center. Along the same spoke-trenches to these scintillators, three radio antennas with two polarization channels each will be deployed in $35\,$m distance to the center. []{data-label="fig:array"}](IceTopUpgradeMap_StationsLayout_Plan2019.pdf){width="0.99\linewidth"}
Planned Detectors and Array Layout
==================================
The scintillator-radio hybrid array will cover the existing IceTop array above the in-ice detector. It will consist of 32 stations comprised each of 4 pairs of scintillation detectors and 3 dual-polarized radio antennas connected to a local data-acquisition (DAQ) electronics at the center (Fig. \[fig:array\]). To avoid deterioration of the measurement response by snow and allow for easy maintenance access, all detectors, i.e. scintillator panels and antennas, as well as the fieldhub housing the central electronics will be elevated structures. Each scintillation detector contains a SiPM whose signal is digitized locally upon a simple threshold trigger. Local triggers, digital signal amplitudes, and timestamps are then sent to the station DAQ. The radio signals are digitized directly in the station DAQ when receiving multiple coincident scintillation triggers. Finally, the station DAQ transmits the data to the central DAQ in the IceCube Lab where data are stored on disk and transferred once yearly to the North. A limited amount of air-shower data, e.g., for veto or monitoring purposes, can be transferred additionally via satellite almost in real time.
![Detection threshold of the scintillator-radio hybrid array determined by CORSIKA proton simulations. A signal-to-noise ratio > 10 is demanded in at least three antennas triggered by scintillators (noise model and normalization as in [@Balagopal2018]). []{data-label="fig:radioThreshold"}](RadioEfficiencyIceTop_proton_IntermediateAntPos_June2019.pdf){width="0.99\linewidth"}
This is significantly lower than the threshold of IceTop, even before it was covered by snow. The threshold of the radio array will be generally higher and depends on the zenith angle and the angle to the geomagnetic field. Simulation studies assuming Galactic and thermal noise of $300\,$K (noise model and normalization as in [@Balagopal2018]) predict an optimum frequency band including the FM broadcast band (fortunately this band is not in use at the South Pole) or at even higher frequencies. The exact band can be chosen differently for each science case by digital filters during data analysis [@Balagopal2018], e.g., the band of $110 - 190\,$MHz provides the lowest threshold for proton showers of $40^\circ$ zenith angle of around $1\,$PeV (see Fig. \[fig:radioThreshold\]). For this study, a conservative radio threshold of a signal-to-noise ratio of $10$ is assumed motivated by other experiments [@Aab:2015vta; @TunkaRex_NIM], which will be lowered by applying more sophisticated techniques for pulse identification [@Erdmann:2019nie; @Glaser:2019rxw; @TunkaRexPRD2018]. The future implementation of a radio self-trigger will be of particular advantage for the detection of inclined showers, e.g., from the direction of the Galactic Center which is continuously visible at $61^\circ$ zenith angle from the South Pole.
Thanks to the low radio background at the South Pole, a special feature of the radio array will be its sensitivity to showers parallel to the Earth’s magnetic field, whose radio emission is an order of magnitude weaker and purely by the Askaryan effect. This will enable to better study Askaryan emission, which is considered the only relevant emission mechanism for radio detection of neutrinos with in-ice arrays. It will also enable full efficiency and full sky coverage for the scintillator-radio hybrid array at the highest energies around $100\,$PeV and above.
Science Goals
=============
The science case of the scintillator-radio enhancement of IceTop is rich and broad, pursuing several science goals regarding astrophysical neutrinos, cosmic rays and air showers, as well as multi-messenger astroparticle physics (see Table \[tab:my\_label\] and Refs. [@Schroder:2018dvb; @Haungs:2019ylq]). The most important goals are summarized in this section.
More Accurate Air-Shower Measurements
-------------------------------------
The surface enhancement will improve air-shower measurements in several ways by increasing the total accuracy and sky coverage and lowering the detection threshold compared to the present IceTop.
First, the current IceTop suffers from snow accumulating above the tanks and attenuating air-shower particles, mostly the electromagnetic component. The limited understanding of the snow attenuation leads to significant systematic uncertainties in the interpretation of recent measurements. This will be mitigated by a detailed calibration of the snow effect depending on snow height, zenith angle, and lateral distance. Moreover, the attenuation by snow approximately doubled the detection threshold of IceTop since its construction. By deploying the scintillators above the snow, the new detection threshold will be lowered by a factor of two compared to the original IceTop before snow coverage.
Second, hadronic interaction models can be tested much more precisely due to the separate measurements of the different shower components: particles at ground level measured by IceTop and the scintillators, high-energy muons measured by the in-ice detector, and the size of the electromagnetic component at shower maximum measured by the radio antennas. Since the signals of the particle detectors at the surface are in most cases dominated by electromagnetic particles, the different response to electrons and muons by the scintillators and tanks contains additional information on the muon number at ground level. $X_\mathrm{max}$ measurements by radio may be used to identify proton-initiated showers [@Collaboration:2012wt]. The ability to select proton events will enable stricter tests of hadronic models than possible today by studies based on the average mass composition [@Dembinski:2019uta].
Third, the simultaneous measurement of several shower components by different detectors will boost the precision and accuracy. Combining the classical methods of the electron-muon ratio and of $X_\mathrm{max}$ in one single experiment will yield unprecedented measurement accuracy for the mass composition and a per-event classification of the primary particle. Moreover, the combination of radio and muon measurements provides a new promising option for per-event mass classification [@Holt:2019fnj]. Radio antennas can additionally be used to cross-check the absolute energy scale of IceCube with that of other air-shower arrays featuring radio antennas [@Apel:2016gws; @Aab:2016eeq].
These essential improvements in air-shower detection will enable a number of exciting science goals, in particular regarding Galactic cosmic rays and regarding IceCube’s neutrino measurements.
**Air Showers** **Neutrino Science**
------------------------------------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------------
$\bullet$Mitigate and calibrate the snow attenuation of air showers measured by IceTop $\bullet$More accurate estimation of atmospheric backgrounds for astrophysical neutrinos
$\bullet$Better tests of hadronic interaction models $\bullet$Improved surface veto for muon events
$\bullet$Higher accuracy for mass and energy $\bullet$Calibration of in-ice detector by muons
**Galactic Cosmic Rays** **Technical Merits**
$\bullet$Search for PeV photons, in particular from H.E.S.S. source at Galactic Center [@Abramowski:2016mir; @Balagopal2018] $\bullet$Testbed for general infrastructure of IceCube-Gen2: communication, timing, etc.
$\bullet$Understand transition to extragalactic CR $\bullet$Development of elevated surface structures
$\bullet$Mass composition measured more accurately + search for mass-sensitive anisotropy $\bullet$Pathfinder for Gen2 surface array: in-ice calibration, veto aperture and threshold, etc.
: Overview of the main scientific and technical goals of the IceTop upgrade (see text for details).[]{data-label="tab:my_label"}
High-Energy Galactic Cosmic Rays
--------------------------------
The surface enhancement provides an opportunity to solve the puzzle of the origin of the most energetic Galactic cosmic rays (see Fig. \[fig:radio\_sketch\]). Depending on the scenario for the transition from Galactic to extragalactic cosmic rays, the highest energy particles still originating from our Galaxy may be at energies around $100\,$PeV up to a few EeV. This is supported by several observations, e.g., a hardening of the spectrum of light cosmic rays and a softening of heavy cosmic rays around $100\,$PeV [@KGheavyKnee2011; @2013ApelKG_LightAnkle]. Since the maximum acceleration energy depends on the nuclear charge and mass numbers, the most energetic Galactic cosmic rays are presumed to be heavier nuclei, at least from the CNO group, probably from the Fe group. Thus, better measurements of the mass composition will help to determine the transition energy and mechanism.
Anisotropy measurements may contain additional information on the number and distribution of Galactic sources contributing at the transition energy. However, no significant anisotropy has been observed in this energy range, yet, only at lower and higher energies [@Aartsen:2018ppz; @Apel:2019afz; @Aab:2017tyv]. This may be due to different phases of the large-scale Galactic and extragalactic anisotropies overlapping in the energy range of the transition. Hence, the event-by-event mass classification provided by the upgraded surface array can be used to build a data set enriched by heavy nuclei, and thus presumably Galactic, cosmic rays of the highest energies. This will be used to extend the current anisotropy measurements by IceTop [@IceCube:2013mar] to higher energies and may lead to the first ever observation of mass-sensitive anisotropies.
Finally, the enhanced accuracy for the primary particle as well as the improved sky coverage due to the radio array will improve the search for PeV photons. In particular, the pevatron at the Galactic Center discovered by H.E.S.S. [@Abramowski:2016mir] is a promising candidate [@Balagopal2018]. If the photon spectrum extends to higher energies, then the first detection of PeV photons may directly reveal a source, potentially *the* source, of the most energetic Galactic cosmic rays.[^3]
High-Energy Neutrinos
---------------------
Atmospheric neutrinos produced in cosmic-ray air showers and interacting in the in-ice detector are a significant background to the measurement of astrophysical neutrinos. Therefore, this background needs to be understood as accurately as possible. At the moment, significant uncertainties on the flux of atmospheric neutrinos [@Gaisser:2016obt], each of several $10\,\%$, are caused by measurement uncertainties of the cosmic-ray flux versus the energy per nucleon and by deficiencies in the hadronic interaction models. Therefore, better tests of the hadronic models will help to solve the latter issue. The measurement uncertainties on the absolute flux versus energy per nucleon will be reduced by the IceTop enhancement due to more accurate measurements of the mass composition and of the absolute energy scale. Consequently, the boosted accuracy for air-shower measurements will also improve the measurement of high-energy astrophysical neutrinos by improving the estimation of their background.
For the upper hemisphere of the sky, a more effective air-shower veto will enable the identification of additional astrophysical neutrino candidates interacting between IceTop and the in-ice detector [@IceTopVeto_ICRC2019]. At the same time, muons are also an excellent calibration tool for the in-ice detector [@Bai:2007zzm], which will benefit from more accurate measurements of the associated air showers.
Pathfinder for IceCube-Gen2
---------------------------
The scintillator-radio array also serves as a pathfinder for a larger surface array planned for IceCube-Gen2, complementing the enlarged optical and radio in-ice arrays. On the one hand, the IceTop Upgrade provides a testbed for technical solutions such as elevated surface structures, communication and timing infrastructure. On the other hand, detector concepts will be tested, such as an efficient veto for near-vertical showers by scintillators, and for inclined showers by radio antennas.
![Schematic sketch of measurements with the IceTop upgrade: the scintillator enhancement will lower the energy threshold compared to IceTop and increase the precision. The combination with radio antennas will provide a significant further improvement in total accuracy and lower the sky coverage bringing the Galactic Center in the field of view (figure from [@Schroder:2018dvb]).[]{data-label="fig:radio_sketch"}](sketch_radioAtSP.pdf){width="13cm"}
Conclusion
==========
In summary, there are significant synergies in enhancing the IceTop array by scintillators and radio antennas at the same time. In the next year, the existing prototype station will be updated to the new design presented here, with the aim of deployment of the full array in subsequent years. Together with the existing in-ice and surface detectors, this hybrid array will boost the accuracy of IceCube for air showers and enable a broad science case in the era of multi-messenger astronomy.
[^1]: For collaboration list, see PoS(ICRC2019) 1177.
[^2]: Moreover, there are plans to add small air-Cherenkov telescopes (IceAct) to further increase the accuracy for cosmic rays at lower energies [@IceAct_ICRC2019].
[^3]: The sensitivity to PeV photons from the Galactic Center will also open another channel for the search of heavy dark matter by IceCube [@Aartsen:2018mxl].
|
---
abstract: 'Detailed magneto-transport data on dense wires of $MgB_2$ are reported for applied magnetic fields up to 18 T. The temperature and field dependencies of the electrical resistivity are consistent with $MgB_2$ behaving like a simple metal and following a generalized form of Kohler’s rule. In addition, given the generally high $T_c$ values and narrow resistive transition widths associated with $MgB_2$ synthesized in this manner, combined with applied magnetic fields of up to 18 T, an accurate and complete $H_{c2}(T)$ curve could be determined. This curve agrees well with curves determined from lower field measurements on sintered pellets and wires of $MgB_2$. $H_{c2}(T)$ is linear in $T$ over a wide range of temperature (7 K $\le~T~\le$ 32 K) and has an upward curvature for $T$ close to $T_c$. These features are similar to other high $\kappa$, clean limit, boron-bearing intermetallics: $YNi_2B_2C$ and $LuNi_2B_2C$.'
address:
- |
Ames Laboratory, U.S. Department of Energy and Department of Physics and Astronomy\
Iowa State University, Ames, Iowa 50011
- 'National High Magnetic Field Laboratory - Pulse Facility, Los Alamos National Laboratory, MS E536 Los Alamos, NM 87545'
author:
- 'S. L. Bud’ko, C. Petrovic, G. Lapertot[^1], C. E. Cunningham[^2], and P. C. Canfield'
- 'M-H. Jung[^3] and A. H. Lacerda'
title: 'Magnetoresistivity and Complete $H_{c2}(T)$ in $MgB_2$.'
---
Introduction
============
The recent discovery[@jap] of superconductivity in $MgB_2$ has lead to a flurry of activity. Measurements of the boron isotope effect[@budko] show that there is a shift in $T_c$ from 39.2 K to 40.2 K (for $Mg^{11}B_2$ and $Mg^{10}B_2$ respectively), a result consistent with electron phonon mediated BCS superconductivity. Measurements[@budko; @DKF; @canfield; @jap1; @wisc; @caplan] of the upper critical field, $H_{c2}(T)$, and the thermodynamic critical field, $H_c(T)$, as well as the specific heat are all consistent with $MgB_2$ being a fairly typical intermetallic superconductor with an atypically high transition temperature. Although single crystal samples are not yet available, it has recently been found that dense, very high quality, wire samples of $MgB_2$ can be made.[@canfield] These samples have a superconducting transition temperature above that found for $Mg^{11}B_2$, as would be expected based on the natural abundance of $^{10}B$. These samples allow for the direct measurement of electrical resistivity and, given that the width of the superconducting transition is narrower in $MgB_2$ wire than in powder or sintered pellet samples,[@DKF; @canfield] wire samples allow for an accurate determination of the upper critical field $H_{c2}(T)$. In this communication we present data on the magneto-transport of $MgB_2$ wires for applied magnetic fields of up to 18 T and over the temperature range 1.5 - 300 K. By a careful analysis of the resistivity data we are able to conclude that $MgB_2$ behaves like a simple metal in the normal state with all of our magnetoresistance data collapsing onto a single curve in accordance to Kohler’s rule. In addition we are able to construct the full $H_{c2}(T)$ curve. We find that $H_{c2}(T)$ is linear over a much larger temperature range than would be expected,[@WHH] leading to a $H_{c2}(0)~\approx$ 16.4 T.
Experimental methods
====================
$MgB_2$ wire was produced[@canfield] by sealing 100 $\mu$m diameter boron fiber and $Mg$ into a $Ta$ tube with a nominal ratio of $Mg_2B$. Given that $MgB_2$ is the most $Mg$ rich binary $Mg-B$ compound known, it was felt that excess $Mg$ would aid in the formation of the proper, stoichiometric phase. The sealed $Ta$ tube was itself sealed in quartz and then placed into a 950$^\circ$C box furnace for two hours. The reaction ampoule was then removed from the furnace and quenched to room temperature. Whereas the boron fiber has a diameter of 100 $\mu$m, the $MgB_2$ wire has a diameter of approximately 160 $\mu$m. Although the $MgB_2$ wires are somewhat brittle, the integrity of the filament segments is preserved during the exposure to the $Mg$ vapor; i.e. the fibers did not decompose, fragment, or turn into powder. The resulting wire has over 80% the theoretical density of $MgB_2$ and measurements of the temperature dependent resistivity reveal that $MgB_2$ is highly conducting in the normal state. The room temperature resistivity has a value of 9.6 $\mu$Ohm-cm; whereas the resistivity at $T$ = 40 K is 0.38 $\mu$Ohm-cm. The zero field $T_c$ value for the wire sample is higher than that found for $Mg^{11}B_2$, a result consistent with the natural abundance of $^{10}B$. It should be noted that both wire and sintered pellet samples synthesized in this manner tend to have very high and sharp superconducting transitions.
Magnetoresistivity measurements utilizing a 20 T, superconducting magnet were performed at the National High Magnetic Field Laboratory, Pulsed Facility. A standard four-probe ac method was used, utilizing Epotek H20E silver epoxy for making electrical contacts. The contact resistance was approximately 1 Ohm. Given the well-defined geometry of the samples, accurate measurements of resistivity were possible. The ac current was applied along the wire and the magnetic field was applied perpendicular to the current direction. The sample was mounted in a flow cryostat able to regulate the temperature from 1.4 K to room temperature.
Data and analysis
=================
Figure 1 presents temperature dependent electrical resistivity data for a $MgB_2$ wire sample taken at a variety of applied fields for $H~\le$ 18 T. Two features are clearly seen: there is a suppression of the superconducting phase to lower temperatures for increasing applied field, and there is a clear, large magnetoresistivity in the normal state. Looking first at the suppression of superconductivity, the inset to Fig. 1 presents an enlarged view of the low temperature resistivity data. Using these data, three temperatures can be extracted from each curve: onset temperature, temperature of maximum $d\rho/dT$, and completion temperature, where onset and completion temperatures are determined by extending the maximum $d\rho/dT$ line up to the normal state and down to zero resistivity.
Figure 2 presents the $H_{c2}(T)$ curve that we deduce from these data. In addition to the high field data taken at NHMFL (shown as open symbols), data taken in a Quantum Design PPMS system at lower fields on a sample from the same batch are also shown (filled symbols). Several features of this curve are worth noting. First of all it has a large temperature / field range over which it is linear (7 K $\le T \le$ 32 K). Below approximately 7 K $H_{c2}(T)$ starts to roll over and saturate. This leads to a low temperature value of $H_{c2}$(1.5 K) = 16.2 T, which is significantly larger than estimates[@DKF] based on the assumption[@WHH] that the low temperature $H_{c2}$(0) = 0.71 $T_c~[dH_{c2}(T)/dT]$ = 12.5 T. Secondly, at high temperatures ($T~\ge$ 32 K) there is a distinct positive, upward curvature associated with the $H_{c2}(T)$ curve. This is not unique to the current form of our sample but was also seen in resistively determined $H_{c2}(T)$ for sintered pellets of $Mg^{10}B_2$.[@DKF] Taken as a whole, the temperature dependence of $H_{c2}$ for $MgB_2$ is remarkably similar to that recently found for other non-magnetic, intermetallic, boride superconductors: $LuNi_2B_2C$ and $YNi_2B_2C$.[@boro1; @boro2; @boro3] In these cases $H_{c2}(T)$ is linear over an extended region of $T$ and near $T_c$ there is a distinct upward curvature. In both the case of $MgB_2$ as well as in the case of $Y/LuNi_2B_2C$ the material is a high $\kappa$, type-II superconductor and in both cases the as grown compounds are well within the clean limit.[@DKF; @canfield; @boro1; @boro2; @boro3]
Turning to the normal state magnetoresistivity, Fig. 3 shows $\Delta\rho/\rho_0$ vs $H/\rho_0$ on a $log-log$ plot to demonstrate that all of the data presented in Fig. 1, as well as the two isothermal $\rho(H)$ plots shown in the inset of Fig. 3, are broadly consistent with the generalized form of Kohler’s rule. The fact that all of these data fall (roughly) onto a single curve implies that there is a single salient scattering time in the normal state transport of $MgB_2$.[@pippard] This is what would be anticipated for a simple nonmagnetic intermetallic sample. It is worth noting that such a clear magnetoresistance would be much harder to detect in samples with enhanced impurity or defect scattering.[@korea]
The isothermal $\Delta\rho(H)/\rho_0$ data shown in the inset of Fig. 3 can be fit to $\Delta\rho(H)/\rho_0~\propto~H^{\alpha}$ with $\alpha$ = 1.4-1.5. In addition, the temperature dependent normal state resistance of sintered $Mg^{10}B_2$ pellets as well as the resistivity of wire samples in zero field can be fit to a power law between $T^{2.6}$ and $T^3$ at low temperatures ($T~\le$ 200 K).[@DKF; @canfield]
Finally, there is a slight upturn observed in the low-temperature $\rho(T)$ data taken in high applied magnetic fields (Figure 1). Although the origin of this feature is not completely understood, similar features have been observed for other high purity, nonmagnetic intermetallic compounds.[@dia1; @dia2]
Conclusions
===========
In this communication we present detailed magetoresistivity data on dense (over 80%), high quality samples of $MgB_2$. In the normal state we find that $MgB_2$ has a temperature and field dependent resistivity that is consistent with $MgB_2$ being a highly conducting intermetallic compound that can be synthesized so as to achieve very low residual resistivities. This allows the large values of $\Delta\rho/\rho_0$ to reveal themselves. The fact that the magnetoresistivity data follow Kohler’s rule is consistent with $MgB_2$ behaving like a simple metal with one dominant scattering time.[@pippard]
In the superconducting state we find that $MgB_2$ has a relatively linear $H_{c2}(T)$ curve with slight deviations from linearity at both high and low temperatures. This leads to a relatively high value of $H_{c2}$(0) = 16.4 T. The linear behavior of $H_{c2}(T)$ as well as the upward curvature in $H_{c2}(T)$ for $T$ near $T_c$ is similar to behavior seen in other boron bearing intermetallic compounds ($YNi_2B_2C$ and $LuNi_2B_2C$) that have large $\kappa$ values and are within the clean limit. This similarity brings up the obvious question of whether such temperature dependencies of $H_{c2}(T)$ are a generic feature associate with this subclass of intermetallic superconductors.
Acknowledgments
===============
We would like to thank D. K. Finnemore for useful discussions. Ames Laboratory is operated for the U. S. Department of Energy by Iowa State University under Contract No. W-7405-Eng.-82. This work was supported by the director for Energy Research, Office of Basic Energy Sciences. Work performed at the National High Magnetic Field Laboratory was supported by the National Science Foundation, The State of Florida and the U. S. Department of Energy. One of us (M-H J) acknowledges partial support form LANSCE - LANL.
J. Akimiitsu, Symposium on Transition Metal Oxides, Sendai, January 10, 2001; J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu (to be published). S. L. Bud’ko, G. Lapertot, C. Petrovic, C. E. Cunningham, N. Anderson, and P. C. Canfield, Phys. Rev. Lett. [**86**]{}, 1877 (2001). D. K. Finnemore, J. E. Ostenson, S. L. Bud’ko, G. Lapertot, and P. C. Canfield, cond-mat/0102114. P. C. Canfield, D. K. Finnemore, S. L. Bud’ko, J. E. Ostenson, G. Lapertot, C. E. Cunningham, and C. Petrovic, cond-mat/0102289. Y. Takano, H. Takeya, H. Fujii, T. Hatano, K. Togano, H. Kito, and H. Ihara, cond-mat/0102167. D. C. Larbalestier, M. Rikel, L. D. Cooley, A. A. Polyanskii, J. Y. Jiang, S. Patnaik, X. Y. Cai, D. M. Feldmann, A. Gurevich, A. A. Squitier, M. T. Naus, C. B. Eom, E. E. Hellstrom, R. J. Cava, K. A. Regan, N. Rogado, M. A. Hayward, T. He, J. S. Slusky, P. Khalifah, K. Inumaru, and M. Hass, cond-mat/0102216. Y. Bugoslavsky, G. K. Perkins, X. Qi, L. F. Cohen, and A. D. Caplin, cond-mat/0102353. N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Phys. Rev. [**147**]{}, 295 (1966) and refs. therein. K. D. D. Rathnayaka, A. K. Bhatnagar, A. Parasiris, D. G. Naugle, P. C. Canfield, and B. K. Cho, Phys. Rev. B [**55**]{}, 8506 (1997). V. Metlushko, U. Welp, A. Koshelev, I. Aranson, G. W. Crabtree, and P. C. Canfield, Phys. Rev. Lett. [**79**]{}, 1738 (1997). S. V. Shulga, S.-L. Drechsler, G. Fuchs, K.-H. Muller, K. Winzer, M. Heinecke, and K. Krug, Phys. Rev. Lett. [**80**]{}, 1730 (1998). see for example: A. B. Pippard, Magnetoresistance in metals (Cambridge University Press, Cambridge, England, 1989). C. U. Jung, Min-Seok Park, W. N. Kang, Mun-Seog Kim, S. Y. Lee, and Sung-Ik Lee, cond-mat/0102215. S. L. Bud’ko, P. C. Canfield, C. H. Mielke, and A. H. Lacerda, Phys. Rev. B [**57**]{}, 13624 (1998). K. D. Myers, S. L. Bud’ko, I. R. Fisher, Z. Islam, H. Kleinke, A. H. Lacerda, and P. C. Canfield, J. Magn. Magn. Mater., [**205**]{}, 27 (1999).
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[^1]: On leave from Commissariat a l’Energie Atomique, DRFMC-SPSMS, 38054 Grenoble, France
[^2]: On leave from Dept. of Physics, Grinnell College, Grinnell, IA 50112
[^3]: also at Physics Dept., New Mexico State University, Las Cruces, New Mexico
|
---
abstract: 'A two-parameter deformed superoscillator system with $SU_{q_{1}/q_{2}}$**(**$n\mid m$**)**-covariance is presented and used to construct a two-parameter deformed $N=2$ SUSY algebra. The Fock space representation of the algebra is discussed and the deformed Hamiltonian for such generalized superoscillators is obtained.'
author:
- 'Abdullah Algin$^{1},$ Metin Arik$^{2},$ Ali S. Arikan$^{2}$'
- '$^{1}$[Department of Physics, Osmangazi University, Meşelik, Eskişehir, Turkey]{}'
- '$^{2}$[Department of Physics, Boğaziçi University, Bebek, Istanbul, Turkey]{}'
title: |
**Two-parameter deformed supersymmetric oscillators with SU**$%
_{q_{1}/q_{2}}$**(**$n\mid m$**)-covariance**
---
Introduction
============
A great deal of effort has recently been spent to the study of many aspects of quantum groups and their associated algebras, which are specific deformations of the usual Lie groups and Lie algebras with some deformation parameter $q$[@a]-[@d]. Many of such studies can be mentioned within a wide spectrum of research of theoretical physics such as noncommutative geometry[@e], two-dimensional conformal field theories[@f], quantum mechanics[@g].
After the realization of intimate relationship between $q$-deformed bosonic as well as fermionic oscillators and quantum groups (and their algebras)[@h]-[@j], these relations have subsequently been extended to two-parameter deformed versions of such oscillator algebras[@k] and quantum group structures[@l]. Meanwhile, several deformed forms of superalgebras and supergroups have extensively been constructed by a natural association of one or two-parameter deformed bosonic and fermionic oscillator algebras[@m]-[@o].
As is well known that the ordinary $N=2$ SUSY algebra developed by Witten[@p] combines undeformed bosons with undeformed fermions. This algebra has the following form:
$$\left\{ Q,Q^{\star }\right\} =H,\qquad Q^{2}=(Q^{\star })^{2}=0,\qquad \left[H,Q\right] =\left[ H,Q^{\star }\right] =0,$$
where $Q$ and $Q^{\star }$ are odd generators called supercharges, $H$ is an even generator called Hamiltonian. These generators are assumed to be well defined on the relevant Hilbert space. After this $N=2$ SUSY algebra has been constructed, several $q$-deformed versions of this algebra have been proposed by using either $q$-deformed boson operators or $q$-deformed fermionic operators[@r]-[@u]. These studies have been done by mutually commuting $q$-deformed bosonic and $q$-deformed fermionic oscillator variables. Recently, such studies have also been extended to $q$-oscillator systems covariant under some quantum supergroup transformations[@m],[@v].
The present paper investigates an interesting generalization for the $%
\noindent N=2$ SUSY algebra. In our generalization, the superoscillators system is accomplished by two independent real deformation parameters $%
(q_{1},q_{2})$ and has a covariance under the two-parameter deformed quantum supergroup $SU_{q_{1}/q_{2}}(n\mid m).$ Moreover, our generalization gives mutually non-commuting two-parameter deformed bosons and fermions.
We should also mention that another example of such a mutually non-commuting bosons and fermions property with $sl_{q}(n\mid
n)$-covariance was recently introduced using the one-parameter deformed oscillator variables by Chung[@v]. On the other hand, different algebraic forms of SUSY structure such as fractional supersymmetry and parasupersymmetry have been studied by taking deformation parameter $q$ being a root of unity[@w]-[@y].
In this paper, our aim is to construct a two-parameter deformed $N=2$ SUSY algebra by using the $SU_{q_{1}/q_{2}}(n\mid m)$-covariant $(q_{1},q_{2})$-deformed bosonic and $(q_{1},q_{2})$-deformed fermionic oscillator system. However, our superoscillator algebra construction serves as a generalization related to the studies on the deformation of the conventional $N=2$ SUSY algebra.
The paper is organized as follows: In section 2, the two-parameter deformed superoscillator algebra and its $SU_{q_{1}/q_{2}}(1\mid
1)$-covariance are shown. In section 3, we construct the $(q_{1},q_{2})$-deformed SUSY quantum mechanics for $SU_{q_{1}/q_{2}}(n\mid m)$-covariant $(q_{1},q_{2})$-deformed boson and $(q_{1},q_{2})$-deformed fermion system. In section 4, we analyze the Fock space representation of the $(q_{1},q_{2})$-deformed $N=2$ SUSY algebra and obtain the deformed Hamiltonian of the two-parameter deformed superoscillator system. Finally, we give our conclusions in section 5.
$SU_{q_{1}/q_{2}}(1\mid 1)$ -covariant two-parameter deformed superoscillator algebra
=====================================================================================
In this section, we consider a system of one $(q_{1},q_{2})$ -deformed bosonic and one $(q_{1},q_{2})$-deformed fermionic oscillators, and show their covariance under the quantum supergroup $SU_{q_{1}/q_{2}}(1%
\mid 1)$. For this aim, we introduce the following $(q_{1},q_{2})$-deformed superoscillator algebra:$$\begin{aligned}
AB &=&\frac{q_{1}}{q_{2}}BA, \\
AB^{\star } &=&q_{1}q_{2}B^{\star }A, \\
AA^{\star }-q_{1}^{2}A^{\star }A &=&q_{2}^{2(N_{b}+N_{f})}, \\
B^{2} &=&(B^{\star })^{2}=0, \\
BB^{\star }+q_{2}^{2}B^{\star }B
&=&q_{2}^{2(N_{b}+N_{f})}+(q_{1}^{2}-q_{2}^{2})A^{\star
}A=q_{1}^{2N_{b}}q_{2}^{2N_{f}},\end{aligned}$$ where $A,\,A^{\star }$ and $B,B^{\star }$ are the deformed bosonic and fermionic annihilation and creation operators, respectively. By using the tensor product, the annihilation operators which satisfy the above algebraic relations can be written as $$\begin{aligned}
A &=&a\otimes q_{2}^{N_{f}} \\
B &=&q_{1}^{N_{b}}\otimes c\end{aligned}$$ where $a$ is the deformed bosonic annihilation operator and $c$ is the deformed fermionic annihilation operator. $q_{1},q_{2}\in \mathbf{R}^{+}.$ $%
N_{b}$ and $N_{f}$ are the bosonic and fermionic number operators, respectively. It is important to notice that, in this consideration, the two-parameter deformed bosonic and fermionic oscillators do not commute with each other (Eqs. (2) and (3)), and the system still satisfies the Pauli exclusion principle (Eq. (5)). Eq. (4) gives deformed bosonic commutation relation whereas Eq. (6) gives the deformed fermionic anticommutation one.
By considering the limit $q_{2}=1$, one can easily realize that above algebraic relations take the $SU_{q_{1}}(1\mid 1)$ covariant form[@m]. It is helpful to remember this limiting case when one tries to find the transformation matrix $T$ which leaves invariant the system defined in Eqs. (2)-(6). In the light of these facts, let us consider the following transformation[@v]: $$\left(
\begin{array}{c}
A^{^{\prime }} \\
B^{^{\prime }}
\end{array}
\right) =T\,\left(
\begin{array}{c}
A \\
B
\end{array}
\right) =\left(
\begin{array}{cc}
a & \beta \\
-(a^{\star })^{-1}\beta ^{\star }(a^{\star })^{-1} & (a^{\star
})^{-1}
\end{array}
\right) \left(
\begin{array}{c}
A \\
B
\end{array}
\right) ,$$ where $T$ is the $2\times 2$ quantum super-matrix with two even $(a,a^{\star })$ and two odd $(\beta ,\beta ^{\star })$ generators. Since these generators satisfy the following algebraic relations: $$\begin{aligned}
a\beta &=&\frac{q_{1}}{q_{2}}\beta a, \nonumber \\
a\beta ^{\star } &=&\frac{q_{1}}{q_{2}}\beta ^{\star }a, \\
\beta ^{2} &=&(\beta ^{\star })^{2}=0,\qquad \ \beta \beta ^{\star }+\frac{%
q_{1}^{2}}{q_{2}^{2}}\beta ^{\star }\beta =0, \nonumber \\
aa^{\star }+\beta \beta ^{\star } &=&1,\qquad \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ aa^{\star }-a^{\star }a=\left(
\frac{q_{1}^{2}}{q_{2}^{2}}-1\right) \beta ^{\star }\beta .
\nonumber\end{aligned}$$ the matrix $T$ is an element of $ SU_{q_{1}/q_{2}}(1\mid 1).$ It is noticed that although the relations involving the oscillator creation and annihilation operators depend on the two deformation parameters $q_{1}$ and $%
q_{2}$, the relations containing the matrix elements of $T$ effectively involve a single parameter $r=q_{1}/q_{2}.$ This is different from Celik’s study[@z] where the quantum matrix which leaves invariant another two-parameter superoscillator algebra also depends on the two deformation parameters separately.
According to Eq. (7), one can easily write the transformed annihilation and creation operators as
$$\begin{aligned}
A^{^{\prime }} &=&a\otimes A+\beta \otimes B, \nonumber \\
B^{^{\prime }} &=&-(a^{\star })^{-1}\beta ^{\star }(a^{\star
})^{-1}\otimes
A+(a^{\star })^{-1}\otimes B, \nonumber \\
(A^{^{\prime }})^{\star } &=&a^{\star }\otimes A^{\star }-\beta
^{\star
}\otimes B^{\star }, \\
(B^{^{\prime }})^{\star } &=&-a^{-1}\beta a^{-1}\otimes A^{\star
}+a^{-1}\otimes B^{\star }. \nonumber\end{aligned}$$
In order to see all relations in Eqs. (2)-(6) remain unchanged for transformed operators, we should also use braided tensor product rule which can be shown as $$(x\otimes f_{1})(f_{2}\otimes y)=-xf_{2}\otimes f_{1}y\,,$$ where $f_{1}$ and $f_{2}$ denote fermionic operators.
With the generalization of the above system to the $(n+m)$-dimensional case, one can write the bosonic and fermionic generators as $$\begin{aligned}
A_{i}
&=&\prod_{k=1}^{i-1}q_{1}^{(N_{b})_{k}}\,a_{i}\,%
\prod_{k=i+1}^{n}q_{2}^{(N_{b})_{k}}\,\prod_{k=1}^{m}q_{2}^{(N_{f})_{k}}, \\
B_{j}
&=&\prod_{k=1}^{n}q_{1}^{(N_{b})_{k}}\,%
\prod_{k=1}^{j-1}(-q_{1})^{(N_{f})_{k}}\,c_{j}\,%
\prod_{k=j+1}^{m}q_{2}^{(N_{f})_{k}},\end{aligned}$$ respectively. Here $i$ stands for $1$ to $n$ whereas $j$ stands for $1$ to $m $. These generators satisfy the following algebraic relations: $$\begin{aligned}
A_{i}A_{j} &=&\frac{q_{1}}{q_{2}}A_{j}A_{i},\qquad \ \ i<j, \\
\qquad A_{i}A_{j}^{\star } &=&q_{1}q_{2}A_{j}^{\star }A_{i},\qquad
i\neq j,
\\
A_{i}A_{i}^{\star }-q_{1}^{2}A_{i}^{\star }A_{i}
&=&q_{2}^{2N}+(q_{1}^{2}-q_{2}^{2})\sum_{j=1}^{i-1}A_{j}^{\star }A_{j}, \\
A_{i}B_{k} &=&\frac{q_{1}}{q_{2}}B_{k}A_{i},\qquad \ \ \ \qquad \\
A_{i}B_{k}^{\star } &=&q_{1}q_{2}B_{k}^{\star }A_{i},\qquad \\
B_{k}B_{l} &=&-\frac{q_{1}}{q_{2}}B_{l}B_{k},\qquad \ \ \ k<l, \\
B_{k}B_{l}^{\star } &=&-q_{1}q_{2}B_{l}^{\star }B_{k},\qquad k\neq l, \\
B_{k}^{2} &=&(B_{k}^{\star })^{2}=0, \\
B_{k}B_{k}^{\star }+q_{2}^{2}B_{k}^{\star }B_{k}
&=&q_{2}^{2N}+(q_{1}^{2}-q_{2}^{2})\sum_{l=1}^{n}A_{l}^{\star
}A_{l}+(q_{1}^{2}-q_{2}^{2})\sum_{l=1}^{k-1}B_{l}^{\star }B_{l},\end{aligned}$$ such that they are invariant under $SU_{q_{1}/q_{2}}(n\mid m)$ transformation. It is noted that in Eqs. (12) and (18), $N$ represents the total number operator which is nothing but the addition of the fermionic and bosonic number operators. It is clear that, in the limit $q_{2}=1,$ above system reduce the one parameter deformed $SU_{q_{1}}(n\mid m)$-covariant superoscillator algebra[@m].
In the next section, the above deformed superoscillator algebra will be used to construct a two-parameter deformed $N=2$ SUSY algebra.
The construction of a two-parameter deformed $N=2$ SUSY algebra for $SU_{q_{1}/q_{2}}(n\mid
m)$-covariant $(q_{1},q_{2})$-deformed bosons and fermions
============================================================================================
In this section, we construct the $(q_{1},q_{2})$-deformed SUSY quantum mechanics for $n$ $(q_{1},q_{2})$-deformed bosonic and $m$ $%
(q_{1},q_{2})$-deformed fermionic oscillators covariant under the quantum supergroup $SU_{q_{1}/q_{2}}(n\mid m).$ For the sake of simplicity, we begin with the $n=m=1$ case. We have the following deformed supercharges constructed from above deformed superoscillators variables: $$Q=A^{\star }\,B,\qquad Q^{\star }=B^{\star }A,$$ where the operators $A,B$ satisfy the commutation relations given in Eqs. (2)-(6). The odd generators $Q,Q^{\star }$ satisfy the nilpotency condition: $$Q^{2}=(Q^{\star })^{2}=0,$$ which can be obtained from Eq. (3), (5). From all considerations above, we have the following two-parameter deformed $N=2$ SUSY algebra with $%
SU_{q_{1}/q_{2}}(1\mid 1)$-covariance$:$$$\left\{ Q,Q^{\star }\right\} _{\frac{q_{2}^{4}}{q_{1}^{4}}} =\widetilde{H}%
=q_{1}^{-2}\left\{ q_{2}^{2N}\left[ A^{\star }A+\frac{q_{2}^{2}}{q_{1}^{2}}%
B^{\star }B\right] +(q_{1}^{2}-q_{2}^{2})(A^{\star
}A)^{2}\right\},$$ $$\begin{aligned}
\left[ Q,\widetilde{H}\right] _{q_{2}^{4}/q_{1}^{4}} &=&0, \\
\left[ Q^{\star },\widetilde{H}\right] _{q_{1}^{4}/q_{2}^{4}} &=&0, \\
Q^{2} =(Q^{\star })^{2}&=&0,\end{aligned}$$ where $N=N_{b}+N_{f},$ and also $\left\{ A,B\right\} _{r}=AB+rBA$ and $\left[ A,B\right] _{r}=AB-rBA.$ The deformed Hamiltonian $\widetilde{H}$ in Eq. (21) gives a two-parameter generalization of the Hamiltonian for the supersymmetric oscillator in quantum mechanics[@ab].
The algebra constructed in Eqs. (21)-(24) has some interesting limiting cases: In the limit $q_{2}=1,$ we find the one-parameter deformed $N=2$ SUSY algebra[@s],[@v]. The conventional $N=2$ SUSY algebra in Eq. (1) can be recovered in the limit $q_{1}=q_{2}=1.$ For arbitrary indices of deformed boson and fermion variables, we have the following $2(n+m)$ supercharges: $$Q_{i}=A_{i}^{\star }\,B_{i},\qquad Q_{i}^{\star }=B_{i}^{\star
}A_{i}.$$ These supercharges are also nilpotent: $$Q_{i}^{2}=(Q_{i}^{\star })^{2}=0,$$ which can be obtained from Eqs. (14) and (17). Thus the generalized two-parameter deformed $N=2$ SUSY algebra for $SU_{q_{1}/q_{2}}(n\mid m)$-covariant $(q_{1},q_{2})$-deformed bosonic and $(q_{1},q_{2})$-deformed fermionic oscillators is constructed by the following deformed commutation and anti-commutation relations: $$\begin{aligned}
\left\{ Q_{i},Q_{j}\right\} &=&0, \\
\left\{ Q_{i},Q_{j}^{\star }\right\} _{q_{2}^{2}/q_{1}^{2}} &=&0,\end{aligned}$$ $$\begin{aligned}
\left\{ Q_{i},Q_{i}^{\star }\right\} _{\frac{q_{2}^{4}}{q_{1}^{4}}} &=&
\widetilde{H}_{i}=q_{1}^{-2}\left\{ q_{2}^{2N}\left[ A_{i}^{\star }A_{i}+(
\frac{q_{2}^{2}}{q_{1}^{2}})B_{i}^{\star }B_{i}\right]
+(q_{1}^{2}-q_{2}^{2})(\frac{q_{2}}{q_{1}})^{2}B_{i}^{\star
}B_{i}(\sum_{j=1}^{i-1}A_{j}^{\star }A_{j})\right\} \nonumber \\
&+&q_{1}^{-2}\left\{(q_{1}^{2}-q_{2}^{2})\left( A_{i}^{\star }A_{i}\right)
\left[\sum_{j=1}^{n}(A_{j}^{\star }A_{j})+\sum_{j=1}^{i-1}(B_{j}^{\star }B_{j})%
\right]\right\},\end{aligned}$$ $$\begin{aligned}
\left[ Q_{j},\widetilde{H}_{i}\right] _{q_{2}^{2}/q_{1}^{2}} &=&0,\qquad \\
\left[ Q_{i},\widetilde{H}_{i}\right] _{q_{2}^{4}/q_{1}^{4}}
&=&0,\qquad\end{aligned}$$ where $N=N_{b}+N_{f}.$ It is important to note that one can recover the one-parameter deformed $N=2$ SUSY algebra with $SU_{q_{1}}(n\mid m)$-covariance[@v] in the limit $q_{2}=1.$
Fock space representation of the two-parameter deformed $N=2$ SUSY algebra with\
$SU_{q_{1}/q_{2}}(n\mid m)$-covariance
=================================================================================
We now discuss the Fock space representation of the two-parameter deformed $N=2$ SUSY algebra with $SU_{q_{1}/q_{2}}(n\mid m)$-covariance. We first consider the simplest case with $n=m=1.$ The bosonic and fermionic number operators $N_{b}$ and $N_{f}$ satisfy the following commutation relations: $$\begin{aligned}
\left[ A,N_{b}\right] &=&A,\qquad \left[ A^{\star },N_{b}\right]
=-A^{\star
}, \nonumber \\
\left[ B,N_{f}\right] &=&B,\qquad \left[ B^{\star },N_{f}\right]
=-B^{\star }.\end{aligned}$$ We introduce the Fock basis $\left| n_{b},n_{f}\right\rangle $ and the number operators also satisfy the following relations: $$\begin{aligned}
N_{b}\left| n_{b},n_{f}\right\rangle &=&n_{b}\left|
n_{b},n_{f}\right\rangle
,\qquad n_{b}=0,1,2,..., \nonumber \\
N_{f}\left| n_{b},n_{f}\right\rangle &=&n_{f}\left|
n_{b},n_{f}\right\rangle
,\qquad n_{f}=0,1, \\
N\left| n_{b},n_{f}\right\rangle &=&(n_{b}+n_{f})\left|
n_{b},n_{f}\right\rangle =n\left| n_{b},n_{f}\right\rangle ,
\nonumber\end{aligned}$$ The representations of the operators $A,\,A^{\star
},\,B,\,B^{\star }$ are $$\begin{aligned}
A\left| n_{b},n_{f}\right\rangle &=&q_{2}^{n_{f}}\sqrt{\left[ n_{b}\right] }%
\,\left| n_{b}-1,n_{f}\right\rangle , \\
\qquad \ \ \ A^{\star }\left| n_{b},n_{f}\right\rangle &=&q_{2}^{n_{f}}\sqrt{%
\left[ n_{b}+1\right] }\,\left| n_{b}+1,n_{f}\right\rangle , \\
B\left| n_{b},0\right\rangle &=&0,\qquad B\left|
n_{b},1\right\rangle
=q_{1}^{n_{b}}\,\left| n_{b},0\right\rangle , \\
\qquad B^{\star }\left| n_{b},1\right\rangle &=&0,\qquad B^{\star
}\left| n_{b},0\right\rangle =q_{1}^{n_{b}}\,\left|
n_{b},1\right\rangle ,\end{aligned}$$ where $$\begin{aligned}
A^{\star }A &=&\left[ N_{b}\right] \,q_{2}^{2N_{f}}=\left( \frac{%
q_{2}^{2N_{b}}-q_{1}^{2N_{b}}}{q_{2}^{2}-q_{1}^{2}}\right) \,q_{2}^{2N_{f}}\,, \\
B^{\star }B &=&N_{f}\,\,q_{1}^{2N_{b}},\end{aligned}$$ which can be deduced from Eqs. (4) and (6). Acting the supercharges on the Fock basis $\left| n_{b},n_{f}\right\rangle ,$ we have $$\begin{aligned}
Q\left| n_{b},0\right\rangle &=&0,\qquad \ \ \ \ \ \ Q^{\star
}\left|
n_{b},1\right\rangle =0, \\
Q\left| n_{b},1\right\rangle &=&q_{1}^{n_{b}}\sqrt{\left[ n_{b}+1\right] }%
\,\left| n_{b}+1,0\right\rangle , \\
Q^{\star }\left| n_{b},0\right\rangle
&=&q_{1}^{n_{b}-1}\sqrt{\left[ n_{b}\right] }\,\left|
n_{b}-1,1\right\rangle .\end{aligned}$$ The energy eigenvalues for the deformed Hamiltonian in Eq. (21) are $$\begin{aligned}
H\,\left| n_{b},1\right\rangle &=&\left(
\frac{q_{2}}{q_{1}}\right)
^{4}q_{1}^{2n_{b}}\left[ n_{b}+1\right] \,\left| n_{b},1\right\rangle , \\
H\,\left| n_{b},0\right\rangle &=&q_{1}^{2(n_{b}-1)}\left[
n_{b}\right] \,\left| n_{b},0\right\rangle .\end{aligned}$$
We now discuss a generic case for the Fock space representation of the two-parameter deformed superoscillator algebra generators $%
A_{i},\,A_{i}^{\star }$ and $B_{i},\,B_{i}^{\star }.$ We introduce the Fock basis $\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{n}%
_{f}\right\} \right\rangle $ as follows: $$\begin{aligned}
(N_{b})_{i}\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{n}%
_{f}\right\} \right\rangle &=&(n_{b})_{i}\,\left| \left\{ \widetilde{n}%
_{b}\right\} ,\left\{ \widetilde{n}_{f}\right\} \right\rangle
,\qquad
(n_{b})_{i}=0,1,2,..., \\
(N_{f})_{i}\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{n}%
_{f}\right\} \right\rangle &=&(n_{f})_{i}\,\left| \left\{ \widetilde{n}%
_{b}\right\} ,\left\{ \widetilde{n}_{f}\right\} \right\rangle
,\qquad
(n_{f})_{i}=0,1, \\
N_{i}\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{n}%
_{f}\right\} \right\rangle &=&(N_{b}+N_{f})_{i}\,\left| \left\{ \widetilde{n}%
_{b}\right\} ,\left\{ \widetilde{n}_{f}\right\} \right\rangle
=n_{i}\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{
\widetilde{n}_{f}\right\} \right\rangle ,\end{aligned}$$ where we have used the abbreviation for the Fock basis
$\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{n}%
_{f}\right\} \right\rangle =\left|
(n_{b})_{1},(n_{b})_{2},....,(n_{b})_{n};(%
\,n_{f})_{1},(n_{f})_{2},...,(n_{f})_{m}\right\rangle .$ The representations of the operators $A_{i},\,A_{i}^{\star
}$ are $$\begin{aligned}
A_{i}\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{n}%
_{f}\right\} \right\rangle
&=&q_{1}{}^{\sum_{k=1}^{i-1}(n_{b})_{k}}q_{2}{}^{%
\sum_{k=i+1}^{n}(n_{b})_{k}}q_{2}{}^{\sum_{k=1}^{m}(n_{f})_{k}}\sqrt{\left[
(n_b)_{i}\right] }\,\left| \left\{ \widetilde{n}_{b}\right\}
-1,\left\{
\widetilde{n}_{f}\right\} \right\rangle \\
A_{i}^{\star }\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{%
n}_{f}\right\} \right\rangle
&=&q_{1}{}^{\sum_{k=1}^{i-1}(n_{b})_{k}}q_{2}{}^{%
\sum_{k=i+1}^{n}(n_{b})_{k}}q_{2}{}^{\sum_{k=1}^{m}(n_{f})_{k}}\sqrt{\left[
(n_b)_{i}+1\right] }\,\left| \left\{ \widetilde{n}_{b}\right\}
+1,\left\{ \widetilde{n}_{f}\right\} \right\rangle ,\end{aligned}$$ where $$\begin{aligned}
A_{i}^{\star }A_{i}
&=&\,(q_{1}^{2})^{\sum_{k=1}^{i-1}(n_{b})_{k}}\,\left[
(n_b)_{i}\right] \,(q_{2}^{2})^{\sum_{k=i+1}^{n}(n_{b})_{k}}\,(q_{2}^{2})^{%
\sum_{k=1}^{m}(n_{f})_{k}},\phantom{AA} \\
\left| \left\{ \widetilde{n}_{b}\right\} \mp 1,\left\{ \widetilde{n}%
_{f}\right\} \right\rangle &=&\left|
(n_{b})_{1},...,(n_{b})_{i}\mp
1,...,(n_{b})_{n};(\,n_{f})_{1},...,(n_{f})_{m}\right\rangle ,
\nonumber\end{aligned}$$ and $\left[ (n_b)_{i}\right] $ is defined by Eq. (38).
For the fermionic sector, we have the following number operator for a generic case: $$B_{i}^{\star
}B_{i}=(n_{f})_{i}\,(q_{1}^{2})^{\sum_{k=1}^{n}(n_{b})_{k}}\,(q_{2}^{2})^{%
\sum_{k=i+1}^{m}(n_{f})_{k}}\,(-q_{1})^{\sum_{k=1}^{i-1}2(n_{f})_{k}},$$ which can be deduced from the defining relation of $B_{i}$ in section 2. Therefore, the representations of the operators $B_{i},\,B_{i}^{\star }$ are $$B_{i}\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{n}%
_{f}\right\} \right\rangle
=\left\{
\begin{array}{l}
0\qquad\qquad \qquad \qquad\qquad \qquad\qquad \qquad \qquad \mbox{if}\quad
(n_{f})_{i}=0, \\
q_{1}{}^{\sum_{k=1}^{n}(n_{b})_{k}}\,q_{2}{}^{\sum_{k=i+1}^{m}(n_{f})_{k}}%
\,(-q_{1}){}^{\sum_{k=1}^{i-1}(n_{f})_{k}}\,\left| \left\{ \widetilde{n}%
_{b}\right\} ,\left\{ \widetilde{n}_{f}\right\} -1\right\rangle \\
\phantom{0}\qquad\qquad \qquad\qquad\qquad \qquad\qquad \qquad
\qquad \mbox{if}\quad (n_{f})_{i}=1,
\end{array}
\right.$$
$$B_{i}^{\star }\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{%
n}_{f}\right\} \right\rangle =\left\{
\begin{array}{l}
0\qquad\qquad \qquad\qquad\qquad\qquad \qquad\qquad \qquad
\mbox{if}\quad
(n_{f})_{i}=1,\\
q_{1}{}^{\sum_{k=1}^{n}(n_{b})_{k}}\,q_{2}{}^{\sum_{k=i+1}^{m}(n_{f})_{k}}%
\,(-q_{1}){}^{\sum_{k=1}^{i-1}(n_{f})_{k}}\,\left| \left\{ \widetilde{n}%
_{b}\right\} ,\left\{ \widetilde{n}_{f}\right\} +1\right\rangle \\
\phantom{0}\qquad\qquad\qquad \qquad\qquad\qquad \qquad\qquad \qquad \mbox{if}
\quad(n_{f})_{i}=0, \\
\end{array}
\right.$$
where $\,\left| \left\{ \widetilde{n}_{b}\right\} ,\left\{ \widetilde{n}%
_{f}\right\} \mp 1\right\rangle =\left|
(n_{b})_{1},...,(n_{b})_{n};(\,n_{f})_{1},...,(n_{f})_{i}\mp
1,...,(n_{f})_{m}\right\rangle .$
Conclusions
===========
In this paper, we defined a two-parameter deformed superoscillator algebra with $SU_{q_{1}/q_{2}}(n\mid
m)$-covariance. By means of such generalized superoscillator system, we constructed a two-parameter deformed $%
N=2$ SUSY algebra covariant under the quantum supergroup $%
SU_{q_{1}/q_{2}}(n\mid m).$ We explicitly studied for the case of one $%
(q_{1},q_{2})$-deformed boson and one $(q_{1},q_{2})$-deformed fermion system with $SU_{q_{1}/q_{2}}(1\mid 1)$-covariance. For this system, we particularly discussed the Fock space properties and found the energy eigenvalues for the deformed Hamiltonian in terms of two deformation parameters. The two-parameter deformed $N=2$ SUSY algebra constructed here has some important limiting cases: The one-parameter deformed $N=2$ SUSY algebra[@v] can be recovered in the limit $q_{2}=1.$ The limit $q_{1}=q_{2}=q$ gives the $SU(n\mid m)$-covariant one-parameter deformed $N=2$ SUSY algebra constructed from the $q$-deformed bosonic and fermionic Newton oscillators[@u]. The conventional $N=2$ SUSY algebra in Eq. (1) can be obtained in the limit $q_{1}=q_{2}=1.$ This work was supported by Bogazici University Research Fund, project number $02$B$301$.
[99]{} M.Jimbo, Lett. Math. Phys. **11**, 247 (1986) V.G. Drinfeld, in Quantum groups Proc. of Int. Congress of Mathematicians (Berkeley, CA), 1987, edited by A M Gleason (Providence, RI: American Mathematical Society)p.798 L.D.Faddeev , N Yu Reshetikhin, L.A. Takhtajan, Algebr. Anal.**1**, 129 (1988) S.L. Woronowicz, Commun. Math. Phys. **111**, 613 (1987) Yu I. Manin, Commun.Math. Phys. **123**, 163 (1989)
J.Wess, B.Zumino, Nucl. Phys. B (Proc. Suppl.) **18**, 302 (1990) L.Alvarez-Gaume , C.Gomez , G.Sierra, Nucl. Phys. B **319**, 155 (1989) M.R. Ubriaco, Phys. Lett. A **163**, 1 (1992)
S. Shabanov, J. Phys. A: Math. Gen. **26**, 2583 (1993)
T. Kobayashi and T.Suzuki, J. Phys. A: Math. Gen. **26**, 6055 (1993)
T.T. Truong, J. Phys. A: Math. Gen. **27**, 3829 (1994)
A. Lorek, J.Wess, Z. Phys. C **67**, 671 (1995)
N.Aizawa, J. Phys. A: Math. Gen. **28**, 4553 (1995) L.C. Biedenharn, J.Phys. A:Math. Gen. **22**, L873 (1989) A.J. Macfarlane, J.Phys. A: Math. Gen. **22**, 4581 (1989) W. Pusz and S.L. Woronowicz, Rep. Math. Phys. **27**, 231 (1989)
T. Hayashi, Commun. Math. Phys. **127**, 129 (1990)
S. Jing, J.J. Xu, J. Phys. A: Math. Gen. **24**, L891 (1991) R. Chakrabarti , R.Jagannathan, J. Phys. A: Math. Gen. **24**, L711 (1991)
M. Arik, E. Demircan, T. Turgut , L. Ekinci , M. Mungan , Z. Phys. C **55**, 89 (1992)
A. Algin, M.Arik, A.S. Arikan, Eur. Phys. J. C **25**, 487 2002 N.Y. Reshetikhin, Lett. Math. Phys. **20**, 331 (1990)
D.B. Fairlie, C.K. Zachos, Phys. Lett. B **256**, 43 (1991)
A. Schirrmacher , J. Wess, B.Zumino, Z. Phys. C **49**, 317 (1991)
A. Schirrmacher, Z. Phys. C **50**, 321 (1991) M. Chaichian , P. Kulish , J. Lukierski, Phys. Lett. B **262**, 43 (1991) M. Chaichian, P. Kulish, Phys. Lett. B **234**, 72 (1990)
R. Floreanini , V. Spiridonov , L. Vinet, Phys. Lett. B **242**, 383 (1990) E.H. Tahri , El A. Hassouni, J. Phys. A: Math. Gen. **31**, 2065 (1998)
A. Hegazi , M. Mansour, Chaos, Solitons and Fractals **12**, 445 (2001)
P. Isaev, R. P. Malik, Phys. Lett. B **280**, 219 (1992) E. Witten, Nucl. Phys. B **185**, 513 (1981) R. Parthasarathy, K.S. Viswanathan, J. Phys. A: Math. Gen. **24**, 613 (1991) V. Spiridonov, Mod.Phys. Lett. A **7**, 1241 (1992) W.S. Chung, Prog.Theor. Phys. **94**, 649 (1995) A. Algin, Czech. J.Phys. **52**, 1011 (2002) W.S. Chung, J. Phys.A: Math. Gen. **32**, 2605 (1999) J.A. de Azcarraga , A.J. Macfarlane, J. Math. Phys. **37**, 1115 (1996)
R.S. Dunne , A.J. Macfarlane , J.A. de Azcarraga , J.C. Pérez Bueno, Int. J. Mod. Phys. A **12**, 3275 (1997) W.S. Chung, Prog. Theor. Phys. **95**, 697 (1996) S. Celik , S .A. Celik , M. Arik, Mod. Phys. Lett. A **13**, 1645 (1998) M. De Crombrugghe, V.Rittenberg, Ann. Phys. **151**, 99 (1983)
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abstract: 'Self-reported diagnosis statements have been widely employed in studying language related to mental health in social media. However, existing research has largely ignored the temporality of mental health diagnoses. In this work, we introduce [<span style="font-variant:small-caps;">RSDD</span>-Time]{}: a new dataset of 598 manually annotated self-reported depression diagnosis posts from Reddit that include temporal information about the diagnosis. Annotations include whether a mental health condition is present and how recently the diagnosis happened. Furthermore, we include exact temporal spans that relate to the date of diagnosis. This information is valuable for various computational methods to examine mental health through social media because one’s mental health state is not static. We also test several baseline classification and extraction approaches, which suggest that extracting temporal information from self-reported diagnosis statements is challenging.'
bibliography:
- 'references.bib'
title: '[RSDD-Time]{}: Temporal Annotation of Self-Reported Mental Health Diagnoses'
---
Conclusion
==========
In this paper, we explained the importance of temporal considerations when working with language related to mental health conditions. We introduced [<span style="font-variant:small-caps;">RSDD</span>-Time]{}, a novel dataset of manually annotated self-reported depression diagnosis posts from Reddit. Our dataset includes extensive temporal information about the diagnosis, including when the diagnosis occurred, whether the condition is still current, and exact temporal spans. Using [<span style="font-variant:small-caps;">RSDD</span>-Time]{}, we applied rule-based and machine learning methods to automatically extract these temporal cues and predict temporal aspects of a diagnosis. While encouraging, the experiments and dataset allow much room for further exploration.
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---
abstract: 'Data transmission over the millimeter wave (mmWave) in fifth-generation wireless networks aims to support very high speed wireless communications. A substantial increase in spectrum efficiency for mmWave transmission can be achieved by using advanced hybrid analog-digital precoding, for which accurate channel state information (CSI) is the key. Rather than estimating the entire channel matrix, it is now well-understood that directly estimating subspace information, which contains fewer parameters, does have enough information to design transceivers. However, the large channel use overhead and associated computational complexity in the existing channel subspace estimation techniques are major obstacles to deploy the subspace approach for channel estimation. In this paper, we propose a sequential two-stage subspace estimation method that can resolve the overhead issues and provide accurate subspace information. Utilizing a sequential method enables us to avoid manipulating the entire high-dimensional training signal, which greatly reduces the computational complexity. Specifically, in the first stage, the proposed method samples the columns of channel matrix to estimate its column subspace. Then, based on the obtained column subspace, it optimizes the training signals to estimate the row subspace. For a channel with $N_r$ receive antennas and $N_t$ transmit antennas, our analysis shows that the proposed technique only requires $\mathcal{O}(N_t)$ channel uses, while providing a guarantee of subspace estimation accuracy. By theoretical analysis, it is shown that the similarity between the estimated subspace and the true subspace is linearly related to the signal-to-noise ratio (SNR), i.e., $\mathcal{O}(\text{SNR})$, at high SNR, while quadratically related to the SNR, i.e., $\mathcal{O}(\text{SNR}^2)$, at low SNR. Simulation results show that the proposed sequential subspace method can provide improved subspace accuracy, normalized mean squared error, and spectrum efficiency over existing methods.'
author:
- 'Wei Zhang, , Taejoon Kim, , and Shu-Hung Leung [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'reference.bib'
title: A Sequential Subspace Method for Millimeter Wave MIMO Channel Estimation
---
[1.0]{}
Channel estimation, compressed sensing, millimeter wave communication, multi-input multi-output, subspace estimation.
Introduction {#sec:introduction}
============
Wireless communications using the millimeter wave (mmWave), which occupies the frequency band (30–300 GHz), address the current scarcity of wireless broadband spectrum and enable high speed transmission in fifth-generation (5G) wireless networks [@rappaport]. Due to the short wavelength, it is possible to employ large-scale antenna arrays with small-form-factor [@heath2016; @Torkildson2011; @Hur13]. To reduce power consumption and hardware complexity, the mmWave systems exploit hybrid analog-digital multiple-input multiple-output (MIMO) architecture operating with a limited number of radio frequency (RF) chains [@heath2016]. Under the perfect channel state information (CSI), [it has been shown that hybrid precoding can achieve nearly optimal performance as fully-digital precoding [@heath2016; @Torkildson2011; @spatially].]{} In practice, accurate CSI must be estimated via channel training in order to have effective precoding for robust mmWave MIMO transmission. However, extracting accurate CSI in the mmWave MIMO poses new challenges due to the limited number of RF chains that limits the observability of the channel and greatly increases the channel use overhead.
To reduce the channel use overhead, initial works focused on the beam alignment techniques [@Wang09; @BeamSteer] utilizing beam search codebooks. By exploiting the fact that mmWave propagation exhibits low-rank characteristic, recent researches formulated the channel estimation task as a [sparse signal reconstruction problem]{} [@OMPchannel; @SBR_channel] [and]{} [low-rank matrix reconstruction problem]{} [@ZhangSD; @AlternatingMin; @recht; @jointSparse; @zhangSP; @zhangSparse]. By using the knowledge of sparse signal reconstruction, orthogonal matching pursuit (OMP) [@OMPchannel] and sparse Bayesian learning (SBL) [@SBR_channel] were motivated to estimate the sparse mmWave channel in angular domain. Alternatively, if the channel is rank-sparse, it is possible to directly extract sufficient channel subspace information for the precoder design [@hadi2015; @ZhangSD; @AlternatingMin]. These subspace-based methods employ the Arnoldi iteration [@hadi2015] to estimate the channel subspaces and knowledge of matrix completion [@ZhangSD; @AlternatingMin] to estimate the low-rank mmWave channel information.
Though the sparse signal reconstruction [@OMPchannel; @SBR_channel] and matrix completion [@ZhangSD; @AlternatingMin] techniques can reduce the channel use overhead compared to traditional beam alignment techniques, the training sounders of these techniques [@OMPchannel; @SBR_channel; @ZhangSD; @AlternatingMin] are pre-designed and high-dimensional, which leads to the fact that these works suffer from explosive computational complexity as the size of arrays grows. To reduce the computational complexity, the adaptive training techniques have been investigated in [@Hur13; @hadi2015; @alk], where the training sounders can be adaptively designed based on the feedback or two-way training. But these adaptive training techniques could not guarantee the performance on mean squared error (MSE) and/or subspace estimation accuracy. [Moreover]{}, the techniques provided in [@Hur13; @hadi2015; @alk] will introduce additional channel use overhead due to the required feedback and two-way training.
To resolve the feedback overhead and maintain the benefit of adaptive training, in this paper, we present a two-stage subspace estimation approach, which sequentially estimates the column and row subspaces of the mmWave MIMO channel. Compared to the existing channel estimation techniques in [@OMPchannel; @SBR_channel; @ZhangSD; @AlternatingMin], the training sounders of the proposed approach are adaptively designed to reduce the channel use overhead and computational complexity. Moreover, the proposed approach is open-loop, thus it has no requirements of feedback and two-way channel sounding compared to priori adaptive training techniques [@Hur13; @hadi2015; @alk]. The main contributions of this paper are described as follows:
- We propose a two-stage subspace estimation technique called a sequential and adaptive subspace estimation (SASE) method. In the channel estimation of the proposed SASE, the column and row subspaces are estimated sequentially. Specifically, in the first stage, we sample a small fraction of columns of the channel matrix to obtain an estimate of the column subspace of the channel. In the second stage, the row subspace of the channel is estimated based on the obtained column subspace. In particular, by using the estimated column subspace obtained in the first stage, the receive training sounders of the second stage are optimized to reduce the number of channel uses. Compared to the existing works with fixed training sounders, where the entire high-dimensional training signals are utilized to obtain the CSI, the proposed adaptation has the advantage that the dimension of signals being processed in each stage is much less than that of the entire training signal, greatly reducing the computational complexity. Thus, the proposed SASE has much less computational complexity than those of the existing methods.
- We analyze the subspace estimation accuracy, which guarantees the performance of the proposed SASE technique. Through extensive analysis, it is shown that the subspace estimation accuracy of the SASE is linearly related to the signal-to-noise ratio (SNR), i.e., $\mathcal{O}(\text{SNR})$, at high SNR, and quadratically related to the SNR, i.e., $\cO(\text{SNR}^2)$, at low SNR. Moreover, simulation results show that the proposed SASE improves estimation accuracy over the prior arts.
- After obtaining the estimated column and row subspaces, an efficient method is developed for estimating the high-dimensional but low-rank channel matrix. Specifically, given the subspaces estimated by the proposed SASE, the mmWave channel estimation task can be simplified to solving a low-dimensional least squares problem, whose computation is much lower. Simulation results show that the proposed channel estimation method has lower normalized mean squared error and higher spectrum efficiency than those of the existing methods.
This paper is organized as follows, in Section II, we introduce the mmWave MIMO system model. In Section III, the proposed SASE is developed and analyzed. The channel use overhead, computational complexity, and an extension of the proposed SASE are discussed in Section IV. Finally, the simulation results and the conclusion remarks are provided in Sections V and VI, respectively.
*Notation*: Bold small and captial letters denote vectors and matrices, respectively. $\bA^T,\bA^H, \bA^{\!-1}$, $| \bA |$, $\| \bA \|_F$, $\tr(\bA)$, and $\|\ba\|_2$ are, respectively, the transpose, conjugate transpose, inverse, determinant, Frobenius norm, trace of $\bA$, and $l_2$-norm of $\ba$. $[\bA]_{:,i}$, $[\bA]_{i,:}$, and $[\bA]_{i,j}$ are, respectively, the $i$th column, $i$th row, and $i$th row $j$th column entry of $\bA$. $\vec(\bA)$ stacks the columns of $\bA$ and forms a column vector. $\diag(\ba)$ denotes a square diagonal matrix with vector $\ba$ as the main diagonal. $\sigma_L(\bA)$ denotes the $L$th largest singular value of matrix $\bA$. $\bI_M \! \in \! \R^{M\times M}$ is the identity matrix. [The $\mathbf{1}_{M,N} \! \in \! \R^{M\times N}$ , $\mathbf{0}_{M}\! \in \! \R^{M\times 1} , \mathbf{0}_{M,N} \! \in \! \R^{M\times N}$ are the all one matrix, zero vector, and zero matrix, respectively.]{} $\mathop{\mathrm{col}}(\bA)$ denotes the column subspace spanned by the columns of matrix $\bA$. The operator $(\cdot)_+$ denotes $\max\{0,\cdot \}$. The operator $\otimes$ denotes the Kronecker product.
MmWave MIMO System Model {#system section}
========================
![The mmWave MIMO channel sounding model[]{data-label="system"}](system_diagram.pdf){width="3.5in"}
Channel Sounding Model
----------------------
The mmWave MIMO channel sounding model is shown in Fig. \[system\], where the transmitter and receiver are equipped with $N_t$ and $N_r$ antennas, respectively. There are $N_{RF}\ge 2$ and $M_{RF} \ge 2$ RF chains at the transmitter and receiver, respectively. Without loss of generality, we assume $N_t$ is an integer multiple of $N_{RF}$, and $N_r$ is also an integer multiple of $M_{RF}$. In the considered mmWave channel sounding framework, one sounding symbol is transmitted over a unit time interval from the transmitter, which is defined as one channel use. It is assumed that the system employs $K$ channel uses for channel sounding. The received signal $\by_{(k)} \in \C^{M_{RF} \times 1}$ at the $k$th channel use is given by \_[(k)]{}= \_[(k)]{}\^H\_[(k)]{}+\_[(k)]{}\^H \_[(k)]{}, k=1,…,K, \[training\] where $ \bW_{(k)}\!\!= \!\! \bW_{A,k} \bW_{D,k} \in \C^{N_r \times M_{RF}}$ is the receive sounder composed of receive analog sounder $\bW_{A,k} \in \C^{N_r \times M_{RF}}$ and receive digital sounder $\bW_{D,k}\in \C^{M_{RF} \times M_{RF}}$ in series, $\bff_{(k)}\!\!= \!\! \bF_{A,k} \bF_{D,k} \bs_k \!\!\in\!\! \C^{N_t \times 1}$ is the transmit sounder composed of transmit analog sounder $\bF_{A,k} \in \C^{N_t \times N_{RF}}$ and transmit digital sounder $\bF_{D,k}\in \C^{N_{RF} \times N_{RF}}$ in series with transmitted sounding signal $\bs_k$, and $\bn_{(k)}\! \in \!\C^{{N_r} \times 1}$ is the noise.
Considering that the transmitted sounding signal $\bs_{k}$ is included in $\bff_{(k)}$, for convenience, we let $\bs_k \!\!= \!\!\frac{1}{\sqrt{N_{RF}}}[1, \ldots, 1]^T \in \R^{N_{RF}\times 1}$, which enables us to focus on the design of $\bF_{A,k}$ and $\bF_{D,k}$. It is worth noting that the analog sounders are constrained to be constant modulus, that is, $|[\bW_{A,k}]_{i,j}|={1}/{\sqrt{N_r}}$, and $|[\bF_{A,k}]_{i,j}|={1}/{\sqrt{N_t}}, \forall i,j$. Without loss of generality, we assume the power of the transmit sounder is one, that is, $\| \bff_{(k)}\|_2^2 = 1$. The noise $\bn_{(k)}$ is an independent zero mean complex Gaussian vector with covariance matrix $\sig^2 \bI_{N_r}$. Due to the unit power of transmit sounder, we define the signal-to-noise-ratio (SNR) as $1/{\sigma^2}$.[^2] The details of designing the receive and transmit sounders for facilitating the channel estimation will be discussed in Section III.
To model the point-to-point sparse mmWave MIMO channel, we assume there are $L$ clusters with $L \ll \min\{N_r, N_t \}$, and each constitutes a propagation path. The channel model can be expressed as [@RobertOver; @brady], = \_[l=1]{}\^[L]{} h\_l \_r(\_[r,l]{}) \_t\^H(\_[t,l]{}). \[channel model\] where $\ba_r(\theta_{r,l})\! \in \! \C^{N_r \! \times 1}\!$ and $\ba_t(\theta_{t,l}) \! \in \!\C^{N_t \! \times 1}\!$ are array response vectors of the uniform linear arrays (ULAs) at the receiver and transmitter, respectively. We extend it to the channel model with 2D uniform planar arrays (UPAs) in Section \[extension sec\]. In particular, $\ba_r(\theta_{r,l})$ and $\ba_t(\theta_{t,l})$ are expressed as &\_r(\_[r,l]{})=\[ 1, e\^[-jd \_[r,l]{}]{} , ,e\^[-jd(N\_r-1) \_[r,l]{}]{} \]\^T,\
&\_t(\_[t,l]{})=\[ 1, e\^[-jd \_[t,l]{}]{} , ,e\^[-jd(N\_t-1) \_[t,l]{}]{} \]\^T,where $\lambda$ is the wavelength, $d = 0.5\lambda$ is the antenna spacing, $\theta_{r,l}$ and $\theta_{t,l}$ are the angle of arrival (AoA) and angle of departure (AoD) of the $l$th path uniformly distributed in $[-\pi/2,\pi/2)$, respectively, and $h_l \sim \cC \cN(0,\sigma_{h,l}^2)$ is the complex gain of the $l$th path.
The channel model in can be rewritten as =\_r () \_t\^H, \[matrix expression\] where $\!\!\bA_r\!\!=\!\![\ba_r(\theta_{r,1} ),\ldots,\ba_r(\theta_{r,L})] \!\!\in\!\! \C^{N_r \times L}$, $\bA_t\!\!=\!\![\ba_t(\theta_{t,1} ),\ldots,\ba_t(\theta_{t,L})]\!\in\! \C^{N_t \times L}$
, and $\bh \!\!= \!\![h_1,\cdots,h_L]^T \!\!\in\!\! \C^{L \times 1}$. The channel estimation task is to obtain an estimate of $\bH$, i.e., $\widehat{\bH}$, from $\by_{(k)}$, $\bW_{(k)}$, and $\bff_{(k)}$, $k \!=\!\!1,\cdots,K$ in .
Performance Evaluation of Channel Estimation
--------------------------------------------
To evaluate the channel estimation performance, the achieved spectrum efficiency by utilizing the channel estimate $\widehat{\bH}$ is discussed in the following. Conventionally, the precoder $\widehat{\bF}\! \in \! \C^{N_t \times N_d}$ and combiner $\widehat{\bW}\! \in \! \C^{N_r \times N_d}$ are designed, based on the estimated $\widehat{\bH}$, where $N_d$ is the number of transmitted data streams with $N_d\! \le \! \min\{ N_{RF},M_{RF} \}$. Here, when evaluating the channel estimation performance, it is assumed the number of transmitted data streams is equal to the number of dominant paths, i.e., $N_d=L$. After the design of precoder and combiner, the received signal for the data transfer is given by = \^H + \^H, \[receiver form\] where the signal follows $\bs\sim\cC\cN(\mathbf{0}_L, \frac{1}{L}\bI_{L})$ and $\bn \sim \cC \cN(\mathbf{0}_{N_r},\sigma^2 \bI_{N_r})$. It is worth noting that is for data transmission, while is for channel sounding. The spectrum efficiency achieved by $\widehat{\bW}$ and $\widehat{\bF}$ in is defined in [@MIMO_capacity] as, R = \_2 |\_[L]{} + \_n\^[-1]{} \_e [\_e]{}\^H | , \[spectrum efficiency f\] where $\bH_e \= \widehat{\bW}^H \bH \widehat{\bF} \in \C^{{L} \times {L}} $ and $\bR_n \= \widehat{\bW}^H \widehat{\bW}\in \C^{{L} \times {L}} $. In this work, we assume that the precoder and combiner are unitary, such that $\widehat{\bW}^H\widehat{\bW}=\bI_{L}$ and $\widehat{\bF}^H\widehat{\bF}=\bI_{L}$. Under this assumption, we have $\bR_n = \bI_{L}$ in .
It is worth noting that the spectrum efficiency in is invariant to the right rotations of the precoder and combiner, i.e., substituting $\widetilde{\bF} = \widehat{\bF} \bR_\bF$ and $\widetilde{\bW} = \widehat{\bW}\bR_\bW$ into , where $\bR_\bF \in \C^{L \times L}$ and $\bR_\bW \in \C^{L \times L}$ are unitary matrices, does not change the spectrum efficiency. Thus, the $R$ in is a function of subspaces spanned by the precoder and combiner, i.e., $\mathop{\mathrm{col}}(\widehat{\bF})$ and $\mathop{\mathrm{col}}(\widehat{\bW})$. Moreover, the highest spectrum efficiency can be achieved when $\mathop{\mathrm{col}}(\widehat{\bF})$ and $\mathop{\mathrm{col}}(\widehat{\bW})$ respectively equal to the row and column subspaces of $\bH$.
Apart from the spectrum efficiency achieved by the signal model in , we consider the effective SNR at the receiver, = = . \[rec SNR\] The received SNR $\gamma$ in has the same rotation invariance property as the spectrum efficiency. In other words, the $\gamma$ in is a function of the estimated column and row subspaces. The maximum of the $\gamma$ is also achieved when $\widehat{\bW}$ and $\widehat{\bF}$ span the column and row subspaces of $\bH$, respectively.
![Illustration of SASE Algorithm[]{data-label="algorithm_diagram"}](algorithm_diagram.pdf){width="3.3in"}
Inspired by the definition in , in this paper, the accuracy of subspace estimation is defined as the ratio of the power captured by the transceiver matrices [@mmvSubspace] $\widehat{\bW}$ and $\widehat{\bF}$ to the power of the channel, ( ,) = . \[subspace metric\] Similarly, the measures for the accuracy of column subspace and row subspace estimation, i.e., $\eta_c(\widehat{\bW})$ and $\eta_r(\widehat{\bF})$, are respectively defined as the ratio of the power captured by $\widehat{\bW}$ and $\widehat{\bF}$ to the power of the channel in the following, \_c() &= &, \[def eta c\]\
\_r()& = &. \[def eta r\] Moreover, $\eta_c$ and $\eta_r$ are also rotation invariant. When the values of $\eta_c$ or $\eta_r$ are closed to one, the corresponding $\widehat{\bW}$ or $\widehat{\bF}$ can be treated accurate subspace estimates.
The illustration of the proposed SASE algorithm is shown in Fig. \[algorithm\_diagram\]. It consists of two stages: one is column subspace estimation and the other is row subspace estimation. In particular, the training sounders of the second stage can be optimized by fully adapting them to the estimated column subspace, which would reduce the number of channel uses and improve the estimation accuracy.
Sequential and Adaptive Subspace Estimation
===========================================
Estimate the Column Subspace {#Section_column}
----------------------------
In this subsection, we present the design of transmit and receive sounders along with the method for obtaining the column subspace of the mmWave channel. [To begin with, the following lemma shows that under the mmWave channel model in (3), the column subspaces of $\bH$ and sub-matrix $\bH_S$ are equivalent.]{}
\[lemma1\] Let $\bH_S = \bH\bS\in \C^{N_r \times m}$ be a sub-matrix that selects the first $m$ columns of $\bH$ with $m \ge L$, where $\bS $ is expressed as =
\_[m]{}\
\_[N\_t-m , m]{}
\^[N\_t m]{}. For the mmWave channel model in , if all the values of angles $\{\theta_{t,l} \}_{l=1}^L$ and $\{\theta_{r,l} \}_{l=1}^L$ are distinct, the column subspaces of $\bH$ and $\bH_S$ will be equivalent, i.e., $\mathop{\mathrm{col}}(\bH_S) = \mathop{\mathrm{col}}(\bH)$.
See Appendix \[appendix1\].
Because $\{\theta_{t,l} \}_{l=1}^L$ and $\{\theta_{r,l} \}_{l=1}^L$ are continuous random variables (r.v.s) in $[-\pi/2,\pi/2)$, hence, they are distinct almost surely (i.e., with probability $1$).
Lemma \[lemma1\] reveals that when $\mathop{\mathrm{col}}(\bH_S) = \mathop{\mathrm{col}}(\bH)$, to obtain the column subspace of $\bH$, it suffices to sample the first $m$ columns of $\bH$, i.e., $\bH_S$, which reduces the number of channel uses. However, the mmWave hybrid MIMO architecture can not directly access the entries of $\bH$ due to the analog array constraints. This can be overcome by adopting the technique proposed in [@hadi2015]. Specifically, to sample the $i$th column of $\bH$, i.e., $[\bH]_{:,i}$, the transmitter needs to construct the transmit sounder [$\bff_{(i)} = \be_i \in \C^{N_t \times 1}$]{}, where $\be_i$ is the $i$th column of $\bI_{N_t}$. [This is possible due to the fact that any precoder vector can be generated by $N_{RF} \ge 2$ RF chains [@xzhang]. To be more specific, there exists $\bF_{A,i}$, $\bF_{D,i}$, and $\bs_i$ such that $\be_{i}=\bF_{A,i} \bF_{D,i}\bs_i$, \_[i]{} = \_[\_[A,i]{} ]{} \_[\_[D,i]{}]{} \_[\_i]{}, where $\bF_{A,i} \!\!= \!\!\frac{1}{\sqrt{N_t}}\mathbf{1}_{N_t, N_{RF}}$ except for $[\bF_{A,i}]_{i,2}\!\!=\!\!-\frac{1}{\sqrt{N_t}}$, the $\bF_{D,i} \!\!=\!\! \mathbf{0}_{N_{RF}, N_{RF}}$ except for $[\bF_{D,i}]_{1,1}\!\!=\!\!\frac{\sqrt{N_{RF}N_t}}{2},[\bF_{D,i}]_{2,1}\!\!=\!\!-\frac{\sqrt{N_{RF}N_t}}{2}$, and $\bs_i \!\!=\!\! \frac{1}{\sqrt{N_{RF}}}[1, \ldots, 1]^T \in \R^{N_{RF}\times 1}$.]{}
At the receiver side, we collect the receive sounders of $N_r/M_{RF}$ channel uses to form the full-rank matrix, = \[\_[(i,1)]{}, \_[(i,2)]{},,\_[(i,N\_r/M\_[RF]{})]{}\] \^[N\_rN\_r]{}, \[Wij expression\] where $\bW_{(i,j)}\in \C^{N_r \times M_{RF}},~ j=1, \ldots ,N_r/M_{RF}$, denotes the $j$th receive sounder corresponding to transmit sounder $\be_i$. In order to satisfy the analog constraint where the entries in analog sounders should be constant modulus, we let the matrix $\bM$ in be the discrete Fourier transform (DFT) matrix. Specifically, the analog and digital receive sounders associated with $\bW_{(i,j)}$ in are expressed as follows \_[(i,j)]{} = \_ \_.
Thus, the received signal $ \by_{(i,j)} \! \in \! \C^{M_{RF} \times 1}$ under the transmit sounder $\be_i$ and receive sounder $\bW_{\!(i,j)}$ is expressed as follows \_[(i,j)]{} =\_[(i,j)]{}\^H \_i + \_[(i,j)]{}\^H\_[(i,j)]{} , where $\bn_{(i,j) }\in \C^{N_r \times 1}$ is the noise vector with $\bn_{(i,j)} \sim \cC \cN(\boldsymbol{0}_{N_r},\sigma^2\bI_{N_r})$. Then we stack the observations of $N_r/M_{RF}$ channel uses as $\by_{i} = [\by_{(i,1)}^T, \cdots,\by^T_{(i,N_r/M_{RF})} ]^T \in \C^{N_r \times 1}$, \_[\_i]{} &=& \_[\^H]{} \_[\_[:,i]{}]{} + \_[\_i]{}\
&=& \^H \[\]\_[:,i]{} + \_i \[H1 observation\], where $\tilde{\bn}_i \in \C^{N_r \times 1}$ is the effective noise vector after stacking, whose covariance matrix is expressed as, & =& \^2
\_[(i,1)]{}\^H\_[(i,1)]{} & & \_[(i,1)]{}\^H\_[(i,)]{}\
& &\
\_[(i,)]{}\^H\_[(i,1)]{} & & \_[(i,)]{}\^H\_[(i,)]{}
.\[cov effective n\] Because the DFT matrix $\bM$ in satisfies $\bM^H \bM = \bM \bM^H = \bI_{N_r}$, the following holds \_[(i,j)]{}\^H \_[(i,k)]{} = {
[rcl]{} \_[M\_[RF]{}]{} & & [j = k]{},\
\_[M\_[RF]{}]{} & & [j k]{}.
. \[property cov\] Substituting into , we can verify that $\E[\tilde{\bn}_i \tilde{\bn}_i^H] = \sigma^2 \bI_{N_r}$, and precisely, $\tilde{\bn}_i \sim \cC\cN(\mathbf{0}_{N_r},\sigma^2 \bI_{N_r})$. Moreover, by denoting $\widetilde{\bN} = [\tilde{\bn}_1,\cdots,\tilde{\bn}_m] \in \C^{N_r \times m}$, it is straightforward that the entries in $\widetilde{\bN}$ are independent, identically distributed (i.i.d.) as $\cC \cN(0, \sigma^2)$. Here, for convenience, we denote $ \widetilde{\bY}_S = [\by_{1},\cdots, \by_{m}] \in \C^{N_r \times m}$ where $\by_i$ is defined in . Then, we apply DFT to the collected observation $\widetilde{\bY}_S$, and obtain $\bY_S = \bM\widetilde{\bY}_S \in \C^{N_r \times m}$ as \_S =\_S + \_S, \[colun observation\] where $\bN_S \! =\! \bM \widetilde{\bN} \in \C^{N_r \times m}$ and $\bH_S\! =\! [\bH]_{:,1:m} \in \C^{N_r \times m}$. Before talking about the noise part $\bN_S$ in , the following lemma is a preliminary which gives the distribution of entries in the product of matrices.
\[lemma2\] Given a semi-unitary matrix $\bA\in \C^{d \times N}$ with $\bA \bA^H = \bI_{d}$, and a random matrix $\bX \in \C^{N \times m}$ with i.i.d. entries of $\cC\cN(0,\sigma^2)$, the product $\bY = \bA \bX \in \C^{d \times m}$ also has i.i.d. entries with distribution of $\cC\cN(0,\sigma^2)$.
See Appendix \[lemma2\_proof\].
Therefore, considering the noise part in , i.e., $\bN_S = \bM\widetilde{\bN}$, where $\bM$ is unitary and $\widetilde{\bN}$ has i.i.d. $\cC \cN(0, \sigma^2)$ entries, the conclusion of Lemma \[lemma2\] can be applied, which verifies that the entries of $\bN_S$ in are i.i.d. as $\cC \cN(0, \sigma^2)$.
Given the expression in , the column subspace estimation problem is formulated as, = \_[\^[N\_r L]{}]{} \^H \_S \_F\^2 \^H = \_L, \[optimal column subpace 1\] where one of the optimal solutions of can be obtained by taking the dominant $L$ left singular vectors of $\bY_S$. Here, the number of paths, $L$, is assumed to be known as a priori. In practice, it is possible to estimate $L$ by comparing the singular values of $\bY_S$ [@eigenValueEst]. Because $\bY_S = \bH_S+\bN_S$ and $\rank(\bH_S)=L$, there will be $L$ singular values of $\bY_S$ whose magnitudes clearly dominate the other singular values. Alternatively, we can set it to $L_\text{sup}$, which is an upper bound on the number of dominant paths such that $L\leq L_\text{sup}$.[^3]
Now, we design the receive combiner $\widehat{\bW}$ in for data transmission to approximate the estimated $\widehat{\bU} \in \C^{N_r \times L}$ in . Specifically, we design the analog combiner $\widehat{\bW}_A \in \C^{N_r \times M_{RF}}$ and digital combiner $\widehat{\bW}_D \in \C^{M_{RF} \times L}$ at the receiver by solving the following problem (\_A, \_D) = \_[\_A,\_D]{} -\_A\_D \_F,\
\_[i,j]{}= .\[receiver sounder\] The problem above can be solved by using the OMP algorithm [@spatially] or alternating minimization method [@Alternating_Min]. The designed receive combiner is given by $\widehat{\bW} = \widehat{\bW}_A \widehat{\bW}_D \in \C^{N_r \times L}$ with $\widehat{\bW}^H\widehat{\bW}=\bI_L$. The methods in [@spatially; @Alternating_Min] have shown to guarantee the near optimal performance, such as $\mathop{\mathrm{col}}(\widehat{\bW}) \approx \mathop{\mathrm{col}}(\widehat{\bU})$. The details of our column subspace estimation algorithm are summarized in Algorithm \[alg\_column\].
In general, $\mathop{\mathrm{col}}(\widehat{\bW})$ is not equal to the column subspace of $\bH$, i.e., $\mathop{\mathrm{col}}( \bU)$ with $\bU \in \C^{N_r \times L}$, due to the noise $\bN_S$ in . To analyze the column subspace accuracy $\eta_c(\widehat{\bW})$ defined in , we introduce the theorem [@cai2018] below.
\[theorem1\] Suppose $\bX \in \C^{M \times N} (M \ge N)$ is of rank-$r$, and $\widehat{\bX} = \bX + \bN$, where $[\bN]_{i,j}$ is i.i.d. with zero mean and unit variance (not necessarily Gaussian). Let the compact SVD of $\bX$ be = \^H, where $\bU \in \C^{M \times r}$, $\bV \in \C^{N \times r}$, and $\bSig \in \C^{r \times r}$. We assume the singular values in $\bSig$ are in descending order, i,e, $\sig_1(\bX) \geq \cdots \geq \sig_r(\bX)$. Similarly, we partition the SVD of $\widehat{\bX}$ as =
&\_
\_1 &\
& \_2
\^H\
\_\^H
, where $\widehat{\bU} \in \C^{M \times r}$, $\widehat{\bU}_{\perp} \in \C^{M \times (M-r)}$, $\widehat{\bV} \in \C^{N \times r}$, $\widehat{\bV}_{\perp} \in \C^{N \times (N-r)}$, $\widehat{\bSig}_1 \in \C^{r \times r}$, and $\widehat{\bSig}_2 \in \C^{(M-r) \times (N-r)}$. Then, there exists a constant $C > 0$ such that ( 1- )\_[+]{},\
(1- )\_[+]{}, where the expectation is taken over the random noise $\bN$. In particular, when the noise is i.i.d. $\cC\cN(0,1)$, it has $C=2$.
\[t\]
Input: channel dimension: $N_r$, $N_t$; number of RF chains at receiver: $M_{RF}$; channel paths: $L$; parameter: $m$. Initialization: channel use index $k=1$. Set transmit sounder as $\bff_{(i)} = \be_{i}$.
Design receive training sounder as $\bW_{(i,j)} = [\bM]_{:, (j-1)M_{RF}+1:jM_{RF}} \bI_{M_{RF}}$. Obtain the received signal $\by_{(i,j)}\!\!\!\!=\!\!\!\!\bW_{(i,j)} ^H \bH \bff_{(i)} \!\!+ \!\!\bW_{(i,j)} ^H \bn_{(i,j)}$. Update $k=k+1$. $\by_{i} = \left[\by_{(i,1)}^T, \cdots,\by^T_{(i,N_r/M_{RF})} \right]^T$. $\bY_S = \bM \left[\by_{1},\cdots, \by_{m}\right]$. Column subspace $\widehat{\bU}$ is obtained by the dominant $L$ left singular vectors of $\bY_S$. Design $\widehat{\bW}$ based on $\widehat{\bU}$ by solving . Output: Column subspace estimation $\widehat{\bW}$.
We have the following proposition for the accuracy of the column subspace estimation in Algorithm \[alg\_column\].
\[prop1\] If the Euclidean distance $\| \widehat{\bW} - \widehat{\bU}\|_F \le \delta_1$ in , then the accuracy of the estimated column subspace matrix $\widehat{\bW}$ obtained from Algorithm \[alg\_column\] is lower bounded as \_L(\^H ) -\_1, \[mid eq1\] where $\bU \in \C^{N_r \times L}$ is the matrix composed of $L$ dominant left singular vectors of $\bH$. In particular, if $\delta_1 \rightarrow 0$, we have && \_L\^2(\^H )\
&& ( 1- )\_+, \[column subspace ac\] where the $\sig_L(\bH_S)$ is the $L$th largest singular value of $\bH_S$.
See Appendix \[appendix2\]. Based on the definition of $\eta_c(\widehat{\bW})$ in , it has &=&\
&=&\
& & -\
&= & -\
& & \_L(\^H ) - -\_2,\
& & \_L(\^H ) -\_1, \[mid eq1 1\] where the inequality $(a)$ holds from the triangle inequality, and the inequality $(b)$ comes from the fact that for $\bA\in \C^{n \times n}$ with $\rank(\bA)=n$ and $\bB\in \C^{n \times k}$, $\lA \bA \bB\rA_F^2 \geq \sig_n^2(\bA) \| \bB \|_F^2$, where the latter follows by $\| \bA \bB\|_F^2 = \sum_{i=1}^{k} \lA \bA [\bB]_{:,i} \rA_2^2 \ge \sum_{i=1}^{k} \sigma_{n}^2(\bA) \lA [\bB]_{:,i} \rA_2^2 =\sigma_{n}^2(\bA) \lA \bB \rA_F^2$. Thus, this concludes the proof for the inequality in .
Then, by letting $\delta_1 \rightarrow 0$ in , we take expectation of squares of both sides in , then it has the following &&\
& &( 1- )\_+. \[final col ac\] where the inequality $(c)$ holds from Theorem \[theorem1\], and this concludes the proof.
From , the larger the value of $m$ is, the more accurate the column subspace estimation. Thus, when more columns are used for the column subspace estimation, the estimated column subspace will be more reliable. In particular, when the noise level is low such that $\sig_L^2(\bH_S)\!\! \gg \! m\sig^2 $ in , we have ( 1- )\_+. It means that the column subspace estimation accuracy is linearly related to the value of $\sigma^2\!\! / \! \sig_L^2(\bH_S)$, i.e., $\cO(\text{SNR})$. On the other hand, when the noise level is high such that $\sig_L^2(\bH_S) \! \ll\! m\sig^2 $, the bound in can be written as ( 1- )\_+. At low SNR, the column subspace estimation accuracy is quadratically related to $\sigma^4 /\sig_L^4(\bH_S)$, i.e., $\cO(\text{SNR}^2)$.
When the number of paths, $L$, increases, the value of $\sigma_L(\bH_S)$ in will decrease, which can be interpreted as follows. When $m, N_r \! \rightarrow \! \infty$, the entries in $\bH_S \! \in \! \C^{N_r \times m}$ can be generally approximated as standard Gaussian r.v.s [@probability2010]. Moreover, it has been shown in [@wei2017upper; @SVDbound] that the $L$th largest singular value of $\sigma_L (\bH_S ) \! \propto \! \frac{N_r+1-L}{\sqrt{N_r}}$ with high probability. As a result, the accuracy of column subspace estimation will be decreased as $ L $ increases due to of Proposition \[prop1\].
Estimate the Row Subspace {#section row}
-------------------------
In this subsection, we present how to learn the row subspace by leveraging the estimated column subspace matrix $\widehat{\bW}$. Because we have already sampled the first $m$ columns of $\bH$ in the first stage, we only need to sample the remaining $N_t-m$ columns to estimate the row subspace as shown in Fig. \[algorithm\_diagram\].
At the $k$th channel use of the second stage, we observe the $(m+k)$th column of $\bH$, $k=1,\ldots,(N_t-m)$. To achieve this, we employ the transmit sounder as \_[(k)]{}=\_[m+k]{} \[s2 tran sounder\]. For the receive sounder, given the estimated column subspace matrix $\widehat{\bW}$ in the first stage, we just let the receive sounder of the second stage be $\widehat{\bW} \in \C^{N_r \times L}$.[^4] It is worth noting $\widehat{\bW}$ is trivially applicable for hybrid precoding architecture since $\widehat{\bW}$ is obtained from . Therefore, under the transmit sounder $\bff_{(k)}$ in and receive sounder $\widehat{\bW}$ in , the observation $\by_{(k)} \in \C^{L \times 1}$ at the receiver can be given by \_[(k)]{}&=& \^H\_[(k)]{} +\^H\_[(k)]{}\
&=& \^H\[\]\_[:,m+k]{}+ \^H\_[(k)]{}, \[second sample\] where $\bn_{(k) }\in \C^{N_r \times 1}$ is the noise vector with $\bn_{(k)} \sim \cC \cN(\boldsymbol{0}_{N_r},\sigma^2\bI_{N_r})$. Then, the observations $k=1, \ldots,(N_t-m)$ in are packed into a matrix [$\widehat{\bQ}_C \in \C^{L \times (N_t-m)}$]{} as \_C &=& \[\_[(1)]{},\_[(2)]{}, , \_[(N\_t-m)]{}\]\
&=& \^H(\_C + \_C), \[Qc expression\] where $\bH_C = \left[[\bH]_{:,m+1},\ldots, [\bH]_{:,N_t}\right] \in \C^{N_r \times (N_t-m)}$, and $\bN_C = [\bn_{(1)}, \ldots, \bn_{(N_t-m)}] \in \C^{N_r \times (N_t-m)}$.
\[t\]
Input: channel dimension: $N_r$, $N_t$; channel paths: $L$; estimated column subspace: $\widehat{\bW}$; observations of first stage: $\bY_S$; parameter: $m$. Set the receive training sounder as $\widehat{\bW}$. Set the transmit training sounder as $\bff_{(k)}= \be_{m+k}$.
Obtain the received signal: $\by_{(k)}=\widehat{\bW}^H \bH \bff_{(k)} + \widehat{\bW}^H \bn_{(k)}$. Stack all the observations and : $\widehat{\bQ}_C = [\by_{(1)},\by_{(2)}, \cdots, \by_{(N_t-m)}] $. Calculate $\widehat{\bQ}$: $\widehat{\bQ}=\left [\widehat{\bW}^H\bY_S,\widehat{\bQ}_C\right]$. Row subspace matrix $\widehat{\bV}$ is obtained by the dominant $L$ right singular vectors of $\widehat{\bQ}$. Design $\widehat{\bF}$ based on $\widehat{\bV}$ by solving . Output: row subspace estimation $\widehat{\bF}$.
In addition, given the receive sounder $\widehat{\bW}$ and observations $\bY_S$ of the first stage in , we define $\widehat{\bQ}_S \in \C^{L \times m}$ as, \_S = \^H \_S = \^H (\_S + \_S). \[coe 1\] Combining and yields [$\widehat{\bQ}\in \C^{L \times N_t}$]{} expressed as, &=&\
&=&\
&=& \_[|]{} +\_[|]{}, \[Q exp\] where $\bN=[\bN_S,\bN_C] \! \!\in \C^{N_r \times N_t}$, $\bH=[\bH_S,\bH_C] \in \C^{N_r \times N_t}$, $\bar{\bQ} = \widehat{\bW}^H \bH \in \C^{L \times N_t}$, and $\bar{\bN} = \widehat{\bW}^H \bN \in \C^{N_r \times N_t}$. Meanwhile, since $\widehat{\bW}$ is semi-unitary and the entries in ${\bN}$ are i.i.d. with distribution $\cC \cN(0,\sigma^2)$, according to Lemma \[lemma2\], the entries in $\bar{\bN}$ are also i.i.d. with distribution $\cC \cN(0,\sigma^2)$.
\[N bar iid\] If we denote the $i$th column of $\bar{\bN}$ in as $\bar{\bn}_i$, then the covariance matrix of $\bar{\bn}_i \in \C^{L \times 1}$ is && \^2 \^H\
&&\^2 \_[L]{}. where the equality (a) holds because the entries in $\bN$ are i.i.d Gaussian r.v.s with distribution $\cC \cN(0,\sigma^2)$, and the equality (b) holds because $\widehat{\bW}$ is semi-unitary. Therefore, the elements in $ \bar{ \bN}$ are i.i.d. with each entry being $\cC \cN(0,\sigma^2)$.
Now, given the expression $\widehat{\bQ}$ in , the row subspace estimation problem is formulated as, = \_[\^[N\_t L]{}]{} \_F\^2 \^H = \_L, where the estimated row subspace matrix $\widehat{\bV} \in \C^{N_t \times L}$ is obtained as the dominant $L$ right singular vectors of $\widehat{\bQ}$. Similarly, in order to design the precoder $\widehat{\bF}$ in for data transmission, we need to approximate the estimated row subspace matrix $\widehat{\bV}$ under the hybrid precoding architecture. Specifically, we design the analog precoder $\widehat{\bF}_A \in \C^{N_t \times N_{RF}}$and digital precoder $\widehat{\bF}_D \in \C^{N_{RF} \times L}$ by solving the following problem (\_A, \_D) = \_[\_A,\_D]{} -\_A\_D \_F,\
\_[i,j]{}= .\[transmit precoder design\] Therefore, the transmit precoder is given by $\widehat{\bF} = \widehat{\bF}_A \widehat{\bF}_D \in \C^{N_t \times L}$ with $\widehat{\bF}^H\widehat{\bF}=\bI_L$. Similarly, the method on solving in [@spatially] can guarantee $\text{col}(\widehat{\bF}) \approx \text{col}(\widehat{\bV})$. The details of our row subspace estimation algorithm are shown in Algorithm \[alg\_row\]. We have the following proposition about the estimated row subspace accuracy for Algorithm \[alg\_row\].
\[lemma row\] If the Euclidean distance $\| \widehat{\bF} - \widehat{\bV}\|_F \le \delta_2$ in , then the accuracy of the estimated row subspace matrix $\widehat{\bF}$ obtained from Algorithm \[alg\_row\] is lower bounded as \_L(\^H ) -\_2, \[lemma row mid1\] where $\bV \in \C^{N_t \times L}$ is the matrix composed of the $L$ dominant right singular vectors of $\bH$. In particular, if $\delta_2 \rightarrow 0$, we have && \_L\^2(\^H )\
&& ( 1- )\_+, \[coefficient matrix\] where $\sig_L(\bar{\bQ})$ is the $L$th largest singular value of $\bar{\bQ}$ in .
See Appendix \[appendix3\]. Recall that the row subspace matrix $\widehat{\bV}$ is given by the right singular matrix of $\widehat{\bQ}=\bar{\bQ} + \bar{\bN}$ in , and the elements in $ \bar{ \bN}$ are i.i.d. with each entry being $\cC \cN(0,\sigma^2)$. Thus, by using the conclusion of Theorem \[theorem1\], we have ( 1- )\_+. \[row 2\] Then, based on the subspace accuracy metric in , it has &=&\
&=&\
& & -\
&= & -\
& & \_L(\^H ) - -\_2,\
& & \_L(\^H ) -\_2. \[lemma row mid1 1\] Thus, the inequality is proved. Moreover, under the condition $\delta_2 \! \rightarrow \! 0$, taking expectation of squares of both sides in yields &&\
&& ( 1- )\_+, where the inequality $(c)$ holds from . This concludes the proof for the row estimation accuracy bound in .
Similar as the column subspace estimation, the row subspace accuracy linearly increases with the SNR, i.e., $\mathcal{O}(\text{SNR})$ at high SNR, and quadratically increases with SNR, i.e., $\cO(\text{SNR}^2)$, at low SNR. Also, the accuracy of row subspace estimation decreases with the number of paths, $L$. By comparing the row subspace accuracy in with the column subspace accuracy in , apart from the dimension parameters, the difference is the $\sigma_L(\bar{\bQ})$ in rather than $\sigma_L(\bH_S)$ in . As the value of $\sig_L(\bar{\bQ})$ in grows, we can have a more accurate row subspace estimation. Moreover, considering $\bar{\bQ} =\widehat{\bW}^H \bH$, it is intuitive that the estimated column subspace matrix $\widehat{\bW}$ will affect the value of $\sig_L(\bar{\bQ})$, and then affect the accuracy of row subspace estimation. Specifically, when $\mathop{\mathrm{col}}(\widehat{\bW}) = \mathop{\mathrm{col}}({\bU}) $, we will have $\sig_L(\bar \bQ) = \sig_L(\bH)$, which attains the maximum. In the following, we further discuss the relationship between $\sig_L(\bar{\bQ})$ and $\sig_L(\bH)$.
With the SVD of $\bH$, i.e., $\bH = \bU\bSig\bV^H$, we have $\bar{\bQ} = \widehat{\bW}^H\bH = \widehat{\bW}^H \bU \bSig \bV^H$. Then, the following relationship is true due to the singular value product inequality, \_L(| ) && \_L( \^H ) \_L(\^H )\
&=& \_L(\^H ) \_L() . \[column affect\] Therefore, $\sigma_L(\bar{\bQ} )$ is lower bounded by the product of the $L$th largest singular values of $\widehat{\bW}^H \bU $ and $\bH $. When the estimation of the column subspace becomes accurate, the $\sigma_L(\widehat{\bW}^H \bU )$ will approach to one. As a result, the value of $\sigma_L(\bar{\bQ} ) $ is approximately equal to $\sigma_L(\bH )$, resulting in a further enhanced row subspace estimation. The inequality in reveals that the column subspace estimation affects the accuracy of the row subspace estimation.
Given the estimated column subspace $\widehat{\bW}$ in Algorithm \[alg\_column\] and row subspace $\widehat{\bF}$ in Algorithm \[alg\_row\], the following lemma shows the subspace estimation accuracy of the proposed SASE, i.e., $\eta(\widehat{\bW},\widehat{\bF})$ defined in .
\[subspace proof\] If we assume $\delta_1 \rightarrow 0 $ and $\delta_2 \rightarrow 0$ in and , the subspace estimation accuracy defined in associated with $\widehat{\bW}$ and $\widehat{\bF}$ is lower bounded as (,)\_L\^2(\^H )\_L\^2(\^H ).
Using the definition of $\eta(\widehat{\bW},\widehat{\bV})$ in , we have the following expressions, (,) &= & [ \^H \_F\^2]{}/[(\^H )]{}\
& & [ \^H \_F\^2]{}/[(\^H )]{}\
&= & [ \^H \^H \_F\^2]{}/[(\^H )]{}\
& & \_L\^2(\^H )\_L\^2(\^H ),where the equality $(a)$ holds for $\delta_1 \rightarrow 0 $ and $\delta_2 \rightarrow 0$, and the inequality $(b)$ holds based on the singular value product inequality.
Lemma \[subspace proof\] tells that the power captured by $\widehat{\bW}$ and $\widehat{\bF}$ is lower bounded by the product of $\sigma_L^2(\widehat{\bU}^H \bU)$ and $\sigma_L^2(\widehat{\bV}^H \bV)$. These two parts denotes the two stages in the proposed SASE, which are column subspace estimation and row subspace estimation, respectively. Ideally, when $\text{col}(\widehat{\bU})=\text{col}({\bU})$ and $\text{col}(\widehat{\bV})=\text{col}({\bV})$, we have $\eta(\widehat{\bW},\widehat{\bF})=1$. Nevertheless, the proposed SASE can still achieve nearly optimal $\eta(\widehat{\bW},\widehat{\bF})$. This is because $\sigma_L^2(\widehat{\bU}^H \bU)$ and $\sigma_L^2(\widehat{\bV}^H \bV)$ are close to one according to the bounds provided in and , respectively.
Channel Estimation Based on the Estimated Subspaces {#SASE channel result}
---------------------------------------------------
In this subsection, we introduce a channel estimation method based on the estimated column subspace $\widehat{\bW} \in \C^{N_r \times L}$ and row subspace $\widehat{\bF} \in \C^{N_t \times L}$. Let the channel estimate be expressed as = \^H, \[rep of est\] where $\widehat{\bR} \in \C^{L \times L}$. Now, given $\widehat{\bW}$ and $\widehat{\bF}$, it only needs to obtain $\widehat{\bR}$ in an optimal manner.
Recalling the column subspace estimation in Section \[Section\_column\] and row subspace estimation in Section \[section row\], the corresponding received signals are expressed as \_S &=&\_S + \_S\
\_C &=& \^H\_C + \^H \_C. It is worth noting that the entries in $\bN_S$ and $\widehat{\bW}^H \bN_C$ are both i.i.d with distribution $\cC\cN(0,\!\sigma^2)$. Based on the expression of $\widehat{\bH}$ in , the maximum likelihood estimation of $\widehat{\bR}$ in can be obtained through the following least squares problem, && \_[\^[L L]{}]{} \_S - \_S \_F\^2+ \_C-\^H \_C\_F\^2\
&& \_S = \[\^H\]\_[:,1:m]{}, \_C = \[\^H\]\_[:,m+1:N\_t]{}. \[R estimation\] Before discussing how to solve the problem in , for convenience, we define &=&()\^[L\^2 1]{},\
\_S &=&(\_S)\^[m N\_r 1]{} ,\
\_C &=&(\_C)\^[(N\_t-m)L 1]{},\
\_1 &=&(\[\]\_[:,1:m]{}\^H)\^T \^[m N\_r L\^2]{},\
\_2 &=&(\[\]\_[:,m+1:N\_t]{}\^H)\^T \_L \^[(N\_t-m)L L\^2]{}. Using the definitions above, the minimization problem in can be rewritten as \_[\^[L\^2 1]{}]{} \_S - \_1 \_2\^2 + \_C - \_2 \_2\^2. \[form r\] The following lemma provides the solution of problem .
\[lemma r\] Given the problem below \_[\^[L\^2 1]{}]{} \_S -\_1 \_2\^2 + \_C-\_2 \_2\^2, the optimal solution is given by = (\_1\^H \_1+\_2\^H \_2)\^[-1]{}(\_1\^H\_S+\_2\^H\_C). \[sol r\]
The problem is convex with respect to $\br$. Thus, the optimal solution can be obtained by setting the first order derivative of the objective function to zero as \_1\^H(\_1-\_S) + \_2\^H(\_2- \_C) =. \[r equation\] The solution of is exactly the result in , which concludes the proof.
It is worth noting that after we have obtained the column and row subspace estimates, i.e., $\widehat{\bW}$ and $\widehat{\bF}$, the channel estimation is simply to compute $\widehat{\br}=\vec(\widehat{\bR})$ in . Since the dimension of $\widehat{\bR}$ is much lower than that of $\bH$, the channel estimation complexity is substantially reduced as shown in Lemma \[lemma r\].
Discussion of Algorithm
=======================
In this section, we analyze the complexity of the proposed SASE method in terms of the channel use overhead and computational complexity. Moreover, we discuss the application of the SASE in other channel scenarios.
Channel Use Overhead
--------------------
Algorithms Number of Channel Uses
---------------------- ------------------------------------
SASE ${m N_r}/{M_{RF}} + (N_t-m)$
MF [@AlternatingMin] $\mathcal{O}(L(N_r+N_t)/{M_{RF}})$
SD [@ZhangSD] $\mathcal{O}(L(N_r+N_t)/{M_{RF}})$
Arnoldi [@hadi2015] $2qN_r/{M_{RF}}+2qN_t/{N_{RF}}$
OMP [@OMPchannel] $\mathcal{O}(L\ln (G^2)/{M_{RF}})$
SBL [@SBR_channel] $\mathcal{O}(L\ln (G^2)/{M_{RF}})$
ACE [@alk] $s^2L^3\log_s (N_m/L)/{M_{RF}}$
: Channel Uses of Algorithms
\[table use\]
Considering the channel uses in each stage, the total number of channel uses for the SASE is given by K\_ &=& [m N\_r]{}/[M\_[RF]{}]{} + (N\_t-m). \[channel uses\] Therefore, the number of channel uses grows linearly with the channel dimension, i.e., $\mathcal{O}(N_t)$. In particular, when we let $m=L$, the number of channel uses in will be ${L N_r}/{M_{RF}} + (N_t-L)$. Considering that each channel use contributes to $M_{RF}$ observations in the first stage, and $L$ observations in the second stage, the total number of the observations is $LN_r + L(N_t-L)$, which is equivalent to the degrees of freedom of $\rank$-$L$ matrix $\bH \in \C^{N_r \times N_t}$ [@matrixCom].
The numbers of channel uses of the proposed SASE and other benchmarks [@OMPchannel; @SBR_channel; @alk; @ZhangSD; @AlternatingMin; @hadi2015] are compared in Table \[table use\]. For the angle estimation methods in [@OMPchannel; @SBR_channel; @alk], the number of required channel uses for the OMP [@OMPchannel] and SBL [@SBR_channel] is $K_\text{OMP}=K_\text{SBL} = \mathcal{O}(L\ln (G^2)/{M_{RF}})$, where $G$ is the number of grids with $G \ge \max\{N_r,N_t \}$. The number of channel uses for adaptive channel estimation (ACE) [@alk] is $K_\text{ACE}=s^2L^3\log_s (N_m/L)/{M_{RF}}$, where $2\pi/N_m$ with $N_m \ge \max\{N_r,N_t \}$ is the desired angle resolution for the ACE, and $s$ is the number of beamforming vectors in each stage of the ACE. For the subspace estimation methods in [@ZhangSD; @AlternatingMin; @hadi2015], the numbers of required channel uses for subspace decomposition (SD) [@ZhangSD] and matrix factorization (MF) [@AlternatingMin] are $K_\text{SD}=K_\text{MF}=\mathcal{O}(L(N_r+N_t)/{M_{RF}})$, while it requires $K_\text{Arnoldi} =2qN_r/{M_{RF}}+2qN_t/{N_{RF}}$ channel uses where $q\ge L$ for Arnoldi approach [@hadi2015]. Because the number of estimated parameters of the angle estimation methods such as OMP, SBL, and ACE, is less than that of the proposed SASE, they require slightly fewer channel uses than SASE. Nevertheless, the proposed SASE consumes fewer channel uses than those of the existing subspace estimation methods [@ZhangSD; @AlternatingMin; @hadi2015] as shown in Table \[table use\].
Computational Complexity
------------------------
For the proposed SASE, the computational complexity of the first stage comes from the SVD of ${\bY}_S$, which is $\mathcal{O}(m^2 N_r )$ \[28\]. The complexity of the second stage is dominated by the design of $\widehat{\bW}$ in , which is $\mathcal{O}( L D N_r )$, where $D\ge N_r$ denotes the cardinality of an over-complete dictionary. Hence, the overall complexity of the proposed SASE algorithm is $\mathcal{O}(m^2 N_r + L D N_r ) = \mathcal{O}(L D N_r )$. The computational complexities of benchmarks, i.e., the angle estimation methods OMP [@OMPchannel], SBL [@SBR_channel], and ACE [@alk] along with the subspace estimation methods Arnoldi [@hadi2015], SD [@ZhangSD], and MF [@AlternatingMin] are compared in Table II, where $K$ denotes the number of channel uses. For a fair comparison, when comparing the computational complexity, we assume the number of channel uses, $K$, is equal among the benchmarks. As we can see from Table \[table com\], the proposed SASE has the lowest computational complexity.
Algorithms Computational Complexity
---------------------- ---------------------------------------------------------------
SASE $\mathcal{O}(L D N_r )$
MF [@AlternatingMin] $\mathcal{O}(K M_{RF} L^2(N_r^2+N_t^2))$
SD [@ZhangSD] $\mathcal{O}(K M_{RF} L^2(N_r^2+N_t^2))$
Arnoldi [@hadi2015] $\mathcal{O}(K^2M_{RF}^2/(N_r+N_t))$
OMP [@OMPchannel] $\mathcal{O}(L KM_{RF}G^2)$
SBL [@SBR_channel] $\mathcal{O}(G^6)$
ACE [@alk] $\mathcal{O} (K M_{RF}^2 D N_r/(sL) +K N_{RF}^2 D N_t/(sL) )$
: Computational Complexity of Algorithms
\[table com\]
Extension of SASE {#extension sec}
-----------------
In this subsection, we extend the proposed SASE to the 2D mmWave channel model with UPAs. There are $N_{cl}$ clusters, and each of cluster is composed of $N_{ray}$ rays. For this model, the mmWave channel matrix is expressed as [@BaiChannel; @IntroductionM; @spatially] = \_[i=1]{}\^[N\_[cl]{}]{} \_[j=1]{}\^[N\_[ray]{}]{} h\_[ij]{} \_r(\^r\_[ij]{}, \^r\_[ij]{}) \_t\^H(\^t\_[ij]{},\^t\_[ij]{}), \[channel model SC\] where $h_{ij}$ represents the complex gain associated with the $j$th path of the $i$th cluster. The $\ba_r(\phi^r_{ij}, \theta^r_{ij}) \in \C^{N_r \times 1}$ and $ \ba_t(\phi^t_{ij},\theta^t_{ij}) \in \C^{N_t \times 1}$ are the receive and transmit array response vectors, where $\phi^r_{ij} (\phi^t_{ij})$ and $ \theta^r_{ij}(\theta^t_{ij})$ denote the azimuth and elevation angles of the receiver (transmitter). Specifically, the $\ba_r(\phi^r_{ij}, \theta^r_{ij}) $ and $ \ba_t(\phi^t_{ij},\theta^t_{ij}) $ are expressed as \_r(\^r\_[ij]{}, \^r\_[ij]{}) = \[1,, e\^[jd(m\_r \^r\_[ij]{} \^r\_[ij]{} + n\_r \^r\_[ij]{} )]{} ,\
,e\^[jd((-1)\^r\_[ij]{} \^r\_[ij]{} + (-1) \^r\_[ij]{} )]{}\],\
\_t(\^t\_[ij]{}, \^t\_[ij]{}) = \[1,, e\^[jd(m\_t \^t\_[ij]{} \^t\_[ij]{} + n\_t \^t\_[ij]{} )]{},\
,e\^[jd((-1)\^t\_[ij]{} \^t\_[ij]{} + (-1) \^t\_[ij]{} )]{}\], where $d$ and $\lambda$ are the antenna spacing and the wavelength, respectively, $0 \le m_r, n_r <\sqrt{N_r}$ and $0 \le m_t, n_t < \sqrt{N_t}$ are the antenna indices in the 2D plane.
For the channel model in , it is worth noting that the rank of $\bH$ is at most $N_{cl} N_{ray}$. Using the similar derivations as the proof of Lemma \[lemma1\], we can verify that when $m\ge N_{cl} N_{ray}$, the sub-matrix $\bH_S=[\bH]_{:,1:m}\in \C^{N_r \times m}$ satisfies $\rank(\bH_S) = \rank(\bH)$. Therefore, it is possible to sample the first $m$ columns of $\bH$ in to obtain column subspace information, and sample the remaining columns to obtain the row subspace information. This means that the proposed SASE can be extended directly to the channel model given in .
In summary, the proposed SASE has no strict limitations to be applied to other channel models if the channel matrix $\bH$ experiences sparse propagation and $\text{col}(\bH_S) = \text{col}(\bH)$. Moreover, because the proposed SASE is an open-loop framework, it can be easily extended to multiuser MIMO downlink scenarios.
Simulation Results
==================
In this section, we evaluate the performance of the proposed SASE algorithm by simulation.
Simulation Setup
----------------
In the simulation, we consider the numbers of the receive and transmit antennas are $N_r = 36$, and $N_t=144$, respectively, and the numbers of the RF chains at the receiver and transmitter are $M_{RF} =6$ and $N_{RF} =8$, respectively. Without lose of generality, it is assumed that the variance of the complex gain of the $l$th path is $\sigma_{h,l}^2=1,\forall l$. We consider three subspace-based channel estimation methods as the benchmarks, i.e., SD [@ZhangSD] and MF [@AlternatingMin], and [Arnoldi [@hadi2015]]{}, where SD and MF aim to recover the low-rank mmWave channel matrix, and Arnoldi is to estimate the dominant singular subspaces of the mmWave channel. For a fair comparison, the considered benchmarks are to estimate the subspace rather than the parameters such as the angles of the paths.
Numerical Results
-----------------
![$\rank(\bH_S)$ versus $m$ ($N_t = 144; N_r = 36; L = 4$)[]{data-label="figure_rank"}](Figure_rank.eps){width="3.3in" height="2.2in"}
In order to evaluate the subspace accuracy of different methods, we compute the subspace accuracy $\eta(\widehat{\bW},\widehat{\bF})$ in , column subspace accuracy $\eta_c(\widehat{\bW})$ in , and row subspace accuracy $\eta_r(\widehat{\bF})$ in for comparison. We also evaluate the normalized mean squared error (NMSE) and spectrum efficiency. The NMSE is defined as $\text{NMSE}=\E[{\|\bH - \widehat{\bH} \|_F^2}/{\|\bH \|_F^2}]$, where $\widehat{\bH}$ denotes the channel estimate. In particular, the channel estimate of the SASE is obtained by the method derived in Section \[SASE channel result\]. The spectrum efficiency in is calculated with the combiner $\widehat{\bW}$ and precoder $\widehat{\bF}$, which are designed according to the precoding design techniques provided in [@spatially] with the obtained channel estimate $\widehat{\bH}$ via channel estimation.
[@c@]{}\
(a)\
\
(b)\
\
(c)\
### Equivalence of Subspace
It is worth noting that the column subspace estimation in Section \[Section\_column\] depends on the fact of subspace equivalence between $\bH_S $ and $\bH$ in . We illustrate in Fig. \[figure\_rank\] the rank of $\bH_S$ with different $m$. In this simulation, we set $L=4$ and $m=\{1L,2L,\ldots,10L\}$. It can be seen in Fig. \[figure\_rank\] that the rank of $\bH_S$ is equal to $L$ for all the values of $m$, i.e., the rank of $\bH_S$ is equal to the rank of $\bH$, for $m\ge L$. This validates the fact that $\text{col}(\bH_S)=\text{col}(\bH)$.
### Performance versus Signal-to-Noise Ratio
In Fig. \[SNR performance\] and Fig. \[SNR performance NMSE\], we compare the performance versus SNR of the proposed SASE algorithm to SD, MF and [Arnoldi]{} methods. The number of paths is set as $L=4$. For a fair comparison, the numbers of channel uses for the benchmarks are kept approximately equal, i.e., $K=244$.
[@c@]{}\
(a)\
\
(b)\
In Fig. \[SNR performance\](a), the column subspace accuracy $\eta_c$ of the proposed SASE is compared with the benchmarks. As we can see, the SASE and SD methods obtain nearly similar column subspace accuracy, and they outperform over the MF and Arnoldi. It means that sampling the sub-matrix $\bH_S$ of the channel $\bH$ can provide a robust column subspace estimation. In Fig. \[SNR performance\](b), the row subspace accuracy $\eta_r$ versus SNR is plotted. We found that the proposed SASE outperforms over the others. It verifies that adapting the receiver sounders to the column subspace can surely improve the accuracy of row subspace estimation. In Fig. \[SNR performance\](c), the subspace accuracy $\eta$ defined in of the proposed SASE is evaluated. As can be seen that the proposed SASE achieves the most accurate subspace estimation over the other methods. For the SASE, MF and SD, the nearly optimal subspace estimation, i.e., $\eta \approx 1$, can be achieved in the high SNR region ($10 \text{dB}\sim 20 \text{dB}$). Since the performance of the Arnoldi highly depends on the number of available channel uses, its accuracy is degraded and saturated at high SNR due to the limited channel uses ($K=244$). Thus, the ideal performance of the Arnoldi relies on a large number of channel uses or enough RF chains [@hadi2015].
In Fig. \[SNR performance NMSE\](a), the NMSE of the proposed SASE is decreased as the SNR increases. It has similar characteristis as that of the MF, but has much lower value. The NMSE of the SD is almost constant in the low SNR region and decreases in higher SNR region. Overall, the SASE outperforms the SD when $\text{SNR}\ge -15 \text{dB}$. In Fig. \[SNR performance NMSE\](b), the spectral efficiency of the SASE is plotted. The curve for perfect CSI with fully digital precoding is plotted for comparison. The proposed SASE achieves the nearly optimal spectrum efficiency among all the methods. It is observed that the spectrum efficiency of the SASE has a different trend from the NMSE in Fig. \[SNR performance NMSE\](a), while it has similar characteristic as the subspace accuracy in Fig. \[SNR performance\](c). The evaluation validates the effectiveness of the SASE in channel estimation to provide good spectrum efficiency.
[@c@]{}\
(a)\
\
(b)\
### Performance versus Number of Channel Uses
In Fig. \[Uses performance\], we show the channel estimation performance of the SASE for different numbers of channel uses. The simulation setting is $L=4, \text{SNR}=5, 20\text{dB}$. The value of $m$ in is in the set of $\{4,8,\cdots,48\}$. Accordingly, the set of the numbers of channel uses is $K=\{164,184,\cdots,384\}$.
[@c@]{}\
(a)\
\
(b)\
Fig. \[Uses performance\](a) shows the subspace estimation performance versus the number of channel uses. As the number of channel uses increases, the subspace accuracy of all the methods is increased monotonically. It is worth noting that when $K=164$ ($m=4$), the subspace accuracy of the SASE is slightly lower than that of the SD. This is because there are only $m=L=4$ columns sampled for column subspace estimation that affects the column subspace accuracy of the SASE slightly. Nevertheless, when $m\ge 8$, i.e., $K\ge 184$, the SASE obtains the most accurate subspace estimation, i.e., $\eta\approx 1$, among all the methods. In particular, when the SNR is moderate, i.e., SNR$=5$dB, the SASE clearly outperforms over the other methods. This means that the SASE requires less channel uses to provide a robust subspace estimation.
Fig. \[Uses performance\](b) shows the spectrum efficiency versus the number of channel uses. The curve for perfect CSI with fully digital precoding is also plotted for comparison. Again, the SASE achieves nearly optimal spectrum efficiency compared to the other methods. The performance gap between the SASE and the other methods are more noticeable at SNR$=5$dB. In particular, as seen in the figure, when the number of channel uses, $K\ge 244$, the performance gap between the SASE and perfect curve at SNR$=5$dB is less than $1.5$bits/s/Hz.
### Performance versus Number of Paths
In Fig. \[Paths performance\], we evaluate the estimation performance of the SASE for different numbers of paths, $L$. The number of channel uses is $K=244$ and $\text{SNR}=5,20\text{dB}$. Due to the limited number of channel uses, the Anorldi method can not perform the channel estimation for $L\ge 5$. Thus, we only show the performance of the Arnoldi for $L\le 4$.
In Fig. \[Paths performance\](a), the subspace accuracy $\eta$ of different methods versus number of paths, $L$, is illustrated. As we can see, the SASE, SD and MF achieve a more accurate subspace estimation compared to the Arnoldi. It is seen that the Arnoldi has a sharp decrease in the accuracy for $L>2$. It means that the Arnoldi can provide a good channel estimate only for $L\le 2$ with the use of $K=244$ channel uses. When SNR$=5$dB, the SASE outperforms over the other methods. When the SNR is high, i.e., SNR$=20$dB, for the proposed SASE, the subspace accuracy decreases slightly with the number of paths, $L$, which verifies our discussion about the effect of $L$ in Remark 2 of Section III.
[@c@]{}\
(a)\
\
(b)\
\
(c)\
In Fig. \[Paths performance\](b), the spectrum efficiency versus number of paths, $L$, is shown. Apart from the Arnoldi, the spectrum efficiency achieved by the SASE, MF and SD increases with the number of paths. When the SNR is high, i.e., SNR$=20$dB, the SASE, MF and SD can achieve nearly optimal performance. When the SNR is moderate, i.e., SNR$=5$dB, the proposed SASE achieves the highest spectrum efficiency among all the methods. Moreover, for the SASE, MF and SD, their performance gaps with the curve of perfect CSI is getting wider as $L$ increases. Nevertheless, the spectrum efficiency of the SASE is more closer than the other methods, which implies that the SASE can leverage the property of limited number of paths in mmWave channels more effectively than the other methods.
### Performance versus Inaccurate Path information
Thus far, in the previous simulations, we have assumed the number of paths, $L$, is known a priori. In Fig. \[different\_LD\], we evaluate the performance of the SASE under the situation that the accurate path information is not available. As discussed in Section III-A, we utilize the upper bound of the number of paths for simplicity, where we let $L_\text{sup}=\{5, 6 \}$ while $L=4$.[^5] For a clear illustration, we also evaluate the performance of proposed SASE by using the lower bound of number of paths, i.e., $L_\text{inf}=3$. As can be seen in Fig. \[different\_LD\], compared to the case of $L_\text{inf}=3$, using the upper bound $L_\text{sup}=\{5,6\}$ for SASE achieves a similar performance as the accurate path information of $L=4$. In particular, it is noted in Fig. \[different\_LD\](a) and Fig. \[different\_LD\](b) that the estimation performance of $L_\text{sup}$ is slightly worse than that of accurate path information when SNR is high, while it is marginally better when SNR is low. This is because using inaccurate path information $L_\text{sup}$ with $L_\text{sup} \ge L$ does not affect the column subspace estimation, but according to Proposition \[lemma row\], it provides worse row subspace estimation at high SNR and more accurate row subspace estimation at low SNR.[^6] This can be interpreted as follows. According to Proposition \[lemma row\], the row subspace accuracy can be bounded as $\mathbb{E}[ \eta_r (\widehat{\bF}) ] \ge \left ( 1-{2 N_t(\sigma^2 \sig_L^2(\bar{\bQ})+L_\text{sup} \sigma^4)}/{\sig_L^4(\bar{\bQ})} \right)_+$. When SNR is low, the subspace accuracy is dominated by the value of $\sigma_L(\bar{\bQ})$. Considering that $\sigma_L(\bar{\bQ})$ increases with $L_\text{sup}$. Nevertheless, in overall, the performance of proposed SASE is not sensitive to the inaccurate path number.
Conclusion
==========
In this paper, we formulate the mmWave channel estimation as a subspace estimation problem and propose the SASE algorithm. In the SASE algorithm, the channel estimation task is divided into two stages: the first stage is to obtain the column channel subspace, and in the second stage, based on the acquired column subspace, the row subspace is estimated with optimized training signals. By estimating the column and row subspaces sequentially, the computational complexity of the proposed SASE was reduced substantially to $\mathcal{O}(LDN_r)$ with $D\ge N_r$. It was analyzed that $\cO(N_t)$ channel uses are sufficient to guarantee subspace estimation accuracy of the proposed SASE. By simulation, the proposed SASE has better subspace accuracy, lower NMSE, and higher spectrum efficiency than those of the existing subspace methods for practical SNRs.
Proof of Lemma \[lemma1\] {#appendix1}
=========================
From the mmWave channel model in , when the angles $\{\theta_{t,l} \}_{l=1}^L$ and $\{\theta_{r,l} \}_{l=1}^L$ are distinct, (\_t) = (\_r)=L, which holds due to the fact that $\bA_t$ and $\bA_r$ are both Vandermonde matrices. Then, $\bH_S = \bH\bS$ can be expressed as \_S = \_r () \_t\^H . Combining the rank inequality of matrix product $\rank(\bH_S) \le \rank(\bH) = L$ and the following lower bound, (\_S) && (\_r ()) +( \_t\^H ) -L\
&= &( \_t\^H ), yields $L \geq \rank(\bH_S) \geq \rank(\bA_t^H \bS)$. Therefore, in order to show $\mathop{\mathrm{col}}(\bH_S) = \mathop{\mathrm{col}}(\bH)$, namely, $\rank(\bH_S) = L$, it suffices to show that $\rank(\bA_t^H\bS)=L$. Considering that $\bA_t^H\bS$ is a Vandermonde matrix, it has $\rank(\bA_t^H\bS)=L$. This completes the proof.
Proof of Lemma \[lemma2\] {#lemma2_proof}
=========================
It is trivial that the entries in $\bY$ follow the identical distribution of $\cC\cN(0,\sigma^2)$. Therefore, it remains to show that all the entries in $\bY$ are independent. Because of the typical property of Gaussian distribution, it suffices to prove that they are uncorrelated. For any $i \neq j $ or $m \neq n $, the following holds, &= &\
&= & 0. Therefore, the entries in $\bY$ are uncorrelated and thus independent, which concludes the proof.
Proof of Proposition \[prop1\] {#appendix2}
==============================
Based on the definition of $\eta_c(\widehat{\bW})$ in , it has &=&\
&=&\
& & -\
&= & -\
& & \_L(\^H ) - -\_2,\
& & \_L(\^H ) -\_1, \[mid eq1 1\] where the inequality $(a)$ holds from the triangle inequality, and the inequality $(b)$ comes from the fact that for $\bA\in \C^{n \times n}$ with $\rank(\bA)=n$ and $\bB\in \C^{n \times k}$, $\| \bA \bB\|_F^2 \geq \sig_n^2(\bA) \| \bB \|_F^2$, where the latter follows by $\| \bA \bB\|_F^2 = \sum_{i=1}^{k} \| \bA [\bB]_{:,i} \|_2^2 \ge \sum_{i=1}^{k} \sigma_{n}^2(\bA) \| [\bB]_{:,i} \|_2^2 =\sigma_{n}^2(\bA) \| \bB \|_F^2$. Thus, this concludes the proof for the inequality in .
Then, by letting $\delta_1 \rightarrow 0$ in , we take expectation of squares of both sides in , then it has the following &&\
& & ( 1- )\_+ , \[final col ac\] where the inequality $(c)$ holds from Theorem \[theorem1\], and this concludes the proof.
Proof of Proposition \[lemma row\] {#appendix3}
==================================
Recall that the row subspace matrix $\widehat{\bV}$ is given by the right singular matrix of $\widehat{\bQ}=\bar{\bQ} + \bar{\bN}$ in , and the elements in $ \bar{ \bN}$ are i.i.d. with each entry being $\cC \cN(0,\sigma^2)$ according to Lemma \[lemma2\]. Thus, Theorem \[theorem1\] is applied, which gives ( 1- )\_+. \[row 2\] Then, based on the subspace accuracy metric in , it has &=&\
&=&\
& & -\
&= & -\
& & \_L(\^H ) - -\_2,\
& & \_L(\^H ) -\_2. \[lemma row mid1 1\] Thus, the inequality is proved. Moreover, under the condition $\delta_2 \! \rightarrow \! 0$, taking expectation of the squares of both sides in yields &&\
&& ( 1- )\_+, where the inequality $(a)$ holds from . This concludes the proof for the row estimation accuracy bound in .
[^1]: [W. Zhang is with the Department of Electrical Engineering, City University of Hong Kong, Hong Kong SAR, China (e-mail: wzhang237-c@my.cityu.edu.hk).]{} [T. Kim is with the Department of Electrical Engineering and Computer Science, The University of Kansas, KS 66045, USA (e-mail: taejoonkim@ku.edu).]{}
[S.-H. Leung is with State Key Laboratory of Terahertz and Millimeter Waves and Department of Electrical Engineering, City University of Hong Kong, Hong Kong SAR, China (e-mail: eeeugshl@cityu.edu.hk).]{}
[^2]: Here, the SNR is the ratio of transmitted sounder’s power to the noise’s power, which is a common practice in the channel estimation literature [@Hur13; @OMPchannel; @SBR_channel; @ZhangSD; @hadi2015; @alk].
[^3]: Due to the limited RF chains, the dimension of channel subspaces for data transmission is less than $\min\{M_{RF}, N_{RF}\}$. Thus, if the path number estimate is larger than $\min\{M_{RF}, N_{RF}\}$, we let it be $\min\{M_{RF}, N_{RF}\}$.
[^4]: It is worth noting that because the estimated column subspace of the first stage is $\widehat{\bW} \in \C^{N_r \times L}$, thus the dimension for receive sounder of second stage is ${N_r \times L}$ rather than ${N_r \times M_{RF}}$ in .
[^5]: If the $L_\text{sup}$ with $L_\text{sup}\ge L$ is utilized for SASE, the estimated subspaces will be $\widehat{\bW}\in \C^{N_r \times L_\text{sup}}$ and $\widehat{\bF}\in \C^{N_t \times L_\text{sup}}$. For a fair comparison, we choose the dominant $L$ modes in $\widehat{\bW}$ and $\widehat{\bF}$ when evaluating the performance.
[^6]: It is worth noting that if $L_\text{sup}$ is utilized for SASE, according to Proposition \[lemma row\], the row subspace accuracy is bounded as $\mathbb{E}[ \eta_r (\widehat{\bF}) ] \ge \left ( 1-{2 N_t(\sigma^2 \sig_L^2(\bar{\bQ})+L_\text{sup} \sigma^4)}/{\sig_L^4(\bar{\bQ})} \right)_+$. The statements can be verified easily through analyzing this row subspace accuracy bound.
|
---
author:
- Alberto Ronzani
- Carles Altimiras
- 'Sophie D’Ambrosio'
- Pauli Virtanen
- Francesco Giazotto
title: 'Phase-driven collapse of the Cooper condensate in a nanosized superconductor'
---
**Superconductivity can be understood in terms of a phase transition from an uncorrelated electron gas to a condensate of Cooper pairs in which the relative phases of the constituent electrons are coherent over macroscopic length scales [@de_gennes_superconductivity_1999]. The degree of correlation is quantified by a complex-valued order parameter, whose amplitude is proportional to the strength of the pairing potential in the condensate. Supercurrent-carrying states are associated with non-zero values of the spatial gradient of the phase. The pairing potential and several physical observables of the Cooper condensate can be manipulated by means of temperature, current bias, dishomogeneities in the chemical composition or application of a magnetic field [@tinkham_introduction_1996]. Here we show evidence of complete suppression of the energy gap in the local density of quasiparticle states (DOS) of a superconducting nanowire upon establishing a phase difference equal to $\pi$ over a length scale comparable to the superconducting coherence length. These observations are consistent with a complete collapse of the pairing potential in the center of the wire, in accordance with theoretical modeling based on the quasiclassical theory of superconductivity in diffusive systems. Our spectroscopic data, fully exploring the phase-biased states of the condensate, highlight the profound effect that extreme phase gradients exert on the amplitude of the pairing potential. Moreover, the sharp magnetic response observed near the onset of the superconducting gap collapse regime can be exploited to realize ultra-low noise magnetic flux detectors [@ronzani_highly_2014].**
![Principle of operation and interferometer design. **a**: Schematic representation of the measurement setup for the transport spectroscopy of a phase-biased superconducting wire. The current-voltage characteristics show the modulation of the local density of states in the latter as a function of the magnetic flux $\Phi$ coupled to a compact superconducting loop in clean contact with the wire. **b**: Conceptual picture of the progressive depairing in the middle of the superconducting wire for increasing phase gradient. The position-dependent value of the complex order parameter $\Delta(x)
\exp\left[\,\mathrm{i} \phi (x)\right]$ is shown as a twisted-wireframe representation of a revolution surface. For a superconducting wire having single-valued current-to-phase relationship, $\Delta(L/2) = 0$ is expected for $\Phi = \Phi_0/2$. **c, d**: Pseudo-colour scanning electron micrographs of, respectively, the interferometer loop and the superconducting wire in a typical device. Here, yellow indicates the $150$-nm-thick “interferometer” and $25$-nm-thick “nanowire” Al layers; the $15$-nm-thick AlOx tunnel probe electrode is shown in purple, realized by oxidizing respectively a Al/Al$_{0.98}$Mn$_{0.02}$ layer to obtain a superconducting/normal-metal electrode. []{data-label="fig:concept"}](squipt_al){width="45.00000%"}
The physics of superconducting boundaries is an invaluable tool for the investigation of the fundamental properties of matter and for fostering the development of novel devices. From tunnel-type contacts [@giaever_electron_1960] to clean galvanic interfaces dominated by proximity effect [@de_gennes_superconductivity_1999], the fabrication and manipulation of superconducting boundaries is the enabling feature for the realization of ultrasensitive magnetometers [@vasyukov_scanning_2013] and electrometers [@schoelkopf_radio-frequency_1998], sub-Kelvin electron thermometers and coolers [@giazotto_opportunities_2006], coherent mesoscopic heat current controllers [@giazotto_josephson_2012; @martinez-perez_rectification_2015], radiation detectors [@day_broadband_2003], parametric amplifiers [@ho_eom_wideband_2012; @macklin_nearquantum-limited_2015], qubits [@clarke_superconducting_2008] and Majorana physics demonstrators [@mourik_signatures_2012].
In superconducting electronics, localized geometrical and compositional inhomogeneities provide preferential pinning points for the establishment of order parameter phase gradients, the distinctive superconducting degree of freedom. This is exemplified by the coherent character of the pair transport between two different Cooper condensates leading to the Josephson effect [@josephson_possible_1962], where the current to phase relation (CPR) is determined by the phase-dependent energy spectrum of weak link-bound states [@golubov_current-phase_2004]. The latter has been resolved spectroscopically down to the individual mesoscopic channel in atomic contacts [@bretheau_exciting_2013], carbon nanotubes [@pillet_andreev_2010] and semiconducting nanowires [@chang_tunneling_2013].
Correspondingly, physical observables in diffusive mesoscopic weak links are dependent on the phase difference between their superconducting boundaries. The phase-driven modulation of the DOS of proximized normal metal weak links is a well known example [@le_sueur_phase_2008]. Analogous observations have been reported for superconductive wires in the long limit, where partial depairing was induced by spatially uniform phase gradient profiles [@anthore_density_2003].
On the other hand, phase-biasing a weak link based on a thin superconducting wire in the short limit (i.e., having comparable geometric and coherence length) allows the observation of the reaction of its Cooper condensate to a spatial phase profile well beyond the linear regime. In particular, provided that the state of the Cooper condensate is a single-valued function of the phase difference applied to this weak link, the latter can be polarized with a phase difference equal to $\pi$, reaching the non-trivial node of its CPR. In this specific state no supercurrent flows in the Cooper condensate and, by virtue of time-reversal symmetry, its order parameter $\Delta \exp \left(\mathrm{i} \phi \right)$ is a real-valued and sign-changing function of the spatial coordinate along the wire. Then, as a consequence of continuity, the pairing potential amplitude $\Delta(x)$ must equal zero in some position inside the wire (e.g., the centre in case of symmetric boundaries). While wide ($w \gtrsim \xi$) weak links can accommodate Abrikosov vortices, quasi-1D wires ($w < \xi$) develop a 1D phase singularity without screening currents [@likharev_superconducting_1979], namely a phase-slip center. A conceptual representation of the mechanism of phase-driven collapse of the order parameter is presented in Figure \[fig:concept\]b.
In our experiment, the phase bias on a narrow nanosized aluminum wire is enforced by virtue of magnetic flux quantization in a closed superconducting loop subjected to a magnetic field applied orthogonally [@doll_experimental_1961; @deaver_experimental_1961]. The phase-dependent DOS inside the wire is probed through standard two-wire charge transport spectroscopy via a tunnel-barrier electrode (refer to the schematic in Figure \[fig:concept\]a). This type of device is the superconducting-wire analogue of the superconducting quantum interference proximity transistor (SQUIPT) [@giazotto_superconducting_2010].
In this design, robust phase-biasing performance is ensured by the adoption of a $150$-nm-thick aluminum loop of micrometric size, a configuration that proved to be effective in SQUIPTs based on normal-metal weak links of comparable size [@ronzani_highly_2014; @dambrosio_normal_2015]. Moreover, the pronounced geometric contrast in the cross section of the ring with respect to the superconducting wire minimizes depairing effects induced by supercurrent concentration at the interfaces between the wire and the thicker superconducting ring [@vijay_approaching_2010].
The modulation of the current-vs-voltage $I(V_b)$ characteristics of a typical (wire length $L=160\,\mathrm{nm}$) normal-metal tunnel probe device (tunnel resistance $R_T \approx 150\,\mathrm{k\Omega}$) as a function of the applied magnetic flux $\Phi$ is presented in Fig. \[fig:nprobe\]. At base temperature ($T=20\,\mathrm{mK}$) increasing the magnetic flux bias from $\Phi = 0$ to $\Phi=\Phi_0/2$ results in a 65% suppression of the energy gap in the quasiparticle DOS compared to its zero-field value (panels a,e). Notably, the low-temperature differential conductance characteristics (see Fig. \[fig:nprobe\]e) recorded for $\Phi/\Phi_0 \lesssim 0.25$ are compatible with data reported for specimens in the constant phase gradient regime [@anthore_density_2003]. This could be expected since in this flux range, the CPR is essentially linear. However, for $\Phi/\Phi_0 \approx 0.5$, a peculiar concentration of quasiparticle states at the edges of the residual energy gap can be inferred from the experimental data. The latter feature, absent in short phase-biased normal-metal wires [@dambrosio_normal_2015], appears reproducibly between different samples, provided a sufficient phase difference is applied to the short superconducting wire. By increasing the temperature ($T=300,\,500,\,700\, \mathrm{mK}$, panels b-d) the $I(V_b)$ curves show evidence of the progressive suppression of the residual energy gap at $\Phi = \Phi_0/2$. The magnetic modulation of the differential conductance recorded at $T=650\,\mathrm{mK}$ (panel f) shows the transition between a superconductor/insulator/normal-metal-like response at zero field to an almost ohmic response at $\Phi=\Phi_0/2$.
These observations can be understood by considering the temperature response [@likharev_superconducting_1979] of the CPR of a weak link based on a superconducting wire in contact with rigid superconducting electrodes. The physical observables of the latter have been calculated in the quasiclassical framework by solving the Usadel equations self-consistently with the pairing amplitude profile. Figure \[fig:theo\] shows a synopsis of modeled physical quantities obtained from a parameter set chosen in accordance with experimental data (refer to the Methods section for details).
As the temperature increases, the current-to-phase relation of the weak link progressively shifts from a multi-valued $I_s(\Phi)$ at low-temperature \[characterized by metastable $\Phi=\Phi_0 (n+1/2)$ nodes\] to a single-valued CPR functional form reached at $T=700\, \mathrm{mK}$. In the latter regime the amplitude of the pairing potential in the centre of the wire can be completely suppressed by applying a phase difference equal to $\pi$ (panel c).
Simultaneously, the differential conductance as a function of voltage bias, applied magnetic flux and temperature can be computed from the corresponding DOS. In Figure \[fig:theo\]d–g, the calculated differential conductance maps are juxtaposed for comparison with data measured at different temperatures. The striking correspondence obtained corroborates the physical interpretation of complete pair potential suppression in the superconducting wire for $\Phi/\Phi_0 = 0.5$.
![Superconducting tunnel probe spectroscopy of the phase-driven collapse of the superconducting gap in the Al nanowire. **a, b**: Normalized differential conductance as a function of the applied magnetic flux and voltage bias, recorded at lattice temperature $T = 0.9,\, 1.0\, \mathrm{K}$, respectively. The region showing negative differential conductance is indicated in magenta. **c, d**: Current-vs-voltage characteristic curves recorded for $\Phi/\Phi_0 = 0.48,\, 0.49,\,0.5$ and for lattice temperature $T = 0.9,\, 1.0\, \mathrm{K}$. []{data-label="fig:sprobe"}](fig4)
This above interpretation is further confirmed by observations focused on the transition from the unstable to stable $\pi$ phase bias regime in devices equipped with a superconducting tunnel electrode. The latter, realized by a $15$-nm-thick oxidized aluminum film, features a BCS-like DOS characterized by a sizeable superconducting gap $\Delta_{pr} \approx 250\,\mathrm{\mu eV}$, typical of thin Al films. As a consequence, the spectroscopic sampling of the DOS in the phase-biased wire does not suffer from the loss of energy resolution due to thermal broadening typical of normal-metal probes. In particular, this setup allows for a direct estimate of the energy gap $\varepsilon_g(\Phi)$ in the probed DOS. At finite temperature, the latter quantity can be derived [@meschke_tunnel_2011] from the difference between voltage bias values relative to the direct and thermally-activated conductance peaks (found, respectively, at $e\, V_b = \Delta_{pr} \pm
\varepsilon_g$, similarly to the well-known case of quasiparticles tunneling between different superconductors at nonzero temperature [@tinkham_introduction_1996]).
Panels a,b in Figure \[fig:sprobe\] show normalized differential conductance maps recorded for a representative device characterized by a $L=210\,\mathrm{nm}$ superconducting wire in contact with a $R_T = 15\,\mathrm{k\Omega}$ superconducting tunnel electrode. The mapping is focused on voltage bias values corresponding to the superconducting gap in the probe ($e\,V_b \approx
\Delta_{pr}$) and with coupled magnetic flux applied in a minute range centered around $\Phi_0/2$. By inspecting the flux modulation of the direct ($e\,V_b > \Delta_{pr}$) and thermally-activated ($e\,V_b < \Delta_{pr}$) conductance peaks, an incomplete suppression of $2\, \varepsilon_g \simeq
40\,\mathrm{\mu eV}$ can be inferred from data recorded at $T=0.9\mathrm{K}$ (panel a). Within a $100\,\mathrm{mK}$ temperature increase, we observe the merging of the direct and thermally-activated peaks at $\Phi/\Phi_0 = 0.5$ and $e V_b = \Delta_{pr}$, the direct evidence of the full suppression of the energy gap in the probed quasiparticle DOS. Equivalently, the latter is associated with a smooth monotonic $I(V_b)$ characteristic curve at $T=1\mathrm{K}$ for $\Phi/\Phi_0 = 0.5$ (panel d), whearas the corresponding curve at $T=0.9\mathrm{K}$ (panel c) displays a $\simeq
40\,\mathrm{\mu V}$-wide plateau.
We interpret these observations as the confirmation that the increase in temperature has driven the CPR of the weak link to the single-valued regime, leading to a complete collapse of the amplitude of the pairing potential in the center of the wire for $\Phi/\Phi_0 = 0.5$. In this case, the temperature value for this transition is higher than for the normal-probe interferometer (Figures \[fig:nprobe\] and \[fig:theo\]) as expected from the difference in the respective lengths of the wires, in agreement with the theory. Notably, while the value $T=1\mathrm{K}$ is arguably sizeable, is also significantly smaller than the critical temperature ($T_{c,w} = 1.4\mathrm{K}$, see Methods) of the 25-nm-thick Al layer the wire consists of.
![Magneto-electric response of a typical superconducting tunnel probe device. **a, b**: Respectively, voltage response (arbitrary offset) and corresponding flux-to-voltage transfer function for $\Phi \approx \Phi_0/2$, recorded at temperature $T= 1.0\, \mathrm{K}$. The device is here operated under fixed current bias values $I_b = 4.30,\, 4.35,\, 4.40\, \mathrm{nA}$. []{data-label="fig:smag"}](fig5)
The steep character of the magnetic flux dependence of the pairing potential suppression suggests to exploit these devices for highly-sensitive magnetometry applications. Inspection of voltage traces recorded at $T=1\mathrm{K}$ under constant current bias $I_b$ from the representative superconducting-probe device (Figure \[fig:smag\], panel a) reveals abrupt but continuous voltage response in a minute magnetic flux range close to $\Phi/\Phi_0 = 0.5$. Here, the different traces indicate $I_b \in \left[ 4.3,\, 4.4\right]
\,\mathrm{nA}$, a range corresponding to the quasiparticle current measured in the voltage-biased setup, with $V_b \approx \Delta_{pr}/e = 250\,\mathrm{\mu V}$. The corresponding flux-to-voltage responsivity characteristics (panel b) obtain values as large as $ \sim 27\, \mathrm{mV}/\Phi_0$, which are unparalleled in this class of devices [@ronzani_highly_2014; @dambrosio_normal_2015]. In a simple DC-biased setup, magnetic flux resolution figures as low as $
250\,\mathrm{n\Phi_0/\sqrt{Hz}}$, limited by shot noise intrinsic to the tunnel-probe readout, have been measured in the $0.1 - 100\,
\mathrm{Hz}$ range (see Supplementary material).
In summary, we have presented a robust and reproducible means of suppressing the order parameter amplitude of the Cooper condensate inside a nanosized superconductor. Our observations are consistent with established theory. Reaching the complete flux modulation of the energy gap in the DOS inside the superconducting wire marks the transition between the *intrinsic*-like and *Josephson*-like regime. The corresponding temperature-dependent modulation of the CPR from a multi-valued locus to a proper single-valued form entails, at the cross-over, a strong dependence of the physical observables on the applied magnetic flux for $\Phi \approx \Phi_0/2$. Yet, this property has been applied for the realization of ultra-sensitive small-area quantum magnetometers. More generally, this work provides experimental coverage of the fundamental consequences of the self-consistency requirement on the order paramenter in spatially inhomogeneous superconductors. Finally, our design might be suitable as a testbed for the generation and investigation of a spatially-locked “phase-slip center”. Another exciting perspective is to investigate whether the multivalued regions of the CPR are reachable in pulsed phase experiments.
Acknowledgements
================
We acknowledge funding from the European Research Council under the European Unions Seventh Framework Program (FP7/2007-2013)/ERC Grant agreement No. 615187-COMANCHE, the Italian Ministry of Education, University and Research (MIUR) through the program FIRB-RBFR1379UX and the Tuscany Region through the CNR joint project “PROXMAG”. C.A. acknowledges discussions with H. Pothier, P. Joyez and S. Bergeret at the root of the project.
Methods
=======
Fabrication protocols and experimental set-up
---------------------------------------------
The devices have been fabricated by directional metallic thin film deposition through a self-aligned suspended mask. The latter is obtained by electron beam lithography on a positive tone resist bilayer (1000 nm copolymer / 100 nm poly-methyl metacrylate) spun on an oxidized Si wafer. After the development of the resist bilayer, the substrate is loaded in a ultra-high vacuum (base pressure $\approx 10^{-9}$ Torr) electron beam evaporator, where an initial 15-nm-thick layer of 5N-Al/Al$_{0.98}$Mn$_{0.02}$ is deposited at $40^\circ$ to realize a superconducting/normal-metal probe electrode, respectively. A subsequent controlled exposition (300 s) to a pure O$_2$ atmosphere with pressure values in the $10 \div 100$ mTorr range yields tunnel junctions having specific resistance in the $20 \div 200 \,
\Omega\,\mathrm{\mu m}^2$ range. The superconducting interferometer is evaporated on top of the tunnel electrode as a 25-nm-thick Al layer at $20^\circ$, followed by a 150-nm-thick Al layer at normal incidence.
After chemical lift-off in acetone followed by rinsing in isopropylic alcohol, the samples are inspected in a scanning-electrode microscope. Suitable devices are wire-bonded on a 24 pin dual-in-line ceramic chip carrier, which is then loaded in a filtered $^3$He/$^4$He dilution refrigerator (Oxford Instruments, mod. Triton 200) for the magneto-electric characterization down to $20\, \mathrm{mK}$. In the latter, the magnetic field is applied orthogonal to the substrate by a custom-built superconducting electromagnet driven by a low-noise programmable current source (Keythley model 2600).
Low-noise current / voltage bias sources (a lithium battery in series with a $200\, \mathrm{M\Omega}$ impedance and a Yokogawa GS200, respectively) are coupled with room-temperature voltage / current preamplifiers (NF Corporation LI-75A and DL Instruments 1201, respectively) to realize the DC readout. For normal-metal tunnel electrode devices the differential conductance is recorded with lock-in amplification (Stanford research model 830) with $4\mu\mathrm{V}$ voltage excitation (rms).
Theoretical model
-----------------
We model the properties of the superconducting wire embedded in the ring by the quasiclassical Green function model [@belzig_quasiclassical_1999], assuming a quasi-1D geometry. The spectral angle $\theta$ and phase $\chi$ are solved from the Usadel equations [@virtanen_spectral_2016] $$\begin{gathered}
\label{eq:usadel}
\hbar{}D\partial_x^2\theta = -2iE\sinh\theta + \frac{D(\partial_x\chi)^2}{2}
\sinh2\theta \\\notag\qquad + 2i|\Delta|\cos(\phi-\chi)\cosh\theta \,,
\\ \hbar{}D\partial_x\cdot(\partial_x\chi \sinh^2\theta) =
-2i|\Delta|\sin(\chi-\phi)\sinh\theta \,,\end{gathered}$$ where $D$ is the diffusion constant. We assume the contacts between the wire and the ring have nonzero resistance, and model them with the Kupriyanov–Lukichev boundary conditions [@kupriyanov_influence_1988] $$\begin{aligned}
\label{eq:kl} \mp{}r\sinh\theta\partial_x\chi &=
\sin(\chi-\chi_{\mp})\sinh\theta_{\mp} \,, \\ \mp{}r\partial_x\theta &=
\sinh\theta\cosh\theta_{\mp}
-\cos(\chi-\chi_{\mp})\cosh\theta\sinh\theta_{\mp} \,,\end{aligned}$$ at the left ($-$) and right ($+$) contacts. The parameter $r=R_{I}/r_w$ is the ratio between the interface resistance $R_I$ and the resistance per length $r_w$ of the wire. The values $\theta_\pm=\operatorname{artanh}(|\Delta_r(T)|/E)$, $\chi_-=0$, $\chi_+=\varphi$ are the values of the angles inside the superconducting ring. Here, the $\Delta_r(T)$ value of order parameter in the ring is assumed to be BCS-like, determined by the critical temperature $T_{c,r}$. We neglect proximity effect in the ring, and assume it is unperturbed by the weak link.
The normalized DOS $N(E,x)={\mathop{\mathrm{Re}}}\cosh\theta(E,x)$ and supercurrent $I_s=A\sigma\int_{-\infty}^\infty{\mathrm{d}E\,}{\mathop{\mathrm{Im}}}[\partial_x\chi\sinh^2\theta]\tanh\frac{E}{2
k_B T}$ are obtained from the solutions. The order parameter $\Delta=|\Delta|e^{i\phi}$ is obtained from the self-consistency relation, $$\begin{aligned}
\label{eq:selfcons} |\Delta| \ln\frac{T}{T_{c,w}} = 2\pi i
k_B T\sum_{\omega_n>0}\left[e^{i(\chi-\phi)}\sinh\theta -
\frac{|\Delta|}{E}\right]_{E=i \omega_n} \,,\end{aligned}$$ where $T_{c,w}$ is the critical temperature of the wire, and $\omega_n = \pi k_B T
(2n +1 )$ are the discrete Matsubara energies. Self-consistent determination is required for current conservation.
The solution to Eq. becomes multivalued when $L\gtrsim{}\xi_0$. To trace the solution branch $(\varphi,\Delta(x))$, in this case, we apply the pseudo-arclength continuation method [@keller_numerical_1977]. The method generates the next point $(\varphi_{k+1},\Delta_{k+1}(x))$ based on previous solutions, by requiring that $\Delta_{k+1}(x)$ satisfies Eq. , and that the phase difference $\varphi_{k+1}$, which is also considered unknown, satisfies a pseudo-arclength condition. This condition is an approximate form for requiring the distance from the previous solution, $s=\eta|\varphi_{k+1}-\varphi_k|^2+(2-\eta)||\Delta_{k+1}-\Delta_k||_2^2$ be constant $s\approx{}s_0$, where $0<\eta<2$ is the weight factor and $s_0$ the step size.
In the comparison with experimental data, the relation between applied magnetic flux $\Phi$ and phase difference $\varphi$ is assumed non-linear, with $\Phi/\Phi_0 = \varphi/2\pi + \beta_L I_s(\varphi)/I_c$, where $I_c$ is the low-temperature value of the critical current and $\beta_L = \mathcal{L} I_c/
\Phi_0$ accounts for nonzero ring inductance $\mathcal{L}$. The thermodynamically stable solution is determined as the one that minimizes the free energy $F(\varphi)=\frac{\Phi_0}{2\pi}\int_0^\varphi{\mathrm{d}\varphi'\,}I_s(\varphi') +
\mathcal{L}I_s^2(\varphi)/2$. The normalized differential conductance $R_T \partial I / \partial V_b$ is calculated from the probe quasiparticle current $ I(V_b, \Phi, T) = 1/(e R_T) \int dE\, \hat{N}(E, \Phi, T) [ f_0(E-eV_b) -
f_0(E)] $, where $f_0(E) = [1+\exp(E/k_B T)]^{-1}$ is the Fermi-Dirac distribution function at temperature $T$ and $\hat{N}(E, \Phi, T) = 2/L \int_{L/4}^{3L/4}
N(E, x)_{T, \Phi}\, dx$ is the wire DOS averaged over the typical physical length sampled by the probe.
The datasets in Figure \[fig:theo\] have been generated assuming the critical temperature values $T_{c,w}=1.4\mathrm{K}$ for the thin wire and the bulk value $T_{c,r}=1.25\mathrm{K}$ for the thick ring. The modeled diffusive weak link has normalized length $L/\xi_0 = 1.7$, corresponding to $\xi_0 = \sqrt{\hbar D / \Delta_0} =
95\,\mathrm{nm}$ for a physical length $L=160\,\mathrm{nm}$. The interfaces of the wire are modeled with a non-ideality coefficient $r=0.75$. The values of the set of mutually-independent parameters $T_{c,w},\, \xi_0,\, r$ are chosen on the basis of optimal reproduction of the differential conductance characteristic curve recorded for null magnetic field at $T=20\,\mathrm{mK}$. Notably, data recorded at higher temperatures is also satisfactorily reproduced. Optimal agreement with the observed flux modulation at all temperatures is obtained by letting $\beta_L = 0.03$, consistent with $I_c \approx 18 \,\mathrm{\mu A}$ deduced from the former parameters while assuming $\mathcal{L} = 3.5\, \mathrm{pH}$ (numerical estimate of the inductance of the superconducting loop including both geometric and kinetic contributions, with magnetic penetration depth $\lambda_\bot \approx
60\,\mathrm{nm}$. FastHenry version 3.0wr by S. R. Whiteley, available from <http://wrcad.com>).
|
---
abstract: 'We examine the possibility that the dark matter (DM) interpretation of the GeV scale Fermi gamma-ray excess at the Galactic Center can be realized in a specific framework - secluded singlet fermionic dark matter model with small mixing between the dark and Standard Model sector. Within this framework it is shown that the DM annihilation into bottom-quark pair, Higgs pair, and new scalar pair, shown to give good fits to the Fermi gamma-ray data in various model independent studies, can be successfully reproduced in our model. Moreover unavoidable constraints from the antiproton ratio by the PAMELA and AMS-02, the gamma-ray emission from the dwarf spheroidal galaxies by the Fermi-LAT, and the Higgs measurements by the LHC are also considered. Then we found our best-fit parameters for the Fermi gamma-ray excess without conflicting other experimental and cosmological constraints if uncertainties on the DM density profile of the Milky Way Galaxy are taken into account. Successfully surviving parameters are benchmark points for future study on the collider signals.'
---
IU-HET-610\
KIAS-P16005
[ **Yeong Gyun Kim$^{a,b,}$[^1], Kang Young Lee$^{c,}$[^2], Chan Beom Park$^{b,}$[^3], Seodong Shin$^{d,}$[^4]**]{}\
$^a$[*Department of Science Education, Gwangju National University of Education,Gwangju 61204, Korea*]{}\
$^b$[*School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea*]{}\
$^c$[*Department of Physics Education & Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea*]{}\
$^d$[*Physics Department, Indiana University, Bloomington, IN 47405, USA*]{}
------------------------------------------------------------------------
------------------------------------------------------------------------
Introduction {#sec:intro}
============
The existence of non-baryonic dark matter (DM) in the universe has been supported by a lot of solid evidences observing from its gravitational interactions. On the other hand its particle property is still in a mystery. Among all the candidates Weakly Interacting Massive Particle (WIMP) is the most popular one because of its natural mass and interaction ranges to give the right amount to account for the matter density observed today. Experimental efforts to detect WIMPs are ongoing in direct and indirect searches as well as collider experiments. Direct detection experiments are designed to observe the elastic scattering of WIMPs on the target nuclei through nuclear recoils. On the other hand, indirect detection experiments search for products of the WIMP annihilation or decay processes such as gamma-rays, neutrinos, and charged cosmic rays. Among those products, gamma-rays are often considered as the golden channel for the indirect detection of the DM because we can easily detect them and identify in which part of the universe they came from.
It is intriguing that several independent collaborations have reported a broad excess of the gamma-ray (at energies around few GeV) from the Galactic Center (GC) above the expected astrophysical emission through the analyses of the data accumulated by the Fermi Large Area Telescope (LAT) [@Goodenough; @Hooper:2011ti; @Abazajian:2012pn; @Hooper:2013rwa; @Gordon:2013vta; @Abazajian:2014fta; @Daylan:2014rsa; @Calore:2014xka; @Calore:2014nla], which is confirmed later by the experimental group [@TheFermi-LAT:2015kwa]. The excess might be explained by (unidentified) astrophysical sources such as millisecond pulsars [@Gordon:2013vta; @Abazajian:2014fta; @msp] which can be an important part of unresolved point sources fitting the observed data [@Bartels:2015aea; @Lee:2015fea]. However the DM annihilation still remains as a most viable possibility to account for it [@TheFermi-LAT:2015kwa]. Many collaborations have investigated such a possibility in a model independent way by classifying various scenarios of the DM on the basis of the final states of the annihilation process, and in turn quantitatively obtaining the scales of the mass and the annihilation cross section of the DM that can fit the gamma-ray excess. As a result, it has been shown that the DM annihilations into a pair of $b$ quarks [@Daylan:2014rsa], leptons [@Lacroix:2014eea], the Higgs bosons [@Agrawal:2014oha; @Calore:2014nla] in the Standard Model (SM), or new particles which subsequently decay to $b \bar b$ pairs [@Abdullah:2014lla; @Cline:2015qha; @Elor:2015tva; @Rajaraman:2015xka; @Ko:2015ioa; @Dutta:2015ysa] can give good fits to the data with proper choices of DM mass and the annihilation cross section.[^5]
The model independent studies typically consider the DM mass and the annihilation cross section as free parameters to fit the gamma-ray data for the given annihilation process. In this case, it is obvious that the impact on other experimental and cosmological constraints is subtle. Hence, in the end, one has to consider a specific model that includes a plausible DM candidate and study various other theoretical and experimental bounds in connection with the gamma-ray excess. Preceding model dependent analyses exist with the main annihilation channel for the GeV scale excess; $b$-quark pair [@Alvares:2012qv; @Okada:2013bna; @Modak:2013jya; @Alves:2014yha; @Ipek:2014gua; @Basak:2014sza; @Wang:2014elb; @Ghorbani:2014qpa; @Cao:2014efa; @Ghorbani:2014gka; @Ghorbani:2015baa; @Duerr:2015bea], lepton pair [@Kyae:2013qna; @Kim:2015fpa], and new particles [@Boehm:2014bia; @Berlin:2014pya; @Cerdeno:2015ega; @Cao:2015loa].
In this paper, we examine if the DM interpretation for the gamma-ray excess can be realized in a model with a singlet fermionic dark matter (SFDM) which is originally proposed in [@Kim:2008pp]. In this model, DM interacts with the SM sector only by the mixing between the SM Higgs and a singlet scalar. Particularly we specify the scenario where such a mixing is quite suppressed to avoid the recent bounds from the Higgs measurements at the Large Hadron Collider (LHC) and various direct detection results of WIMP, while keeping the relic density as observed.[^6] Actually this secluded SFDM scenario was previously suggested in [@Kim:2009ke] to provide a viable light WIMP setup.[^7] Here we slightly modify the secluded SFDM scenario by adding a pseudoscalar interaction in the dark sector to easily explain the Fermi gamma-ray excess.
Interestingly, following the analysis in [@Calore:2014nla], we could find the parameters giving good fits to the excess in several DM annihilation channels; $b$ quark pair, Higgs pair, and new scalar pair (without fixing the decay mode of it by hand), depending on the mass hierarchies of particles and couplings. On top of these we further consider unavoidable constraints from the antiproton ratio by the PAMELA [@Adriani:2010rc] and AMS-02 [@AMS02; @Giesen:2015ufa], the gamma-ray emission from the dwarf spheroidal galaxies by the Fermi-LAT [@Ackermann:2015zua], and the Higgs measurements by the LHC [@atlascms]. These bounds are quite strong so the annihilation channel to $b$ quark pair remains viable only around the resonance region and after introducing a mixture of the scalar and pseudoscalar interaction in the dark sector. The surviving parameters in all the channels will be our benchmark points for future study on the collider signals.
The rest of the paper is organized as follows. The description for the SFDM model is given in Sec. \[sec:model\]. Then in Sec. \[sec:gamma\_excess\], we calculate the photon energy spectrum from the SFDM annihilations, and in turn perform the fit to the gamma-ray data by considering other experimental and cosmological constraints. Sec. \[sec:concl\] is devoted to conclusions.
Singlet fermionic dark matter {#sec:model}
=============================
We introduce a real scalar field $S$ and a Dirac fermion field $\psi$, which transform as the singlet under the SM gauge group. In addition to the SM Lagrangian, the dark sector Lagrangian with the renormalizable interactions is given by $$\begin{aligned}
{\mathcal}{L}^{\rm dark} = \bar\psi ({\ensuremath{i\mkern1mu}}\slashed{\partial}
- m_{\psi_0}) \psi + {\frac}{1}{2} \partial_\mu S \partial^\mu S - g_S
(\cos\theta \,\bar\psi \psi + \sin\theta \,\bar\psi {\ensuremath{i\mkern1mu}}\gamma^5 \psi) S
- V_S (S, \, H),
\label{eq:lagrangian}\end{aligned}$$ where $$\begin{aligned}
V_S (S, \, H) = {\frac}{1}{2} m_0^2 S^2 + \lambda_1 H^\dagger H S +
\lambda_2 H^\dagger H S^2 + {\frac}{\lambda_3}{3!} S^3 +
{\frac}{\lambda_4}{4!} S^4 .\end{aligned}$$ The interactions of the singlet sector to the SM sector arise only through the Higgs portal $H^\dagger H$ as given above. Note that we extend the model proposed in [@Kim:2008pp; @Kim:2009ke; @Kim:2006af] by including the pseudoscalar interaction in the singlet sector. The inclusion of the pseudoscalar interaction is helpful for the analysis on the gamma-ray excess since it can conveniently fit the excess while satisfying the other constraints as explained in more detail in the next section.[^8]
Together with the SM Higgs potential, $$\begin{aligned}
V_\mathrm{SM} = -\mu^2 H^\dagger H + \lambda_0 (H^\dagger H)^2,\end{aligned}$$ the Higgs boson acquires a vacuum expectation value (VEV) after electroweak symmetry breaking (EWSB), and it can be written in the unitary gauge as $$\begin{aligned}
H = {\frac}{1}{\sqrt{2}}
\begin{pmatrix}
0 \\ v_h + h
\end{pmatrix}\end{aligned}$$ with $v_h = (\sqrt{2}G_F)^{-1/2} \simeq 246$ GeV. The singlet scalar field also develops a non-zero VEV $v_s$, so we expand the singlet scalar field around the VEV as $S = v_s + s$. The mass parameters $\mu^2$ and $m_0^2$ can be eliminated by using the minimization conditions of the full potential of the scalar fields, $V_S + V_\mathrm{SM}$. The relations are given as follows. $$\begin{aligned}
\mu^2 &= \lambda_0 v_h^2 + (\lambda_1 + \lambda_2 v_s) v_s ,{\nonumber}\\
m_0^2 &= - \l( {\frac}{\lambda_1}{2 v_s} + \lambda_2 \r) v_h^2
- \l( {\frac}{\lambda_3}{2 v_s} + {\frac}{\lambda_4}{6} \r) v_s^2 .\end{aligned}$$ In this setup, the mass term for the scalar fields $\Phi^\mathsf{T} = (h, \, s)$ is $$\begin{aligned}
{\mathcal}{L}_\mathrm{mass} = - {\frac}{1}{2} \Phi^\mathsf{T} {\mathcal}{M}_\Phi^2 \Phi
= -{\frac}{1}{2}
\begin{pmatrix}
h & s
\end{pmatrix}
\begin{pmatrix}
\mu_h^2 & \mu_{hs}^2\\
\mu_{hs}^2 & \mu_s^2
\end{pmatrix}
\begin{pmatrix}
h \\ s
\end{pmatrix}
\label{eq:higgs_mass_matrix}\end{aligned}$$ with $$\begin{aligned}
\mu_{h}^2 &= 2\lambda_0 v_h^2 ,{\nonumber}\\
\mu_{s}^2 &= -{\frac}{\lambda_1 v_h^2}{2 v_s} + {\frac}{(3\lambda_3 +
2\lambda_4 v_s) v_s}{6} ,{\nonumber}\\
\mu_{hs}^2 &= (\lambda_1 + 2\lambda_2 v_s) v_h .
\label{eq:baremass}\end{aligned}$$ Since the off-diagonal term in the mass matrix ${\mathcal}{M}_\Phi^2$ is non-vanishing in general, the physical Higgs states are admixtures of $h$ and $s$. $$\begin{aligned}
\begin{pmatrix}
h_1 \\ h_2
\end{pmatrix} = \begin{pmatrix}
\cos\theta_s & \sin\theta_s\\
-\sin\theta_s & \cos\theta_s
\end{pmatrix} \begin{pmatrix}
h \\ s
\end{pmatrix} ,\end{aligned}$$ where the mixing angle $\theta_s$ is given by $$\begin{aligned}
\tan\theta_s = {\frac}{y}{1 + \sqrt{1 + y^2}}\end{aligned}$$ with $y \equiv 2\mu_{hs}^2 / (\mu_h^2 - \mu_s^2)$. By diagonalizing the mass matrix in (\[eq:higgs\_mass\_matrix\]), we obtain the tree-level Higgs boson masses as follows. $$\begin{aligned}
m_{h_1,\,h_2}^2 = {\frac}{1}{2} \l[ (\mu_h^2 + \mu_s^2) \pm (\mu_h^2 -
\mu_s^2) \sqrt{1 + y^2} \r] .\end{aligned}$$ We assume that $h_1$ corresponds to the SM-like Higgs boson in what follows.
The Lagrangian in (\[eq:lagrangian\]) contains the pseudoscalar interaction in the singlet sector, which is proportional to $\sin\theta$. After the EWSB it can be performed a chiral rotation of the singlet fermion field $\psi$ as $$\begin{aligned}
\psi \to {\ensuremath{e\mkern1mu}}^{{\ensuremath{i\mkern1mu}}\gamma^5 \alpha/2} \psi ,\end{aligned}$$ and made the imaginary mass term of $\psi$ vanish by choosing $$\begin{aligned}
\tan\alpha = {\frac}{-g_S v_s \sin\theta}{m_{\psi_0} + g_S v_s
\cos\theta} .\end{aligned}$$ Then, the mass of the singlet fermion is given as $$\begin{aligned}
m_\psi &= (m_{\psi_0} + g_S v_s \cos\theta) \cos\alpha
- g_S v_S \sin\theta \sin\alpha {\nonumber}\\
&= \pm \sqrt{(m_{\psi_0} + g_S v_s\cos\theta)^2 + g_S^2 v_s^2
\sin^2\theta} .\end{aligned}$$ Note that we can always take the sign of $m_{\psi}$ to be positive by performing the chiral rotation further. By redefining the singlet fermion field using a chiral rotation described above, the interaction terms for the singlet fermion become $$\begin{aligned}
-{\mathcal}{L}^{\rm dark}_\mathrm{int} = g_S \cos\xi \,s \bar\psi \psi + g_S \sin\xi \,s
\bar\psi {\ensuremath{i\mkern1mu}}\gamma^5 \psi ,\end{aligned}$$ where $$\begin{aligned}
\cos\xi &= {\frac}{m_{\psi_0} \cos\theta + g_S v_s}{m_\psi} ,{\nonumber}\\
\sin\xi &= {\frac}{m_{\psi_0} \sin\theta}{m_\psi} .\end{aligned}$$ Consequently, the independent parameters for the singlet fermion are $m_\psi$, $g_S$, and $\xi$. The other six parameters $\lambda_0$, $\lambda_1$, $\lambda_2$, $\lambda_3$, $\lambda_4$, and $v_s$ with $v_h\simeq 246$ GeV in the scalar sector determine the masses $m_{h_1}$ and $m_{h_2}$, the mixing angle $\theta_s$, and self-couplings of the two physical Higgs particles $h_1$ and $h_2$. The cubic self-couplings $c_{ijk}$ for $h_i h_j h_k$ interactions are given as $$\begin{aligned}
c_{111}
=&~ 6 \lambda_0 v_h \cos^3 \theta_s + \l( 3 \lambda_1 + 6 \lambda_2
v_s \r) \cos^2 \theta_s \sin\theta_s + 6 \lambda_2 v_h
\cos\theta_s \sin^2\theta_s + (\lambda_3 + \lambda_4 v_s)
\sin^3\theta_s , {\nonumber}\\
c_{112}
=& -6 \lambda_0 v_h \cos^2 \theta_s \sin\theta_s + 2 \lambda_2 v_h
\l( 2 \cos^2\theta_s \sin\theta_s - \sin^3 \theta_s\r) {\nonumber}\\
& + \l(\lambda_1 + 2 \lambda_2 v_s \r) \l( \cos^3 \theta_s -
2 \cos\theta_s \sin^2\theta_s \r) + \l( \lambda_3 + \lambda_4 v_s
\r) \cos\theta_s \sin^2\theta_s , {\nonumber}\\
c_{122}
=&~ 6 \lambda_0 v_h \cos \theta_s \sin^2 \theta_s + 2 \lambda_2 v_h
\l( \cos^3\theta_s - 2 \cos\theta_s \sin^2 \theta_s\r) {\nonumber}\\
& - \l(\lambda_1 + 2 \lambda_2 v_s \r) \l( 2 \cos^2 \theta_s
\sin\theta_s - \sin^3 \theta_s \r) + \l( \lambda_3 +
\lambda_4 v_s \r) \cos^2\theta_s \sin\theta_s , {\nonumber}\\
c_{222}
=& - 6 \lambda_0 v_h \sin^3 \theta_s + \l( 3 \lambda_1 + 6
\lambda_2 v_s \r) \sin^2 \theta_s \cos\theta_s - 6 \lambda_2 v_h
\sin\theta_s \cos^2\theta_s {\nonumber}\\
& + (\lambda_3 + \lambda_4 v_s) \cos^3\theta_s .
\label{eq:cijk}\end{aligned}$$ Note that $c_{112}$ is practically proportional to $\sin\theta_s$ since $\lambda_1 + 2 \lambda_2 v_s$ is vanishing if $\sin\theta_s = 0$ while the other couplings can remain non-vanishing.
Galactic Center gamma-ray excess {#sec:gamma_excess}
================================
Several collaborations have analyzed the Fermi-LAT data and found statistically significant excesses of gamma-rays at the GC over the predictions of Galactic diffuse emission models [@Goodenough; @Daylan:2014rsa; @Calore:2014xka]. Consistency of the previous results was examined in Ref. [@Calore:2014nla] for the intensity of the excess at energies of 2 GeV as a function of Galactic latitude. It was shown that those excesses typically follow the predictions of a DM profile that is compatible with a generalized Navarro-Frenk-White density distribution, which is given by [@Navarro:1995iw; @Navarro:1996gj] $$\begin{aligned}
\rho(r) = \rho_s \frac{(r/r_s)^{-\gamma}}{(1 + r/r_s)^{3 - \gamma}}~,\end{aligned}$$ where $r$ is the distance from the GC. As canonical profile we choose the scale radius $r_s = 20$ kpc, the slope $\gamma = 1.2$, and fix the scale density $\rho_s$ by requiring that the local DM density $\rho=\rho_\odot = 0.4$ GeV/cm$^3$ at the location of the Solar system $r=r_\odot=8.5$ kpc.
For the present work we adopt the photon energy spectrum of the Fermi GeV excess derived by Calore, Cholis, and Weniger (CCW) in Refs. [@Calore:2014xka; @Calore:2014nla] including systematic and statistical errors. The CCW spectrum rises below 1 GeV, peaking around 2–3 GeV and has a high-energy tail up to 100 GeV. Although the excess could be explained by astrophysical sources like millisecond pulsars [@msp], the DM annihilation still remains as an intriguing explanation.
The gamma-ray differential flux from the annihilation of a non-self-conjugate DM $\chi$ over a solid angle $\Delta \Omega$ is given by $$\begin{aligned}
\frac{{\mathop{}\!\mathrm{d}}N}{{\mathop{}\!\mathrm{d}}E} =\frac{\bar J}{16\pi m_\chi^2}
\sum_{f} \langle \sigma v \rangle_f \frac{{\mathop{}\!\mathrm{d}}N_\gamma^f}{{\mathop{}\!\mathrm{d}}E},\end{aligned}$$ where the sum is extended over all possible annihilation channels into final states $f$. Here, $\langle \sigma v \rangle_f$ is the thermally averaged annihilation cross section and ${\mathop{}\!\mathrm{d}}N_\gamma^f / {\mathop{}\!\mathrm{d}}E$ is the DM prompt gamma-ray spectrum per annihilation to the final state $f$. While the annihilation cross section and the spectrum per annihilation depend on particle properties, the astrophysical factor $\bar J$ is determined from the line-of-sight (l.o.s) integral over the DM halo profile $\rho(r)$ averaged for a Region Of Interest (ROI) $\Delta\Omega$, $$\begin{aligned}
\bar J = \frac{1}{\Delta\Omega} \int_{\Delta\Omega}
\int_\mathrm{l.o.s} \rho^2 (r(s,\,\psi)) {\mathop{}\!\mathrm{d}}s {\mathop{}\!\mathrm{d}}\Omega,\end{aligned}$$ where $\psi$ is the angle from the GC. For the ROI in the CCW analysis ($2^\circ \le \left\vert b \right\vert \le 20^\circ$ for Galactic latitude and $\left\vert \ell \right\vert \le 20^\circ$ for Galactic longitude) with the canonical profile for the DM halo, the value of $\bar J$ is given as $\bar J_\mathrm{canonical} \simeq 2 \times
10^{23}$ $\mathrm{GeV}^2 / \mathrm{cm}^5$. However, it is known that there is a significant uncertainty on the DM density profile near the GC in particular. If the uncertainty on the DM profile is included, the $\bar J$ value varies from about 10% to few times the canonical value. In representing our analysis results we will depict the range between 0.19 and 3 times the canonical one as in Ref. [@Agrawal:2014oha]. In practice, we have extended the allowed range to \[0.17, 5.3\] for numerical calculations to find the best-fit parameter point.
The expected spectra from the DM annihilations in our analysis will be depicted alongside with the systematic uncertainties from the diagonal elements of the covariance matrix given in [@Calore:2014xka; @Calore:2014nla]. However the values of goodness-of-fit $\chi^2$ are calculated using the full covariance matrix which includes the large off-diagonal elements due to the strong correlation of the systematic uncertainties in different energy bins. `LanHEP` [@Semenov:2008jy] has been used to implement the SFDM model described in Sec. \[sec:model\], and the photon spectra from the annihilation of the SFDM are obtained by using `MicrOMEGAs` [@Belanger:2014vza]. To illustrate our analysis results we choose parameters with the minimum $\chi^2$ while providing the DM relic density consistent with the observed value, but by changing the scale factor $\bar J$ explained above.
The annihilation of the SFDM through a pure scalar interaction ($\sin\xi = 0$) is velocity-suppressed. Therefore, it is inevitable to introduce a pseudoscalar interaction in order to explain the Fermi gamma-ray excess from the DM annihilation, taking into account its current velocity at the GC as small as $10^{-3}$. For the pure pseudoscalar interaction ($\sin\xi=1$), the main annihilation channel arises from the $s$-channel pseudoscalar exchange because the contributions from $t$ and $u$-channels and interference terms are still $p$-wave suppressed. Moreover, for the pure pseudoscalar interaction, the elastic scattering of the DM on the target nuclei is velocity-suppressed, and in consequence, the constraints from direct detection experiments can easily be satisfied [@LopezHonorez:2012kv]. In this regard we mainly consider the pure pseudoscalar interaction ($\sin\xi =1$), but include the analysis for the case of mixed scalar and pseudoscalar interactions if it is necessary to fit the gamma-ray excess avoiding other astrophysical constraints.
--------------------------------------------------------- ---------------- -----------------
\[-2mm\] Annihilation process $m_\psi$ (GeV) $m_{h_2}$ (GeV)
\[2mm\]
\[-2mm\] $\psi \bar \psi \to h_2 \to b \bar b$ 49.82 99.416
\[2mm\] $\psi \bar \psi \to h_2 \to h_1 h_1$ 127.5 213.5
\[2mm\] $\psi \bar \psi \to h_2 \to h_2 h_2$, $h_1 h_2$ 127.5 125.7
\[2mm\] $\psi \bar \psi \to h_2 \to h_2 h_2$ 69.2 35.7
\[2mm\]
--------------------------------------------------------- ---------------- -----------------
: Dominant annihilation channels that can contribute to the gamma-ray excess are listed with the best fitted masses of the DM and the singlet-like Higgs boson.[]{data-label="table:processes"}
The detailed analysis is proceeded by finding the parameter space in each scenario of the main annihilation channels, $\psi \bar \psi \to b \bar b$, $h_i h_j (\to 4b)$ for $i, \, j = 1, \, 2$. Note that these decay modes are proven to give good fits in model independent analyses by other groups.[^9] The goodness of fits can be different in this model dependent study partly due to theoretical and experimental bounds that can impose constraints on the model. On the other hand, it often occurs that several processes contribute to the DM annihilation, depending on the mass hierarchies of particles and couplings. The mass values of the SFDM and the singlet-like Higgs boson $h_2$ that turn out to give the best fits are shown in Table \[table:processes\] for each dominant annihilation process, which will be discussed in more detail in the following subsections. We set the mass of SM-like Higgs $m_{h_1} \simeq 125$ GeV throughout our analyses and choose a small mixing angle $\sin\theta_s \lesssim 0.12$ in order to be compatible with the SM-like Higgs properties from the LHC analysis results [@atlascms].
$\psi \bar{\psi} \rightarrow b \bar{b}$ annihilation channel
------------------------------------------------------------
In the model independent study in [@Calore:2014nla], it was shown that the DM annihilation into $b\bar{b}$ gives a good fit ($\chi^2 = 23.9$, $p$-value 0.35) to the gamma-ray excess data if $m_\mathrm{DM} \simeq 48.7$ GeV and $\langle \sigma v \rangle \simeq 1.75 \times 10^{-26} \,\mathrm{cm}^3 \,\mathrm{s}^{-1}$ for a self-conjugate DM. In this subsection we investigate the corresponding parameter space in the SFDM model to see if such a scenario can be realized in the model.
The SM-like Higgs boson $h_1$ would decay dominantly to the $\psi \bar{\psi}$ pair for $m_\psi \simeq 50$ GeV unless the coupling for the decay vertex $g_S \,\sin\theta_s$ is small. The current analysis results on the SM-like Higgs at the LHC [@atlascms] indicate that its invisible branching ratio should be smaller than 13% at 95% C.L., provided that the production of the Higgs boson is not affected much by unknown new physics effect. This experimental constraint on the invisible branching ratio of the Higgs boson implies the upper bound on the coupling $(g_S \,\sin\theta_s)^2
\lesssim 4 \times 10^{-4}$.
The pair annihilation of the SFDM to a $b\bar{b}$ pair can proceed through $s$-channel (singlet-like) Higgs exchange diagram. For the $\psi\bar\psi\rightarrow h_2
\rightarrow b\bar{b}$ process, the annihilation cross section is given by $$\begin{aligned}
\sigma v =
{g_S^2~ \sin^2 2\theta_s \over 32\pi} \l({m_b\over v_h}\r)^2
{s\over (s-m_{h_2}^2)^2+m_{h_2}^2\,\Gamma_2^2}~
\bigg(1-{4 m_b^2 \over s}\bigg)^{3/2} N_c,\end{aligned}$$ where $\sqrt{s}$ is the center of mass energy of the annihilation process, $\Gamma_2$ is the decay width of $h_2$ and $N_c$ is the number of color of the $b$ quark. The upper bound of $(g_S\,\sin\theta_s)^2 \lesssim 4\times 10^{-4}$ from the LHC results in too small annihilation cross section to explain the DM relic density and the Fermi gamma-ray excess, unless there is a resonance effect with $m_{h_2} \simeq 2 m_\psi$. Figure \[fig:bb1\] shows such a resonance effect on the DM relic density $\Omega h^2$ and the thermally averaged annihilation cross section $\langle \sigma v \rangle$ at the present universe. $\Omega h^2$ (red dashed line) and $\langle \sigma v \rangle$ (black solid line) are depicted on different $m_\psi$ values nearby the resonance region. Here we fix the model parameters as follows: For the scalar sector, $\lambda_0 = 0.128816$, $\lambda_1 =36.625338$ GeV, $\lambda_2 = - 0.131185$, $\lambda_3 = - 333.447606$ GeV, $\lambda_4 = 5.648618$, and $v_s = 150.017297$ GeV, which gives $m_{h_1} = 125.3$ GeV, $m_{h_2} = 99.416$ GeV, $\sin\theta_s = - 0.117$. [^10] We also set $g_S = 0.0958$ and $\sin\xi =1$, which corresponds to the pure pseudoscalar interaction. As $m_\psi$ increases from 45 GeV, one can see that the relic density of the DM $\Omega h^2$ drops down, but suddenly boosts up after passing the resonance point $m_\psi = m_{h_2} / 2 \sim 49.6$ GeV. For $m_\psi = 49.82$ GeV, we obtain the DM relic density $\Omega h^2 = 0.122$, which is consistent with the current measured value from Planck [@Ade:2015xua] and the fraction of the annihilation process of $\psi \bar \psi \to b \bar{b}$ reaches 86.8%. However, for the same parameter values, we have the annihilation cross section $\langle \sigma v \rangle = 1.55 \times 10^{-25} \,\mathrm{cm}^3 \,\mathrm{s}^{-1}$, which yields too large gamma-ray flux at the GC if the canonical $\bar
J$ value is used.
Note that the $\langle \sigma v \rangle$ value at the GC can be much larger than the usual thermal annihilation rate at the early universe. It is because a small difference in the center of mass energy of the DM annihilation gives a huge difference on the annihilation cross section in the resonance region. As a consequence, the choice of the parameter values results in a bad fit to the gamma-ray data, as one can see in Fig. \[fig:bb\].[^11] However, as mentioned above, the astrophysical $\bar J$ factor has large uncertainties. Thus in order to have a desired gamma-ray flux at the GC, it is required to have a smaller $\bar J$ value than the canonical one to compensate too large $\langle \sigma v \rangle$. We find that the best fit ($\chi^2=23.53$, $p$-value = 0.37) to the Fermi gamma-ray excess is obtained with ${\cal J} = 0.23$ for $\bar J =
{\cal J} \times \bar{J}_\mathrm{canonical}$. The corresponding gamma-ray spectrum is shown as solid line in Fig. \[fig:bb\], where the gamma-ray spectra with other $\bar J$ values are shown as well.
![$\langle \sigma v \rangle$ (black solid line) and $\Omega h^2$ (red dashed line) as a function of $m_\psi$ near the resonance region ($m_\psi \sim m_{h_2}/2$) in the case of the pure pseudoscalar interaction. See the text for details. For $m_\psi = 49.82$ GeV, $\Omega h^2 = 0.122$ and $\langle \sigma v \rangle = 1.55 \times 10^{-25}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$.[]{data-label="fig:bb1"}](bb1.pdf){width=".6\textwidth"}
![Photon energy spectra for $m_\psi = 49.82$ GeV with different $\cal J$ values. The DM annihilation is dominated by $\psi \bar{\psi} \rightarrow b
\bar{b}$ process (87$\%$). The Higgs masses are $m_{h_1} = 125.3$ GeV, $m_{h_2} = 99.416$ GeV, and $\Omega h^2 = 0.122$, $\langle \sigma v \rangle = 1.55 \times
10^{-25}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$. $\chi^2 = 23.53$ ($p$-value $= 0.37$) in the best-fit parameter point with ${\cal J} = 0.23$.[]{data-label="fig:bb"}](figure-bb.png){width=".6\textwidth"}
Although the Fermi gamma-ray excess at the GC can be explained for a smaller value of the $\bar J$ factor, the annihilation cross section is too large to evade the constraints by the observations of the gamma-ray from the dwarf spheroidal galaxies [@Ackermann:2015zua]. It sets an upper bound $\langle \sigma v \rangle \lesssim 2 \times 10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$ for the non-self-conjugate DM case with $m_\textrm{DM} \sim 50$ GeV if the majority of the annihilation products are $b \bar b$. Furthermore, the bounds from the antiproton ratio measured by PAMELA [@Adriani:2010rc] and AMS-02 [@AMS02; @Giesen:2015ufa] can strongly constrain the parameters for $\psi\bar\psi\rightarrow b\bar{b}$ channel. Even taking into account the uncertainties of the propagation models, the bound should be at least $\langle \sigma v \rangle \lesssim 2 \times 10^{-26} \,\mathrm{cm}^3
\,\mathrm{s}^{-1}$ for the non-self-conjugate DM with $m_\textrm{DM}
\sim 50$ GeV.[^12]
![$\langle \sigma v \rangle$ (black solid line) and $\Omega h^2$ (red dashed line) as a function of $m_\psi$ near the resonance region ($m_\psi \sim m_{h_2}/2$) in the case of the mixed scalar and pseudoscalar interaction. See the text for details. For $m_\psi = 49.706$ GeV, $\Omega h^2 = 0.118$ and $\langle \sigma v \rangle = 1.5 \times 10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$.[]{data-label="fig:bb2"}](bb2.pdf){width=".6\textwidth"}
To resolve this problem we can alternatively consider a mixture of scalar and pseudoscalar interactions between singlet scalar and singlet fermion, [*i.e.*]{}, $\sin\xi < 1$ in order to reduce the magnitude of $\langle \sigma v \rangle$ to an acceptable level while keeping $\Omega h^2 \sim 0.12$ and the direct detection rate of the DM small enough. This is a viable scenario since the annihilation rate and the relic density of the DM depend on the DM velocity in different ways for the scalar and the pseudoscalar interactions. For a demonstration of the effect, we set $\sin\xi = 0.01$ which is an extreme choice making the dark sector Yukawa interaction almost scalar-like and $g_S = 0.055$ while other model parameters unchanged. Then we obtained the annihilation cross section and the relic density as shown in Fig. \[fig:bb2\]. The photon flux explaining the gamma-ray excess is obtained for $m_\psi = 49.706$ giving $\langle \sigma v \rangle = 1.5 \times 10^{-26}$ $\mathrm{cm}^3\,\mathrm{s}^{-1}$ and $\Omega h^2 = 0.118$. This can be seen in Fig. \[fig:bbmixed\] and the best fit is obtained with ${\cal J} = 2.5$. The annihilation cross section is now within an acceptable range satisfying the astrophysical constraints mentioned above. In addition the spin independent cross section of the DM recoiling against neutron or proton is still around $6.3 \times 10^{-48}~{\rm cm}^2$ which is below the bounds from various direct detection experiments. This is because of the small mixing angle $\theta_s$ although we considered mostly scalar-like interaction $s \bar \psi \psi$.
![Photon energy spectra for $m_\psi = 49.706$ GeV with a mixture of scalar and pseudoscalar interactions between singlet scalar and singlet fermion $\sin\xi = 0.01$. Here $\Omega h^2 = 0.118$ and $\langle \sigma v \rangle = 1.5 \times
10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$. $\chi^2 = 23.65$ ($p$-value $= 0.37$) in the best-fit parameter point with ${\cal J} = 2.5$.[]{data-label="fig:bbmixed"}](figure-mixed-sinxi001.png){width=".6\textwidth"}
$\psi \bar{\psi} \rightarrow h_1 h_1$ annihilation channel
----------------------------------------------------------
The annihilation into a non-relativistic pair of the Higgs boson can give a good fit to the gamma-ray excess [@Agrawal:2014oha; @Calore:2014nla]. It has been shown in Ref. [@Calore:2014nla] that the best $\chi^2 =
29.5$ ($p$-value $= 0.13$) is obtained with $m_\psi \simeq m_{h_1}
\simeq 125.7$ GeV and $\langle \sigma v \rangle = 5.33 \times 10^{-26}$ $\mathrm{cm}^3
\,\mathrm{s}^{-1}$ for a self-conjugate DM that annihilates into $h_1
h_1$. Here we investigate if this scenario can be realized in the SFDM model. Diagrams for the annihilation processes $\psi\bar\psi
\rightarrow h_i h_j$ are shown in Fig. \[fig:diagram\].
For the pure scalar interaction, [*i.e.*]{}, $\sin\xi= 0$, the annihilation cross section vanishes in the zero-velocity limit. On the other hand, for the pure pseudoscalar interaction [*i.e.*]{}, $\sin\xi = 1$, only $s$-channel diagram contributes to the annihilation cross section in the zero-velocity limit. Therefore, magnitudes of cubic couplings $c_{ijk}$ of the Higgses given in (\[eq:cijk\]) play important roles for the processes $\psi\bar\psi
\rightarrow h_i h_j$ in the zero-velocity limit. The annihilation cross section for $\psi\bar\psi \rightarrow h_2
\rightarrow h_1 h_1$, which would provide the most important contribution to the $h_1 h_1$ channel, is given by $$\begin{aligned}
\sigma v
= {\frac}{g_S^2}{32\pi} \sqrt{1-{\frac}{4m_{h_1}^2}{s}} \,
{\frac}{c_{112}^2 \cos^2 \theta_s}{(s - m_{h_2}^2)^2 + m_{h_2}^2 \Gamma_{2}^2}.\end{aligned}$$
![Diagrams for $\psi\psi \to h_i h_j$ annihilation processes. $i$, $j$, $k = 1$, 2.[]{data-label="fig:diagram"}](schan.pdf "fig:"){width="32.00000%"} ![Diagrams for $\psi\psi \to h_i h_j$ annihilation processes. $i$, $j$, $k = 1$, 2.[]{data-label="fig:diagram"}](tchan.pdf "fig:"){width="32.00000%"} ![Diagrams for $\psi\psi \to h_i h_j$ annihilation processes. $i$, $j$, $k = 1$, 2.[]{data-label="fig:diagram"}](uchan.pdf "fig:"){width="32.00000%"}
If $\psi\bar\psi$ annihilation into $h_1 h_1$ channel opens, the annihilation rates into $WW$ and $ZZ$ channels can be sizable as well. The channels with SM gauge bosons are known to yield relatively a bad fit to the Fermi gamma-ray excess [@Calore:2014nla]. Thus, good fits can be obtained if the DM annihilation rate into $h_1 h_1$ is dominant over $WW$ and $ZZ$ by having a large value of the cubic coupling $c_{112}$ for given masses $m_{h_1}$, $m_{h_2}$, and the mixing angle $\theta_s$.
As a specific example, we choose the parameter values as follows: $m_\psi = 127.5$ GeV, $g_S = 0.098$, and $\sin\xi=1$ for interactions of the SFDM, and $\lambda_0 =0.1315$, $\lambda_1 = 1237.8$ GeV, $\lambda_2 = - 2.0$, $\lambda_3 = - 820.5$ GeV, $\lambda_4 = 9.39$, and $v_s = 306.15$ GeV for the scalar sector, which yield Higgs masses $m_{h_1}=124.9$ GeV, $m_{h_2}=213.5$ GeV, and the mixing angle $\sin\theta_s=-0.11$ with the cubic couplings $c_{111}=149.0$ GeV and $c_{112}=268.8$ GeV. With this parameter choice, $\psi\bar\psi \rightarrow h_1 h_1$ becomes the most dominant annihilation process ($\simeq 96\%$ for the annihilation at the GC), and we have $\Omega h^2 = 0.12$ and $\langle \sigma v \rangle = 2.11 \times 10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$. The $\langle \sigma v \rangle$ value is rather smaller than desired, so a large $\mathcal J$ factor is necessary to explain the Fermi gamma-ray excess. For $\mathcal{J}=4$, we obtain the best fit ($\chi^2 = 31.3$, $p$-value $= 0.09$) and the corresponding gamma-ray spectrum is shown in Fig. \[fig:h1h1\].
The important astrophysical bounds to be considered in this scenario are the observations of gamma-rays from the dwarf spheroidal galaxies and the antiproton ratio. The annihilation cross section value that we obtained is below the upper bound from the dwarf spheroidal galaxies for a non-self-conjugate DM, $5.2 \times
10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$ around $m_\psi \simeq
125$ GeV, assuming that the dominant annihilation process is $\psi \bar \psi \to
b \bar b$ [@Ackermann:2015zua]. For the four-body final states, [*i.e.*]{}, $\psi \bar \psi \to
b \bar b b \bar b$, the authors of Ref. [@Dutta:2015ysa] extracted rough bounds, but they tend to be less constrained than the two-body case. Therefore, the best-fit parameter in our analysis is safe from the gamma-ray bound from the dwarf spheroidal galaxies. On the other hand, the $4b$ final state has been included in the analysis of [@Cline:2015qha] in light of the search results on the antiproton excess combined from BESS [@BESS], CAPRICE [@Boezio:2001ac], and PAMELA [@Adriani:2010rc]. The value of the annihilation cross section for $m_\psi \simeq 125$ GeV that we obtained is below this bound if the uncertainties of the propagation models are included.
![Photon energy spectra for $m_\psi = 127.5$ GeV with different $\mathcal{J}$ values. The DM annihilation is dominated by $\psi \bar{\psi} \to h_1 h_1$ process (96%). The Higgs masses are $m_{h_1} = 124.9$ GeV, $m_{h_2} = 213.5$ GeV, and $\Omega h^2 = 0.12$, $\langle \sigma v \rangle = 2.11 \times
10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$. $\chi^2 = 31.32$ ($p$-value $= 0.09$) in the best-fit parameter point with ${\cal J} = 4.0$.[]{data-label="fig:h1h1"}](figure-h1h1.png){width=".6\textwidth"}
$\psi \bar\psi \to h_1 h_1$/$h_1 h_2$/$h_2 h_2$ annihilations in the mass-degenerate case
-----------------------------------------------------------------------------------------
Another interesting possibility that is closely related to the $h_1 h_1$ channel in the previous subsection arises if two Higgses $h_1$ and $h_2$ are almost degenerate in mass. Then all the annihilation modes $\psi \bar \psi \to h_{1,2} \to h_1
h_1$/$h_1 h_2$/$h_2 h_2$ have no differences in the phase space and provide the same spectral shape for the photon energy spectrum.
We find the parameter choice giving one of the best fit for the galactic gamma-ray excess at $m_\psi =127.5$ GeV is $\lambda_0 = 0.13$, $\lambda_1 = 112.49$ GeV, $\lambda_2 = -0.20$, $\lambda_3 = -898.97$ GeV, $\lambda_4 = 5.97$, $v_s = 277.01$ GeV, and $g_S = 0.085$, which gives $m_{h_1} = 125.5$ GeV, $m_{h_2} = 125.7$ GeV, $\sin\theta_s = -0.11$, $c_{222} = 705.8$ GeV, and $c_{122} = -177.8$ GeV. In this case the process $\psi \bar \psi \to h_2 \to h_2 h_2$ is the most dominant since the amplitude is not suppressed by the smallness of the mixing angle $\sin\theta_s$ and the magnitude of $c_{222}$ is much larger than that of $c_{122}$. Note that the values of $c_{222}$ (and $c_{122}$) can be arbitrarily given without affecting the scalar masses and mixing angle. The annihilation cross section for $\psi\bar\psi \rightarrow h_2
\rightarrow h_2 h_2$ is given by $$\begin{aligned}
\sigma v
= {\frac}{g_S^2}{32\pi} \sqrt{1-{\frac}{4m_{h_2}^2}{s}} \,
{\frac}{c_{222}^2 \cos^2 \theta_s}{(s - m_{h_2}^2)^2 + m_{h_2}^2
\Gamma_{2}^2}.
\label{eq:ann_h2h2}\end{aligned}$$ With those parameters we obtain the DM relic density $\Omega h^2 =
0.12$ and the total annihilation cross section $\langle \sigma v \rangle = 1.71 \times
10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$. The fraction of the DM annihilation rate to $h_2 h_2$ is 88.2% while that to $h_1 h_2$ is 11.6%. The annihilation cross section value is in the allowed region for the constraints from the dwarf spheroidal galaxies [@Dutta:2015ysa] and also from the antiproton measurements [@Cline:2015qha], but $\mathcal{J} = 4.822$ is required to explain the Fermi gamma-ray excess. We obtain an acceptable fit ($\chi^2 = 30.8$, $p$-value = 0.1) with this large $\mathcal J$ factor. The corresponding gamma-ray spectrum is shown in Fig. \[fig:h1h2deg\].
![Photon energy spectra for $m_\psi = 127.5$ GeV with different $\mathcal{J}$ values. The DM annihilation is dominated by $\psi \bar{\psi} \to h_i h_j$ $(i, \, j = 1,\, 2)$ process ($\simeq 100\%$). The Higgs masses are $m_{h_1} = 125.5$ GeV, $m_{h_2} = 125.7$ GeV, and $\Omega h^2 = 0.12$, $\langle \sigma v \rangle = 1.71 \times
10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$. $\chi^2 = 30.8$ ($p$-value $= 0.1$) in the best-fit parameter point with ${\cal J} = 4.822$.[]{data-label="fig:h1h2deg"}](figure-h1h2degenerate.png){width=".6\textwidth"}
$\psi \bar{\psi} \rightarrow h_2 h_2$ annihilation channel
----------------------------------------------------------
One of interesting features of the SFDM model is that there is a mediator $h_2$, which is an important new particle for realizing what is called the secluded WIMP scenario in our setup [@pospelov:2008; @Kim:2009ke]. Various mediator particles have been introduced in several model independent studies to explain the Fermi gamma-ray excess from the production of a pair of the light mediator particle with its subsequent cascade decay into SM fermions [@Abdullah:2014lla; @Berlin:2014pya; @Cline:2015qha; @Rajaraman:2015xka; @Dutta:2015ysa; @Fortes:2015qka; @Elor:2015tva]. In the model independent study of Ref. [@Dutta:2015ysa], it was shown that the DM annihilation into a pair of new particles ($\phi\phi$) with subsequent $\phi$ decay to $b\bar{b}$, gives a good fit ($\chi^2 = 23.1$) if $m_\mathrm{DM} = 65$ GeV, $m_\phi=m_\mathrm{DM}/2$ and $\langle \sigma v \rangle = 2.45 \times 10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$ for a self-conjugate DM. In this subsection we consider the corresponding channel in our model and find the best-fit parameters by varying the masses.
The fraction of the DM annihilation rate for $\psi \bar \psi \to h_2 h_2$ with $m_\psi \simeq 70$ GeV and $m_{h_2} < m_\psi$ can easily become as large as 100% by taking suitable parameter values of the model. We also found that the best-fit spectrum is obtained if $m_{h_2} \sim m_\psi / 2$, as pointed out in the model independent study [@Dutta:2015ysa]. Finding parameters for the good fits, we further consider the bound from the search of exotic Higgs decays [@atlascms] due to the decay mode $h_1 \to h_2 h_2 \to 4b$. Our choice of model parameters for the best fit is as follows: $m_\psi = 69.2$ GeV, $g_S=0.056$, $m_{h_1}=125.1$ GeV, $m_{h_2} = 35.7$ GeV, $\sin\theta_s = 0.025$, and $c_{222} = 215.1$ GeV from $\lambda_0 = 0.13$, $\lambda_1 = 4.5$ GeV, $\lambda_2 = -0.0055$, $\lambda_3 = -391.51$ GeV, $\lambda_4 = 2.20$, and $v_s = 276.21$ GeV. With these parameters the dominant contribution for the DM annihilation comes from the $\psi\bar\psi \rightarrow h_2 \rightarrow h_2 h_2$ process. See Eq. (\[eq:ann\_h2h2\]) for the corresponding annihilation cross section formula. Here the relic density $\Omega h^2 = 0.121$ and $\langle \sigma v \rangle = 2.26 \times 10^{-26}$ $\mathrm{cm}^3\,s^{-1}$. We find that a good value of $\chi^2 = 23.19$ ($p$-value $= 0.39$) can be obtained with a moderate $J$ factor value, $\mathcal{J}=2.2$. The corresponding gamma-ray spectrum is shown in Fig. \[fig:h2h2\].
The astrophysical bounds can be important like the previous scenarios. The analysis results on the gamma-ray search coming from the dwarf spheroidal galaxies in [@Dutta:2015ysa] show that the upper bound of the annihilation rate of $\psi\bar\psi \rightarrow h_2 h_2 \rightarrow 4 b$ is expected to be at least $3.3 \times 10^{-26}~\mathrm{cm}^3\,\mathrm{s}^{-1}$ with $m_\psi =70$ GeV for a non-self-conjugate DM. Therefore, our $\langle \sigma v \rangle$ value from the best-fit parameters is below the current upper bound. Following the antiproton bound for the $4b$ final-state analyzed in [@Cline:2015qha], as commented in previous subsections, $\langle \sigma v \rangle = 2.26 \times 10^{-26}$ $\mathrm{cm}^3\,\mathrm{s}^{-1}$ at $m_\psi \simeq 70$ GeV is below the bound if the uncertainties in the propagation models are taken into consideration.
![Photon energy spectra for $m_\psi = 69.2$ GeV with different $\mathcal{J}$ values. The DM annihilation is dominated by $\psi \bar{\psi} \to h_2 h_2$ process ($\simeq 100$%). The Higgs masses are $m_{h_1} = 125.1$ GeV, $m_{h_2} = 35.7$ GeV, and $\Omega h^2 = 0.121$, $\langle \sigma v \rangle = 2.26 \times
10^{-26}$ $\mathrm{cm}^3 \,\mathrm{s}^{-1}$. $\chi^2 = 23.19$ ($p$-value $= 0.39$) in the best-fit parameter point with ${\cal J} = 2.2$.[]{data-label="fig:h2h2"}](figure-h2h2.png){width=".6\textwidth"}
Conclusions {#sec:concl}
===========
In this paper we considered a model with the SFDM, in which the mechanism of the thermal freeze-out is secluded from its observations in the direct detection and collider experiments. In addition to suppressing the mixing angle between the SM Higgs boson and the singlet scalar we introduced a pseudoscalar interaction at the singlet sector to amplify the secludedness. In this type of model the DM search is generically difficult in the direct detection and collider experiments due to the secludedness. Nonetheless, various indirect detection results can shed light on the parameter space to be probed. As an observational guide, we applied the model to the recent results on a few GeV level gamma-ray excess at the GC revealed by the analyses on the Fermi-LAT data, which has been a hot issue in both theoretical and experimental sides to date.
As a concrete analysis we adopted the results by CCW and applied the systematic uncertainties estimated by them. Then we categorized the annihilation processes depending on the final states, $\psi \bar \psi \to b \bar{b}$, $h_1 h_1$, $h_1 h_1$/$h_1 h_2$/$h_2
h_2$, or $h_2 h_2$, where the latter three channels are cascade processes producing multiple SM fermions or gauge bosons. The direction of our paper is not just explaining the gamma-ray excess but finding the model parameter values preferred by the observation and the constraints for the future study of the SFDM model. In this regard, other astrophysical constraints such as gamma-ray bounds from the dwarf spheroidal galaxies (by Fermi-LAT) and searches of antiproton excesses (by PAMELA and AMS-02) together with LHC bounds on the Higgs boson are taken into account in our study. In order to satisfy these bounds we keep the value of the observed relic density of the DM while adopting the large uncertainties of the DM density profile near the GC to obtain the best-fit parameter point in each channel. Our analysis found that the excess can be obtained with the similar level of $\chi^2$ values as those in various model independent searches, particularly for $\psi \bar \psi \to h_2 \to b \bar{b}$ and $h_2 h_2$ channels with $(m_\psi, \, m_{h_2}) = (49.82~\mathrm{GeV}, \,
99.416~\mathrm{GeV})$, $(69.2~\mathrm{GeV}, \, 35.7~\mathrm{GeV})$, respectively considering the pure pseudoscalar interaction in the dark sector. However the former case is again strongly constrained from astrophysical and collider bounds commented above so a mixture of the singlet and pseudoscalar interaction in the dark sector is needed.
Although it is not easy to find the signals of the secluded SFDM model at the current level of the LHC, we may observe those in the future high luminosity LHC or next generation colliders. In particular various channels by trilinear Higgs interactions can provide interesting signals. We will proceed the collider analyses for the parameter space found in this work, targeting their signatures at the LHC and future colliders [@future].
Acknowledgements {#acknowledgements .unnumbered}
================
YGK is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Korean Ministry of Education, Science and Technology (NRF-2013R1A1A2012392). KYL is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (Grant No. NRF-2015R1A2A2A01004532). SS thanks Jong-Chul Park for useful discussions on various astrophysical constraints. SS also thanks the University of Tokyo and the Korea Institute for Advanced Study for hospitality and support during the completion of this work.
[99]{}
L. Goodenough and D. Hooper, arXiv:0910.2998 \[hep-ph\]; D. Hooper and L. Goodenough, Phys. Lett. B [**697**]{}, 412 (2011) \[arXiv:1010.2752 \[hep-ph\]\].
D. Hooper and T. Linden, Phys. Rev. D [**84**]{}, 123005 (2011) \[arXiv:1110.0006 \[astro-ph.HE\]\]. K. N. Abazajian and M. Kaplinghat, Phys. Rev. D [**86**]{}, 083511 (2012) \[Phys. Rev. D [**87**]{}, 129902 (2013)\] \[arXiv:1207.6047 \[astro-ph.HE\]\]. D. Hooper and T. R. Slatyer, Phys. Dark Univ. [**2**]{}, 118 (2013) \[arXiv:1302.6589 \[astro-ph.HE\]\]. C. Gordon and O. Macias, Phys. Rev. D [**88**]{}, no. 8, 083521 (2013) \[Phys. Rev. D [**89**]{}, no. 4, 049901 (2014)\] \[arXiv:1306.5725 \[astro-ph.HE\]\]. K. N. Abazajian, N. Canac, S. Horiuchi and M. Kaplinghat, Phys. Rev. D [**90**]{}, no. 2, 023526 (2014) \[arXiv:1402.4090 \[astro-ph.HE\]\]. T. Daylan, D. P. Finkbeiner, D. Hooper, T. Linden, S. K. N. Portillo, N. L. Rodd and T. R. Slatyer, arXiv:1402.6703 \[astro-ph.HE\].
F. Calore, I. Cholis and C. Weniger, JCAP [**1503**]{}, 038 (2015) \[arXiv:1409.0042 \[astro-ph.CO\]\].
F. Calore, I. Cholis, C. McCabe and C. Weniger, Phys. Rev. D [**91**]{}, no. 6, 063003 (2015) \[arXiv:1411.4647 \[hep-ph\]\]. M. Ajello [*et al.*]{} \[Fermi-LAT Collaboration\], arXiv:1511.02938 \[astro-ph.HE\]. K. N. Abazajian, JCAP [**1103**]{}, 010 (2011) \[arXiv:1011.4275 \[astro-ph.HE\]\]; Q. Yuan and B. Zhang, JHEAp [**3-4**]{}, 1 (2014) \[arXiv:1404.2318 \[astro-ph.HE\]\].
R. Bartels, S. Krishnamurthy and C. Weniger, arXiv:1506.05104 \[astro-ph.HE\]. S. K. Lee, M. Lisanti, B. R. Safdi, T. R. Slatyer and W. Xue, arXiv:1506.05124 \[astro-ph.HE\]. T. Lacroix, C. Boehm and J. Silk, Phys. Rev. D [**90**]{}, no. 4, 043508 (2014) \[arXiv:1403.1987 \[astro-ph.HE\]\]. P. Agrawal, B. Batell, P. J. Fox and R. Harnik, JCAP [**1505**]{}, 011 (2015) \[arXiv:1411.2592 \[hep-ph\]\]. B. Dutta, Y. Gao, T. Ghosh and L. E. Strigari, Phys. Rev. D [**92**]{}, no. 7, 075019 (2015) \[arXiv:1508.05989 \[hep-ph\]\]. J. M. Cline, G. Dupuis, Z. Liu and W. Xue, Phys. Rev. D [**91**]{}, no. 11, 115010 (2015) \[arXiv:1503.08213 \[hep-ph\]\]. A. Rajaraman, J. Smolinsky and P. Tanedo, arXiv:1503.05919 \[hep-ph\]. P. Ko and Y. Tang, arXiv:1504.03908 \[hep-ph\]. M. Abdullah, A. DiFranzo, A. Rajaraman, T. M. P. Tait, P. Tanedo and A. M. Wijangco, Phys. Rev. D [**90**]{}, 035004 (2014) \[arXiv:1404.6528 \[hep-ph\]\]. G. Elor, N. L. Rodd and T. R. Slatyer, Phys. Rev. D [**91**]{}, 103531 (2015) \[arXiv:1503.01773 \[hep-ph\]\]. D. Kim and J. C. Park, arXiv:1507.07922 \[hep-ph\]. J. D. Ruiz-Alvarez, C. A. de S.Pires, F. S. Queiroz, D. Restrepo and P. S. Rodrigues da Silva, Phys. Rev. D [**86**]{}, 075011 (2012) \[arXiv:1206.5779 \[hep-ph\]\]. N. Okada and O. Seto, Phys. Rev. D [**89**]{}, no. 4, 043525 (2014) \[arXiv:1310.5991 \[hep-ph\]\]. K. P. Modak, D. Majumdar and S. Rakshit, JCAP [**1503**]{}, 011 (2015) \[arXiv:1312.7488 \[hep-ph\]\]. A. Alves, S. Profumo, F. S. Queiroz and W. Shepherd, Phys. Rev. D [**90**]{}, no. 11, 115003 (2014) \[arXiv:1403.5027 \[hep-ph\]\]. S. Ipek, D. McKeen and A. E. Nelson, Phys. Rev. D [**90**]{}, no. 5, 055021 (2014) \[arXiv:1404.3716 \[hep-ph\]\]. T. Mondal and T. Basak, Phys. Lett. B [**744**]{}, 208 (2015) \[arXiv:1405.4877 \[hep-ph\]\]. L. Wang and X. F. Han, Phys. Lett. B [**739**]{}, 416 (2014) \[arXiv:1406.3598 \[hep-ph\]\]. K. Ghorbani, JCAP [**1501**]{}, no. 01, 015 (2015) \[arXiv:1408.4929 \[hep-ph\]\].
J. Cao, L. Shang, P. Wu, J. M. Yang and Y. Zhang, Phys. Rev. D [**91**]{}, no. 5, 055005 (2015) \[arXiv:1410.3239 \[hep-ph\]\]. K. Ghorbani and H. Ghorbani, arXiv:1501.00206 \[hep-ph\]. K. Ghorbani and H. Ghorbani, Phys. Rev. D [**91**]{}, no. 12, 123541 (2015) \[arXiv:1504.03610 \[hep-ph\]\]. M. Duerr, P. Fileviez Perez and J. Smirnov, arXiv:1510.07562 \[hep-ph\]. B. Kyae and J. C. Park, Phys. Lett. B [**732**]{}, 373 (2014) \[arXiv:1310.2284 \[hep-ph\]\]. J. C. Park, S. C. Park and J. Kim, Phys. Lett. B [**752**]{}, 59 (2016) \[arXiv:1505.04620 \[hep-ph\]\]. C. Boehm, M. J. Dolan and C. McCabe, Phys. Rev. D [**90**]{}, no. 2, 023531 (2014) \[arXiv:1404.4977 \[hep-ph\]\]. A. Berlin, P. Gratia, D. Hooper and S. D. McDermott, Phys. Rev. D [**90**]{}, no. 1, 015032 (2014) \[arXiv:1405.5204 \[hep-ph\]\]. D. G. Cerdeno, M. Peiro and S. Robles, Phys. Rev. D [**91**]{}, no. 12, 123530 (2015) \[arXiv:1501.01296 \[hep-ph\]\]. J. Cao, L. Shang, P. Wu, J. M. Yang and Y. Zhang, JHEP [**1510**]{}, 030 (2015) \[arXiv:1506.06471 \[hep-ph\]\]. Y. G. Kim, K. Y. Lee and S. Shin, JHEP [**0805**]{}, 100 (2008) \[arXiv:0803.2932 \[hep-ph\]\]; arXiv:0809.2745 \[hep-ph\]. Y. G. Kim and S. Shin, JHEP [**0905**]{}, 036 (2009) \[arXiv:0901.2609 \[hep-ph\]\]. K. J. Bae, H. D. Kim and S. Shin, Phys. Rev. D [**82**]{}, 115014 (2010) \[arXiv:1005.5131 \[hep-ph\]\]; S. Shin, PoS IDM [**2010**]{}, 094 (2011) \[arXiv:1011.6377 \[hep-ph\]\].
O. Adriani [*et al.*]{} \[PAMELA Collaboration\], Phys. Rev. Lett. [**105**]{}, 121101 (2010) \[arXiv:1007.0821 \[astro-ph.HE\]\]. S. Ting, PoS ICRC 2015 (2015) 036; A. Kounine, PoS ICRC 2015 (2015) 300.
G. Giesen, M. Boudaud, Y. Genolini, V. Poulin, M. Cirelli, P. Salati and P. D. Serpico, JCAP [**1509**]{}, no. 09, 023 (2015) \[arXiv:1504.04276 \[astro-ph.HE\]\].
M. Ackermann [*et al.*]{} \[Fermi-LAT Collaboration\], Phys. Rev. Lett. [**115**]{}, no. 23, 231301 (2015) \[arXiv:1503.02641 \[astro-ph.HE\]\]. The ATLAS and CMS Collaborations, ATLAS-CONF-2015-044; CMS Collaboration, CMS-PAS-HIG-15-002.
S. Orito [*et al.*]{} \[BESS Collaboration\], Phys. Rev. Lett. [**84**]{}, 1078 (2000) \[astro-ph/9906426\]; Y. Asaoka [*et al.*]{}, Phys. Rev. Lett. [**88**]{}, 051101 (2002) \[astro-ph/0109007\]. M. Boezio [*et al.*]{} \[WiZard/CAPRICE Collaboration\], Astrophys. J. [**561**]{}, 787 (2001) \[astro-ph/0103513\].
Y. G. Kim and K. Y. Lee, Phys. Rev. D [**75**]{}, 115012 (2007) \[hep-ph/0611069\].
M. A. Fedderke, J. Y. Chen, E. W. Kolb and L. T. Wang, JHEP [**1408**]{}, 122 (2014) \[arXiv:1404.2283 \[hep-ph\]\].
L. Lopez-Honorez, T. Schwetz and J. Zupan, Phys. Lett. B [**716**]{}, 179 (2012) \[arXiv:1203.2064 \[hep-ph\]\].
J. F. Navarro, C. S. Frenk and S. D. M. White, Astrophys. J. [**462**]{}, 563 (1996) \[astro-ph/9508025\]. J. F. Navarro, C. S. Frenk and S. D. M. White, Astrophys. J. [**490**]{}, 493 (1997) \[astro-ph/9611107\].
A. Semenov, Comput. Phys. Commun. [**180**]{}, 431 (2009) \[arXiv:0805.0555 \[hep-ph\]\]; A. Semenov, arXiv:1412.5016 \[physics.comp-ph\].
G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, Comput. Phys. Commun. [**192**]{}, 322 (2015) \[arXiv:1407.6129 \[hep-ph\]\].
P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], arXiv:1502.01589 \[astro-ph.CO\].
K. Kong and J. C. Park, Nucl. Phys. B [**888**]{}, 154 (2014) \[arXiv:1404.3741 \[hep-ph\]\]. M. Pospelov, A. Ritz and M. B. Voloshin, Phys. Lett. B [**662**]{}, 53 (2008) \[arXiv:0711.4866 \[hep-ph\]\]
E. C. F. S. Fortes, V. Pleitez and F. W. Stecker, Astropart. Phys. [**74**]{}, 87 (2016) \[arXiv:1503.08220 \[hep-ph\]\]. Y. G. Kim, K. Y. Lee, C. B. Park and S. Shin, work in progress.
[^1]: ygkim@gnue.ac.kr
[^2]: kylee.phys@gnu.ac.kr
[^3]: cbpark@kias.re.kr
[^4]: shinseod@indiana.edu
[^5]: See also [@Kim:2015usa] for the contribution by a diphoton production in a dark sector cascade decay with two or more DM candidates whose mass gaps are non-negligible.
[^6]: Additional annihilation channels exist at the freeze-out other than the $s$-channel scalar exchanges.
[^7]: See also [@lightwimp] for the summary of the related issue.
[^8]: See also Refs. [@Fedderke:2014wda; @LopezHonorez:2012kv] for the related study.
[^9]: For example see Refs. [@Daylan:2014rsa; @Calore:2014nla; @Cline:2015qha].
[^10]: In this case $\langle \sigma v \rangle$ and $\Omega h^2$ are highly sensitive on the exact values of the parameters.
[^11]: We have used the code and the data provided in the reference of [@Calore:2014nla] with modifications for our analysis.
[^12]: See [@Kong:2014haa] for various other bounds besides this.
|
=1
Introduction
============
In noncommutative geometry [@IHES; @NCG], $C^*$-dynamical systems $(A, G, \alpha)$ have been long studied from a differentiable point of view starting with extending the basic notions of differential geometry and differential topology to a differential structure on a $C^*$-algebra $A$ endowed with an action $\alpha\colon G \to \operatorname{Aut}(A)$ of a Lie group $G$. That is, the notion of a connection, a vector bundle, and Chern classes were introduced for such a dynamical system, a pseudodifferential calculus was developed and the analog of the Atiyah–Singer index theorem was proved in [@CAlgGeoDiff]. The noncommutative two torus $\mathbb{T}_\theta^2$ has been one of the main motivating examples for these developments. In [@TrSpLieGpWahl], this line of investigation has been taken further focusing on general compact Lie groups and index theory.
Following the seminal work of Connes and Tretkoff on the Gauss–Bonnet theorem for $\mathbb{T}_\theta^2$ [@GaussBonnet-ConnesTretkoff] and its extension in [@GaussBonnet-FK] concerning general translation invariant conformal structures, local differential geometry of non-flat noncommutative tori has been a subject of increasing interest in recent years [@RicciBM; @ModCurvCM; @ScalarCurvNCtorusFK; @DixmierTraceNCtorusFK; @ScalarCurv4NCTFK; @MoritaLM]. A Weyl conformal factor may be used to perturb a flat metric on noncommutative tori, and Connes’ pseudodifferential calculus [@CAlgGeoDiff] can be employed along with noncommutative computational methods to carry out calculation of scalar curvature and to investigate the related differential geometric statements, see also [@AsymmetricDS] for an asymmetric perturbation of the metric.
The idea and the techniques were indeed initiated in a preprint [@PreprintCC], where with the help of complicated modified logarithmic functions and a modular automorphism, an expression for the value $\zeta(0)$ of the spectral zeta function of the Laplacian of a curved metric on $\mathbb{T}^2_\theta$ was written. The vanishing of this expression is interpreted as the Gauss–Bonnet theorem [@GaussBonnet-ConnesTretkoff], which was suggested by the developments in the following intimately related theories. In fact, the spectral action principle [@SpActChC], in particular the related calculations in the presence of a dilaton [@SpActScaleChC], and the theory of twisted spectral triples, which arise naturally in noncommutative conformal geometry [@TrSpTypeIII; @NCConformalPW], indicate independence of $\zeta(0)$ from the conformal factor.
Connes’ index formula for Fredholm modules, which involves cyclic cohomology, is quite broad [@IHES]. It asserts that given a finitely summable Fredholm module over an algebra, the analytic index, given by pairing a $K$-homology and a $K$-theory element of the algebra, coincides with the topological index, which pairs the corresponding elements in periodic cyclic cohomology and homology obtained by the Chern–Connes characters. The local index formula of Connes and Moscovici [@localtrace95] gives a local formula based on residue trace functionals, which is in the same cyclic cohomology class as the Chern–Connes character, and has the advantage that one can perform explicit computations with it (see also [@localHig]). The residue trace functionals are intimately related to the spectral formulation of Wodzicki’s noncommutative residue [@LocalInvarW; @NCResidue1W]. In fact, the formulation of the noncommutative residue as an integration over the cosphere bundle of a manifold also is important for explicit computations with noncommutative geometric spaces, see [@Residue4NCTF; @ScalarCurv4NCTFK; @NCResidueFW] for a related treatment on noncommutative tori.
=1 The notion of a twisted spectral triple introduced by Connes and Moscovici [@TrSpTypeIII] allows to incorporate a variety of new examples, in particular type III examples in the sense of the Murray–von Neumann classification of operator algebras. They have shown that the Chern–Connes character of a finitely summable twisted spectral triple is an ordinary cyclic cocycle and enjoys an index pairing with $K$-theory. Also, they have constructed a local Hochschild cocycle, which indicates that the ground is prepared for extending the local index formula to the twisted case. This was carried out in [@TwsitedMos] for a particular class of twisted spectral triples; the analog of Connes’ character formula was investigated in [@TwistedFatKha] for the examples. For treatments using twisted cyclic theory, in particular for relations of the theory with Cuntz algebra [@CuntzAlg] and quantum groups, we refer to [@KMSCarNeshNes; @TwistedCarPhiRen; @CuntzAlgCarPhiRen], see also [@ModularKaad; @TwistedSU2KaaSen]. More recent works related to the twisted version of spectral triples reveal their connections with the Bost–Connes system, Riemann surfaces and graphs [@TwistedGreMarTeh], and with the standard model of particle physics [@TwistedDevMar]. Twisted spectral triples associated with crossed product algebras are studied in [@TrSpTypeIII; @TwistedIochum; @TwsitedMos], see also [@FatKhaTwistedSymbols] for an algebraic treatment.
Ergodic actions of compact groups on operator algebras are well-studied in the von Neumann setting (see, e.g., [@ErgodActIWassermann; @ErgodActIIWassermann; @ErgodClassSU2Wassermann]) and in a $C^*$-algebraic context. They were first introduced for $C^*$-algebras by E. Størmer [@SpectraErgodTransfStormer] and this initial effort was expanded in various articles. Let us just mention two of them:
- In their article [@ErgodActAHK], Albeverio and Høegh-Krohn investigate in particular ergodic actions on commutative $C^*$-algebras $A = C(X)$ and prove that they correspond to continuous transitive actions on $X$.
- The article [@ErgodCpctGpHKLS] by Høegh-Krohn, Landstad and Størmer proves that if $G$ acts ergodically on a unital $C^*$-algebra $A$, its unique $G$-invariant state is actually a *trace*.
The article [@Rieffel98] was the first to suggest in 1998 that ergodic actions give rise to interesting spectral triples. This article proceeds with studying the metric induced on state spaces by ergodic actions. More recently, the article [@TrSpLieGpGG] produced a detailed construction of a so called *Lie–Dirac operator* on a $C^*$-algebra $A$, based on an ergodic action of a compact Lie group $G$ on $A$. It also investigated the analytic properties of these Lie–Dirac operators, proving in particular that they are finitely summable spectral triples. In the present article, we elaborate on the techniques used in [@TrSpLieGpGG] in order to prove quite different results.
Indeed, in [@TrSpLieGpGG] the focus was on a Dirac operator for a “noncommutative spin manifold”, whereas here the emphasis is on a sort of Hodge–de Rham operator associated with a conformally perturbed metric, construction of twisted spectral triples and the analog of the Chern–Gauss–Bonnet theorem. The Hodge–de Rham operators constructed here are (in general) *not* Lie–Dirac operators in the sense of [@TrSpLieGpGG]. In this previous article, the algebra structure of $A$ played only a minor role in the analytical properties of the spectral triple. Here, the multiplication of $A$ has a central importance.
For a recent approach of Hodge theory using Hilbert modules, we refer to the recent article [@EllipticComplexKrysl]. See also [@HodgeTheoryEllipticKrysl; @HodgeTheoryKrysl].
This article is organized as follows. In Section \[Sec:Reminders\], we recall the necessary statements from representation theory and operator theory, and the notion of ordinary and twisted spectral triples along with their main properties that are used in our arguments and concern our constructions. We associate a complex of noncommutative differential forms to a $C^*$-dynamical system $(A, G, \alpha)$ in Section \[Sec:HodgeOperator\]. In the ergodic case, the analog of the Hodge–de Rham operator is studied when the complex is equipped with a Hermitian structure determined by a metric in the conformal class of the standard metric associated with the unique $G$-invariant trace on $A$.
Inspired by a construction in [@GaussBonnet-ConnesTretkoff], we construct in Section \[Sec:ConfTwistedTrSp\] a spectral triple on $A$ and a twisted spectral triple on the opposite algebra $A^\text{op}$, which encode the geometric information of the conformally perturbed metric. We study the Dirac operator of the perturbed metric carefully and prove that it is selfadjoint and enjoys having the same spectral dimension as the non-perturbed case. It should be stressed that ergodicity plays a crucial role for the latter to hold.
The existence of an analog of the Chern–Gauss–Bonnet theorem is studied in Section \[Sec:CGBTheorem\] by proving a Hodge decomposition theorem for our complex and showing that its Euler characteristic is independent of the conformal factor. Combining this with the McKean–Singer index formula and small time asymptotic expansions, which often exist for noncommutative geometric spaces, we explain how the analog of the Euler class or the Pfaffian of the curvature form can be computed as local geometric invariants of examples that fit into our setting. Indeed, such invariants depend on the behavior at infinity of the eigenvalues of the involved Laplacians and the action of the algebra. Finally, our main results and conclusions are summarized in Section \[Sec:Conclusions\].
Preliminaries {#Sec:Reminders}
=============
We start by some reminders about results and notations from various anterior articles.
Given a strongly continuous action $\alpha$ of a compact group $G$ on a unital $C^*$-algebra $A$, we say that it is *ergodic* if the fixed algebra of $G$-invariants elements is reduced to the scalars, i.e., if $\forall \, g \in G$, $\alpha_g(a) = a$, then $a \in {\mathbb{C}}1_A$.
Among the important results obtained with this notion of ergodic action, let us quote the following [@ErgodCpctGpHKLS Theorem 4.1, p. 82]:
\[Thm:ErgodAct\] Let $A$ be a unital $C^*$-algebra, $G$ a compact group and $\alpha$ a strongly continuous representation of $G$ as an ergodic group of $*$-automorphisms of $A$, then the unique $G$-invariant state $\varphi_0$ on $A$ is a trace.
Another result that will play an important role in our article is [@ErgodCpctGpHKLS Proposition 2.1, p. 76], which we adapt slightly in the following:
\[Prop:multiplicity\] Let $A$ be a unital $C^*$-algebra, $G$ a compact group and $\alpha$ a strongly continuous representation of $G$ as an ergodic group of $*$-automorphisms of $A$. Let $V$ be an irreducible unitary representation of $G$, $A(V)$ the spectral subspace of $V$ in $A$ and $m(V)$ the multiplicity of $V$ in $A(V)$. Then we have $$\begin{gathered}
m(V) \leqslant \dim V.\end{gathered}$$
Among our main results, we prove the finite summability of certain spectral triples (ordinary and twisted), we therefore define those terms:
\[Def:TrSp\] Let $A$ be a unital $C^*$-algebra. An odd (ordinary) *spectral triple*, also called an odd *unbounded Fredholm module*, is a triple $({\mathcal{A}}, {\mathscr{H}}, D)$ where
- ${\mathscr{H}}$ is a Hilbert space and $\pi \colon A\to B({\mathscr{H}})$ a $*$-representation of $A$ as bounded operators on ${\mathscr{H}}$,
- $D$ is a selfadjoint unbounded operator – which we will call the *Dirac operator* – with domain $\operatorname{Dom}(D)$,
such that
1. $(1+D^2)^{-1}$ is a compact operator,
2. the subalgebra ${\mathcal{A}}$ of all $a\in A$ such that $$\begin{gathered}
\pi(a) (\operatorname{Dom}(D)) \ \subseteq \operatorname{Dom}(D)
\qquad
\text{and}
\qquad
[D,\pi(a)] \ \text{ extends to a bounded map on }{\mathscr{H}}\end{gathered}$$ is dense in $A$.
An *even spectral triple* is given by the same data, but we further require that a grading $\gamma$ be given on ${\mathscr{H}}$ such that (i) $A$ acts by even operators, (ii) $D$ is odd.
\[Rk:SymmIdeals\] For a selfadjoint operator $D$, condition (i) of the definition above is actually equivalent to $\exists\, \lambda \in {\mathbb{R}}{\setminus} \{ 0 \}$ s.t. $(D+i \lambda)^{-1}$ is a compact operator.
To define *finitely summable* spectral triples, we now need a brief reminder regarding *trace ideals* (also known as *symmetric ideals*), for which we follow Chapter IV of [@NCG]. For more details concerning symmetrically normed operator ideals and singular traces we refer the reader to [@TraceIdealsSimon] and [@SingularTracesLSZ].
\[Def:DixmierSum\] For $p>1$, the ideal ${\mathscr{L}}^{p^+}$ (also denoted ${\mathscr{L}}^{(p,\infty )}$ in [@NCG] and ${\mathcal{J}}_{p, \omega}$ in [@TraceIdealsSimon p. 21]) consists of all compact operators $T$ on ${\mathscr{H}}$ such that $$\begin{gathered}
\| T\|_{p^+} := \sup_k \frac{ \sigma_k(T)}{k^{(p-1)/p}} < \infty,\end{gathered}$$ where $\sigma_k$ is defined as the supremum of the trace norms of $T E$, when $E$ is an orthonormal projection of dimension $k$, i.e., $$\begin{gathered}
\sigma_k(T) := \sup \{ \| T E \|_1 , \dim E = k \}.\end{gathered}$$ Equivalently, $\sigma_k(T)$ is the sum of the $k$ largest eigenvalues (counted with their multiplicities) of the positive compact operator $|T| := (T^* T)^{1/2}$. The definition extends to the case of $p = 1$: ${\mathscr{L}}^{1^+}$ is the ideal of compact operators $T$ s.t.$$\begin{gathered}
\| T \|_{1^+} := \sup_k \frac{ \sigma_k(T)}{\log k} < \infty .\end{gathered}$$ The elements of ${\mathscr{L}}^{p^+}$ are called *$p^+$-summable* (or $(p, \infty )$-summable – see [@NCG Section IV.2$\alpha$, p. 299 and following]).
A *spectral dimension* for spectral triples is defined as follows
A spectral triple is *$p^+$-summable* if $(1 + D^2)^{-1/2} \in {\mathscr{L}}^{p^+}$.
Finally, we will consider *twisted spectral triples* (also called *$\sigma$-spectral triples*) as introduced in [@TrSpTypeIII Definition 3.1]. This is a spectral triple just like in Definition \[Def:TrSp\], but for a fixed automorphism $\sigma$ of ${\mathcal{A}}$, the bounded commutators condition (denoted (ii) above) is replaced by
1. the subalgebra ${\mathcal{A}}$ of all $a\in A$ such that
- $\pi(a) (\operatorname{Dom}(D)) \subseteq \operatorname{Dom}(D)$,
- $D\pi(a) - \pi(\sigma(a)) D$ extends to a bounded map on ${\mathscr{H}}$
is dense in $A$.
In this paper, we will need the subalgebras ${\mathcal{A}}^k$ for $k \geqslant 0$, corresponding to the $C^k$-differentiable class. Following [@DerivBratteli Section 2.2], we introduce the space $$\begin{gathered}
{\mathcal{A}}^m := \big\{ a \in A\colon g \mapsto \alpha_g(a) \text{ is in }C^m(G, A) \big\}.\end{gathered}$$ Let us fix a basis $(\partial _i)$ of the Lie algebra ${\mathfrak{g}}$. For such a choice of basis, the infinitesimal generators $\partial _i$ act as derivations ${\mathcal{A}}^m \to {\mathcal{A}}^{m-1}$. According to [@DerivBratteli Example 2.2.4, p. 41], ${\mathcal{A}}^m$ equipped with the norm $$\begin{gathered}
\| a \|_m := \|a \| + \sum_{k=1}^m \sum_{i_1=1}^n \cdots \sum_{i_k=1}^n \frac{\| \partial_{i_1} \cdots \partial_{i_k}(a) \|}{k !},\end{gathered}$$ is a Banach algebra with $\|a b \|_m \leqslant \| a \|_m \| b\|_m$. In particular, if $h \in {\mathcal{A}}^1$, then $e^{\lambda h} \in {\mathcal{A}}^1$, for all complex number $\lambda$ – see also Lemma \[Lem:CVExp\] below for a more precise estimate.
Following the density properties established in [@DerivBratteli] (see, e.g., Definition 2.2.15, p. 47), the intersection ${\mathcal{A}}^\infty = \bigcap_{j=0}^\infty {\mathcal{A}}^j$ is a dense $*$-subalgebra of the $C^*$-algebra $A$, which is stable under the derivations $\partial _i$.
Hodge–de Rham Dirac operator and $\boldsymbol{C^*}$-dynamical systems {#Sec:HodgeOperator}
=====================================================================
In this article, we consider a fixed $A$, a $C^*$-algebra with an ergodic action $\alpha$ of a compact Lie group $G$ of dimension $n$. We write ${\mathcal{A}}$ for ${\mathcal{A}}^\infty$, the “smooth subalgebra” of $A$, which can alternatively be defined as $$\begin{gathered}
{\mathcal{A}}:= \{ a \in A \colon g \mapsto \alpha_g(a) \text{ is in }C^\infty (G, A) \}.\end{gathered}$$ The Chevalley–Eilenberg cochain complex with coefficients in ${\mathcal{A}}$ provides a complex that we interpret as “differential forms” on ${\mathcal{A}}$. For the reader’s convenience and to fix notations, we provide a reminder of this construction. For all $k \in {\mathbb{N}}$, $$\begin{gathered}
\Omega^k := {\mathcal{A}}\otimes \bigwedge^k {\mathfrak{g}}^*,\end{gathered}$$ where ${\mathfrak{g}}^*$ denotes the linear forms on ${\mathfrak{g}}$, the Lie algebra of the Lie group $G$. Given a scalar product on ${\mathfrak{g}}^*$ (e.g., obtained from the Killing form), we can extend it to a scalar product on $\bigwedge^k {\mathfrak{g}}^*$ by setting $$\begin{gathered}
\langle v_1 \wedge \cdots \wedge v_k, w_1 \wedge \cdots \wedge w_k \rangle := \det( \langle v_i, w_j \rangle ),\end{gathered}$$ i.e., the determinant of the matrix of scalar products. We fix an orthonormal basis $(\omega_j)_{j = 1, \ldots , n}$ of ${\mathfrak{g}}^*$ for this scalar product and consider its dual basis $(\partial _j)_{j=1, \ldots , n}$ in ${\mathfrak{g}}$.
Following [@KnappLieGp the model of (4.6), p. 157], we write the exterior derivative of the complex $$\begin{gathered}
d( a \otimes \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K} ) =\sum_{j = 1}^n \partial _j(a) \otimes \omega_j \wedge \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K} \nonumber\\
\qquad{}
- \frac{1}{2} \sum_{k=1}^K \sum_{\alpha, \beta} (-1)^{k+1} c^{i_k}_{\alpha \beta} a \otimes \omega_\alpha \wedge \omega_\beta \wedge \omega_{i_1} \wedge \cdots \wedge \omega_{i_{k-1}} \wedge \omega_{i_{k+1}} \wedge \cdots \wedge \omega_{i_K},\label{Eqn:Defd}\end{gathered}$$ where $[\partial _i, \partial _j] = \sum\limits_{k=1}^n c^k_{i j} \partial _k$ – the $c^{k}_{i j}$ are called the *structure constants* of the Lie algebra ${\mathfrak{g}}$. A lengthy but straightforward computation proves that this exterior derivative satisfies $d^2 = 0$ on $\Omega^\bullet$, therefore $(\Omega^\bullet , d)$ is a complex.
The Chevalley–Eilenberg complex is available even for noncompact groups $G$ and nonergodic actions. In other words, the square $d^2$ actually vanishes even when $G$ is *not* a *compact* Lie group and when the action of $G$ on $A$ is *not* ergodic.
The natural product on $\Omega^\bullet$ is $$\begin{gathered}
\label{Eqn:ProdOmega}
(a \otimes v_1 \wedge \cdots \wedge v_k) \cdot (a' \otimes w_{1} \wedge \cdots \wedge w_{k'}) := a a' \otimes v_1 \wedge \cdots \wedge v_k \wedge w_1 \wedge \cdots \wedge w_{k'},\end{gathered}$$ i.e., the product of a $k$-form with a $k'$-form is a $k+k'$-form. In particular, for all $k$, $\Omega^k$ is an ${\mathcal{A}}$-bimodule. The exterior derivative $d$ is compatible with the right module structure in the following sense $$\begin{gathered}
d( a a'\otimes \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K} ) =\sum_{j = 1}^n \partial _j(a a') \otimes \omega_j \wedge \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K} \\
\qquad\quad{}
- \frac{1}{2} \sum_{k} \sum_{\alpha, \beta} c^{i_k}_{\alpha \beta} a a'\otimes \omega_\alpha \wedge \omega_\beta \wedge \omega_{i_1} \wedge \cdots \wedge \omega_{i_{k-1}} \wedge \omega_{i_{k+1}} \wedge \cdots \wedge \omega_{i_K}
\\
\qquad{}= \left( \sum_{j = 1}^n \partial _j(a) \otimes \omega_j \wedge \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K} \right) a' \\
\qquad\quad{}
+\sum_{j = 1}^n a\partial _j(a') \otimes \omega_j \wedge \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K} \\
\qquad\quad{}
- \frac{1}{2} \left( \sum_{k} \sum_{\alpha, \beta} c^{i_k}_{\alpha \beta} a \otimes \omega_\alpha \wedge \omega_\beta \wedge \omega_{i_1} \wedge \cdots \wedge \omega_{i_{k-1}} \wedge \omega_{i_{k+1}} \wedge \cdots \wedge \omega_{i_K} \right) a'
\\
\qquad{}
= d( a \otimes \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K} ) a'
+(-1)^K (a \otimes \omega_j \wedge \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K}) d(a').\end{gathered}$$ Since we want to treat conformal deformations of the original structure, we follow [@GaussBonnet-ConnesTretkoff] and fix a positive invertible element $e^{h} \in {\mathcal{A}}^1$, where $h$ is a smooth selfadjoint element in ${\mathcal{A}}^1$. Then we define a scalar product on $\Omega^k$ by the formula $$\begin{gathered}
\label{Eqn:ScalProd}
(a \otimes v_1 \wedge \cdots \wedge v_k, a' \otimes w_1 \wedge \cdots \wedge w_k )_\varphi := \varphi_0\big( a^* a' e^{(n/2-k)h}\big) \det( \langle v_i, w_j \rangle ),\end{gathered}$$ where $\varphi_0$ is the unique $G$-invariant state on $A$, which is actually a trace according to [@ErgodActAHK Theorem 3.1, p. 8]. We set the scalar product of two forms of different degrees to vanish. The scalar product obtained for $h = 0$ is the one we call the *natural scalar product* on forms. We define the Hilbert space ${\mathscr{H}}_\varphi$ as the completion of $\Omega^\bullet$ for the scalar product . In the particular case of $h = 0$, we obtain our reference Hilbert space ${\mathscr{H}}$. We will also need the Hilbert spaces ${\mathscr{H}}_{0, \varphi} := \operatorname{GNS}({\mathcal{A}}, \varphi)$ and ${\mathscr{H}}_{0} := \operatorname{GNS}({\mathcal{A}}, \varphi_0)$ as well as the Hilbert spaces ${\mathscr{H}}_k := {\mathscr{H}}_0 \otimes \bigwedge^k {\mathfrak{g}}^*$ – i.e., the completion of $k$-forms – and ${\mathscr{H}}_{k, \varphi}$.
To understand why we choose the form for the conformal deformation, we compare with the commutative case of a $n$-dimensional compact manifold $M$, where we have the following property: if the Riemannian metric is transformed by $g \rightsquigarrow \lambda g$ (for $\lambda > 0$), then the (pointwise) norm of all vectors is multiplied by $\lambda^{1/2}$ and thus the pointwise norm of $1$-forms is multiplied by $\lambda^{-1/2}$. This in turn implies that the pointwise norm of $k$-form is multiplied by $\lambda^{-k/2}$. Finally, the (global) scalar product of $k$-forms is the integral of the pointwise scalar products. Since under the conformal deformation, the total volume of the manifold $M$ is multiplied by $\lambda^{n/2}$, the (global) scalar products of $k$-forms are multiplied by $\lambda^{n/2-k}$. In particular, if $n$ is even and $k = n/2$, then the scalar product on $n/2$-forms is left invariant under the conformal deformation.
In order to study $d$ and its adjoint, we introduce the degree $1$ maps $T_j \colon \bigwedge^\bullet {\mathfrak{g}}^* \to \bigwedge^\bullet {\mathfrak{g}}^*$ defined for all $j \in \{1, \ldots, n\}$ by $$\begin{gathered}
T_j( v_1 \wedge \cdots \wedge v_k) = \omega_j \wedge v_1 \wedge \cdots \wedge v_k.\end{gathered}$$ Let ${\mathcal{R}}_{x}$ denote the right multiplication operator for any $x \in A$: ${\mathcal{R}}_{x}(a) = a x$, and let $B^{i_k}_{\alpha \beta}$ be the bounded operator on $\bigwedge^\bullet {\mathfrak{g}}^*$ defined using the basis $(\omega_j)$ by $$\begin{gathered}
B^{i_k}_{\alpha \beta}( \omega_{i_1} \wedge \omega_{i_2} \wedge \cdots \wedge \omega_{i_K}) = \omega_\alpha \wedge \omega_\beta \wedge \omega_{i_1} \wedge \cdots \wedge \omega_{i_{k-1}} \wedge \omega_{i_{k+1}} \wedge \cdots \wedge \omega_{i_K}.\end{gathered}$$ We can now give an explicit form to the operator $d$ and its (formal) adjoint for the unperturbed metric.
\[Lem:Formed\] With the previous notations, when $h = 0$, the operator $d$ can be written $$\begin{gathered}
d = \sum_j \partial_j \otimes T_j - \frac{1}{2} \sum_{k,\alpha,\beta} c^{i_k}_{\alpha \beta} \otimes B^{i_k}_{\alpha \beta}\end{gathered}$$ and its adjoint $d^*$ is $$\begin{gathered}
d^* = \sum_j \partial_j \otimes T_j^* - \frac{1}{2} \sum_{k,\alpha, \beta} \overline{c^{i_k}_{\alpha \beta}} \otimes \big(B^{i_k}_{\alpha \beta}\big)^*.\end{gathered}$$
The only point that is not self-explanatory is the behavior of $\partial_j$ with respect to the trace $\varphi_0$ $$\begin{gathered}
\langle [\partial_j(a)], [a'] \rangle = \varphi_0( \partial_j(a)^* a') = \varphi_0( -\partial_j(a^*) a') = \varphi_0(a^* \partial_j(a')),\end{gathered}$$ where we used the relations $\partial_j(a)^* = - \partial_j(a^*)$ and $\varphi_0( \partial_j(a)) = 0$.
\[Lem:CVExp\] Let $h$ be an element of ${\mathcal{A}}^1$ and $\partial $ be an infinitesimal generator of $G$, acting as a derivation on ${\mathcal{A}}^1$, then $\partial ( e^{h} )$ is in the $C^*$-algebra $A$ and satisfies $$\begin{gathered}
\big\| \partial \big( e^{h} \big) \big\| \leqslant \| \partial (h) \| e^{\| h\| }.\end{gathered}$$ In particular, for a scalar parameter $v \to 0$, $\partial (e^{v h}) \to 0$.
As an operator from ${\mathcal{A}}^1$ to $A$, the derivation $\partial $ is continuous, therefore we can estimate $\partial (e^h)$ by using an ${\mathcal{A}}^1$-converging sequence, like the partial sums of $e^{h}$.
For this sequence, using the derivation property, we get $$\begin{gathered}
\left\| \partial \left( \sum_{k=0}^N \frac{h^k}{k !} \right) \right\| \leqslant \sum_{k=1}^N k \frac{\| \partial h \| }{k} \frac{\| h \|^{k-1}}{(k-1)!} \leqslant \| \partial h \| e^{\| h \|}.\end{gathered}$$ The property $\partial (e^{v h}) \to 0$ as $v \to 0$ follows immediately.
We call $d_\varphi$ the operator defined on ${\mathscr{H}}_\varphi$ by the formula . Once the scalar product is defined, we want to define an adjoint $d^*_\varphi$ to $d_\varphi$ for this scalar product. The Hodge–de Rham operator that we would like to study *in fine* is $d_\varphi + d_\varphi^*$. However, the unbounded operator $d_\varphi$ is *a priori* arbitrary, so it is not clear that it admits a densely defined adjoint. To clarify the relations between ${\mathscr{H}}$ and ${\mathscr{H}}_\varphi$, we introduce the following lemma:
\[Lem:EquivRep\] For any selfadjoint $h \in {\mathcal{A}}^1$,
- the Hilbert space ${\mathscr{H}}_{\varphi}$ is equipped with a $G$-representation defined on degree $k$ forms by ${\mathbb{V}}_g( [a \otimes v_1 \wedge \cdots \wedge v_k]_\varphi) = [\alpha_g(a) \otimes v_1 \wedge \cdots \wedge v_k]_\varphi$, which leads to a $G$-equivariant left ${\mathcal{A}}$-module structures on ${\mathscr{H}}_{\varphi}$;
- the map $L \colon {\mathscr{H}}_0 \to {\mathscr{H}}_{0,\varphi}$ defined by $L([a]):= [a]_\varphi$ is invertible and intertwines the $G$-equivariant left ${\mathcal{A}}$-module structures on ${\mathscr{H}}_0$ and ${\mathscr{H}}_{0,\varphi}$.
Of course, $L$ extends to $L \otimes \operatorname{Id}\colon {\mathscr{H}}\to {\mathscr{H}}_\varphi$ which is still a continuous and invertible map. We denote its adjoint by $H \colon {\mathscr{H}}_\varphi \to {\mathscr{H}}$, whose explicit form is $$\begin{gathered}
H( [a \otimes v_1 \wedge \cdots \wedge v_k]_\varphi ) =\big [a e^{(n/2-k)h} \otimes v_1 \wedge \cdots \wedge v_k\big].\end{gathered}$$ Finally, ${\mathscr{H}}$ and ${\mathscr{H}}_\varphi$ are related by the unitary map $U \colon {\mathscr{H}}\to {\mathscr{H}}_\varphi$ given on degree $k$ forms by $$\begin{gathered}
U( [a \otimes v_1 \wedge \cdots \wedge v_k] ) = \big[a e^{-(n/2-k)h/2} \otimes v_1 \wedge \cdots \wedge v_k\big]_\varphi.\end{gathered}$$
The map $U$ defined above is unitary, but it does not intertwine the $G$-structures on ${\mathscr{H}}$ and ${\mathscr{H}}_\varphi$.
Since the sum ${\mathscr{H}}_{\varphi} = \bigoplus_k {\mathscr{H}}_{k, \varphi}$ is finite, it suffices to check that ${\mathbb{V}}_g$ is continuous on each ${\mathscr{H}}_{k, \varphi}$ separately. Since ${\mathbb{V}}_g$ does not act on $\bigwedge^\bullet {\mathfrak{g}}^*$, it is enough to prove continuity on $\operatorname{GNS}(A, \tilde{\varphi})$ where $\tilde{\varphi}(a) = \varphi_0(a e^{-h_k})$ and $h_k = -(n/2 -k)h \in {\mathcal{A}}^1$ (corresponding to forms of degree $k$). We get $$\begin{gathered}
\| \alpha_g(a) \|_{\tilde{\varphi}}^2 = \varphi_0\big( \alpha_g(a)^* \alpha_g(a) e^{-h_k} \big) = \varphi_0\big( a^* a \alpha_{g^{-1}}\big(e^{-h_k}\big) \big) \\
\hphantom{\| \alpha_g(a) \|_{\tilde{\varphi}}^2}{}
= \varphi_0\big( a e^{-h_k/2} e^{h_k/2}\alpha_{g^{-1}}\big(e^{-h_k}\big)e^{h_k/2} e^{-h_k/2} a^*\big) \leqslant K \varphi_0\big( a e^{-h_k}a^*\big) = K \| a \|_{\tilde{\varphi}}^2,\end{gathered}$$ for a constant $K = \| e^{h_k/2}\alpha_{g^{-1}}(e^{-h_k})e^{h_k/2} \|$. The above (scalar) inequality follows from the inequality of operators $$\begin{gathered}
a e^{-h_k/2} e^{h_k/2}\alpha_{g^{-1}}\big(e^{-h_k}\big)e^{h_k/2} e^{-h_k/2} a^* \leqslant a e^{-h_k/2} K e^{-h_k/2} a^* = K a e^{-h_k} a^*,\end{gathered}$$ which is valid since $e^{h_k/2}\alpha_{g^{-1}}(e^{-h_k})e^{h_k/2}$ is a positive operator. It then suffices to apply the positive functional $\varphi_0$. Once we know that the map ${\mathbb{V}}_g$ is defined on the full Hilbert space, proving that it is compatible with the left ${\mathcal{A}}$-module structure is a formality.
The process is similar for $L$: it is clear from the definition that if the map $L$ exists, then it intertwines the $G$-equivariant left ${\mathcal{A}}$-module structures on ${\mathscr{H}}$ and ${\mathscr{H}}_\varphi$. It remains to prove that $L$ is well-defined and invertible.
We first evaluate $$\begin{gathered}
\| L( a) \|_{\tilde{\varphi}}^2 = \varphi_0\big( a^* a e^{-h_k}\big) = \varphi_0\big( a e^{-h_k} a^*\big) \leqslant \big\| e^{-h_k} \big\| \varphi_0(a a^*) = \big\| e^{-h_k} \big\| \, \| a \|_{\varphi_0}^2,\end{gathered}$$ by the same argument as above.
To prove that $L$ is invertible, consider the norm of its inverse $$\begin{gathered}
\| a \|_{\varphi_0}^2 = \varphi_0\big( a e^{-h_k/2} e^{h_k} e^{-h_k/2} a^*\big) \leqslant \big\| e^{h_k} \big\| \varphi_0\big( a e^{-h_k} a^*\big) = \big\|e^{h_k} \big\| \, \| a \|_{\tilde{\varphi}}^2.\end{gathered}$$ The evaluation of the adjoint $H$ of $L$ and of the unitary map $U \colon {\mathscr{H}}\to {\mathscr{H}}_\varphi$ is an easy exercise.
With the previous notations, we see that $d_\varphi = (L \otimes \operatorname{Id}) d (L \otimes \operatorname{Id})^{-1}$ and thus (at least formally) $d_\varphi^* = H^{-1} d^* H$. However, in order to facilitate the comparison between $d_\varphi + d^*_\varphi$ acting on ${\mathscr{H}}_\varphi$ and $d + d^*$ acting on ${\mathscr{H}}$, we “push” $d_\varphi + d_\varphi^*$ to ${\mathscr{H}}$ using the unitary $U$. This leads us to the operators $D_u$ studied in the Proposition \[Prop:Du\] below. But first, for $h = h^*$, we need to introduce the operators $K_{u} \colon {\mathscr{H}}\to {\mathscr{H}}$, where $u \in {\mathbb{R}}$, defined by $$\begin{gathered}
\label{Eqn:DefKu}
K_{u}( [a \otimes v_1 \wedge \cdots \wedge v_k ]) = \big[a e^{(n/2 - k)h u} \otimes v_1 \wedge \cdots \wedge v_k\big].\end{gathered}$$ This is a one-parameter group of invertible selfadjoint operators. Moreover, following our remark on ${\mathcal{A}}^1$ at the end of Section \[Sec:Reminders\], for all $u \in {\mathbb{R}}$, the operators $K_u$ preserve the space ${\mathcal{A}}^1 \otimes \bigwedge^\bullet {\mathfrak{g}}^*$.
In the proof below, we consider the orthogonal projections $\Pi_k \colon {\mathscr{H}}\to {\mathscr{H}}_k$ onto the completion of the $k$-forms, for all $k \in \{0, \ldots, n \}$.
\[Prop:Du\] For all $u \in [0,1]$, we consider two unbounded operators defined on the dense domain ${C}= {\mathcal{A}}^1 \otimes \bigwedge^\bullet {\mathfrak{g}}^* \subseteq {\mathscr{H}}$, $$\begin{gathered}
d_u := K_{u} d K_{-u},
\qquad
d_u^* := K_{-u} d^* K_{u}.\end{gathered}$$ We have:
1. if $E_{+,u}$ and $E_{-, u}$ are, respectively, the closures in ${\mathscr{H}}$ of the images of the operators $d_u$ and $d_u^*$, then $E_{+,u}$ and $E_{-, u}$ are orthogonal in ${\mathscr{H}}$; we denote by $\Pi_{+,u}$ and $\Pi_{-,u}$, respectively, the orthogonal projections on these spaces;
2. the operator $$\begin{gathered}
D_{u} = K_{u} d K_{-u} + K_{-u} d^* K_{u},\end{gathered}$$ is essentially selfadjoint on a common core domain ${C}= {\mathcal{A}}^1 \otimes \bigwedge^\bullet {\mathfrak{g}}^*$.
The family of operators $D_u$ satisfies the estimate $$\begin{gathered}
\label{Eqn:EstimateDu}
\| (D_{u+v} - D_u)({\underline{\omega}}) \| \leqslant o_v(1) \| D_u {\underline{\omega}}\| + o_v(1) \| {\underline{\omega}}\|,\end{gathered}$$ where ${\underline{\omega}}$ is any vector in the common selfadjointness domain and following Landau’s notations, $o_v(1)$ stands for functions of $v$ which tend to $0$ for $v \to 0$. We can choose these two functions independently of the parameter $u \in [0,1]$.
Regarding point (1), we start by proving the property for $u = 0$, i.e., for the untwisted case. There, following Lemma \[Lem:EquivRep\] the trace $\varphi_0$ is $G$-invariant and therefore the action ${\mathbb{V}}_g$ of $G$ on ${\mathscr{H}}_0$ is unitary. Consequently, the $G$-representation can be decomposed into a direct sum of finite-dimensional $G$-representations. Let us denote by $V$ one of these finite-dimensional spaces.
It is clear from the definition that both $V \otimes \bigwedge^k {\mathfrak{g}}^* \subseteq {\mathscr{H}}$ and its orthogonal are stable under the action of $d$. Thus the restriction of $d$ to the finite-dimensional space $V \otimes \bigwedge^k {\mathfrak{g}}^*$ is bounded and admits an adjoint $d^*$ whose form is given by Lemma \[Lem:Formed\]. Varying the space $V$, we see that $d^*$ is defined on ${\mathscr{D}}$, the algebraic direct sum of $V \otimes \bigwedge^k {\mathfrak{g}}^*$, which is a dense subset of ${\mathscr{H}}$. If we restrict to the case of forms ${\underline{\omega}}$, ${\underline{\omega}}'$ in the space $V \otimes \bigwedge^k {\mathfrak{g}}^* \subseteq {\mathscr{H}}$, we have $$\begin{gathered}
\langle d {\underline{\omega}}, d^* {\underline{\omega}}' \rangle = \langle d^2 {\underline{\omega}}, {\underline{\omega}}' \rangle = 0,\end{gathered}$$ there are no considerations of domains for $d$ and $d^*$, since we consider finite-dimensional spaces. The same argument applied to different finite vector spaces $V$ proves that $E_{+,0}$ (the image of $d$) and $E_{-,0}$ (the image of $d^*$) are orthogonal.
To treat point (1) for a general $u \in [0,1]$, we note that $E_{+,u} = K_u E_{+,0}$: indeed, if $\xi = \lim d {\underline{\omega}}_n$, then $K_u \xi = \lim K_u d K_{-u} (K_u {\underline{\omega}}_n)$ and vice versa. Similarly, $E_{-,u} = K_{-u} E_{-,0}$. Given $K_u e_+ \in E_{+,u}$ and $K_{-u} e_- \in E_{-,u}$, we have $$\begin{gathered}
\langle K_u e_+, K_{-u} e_- \rangle = \langle e_+, e_- \rangle = 0,\end{gathered}$$ since $K_u$ is selfadjoint and $e_+ \in E_{+,0}$, $e_- \in E_{-,0}$ are orthogonal. This proves the requested orthogonality relation.
Regarding point (2), let us start by giving a sketch of the proof: we first prove that $D_0 = D$ is essentially selfadjoint on the requested domain. Using the estimate , we then apply Kato–Rellich theorem to show that if $D_{u}$ is essentially selfadjoint for the domain ${C}$, then so is the operator $D_{u + v}$ for all $|v| \leqslant \varepsilon$, where $\varepsilon$ is independent of the point $u \in [0,1]$ chosen. As a consequence, all operators $D_u$ are essentially selfadjoint for the fixed domain.
We first prove that $D_0$ is selfadjoint. This is done by using the Peter–Weyl decomposition of ${\mathscr{H}}_0$ for the unitary action ${\mathbb{V}}_g$ of $G$ on ${\mathscr{H}}_0$. As mentioned in point (1), the restriction of $d$ to this finite-dimensional space $V \otimes \bigwedge^k {\mathfrak{g}}^* \subseteq {\mathscr{H}}$ is well-defined, as is its adjoint $d^*$. Varying the space $V$, we consider ${\mathscr{D}}$, the direct sum of $V \otimes \bigwedge^k {\mathfrak{g}}^*$, which is a dense subset of ${\mathscr{H}}$.
In this situation, we can define $D = d + d^*$ on ${\mathscr{D}}$. If we restrict $D$ to a component $V \otimes \bigwedge^k {\mathfrak{g}}^* \subseteq {\mathscr{H}}$, it is formally selfadjoint by definition. It therefore admits an orthonormal basis of eigenvectors with real associated eigenvalues. It follows that $\operatorname{Ran}(D +i)$ and $\operatorname{Ran}(D -i)$ are dense in ${\mathscr{H}}$, and this is enough to prove that $D$ is essentially selfadjoint on the domain ${C}$ (see [@ReedSimon1 Corollary, p. 257]).
Let us now consider an arbitrary $u \in [0,1]$. We want to find $\varepsilon > 0$ uniform in $u$ and small enough so that for all $v$ with $|v| \leqslant \varepsilon$, the operator $D_{u+v}$ is essentially selfadjoint. By definition, $$\begin{gathered}
D_{u+v} = K_{u+v} d K_{-(u+v)} + K_{-(u+v)} d^* K_{u+v}.\end{gathered}$$ If we introduce $R_v = K_v -1$, then we can write $$\begin{gathered}
K_{u+v} = K_u (1 + R_v),
\qquad
K_{-(u+v)} = (1 + R_{-v}) K_{-u}.\end{gathered}$$ It is clear that both $R_v$ and $R_{-v}$ are bounded with $\|R_v \| \to 0$, $\|R_{-v} \| \to 0$ for $v \to 0$ and by definition, for all $v \in {\mathbb{R}}$, $R_v$ commutes with $K_u$ for $u \in {\mathbb{R}}$.
We write $$\begin{gathered}
D_{u+v} = K_{u} (1 + R_v) d (1+ R_{-v}) K_{-u} + K_{-u} (1+R_{-v}) d^* (1+R_v) K_{u} \\
\hphantom{D_{u+v}}{}
= K_u d K_{-u} + K_u d R_{-v} K_{-u} + K_u R_v d K_{-u} + K_u R_v d R_{-v} K_{-u} \\
\hphantom{D_{u+v}=}{}
+ K_{-u} d^* K_{u} + K_{-u} d^* R_v K_u + K_{-u} R_{-v} d^* K_u + K_{-u} R_{-v} d^* R_v K_u.\end{gathered}$$ The sum of the terms $K_u d K_{-u}$ and $K_{-u} d^* K_{u}$ gives back $D_u$. Since $E_{+, u}$ and $E_{-,u}$ are orthogonal, we have $$\begin{gathered}
\label{Eqn:Pyth}
\| D_u {\underline{\omega}}\|^2 = \| \Pi_{+, u}( D_u {\underline{\omega}}) \|^2 + \| \Pi_{-, u}( D_u {\underline{\omega}}) \|^2 = \| d_{u} {\underline{\omega}}\|^2 + \| d_u^* {\underline{\omega}}\|^2.\end{gathered}$$ Both $D_{u+v}$ and $D_u$ are symmetric operators, so their difference (namely the sum $\Sigma$ of the six remaining terms) is a symmetric operator. By hypothesis, $D_u$ is selfadjoint. According to Kato–Rellich theorem as stated in [@ReedSimon2 Theorem X.12, p. 162], it therefore only remains to prove that ${C}$ is also a domain for $\Sigma$ and that for all ${\underline{\omega}}\in {C}$, $$\begin{gathered}
\| \Sigma({\underline{\omega}}) \| \leqslant a \| D_u {\underline{\omega}}\| + b \| {\underline{\omega}}\|,\end{gathered}$$ where both real numbers $a$, $b$ are positive and $a < 1 $. It is clear from the definition of $K_{\pm u}$ and $R_v$ that their actions preserve the core ${C}$ of $C^1$-functions on $G$ and thus $\Sigma({\underline{\omega}})$ has a well-defined meaning for all ${\underline{\omega}}\in {C}$. We decompose ${\underline{\omega}}\in {C}$ into a sum ${\underline{\omega}}= \sum_{k} {\underline{\omega}}_k$ of $C^1$-forms of degree $k$ and start by an estimate of the different terms $\| \Sigma {\underline{\omega}}_k \|$ for any fixed $k$.
Remember from Lemma \[Lem:Formed\] that $d$ can be written $$\begin{gathered}
d = \sum_j \partial_j \otimes T_j - \frac{1}{2} \sum_{k,\alpha,\beta} c^{i_k}_{\alpha \beta} \otimes B^{i_k}_{\alpha \beta},\end{gathered}$$ where the different $B^{i_k}_{\alpha \beta}$ are bounded operators. We remark that the $B^{i_k}_{\alpha \beta}$ commute with right multiplications, like the one appearing in the definition of $K_u$ acting on an element of given degree. For ${\underline{\omega}}= a \otimes {\underline{v}}$ of degree $k$, we write $$\begin{gathered}
K_u d R_{-v} K_{-u} a \otimes {\underline{v}}= \sum \partial _j(a e^{-(n/2-k)h u}(e^{-(n/2-k) hv} -1) )e^{(n/2-(k+1)) h u} \otimes T_j( {\underline{v}}) \\
\hphantom{K_u d R_{-v} K_{-u} a \otimes {\underline{v}}=}{}
- \frac{1}{2} \sum_{k} \sum_{\alpha, \beta} c^{i_k}_{\alpha \beta} a e^{-(n/2-k)h u}(e^{-(n/2-k) hv} -1) e^{(n/2-(k+1)) h u} \otimes B^{i_k}_{\alpha \beta} {\underline{v}}\\
\hphantom{K_u d R_{-v} K_{-u} a \otimes {\underline{v}}}{}
= \sum \partial _j(a e^{-(n/2-k)h u})e^{(n/2-(k+1)) h u} (e^{-(n/2-k) hv} -1) \otimes T_j( {\underline{v}}) \\
\hphantom{K_u d R_{-v} K_{-u} a \otimes {\underline{v}}=}{}
- \frac{1}{2} \sum_{k} \sum_{\alpha, \beta} c^{i_k}_{\alpha \beta} a e^{-h u} (e^{-(n/2-k) hv} -1) \otimes B^{i_k}_{\alpha \beta} {\underline{v}}\\
\hphantom{K_u d R_{-v} K_{-u} a \otimes {\underline{v}}=}{}
+ \sum a e^{-(n/2-k)h u}\partial _j(e^{-(n/2-k) hv} -1) )e^{(n/2-(k+1)) h u} \otimes T_j( {\underline{v}}) \\
\hphantom{K_u d R_{-v} K_{-u} a \otimes {\underline{v}}}{}
= R_{-v}(K_u d K_{-u})(a \otimes {\underline{\omega}}) \! + K_{u} \circ\! \sum_{j} ({\mathcal{R}}_{\partial _j(e^{-(n/2-k) h v})}\! \otimes T_j) \circ K_{-u}(a \otimes {\underline{v}}).\end{gathered}$$ Taking a linear combination to treat the case of a sum ${\underline{\omega}}= \sum {\underline{\omega}}_k$, we get $$\begin{gathered}
K_u d R_{-v} K_{-u} = R_{-v}(K_u d K_{-u}) + K_{u} \circ \bigg( \sum_{j,k} ({\mathcal{R}}_{\partial _j(e^{-(n/2-k) h v})} \otimes T_j) \circ \Pi_k \bigg) \circ K_{-u}.\end{gathered}$$ In this equality, $\sum_{j,k} ({\mathcal{R}}_{\partial _j(e^{-(n/2-k) h v})} \otimes T_j) \circ \Pi_k$ is a finite sum of bounded operators. As a consequence of Lemma \[Lem:CVExp\], the norm of these operators tend to $0$ for $ v \to 0$. We already know that $R_{-v}$ tends to $0$ in norm for $v \to 0$, we therefore get the estimate $$\begin{gathered}
\label{Eqn:EstimateKdRK}
\| K_u d R_{-v} K_{-u}( {\underline{\omega}}) \| \leqslant o_v(1) \| K_u d K_{-u}( {\underline{\omega}}) \| + o_v(1) \| {\underline{\omega}}\|.\end{gathered}$$ The two functions $o_v(1)$ can be taken uniform in $u \in [0,1]$, since $[0,1]$ is a compact.
The term $K_u R_v d K_{-u}$ is easily treated: $K_u R_v d K_{-u} = R_v K_u d K_{-u}$. The term $K_u R_v d R_{-v} K_{-u}$ is processed similarly: $ K_u R_v d R_{-v} K_{-u} = R_v K_u d R_{-v} K_{-u}$ and then the estimate enables us to write $$\begin{gathered}
\| K_u R_v d R_{-v} K_{-u}({\underline{\omega}}) \| \leqslant o_v(1) \big\| \big(K_u d K_u^{-1}\big)({\underline{\omega}}) \big\| + o_v(1) \| {\underline{\omega}}\|.\end{gathered}$$ As a result, we get $$\begin{gathered}
\label{Eqn:Estd}
\| (d_{u+v} - d_u) {\underline{\omega}}\| \leqslant o_v(1) \| d_u {\underline{\omega}}\| + o_v(1) \| {\underline{\omega}}\|.\end{gathered}$$ Lemma \[Lem:Formed\] affords a similar treatment of the term $K_{-u} d^* R_v K_u $, just replacing $T_j$ by $T^*_j$, $B^{i_k}_{\alpha \beta}$ by $(B^{i_k}_{\alpha \beta})^*$ and $c^{i_k}_{\alpha \beta}$ by $\overline{c^{i_k}_{\alpha \beta}}$. We get an estimate $$\begin{gathered}
\label{Eqn:EstdAdj}
\| (d_{u+v}^* - d_u^*) {\underline{\omega}}\| \leqslant o_v(1) \| d_u^* {\underline{\omega}}\| + o_v(1) \| {\underline{\omega}}\|.\end{gathered}$$ Using the equation , which ensures that $\| d_u {\underline{\omega}}\| \leqslant \| D_u {\underline{\omega}}\|$ and $\| d_u^* {\underline{\omega}}\| \leqslant \| D_u {\underline{\omega}}\|$, we can combine and to show that the relation is satisfied.
We can therefore apply the Kato–Rellich theorem for all $u \in [0,1]$ and this proves that all $D_u$ (including $D_1$) have selfadjoint extensions with the same core ${C}= {\mathcal{A}}^1 \otimes \bigwedge^\bullet {\mathfrak{g}}^*$.
\[Rk:Domain\] It appears from the proof of point (2) that we could also take ${\mathcal{A}}^\infty \otimes \bigwedge^\bullet {\mathfrak{g}}^*$ as core for the operator $D_0$ (using the Peter–Weyl decomposition). If we further assume $h \in {\mathcal{A}}^\infty $, the rest of the proof applies *verbatim* and shows that all $D_u$ have a common core, namely ${\mathcal{A}}^\infty \otimes \bigwedge^\bullet {\mathfrak{g}}^*$.
\[Cor:FiniteSumma\] For all selfadjoint elements $h \in {\mathcal{A}}^1$ and all parameters $u \in [0,1]$, the operators $D_u$ are $n^+$-summable.
In the untwisted case, i.e., for $D_0$, we can follow the argument of Theorem 5.5 of [@TrSpLieGpGG] to prove that $d +d^*$ is $n^+$-summable, where $n$ is the dimension of $G$. Indeed, according to Proposition \[Prop:multiplicity\] from [@ErgodCpctGpHKLS], as $G$-vector spaces, we have ${\mathscr{H}}{\hookrightarrow}{\mathscr{H}}_\text{ref}$ where ${\mathscr{H}}_\text{ref} := L^2(G) \otimes \bigwedge^k {\mathfrak{g}}^*$. Moreover, the operator $d +d^* := D_\text{ref}$ on this space is just the Hodge–de Rham operator on $G$ and therefore it is $n^+$-summable. Since $D_\text{ref}$ also preserves the finite-dimensional spaces $V \otimes \bigwedge^k {\mathfrak{g}}^*$ obtained by Peter–Weyl decomposition, the eigenvalues of $|D|$ coincide with those of $|D_\text{ref}|$ except that they may have lower (and possibly zero) multiplicities. Consequently, the same computation as in [@TrSpLieGpGG] proves that $D$ is $n^+$-summable.
To extend this property to all $D_u$ for $u \in [0,1]$, we first note that to prove $D_u$ is $n^+$-summable, it suffices to show that the operator $(D_u + i)^{-1}$ is in the symmetric ideal ${\mathscr{L}}^{n^+}$ – as mentioned in Remark \[Rk:SymmIdeals\]. The existence of the operator $(D_u + i)^{-1}$ is a consequence of Proposition \[Prop:Du\]. The discussion above proves that $(D_0 + i)^{-1}$ is in this ideal.
We then use [@Kato Theorem 1.16, p. 196] to prove that if $(D_u + i)^{-1} \in {\mathscr{L}}^{n^+}$ then for some $\varepsilon > 0$ small enough but independent of $u \in [0,1]$, and for any $v$ in $|v| \leqslant \varepsilon$, then $(D_{u+v} + i)^{-1} \in {\mathscr{L}}^{n^+}$. For all $u$, $v$, $$\begin{gathered}
(D_{u+v} + i ) - (D_u + i) = D_{u+v} - D_u,\end{gathered}$$ and to apply Kato’s stability property, we need to give a relative bound on $D_{u+v} - D_u$, expressed in terms of $D_u +i$. We are going to obtain this using the relation . Indeed, since we know that $D$ is selfadjoint, $\langle D \xi, \xi \rangle = \langle \xi, D \xi \rangle $ and thus $$\begin{gathered}
\| (D + i) \xi \|^2 = \| D \xi \|^2 + \| \xi \|^2.\end{gathered}$$ which shows that $\| D \xi \| \leqslant \| (D + i) \xi \|$. From this fact and , we deduce $$\begin{gathered}
\| (D_{u+v} - D_u)({\underline{\omega}}) \| \leqslant o_v(1) \| (D_u +i) {\underline{\omega}}\| + o_v(1) \| {\underline{\omega}}\|,\end{gathered}$$ which let us apply [@Kato Theorem 1.16, p. 196] to $D_u +i$ and $D_{u+v} - D_u$, leading to the expression $$\begin{gathered}
(D_{u+v} + i)^{-1} = (D_u + i)^{-1} \big(1 + (D_{u+v} - D_u)(D_u +i)^{-1}\big)^{-1},\end{gathered}$$ where both $(D_{u+v} - D_u)(D_u +i)^{-1}$ and $(1 + (D_{u+v} - D_u)(D_u +i)^{-1})^{-1}$ are bounded operators. This expression shows that $(D_{u+v} + i)^{-1}$ is a product of $(D_u + i)^{-1}$ in the ideal ${\mathscr{L}}^{n^+}$ and a bounded operator. It is therefore itself in the ideal ${\mathscr{L}}^{n^+}$ and this completes the proof.
The operator $d_u$ of Proposition \[Prop:Du\] induces a cochain complex:
The operator $d_u := K_u d K_{-u}$, defined from $d = \Pi_+ D$ on the domain of selfadjointness of $D_u$ is closable. Taking its closure, there is a cochain complex ${(d_{u}, {\mathscr{H}}_k)}$ $$\begin{gathered}
\label{Eqn:Complex}
0 \to {\mathscr{H}}_{0} \xrightarrow{d_{u,0}} {\mathscr{H}}_1 \to \cdots \to {\mathscr{H}}_{n-1} \xrightarrow{d_{u, n-1}} {\mathscr{H}}_n \to 0.\end{gathered}$$
In the complex , the map $d_{u,k} \colon {\mathscr{H}}_{k} \to {\mathscr{H}}_{k+1}$ is of course (the closure of) the restriction of $d_u$ to ${\mathscr{H}}_k \cap \operatorname{Dom}(D_u)$, where $\operatorname{Dom}(D_u)$ is the domain of selfadjointness of $D_u$.
We first treat the case of $d$ (for $h = 0$). In this case, if $x_n \to x$ and $y_n \to x$ while both $d x_n$ and $d y_n$ converge, we want to prove that $\lim d x_n = \lim d y_n$. Consider any $z \in {\mathscr{H}}$ which lives in a finite-dimensional vector space $V \otimes \bigwedge^\bullet {\mathfrak{g}}^*$ obtained from the Peter–Weyl decomposition. This ensures that $\Pi_+ z$ is in $V \otimes \bigwedge^\bullet {\mathfrak{g}}^*$ and thus in the domain of $D$. We then have $$\begin{gathered}
\langle z, \Pi_+ D x_n \rangle = \langle D \Pi_+ z, x_n \rangle \to \langle D \Pi_+ z, x \rangle \leftarrow \langle z, \Pi_+ D y_n \rangle.\end{gathered}$$ Since we know that both $d x_n$ and $d y_n$ converge in ${\mathscr{H}}$ and that ${\mathscr{D}}$, the algebraic direct sum of all $V \otimes \bigwedge^\bullet {\mathfrak{g}}^*$ is dense, it is necessary that $\lim d x_n = \lim d y_n$ and this proves that $d$ is closable.
It follows that the kernel $\ker( \overline{d})$ is closed. Since ${\mathcal{A}}^1 \otimes \bigwedge^\bullet {\mathfrak{g}}^*$ is a core for $D$, any $x$ in the domain $\operatorname{Dom}( \overline{d} )$ can be approximated by $x_n \in {\mathcal{A}}^1 \otimes \bigwedge^\bullet {\mathfrak{g}}^*$ such that $x_n \to x$ and $d x_n \to d x$. The density of ${\mathcal{A}}^\infty $ inside ${\mathcal{A}}^1$ (as discussed at the end of Section \[Sec:Reminders\]) then provides an approximation of the original $x \in \dim( \overline{d} )$ by $y_n \in {\mathcal{A}}^\infty \otimes \bigwedge^\bullet {\mathfrak{g}}^* = \Omega^\bullet$. For this sequence $y_n$, we know from Section \[Sec:HodgeOperator\] that $d^2 y_n = 0$. By density, we obtain that is a cochain complex.
Similarly, for $d_u = K_u d K_{-u}$ if $x_n \to x$ and $y_n \to x$ while both $d_u x_n$ and $d_u y_n$ converge, we have $K_{-u} x_n \to K_{-u} x \leftarrow K_{-u} y_n$ and $K_{-u} d_u x_n = d K_{-u} x_n$, $K_{-u} d_u y_n = d K_{-u} y_n$. Since $d$ is closable, we get $\lim K_{-u} d_u x_n = \lim K_{-u} d_u y_n$, which suffices to prove that $d_u$ is also closable. The cochain property then follows from $d_u^2 = K_u d^2 K_{-u} = 0$.
In the rest of this section, we will be interested in the *reduced cohomology* of the complex , namely the cohomology groups $$\begin{gathered}
\label{Eqn:RedCohom}
H^k{(d_{u}, {\mathscr{H}}_k)} := \ker(d_{u,k})/ \overline{\operatorname{Ran}(d_{u,k-1})}.\end{gathered}$$
For any $u \in [0,1]$, let us write $E_{0,u}$ for the kernel of $D_u$. We have the following Hodge decomposition theorem for the conformally perturbed metric:
\[Thm:HodgeDecomp\] Let $G$ be a *compact* Lie group of dimension $n$ acting *ergodically* on a unital $C^*$-algebra $A$. With the notations introduced previously, for any parameter $u \in [0,1]$, there is a decomposition of ${\mathscr{H}}$ into a direct sum of orthogonal Hilbert spaces $$\begin{gathered}
{\mathscr{H}}= E_{-,u} \oplus E_{0,u} \oplus E_{+,u}.\end{gathered}$$
The operator $D_u$ is selfadjoint with compact resolvent, as a consequence of Proposition \[Prop:Du\] and Corollary \[Cor:FiniteSumma\]. Thus, we have an orthogonal sum ${\mathscr{H}}= E_{0,u} \oplus \overline{\operatorname{Ran}(D_u)}$. Following Proposition \[Prop:Du\], $\overline{\operatorname{Ran}(D_u)} = E_{-,u} \oplus E_{+,u}$ and the sum is orthogonal, which proves the result.
We call the restriction of $D_u^2$ to ${\mathscr{H}}_k$ the *Laplacian on ${\mathscr{H}}_k$* and denote it by $\Delta_k$, which is thus an unbounded operator on ${\mathscr{H}}_k$, defined on the domain ${\mathcal{A}}^\infty \otimes \bigwedge^k {\mathfrak{g}}^*$. Note that $\Delta_k$ actually depends on our choice of conformal perturbation $h \in {\mathcal{A}}^1$.
\[Cor:CohomGp\] Let $H^k{(d_{u}, {\mathscr{H}}_k)}$ be the cohomology groups introduced in , they identify naturally with the kernel of $\Delta_k$, i.e., $$\begin{gathered}
\ker(\Delta_k) \simeq H^k{(d_{u}, {\mathscr{H}}_k)}.\end{gathered}$$
\[remarkfinitedimensionalityofcoh\] This Corollary implies in particular that these cohomology groups are finite-dimensional, since $\ker(\Delta_k) = \ker(D_u)$ and $D_u$ has compact resolvent by Corollary \[Cor:FiniteSumma\].
The cohomology group $H^k{(d_{u}, {\mathscr{H}}_k)}$ is defined as $\ker(d_{u, k})/\overline{\operatorname{Ran}(d_{u, k-1})}$. The Hodge decomposition Theorem \[Thm:HodgeDecomp\] can be combined with the projections $\Pi_\pm$ and $\Pi_k$ on ${\mathscr{H}}_k$ to prove that ${\mathscr{H}}_k = \ker(\Delta_k) \oplus \overline{\operatorname{Ran}(d_{u,k-1})} \oplus \overline{\operatorname{Ran}(d_{u, k}^*)}$. We know that $\ker(d_{u, k}) = \operatorname{Ran}(d_{u,k}^*)^\perp$. Therefore $\ker(d_{u,k}) = \overline{\operatorname{Ran}(d_{u,k-1})} \oplus \ker(\Delta_k)$ from which it follows immediately that $H^k{(d_{u}, {\mathscr{H}}_k)} = \ker(d_{u,k})/\overline{\operatorname{Ran}(d_{u,k-1})} \simeq \ker(\Delta_k)$.
\[Prop:Stability\] The cohomology groups $H^k{(d_{u}, {\mathscr{H}}_k)}$ are abstractly isomorphic to the nonperturbed $(h = 0)$ cohomology groups $H^k{(d_{}, {\mathscr{H}}_k)}$.
It is easy to check that $\ker(d_u) = K_u \ker(d)$ and $E_{+, u} = K_u E_{+,0}$. Thus, as abstract vector space, $\ker(d_u)/E_{+,u} = K_u \ker(d_0)/ K_u E_{+,0}$ is finite-dimensional, with the same dimension as $\ker(d_0)/E_{+,0}$.
The dimensions of $\ker(d_u)/E_{+,u}$ and $\ker(d_0)/E_{+,0}$ are the same, but there are not “[concretely]{} isomorphic” for the scalar product we consider. The *concrete realisation* of $\ker(d_u)/E_{+, u}$ is $\{ {\underline{\omega}}\in \ker(d_u)\colon \forall\, {\underline{\omega}}' \in E_{+,u}, \langle {\underline{\omega}}, {\underline{\omega}}' \rangle = 0 \}$. However, $K_u$ does not preserve scalar products and therefore, $\ker(d_u)/E_{+,u}$ is not realised concretely by $K_u E_{0,0}$. In other words, $K_u E_{0,0}$ is *not* the space of harmonic forms for $D_u$.
Conformally twisted spectral triples\
for $\boldsymbol{C^*}$-dynamical systems {#Sec:ConfTwistedTrSp}
========================================
In the following theorem, we use the selfadjoint operator $D_u$ to construct spectral triples for the natural actions of the algebra $A$ (with its left action on ${\mathscr{H}}$) and the algebra $A^\text{op}$ (acting on the right of ${\mathscr{H}}$).
\[Thm:Main\] Let $G$ be a *compact* Lie group of dimension $n$ acting ergodically on a unital $C^*$-algebra $A$, then using the unique $G$-invariant trace $\varphi_0$ of Theorem [\[Thm:ErgodAct\]]{}, we write ${\mathscr{H}}_0 := \operatorname{GNS}(A, \varphi_0)$.
For any fixed $h \in {\mathcal{A}}^1$ and any $u \in [0,1]$, the data $(A, {\mathscr{H}}_0 \otimes \bigwedge^\bullet {\mathfrak{g}}^*, D_u)$ with grading $\gamma$ defines an even $n^+$-summable spectral triple, where
- the representation $\pi$ of $A$ on ${\mathscr{H}}= {\mathscr{H}}_0 \otimes \bigwedge^\bullet {\mathfrak{g}}^*$ is given by restriction of the left multiplication ;
- the unbounded operator $D_u$ is the unique selfadjoint extension of $$\begin{gathered}
D_{u} = K_{u} d K_{-u} + K_{-u} d^* K_{u},\end{gathered}$$ defined on the core ${C}= {\mathcal{A}}^1 \otimes \bigwedge^\bullet {\mathfrak{g}}^*$, the operator $K_u$ being defined by ;
- the grading operator $\gamma$ is defined on degree $k$ forms by $$\begin{gathered}
\gamma( a \otimes v_1 \wedge \cdots \wedge v_k) = (-1)^k ( a \otimes v_1 \wedge \cdots \wedge v_k).\end{gathered}$$
For any fixed $h \in {\mathcal{A}}^1$ and any $u \in [0,1]$, the data $(A^\text{op}, {\mathscr{H}}_0 \otimes \bigwedge^\bullet {\mathfrak{g}}^*, D_u)$ with grading $\gamma$ defines an even $n^+$-summable *twisted* spectral triple, with the automorphism $\beta$ on $A$ given by $\beta(a) = e^{h u} a e^{-h u}$ – we use this $\beta$ to define an automorphism on $A^\text{op}$.
The morphism $\beta$ defined above preserves the multiplication of $A^\text{op}$. It also satisfies the relation *unitarity condition* (see [@TrSpTypeIII equation (3.4)]) that is $\beta\big( (a^\text{op})^* \big) = (\beta^{-1}(a^\text{op}))^*$.
It is clear from the definition of $\pi$ that $A$ is represented on ${\mathscr{H}}$ by bounded operators. The existence and uniqueness of the selfadjoint extension of $D_u$ is proved in Proposition \[Prop:Du\], while the compact resolvent and finite summability properties are shown in Corollary \[Cor:FiniteSumma\].
We now prove that the commutator of $D_u$ with $a \in {\mathcal{A}}^1$ is bounded. To this end, we use the notations of Lemma \[Lem:Formed\] to decompose the operator $D_u$. We call
- Part (0) is the “bounded part” of $D_u$, that is the terms $$\begin{gathered}
- K_u \bigg( \frac{1}{2} \sum_{k,\alpha,\beta} c^{i_k}_{\alpha \beta} \otimes B^{i_k}_{\alpha,\beta} \bigg) K_{-u}
\qquad
\text{and}
\qquad
- K_{-u} \bigg( \frac{1}{2} \sum_{k,\alpha,\beta} \overline{c^{i_k}_{\alpha, \beta}} \otimes (B^{i_k}_{\alpha, \beta})^* \bigg) K_{u}.\end{gathered}$$
- Part (I) consists of the terms $$\begin{gathered}
K_u \bigg( \sum_{j} \partial _j \otimes T_j \bigg) K_{-u}.\end{gathered}$$
- Part (II) consists of the terms $$\begin{gathered}
K_{-u} \bigg( \sum_{j} \partial _j \otimes T_j^* \bigg) K_{u}.\end{gathered}$$
Part (0) commutes with the left multiplication by $a \in {\mathcal{A}}^1$, and thus it does not contribute to the commutator. We therefore only need to estimate Parts (I) and (II) of $D_u( a' {\underline{\omega}})$ for ${\underline{\omega}}= a \otimes v_1 \wedge \cdots \wedge v_k$, that is $$\begin{gathered}
\sum_j \partial_j\big( a' a e^{-(n/2 - k)h u} \big) e^{(n/2 - (k+1))h u} \otimes T_j( v_1 \wedge \cdots \wedge v_k) \\
\qquad\quad{}
+ \sum_j \partial_j\big( a' a e^{(n/2 - k)h u} \big) e^{-(n/2 - (k-1))h u} \otimes T_j^*( v_1 \wedge \cdots \wedge v_k)\\
\qquad{}
= \sum_j \big(\partial_j( a') a e^{-(n/2 - k)h u} + a' \partial_j\big(a e^{-(n/2 - k)h u}\big)\big) e^{(n/2 - (k+1))h u}
\otimes T_j( v_1 \wedge \cdots \wedge v_k) \\
\qquad\quad{}
+ \sum_j \big(\partial_j( a') a e^{(n/2 - k)h u} + a' \partial_j\big(a e^{(n/2 - k)h u}\big)\big) e^{-(n/2 - (k-1))h u} \otimes T_j^*( v_1 \wedge \cdots \wedge v_k).\end{gathered}$$ It follows from these considerations that $$\begin{gathered}
[D_u, a'] {\underline{\omega}}= \sum_j \partial_j(a') a e^{-h u} \otimes (T_j + T_j^*)( v_1 \wedge \cdots \wedge v_k),\end{gathered}$$ which is clearly a bounded function of ${\underline{\omega}}$ for any $a' \in {\mathcal{A}}^1$. Moreover, such $a' \in {\mathcal{A}}^1$ sends the core ${C}$ of our selfadjoint operator $D_u$ to itself and following [@TrSpPiCr-Paterson Proposition A.1, p. 293], this suffices to ensure that $a' \in {\mathcal{A}}^1$ sends the domain of $D_u$ to itself. The algebra ${\mathcal{A}}$ of Definition \[Def:TrSp\] thus contains ${\mathcal{A}}^1$ and is dense in the $C^*$-algebra $A$. This completes the proof that $(A, {\mathscr{H}}, D_u)$ is a $n^+$-summable spectral triple.
=-1 It remains to study its parity: it is clear from the definition that $\gamma$ sends the core ${C}$ to itself and thus it leaves the full domain of the selfadjoint operator $D_u$ stable. Clearly, $\gamma$ distinguishes only between ${\mathscr{H}}_\text{even} := A \otimes \bigwedge^\text{even} {\mathfrak{g}}^*$ and ${\mathscr{H}}_\text{odd} := A \otimes \bigwedge^\text{odd} {\mathfrak{g}}^*$ and $\pi(a)$ leaves both spaces invariant, while $D_u$ is an odd operator. This proves that $(A, {\mathscr{H}}, D_u)$ with $\gamma$ is an even spectral triple.
The parity paragraph above applies *verbatim* to the spectral triple constructed from the right action of $A^\text{op}$. The summability property is also conserved. It remains to investigate the bounded twisted commutators. Notice first that if $a', h \in {\mathcal{A}}^1$ then both right multiplications by $a'$ and by $\beta(a')$ leave the core ${C}$ of $D_u$ invariant and therefore the domain of $D_u$ is also stable under these right multiplication.
Using the decomposition of $D_u$ into Parts (0), (I) and (II), it appears that Part (0) commutes with the right action of $A^\text{op}$ and therefore does not contribute to the commutator. We treat Parts (I) and (II) separately. Keeping only Part (I) in the expression $D_u( {\underline{\omega}}\cdot a')$ for ${\underline{\omega}}= a \otimes v_1 \wedge \cdots \wedge v_k$, we get $$\begin{gathered}
\sum_j \partial_j\big(a a' e^{-(n/2 - k)h u} \big) e^{(n/2 - (k+1))h u} \otimes T_j( v_1 \wedge \cdots \wedge v_k) \\
\qquad{}
= \sum_j \big(\partial_j( a) a' e^{-(n/2 - k)h u} + a \partial_j(a') e^{-(n/2 - k)h u}\big) e^{(n/2 - (k+1))h u}
\otimes T_j( v_1 \wedge \ldots \wedge v_k) \\
\qquad\quad{}
+ \sum_{j} a a' \partial _j\big(e^{-(n/2 - k)h u}\big) e^{(n/2 - (k+1))h u} \otimes T_j(v_1 \wedge \cdots \wedge v_k)\\
\qquad{}
= \sum_{j} \big(\partial _j(a) a' e^{- h u} + a \partial _j(a') e^{-hu} + a a' \partial _j\big(e^{-(n/2 - k)h u}\big) e^{(n/2 - (k+1))h u} \big)\\
\qquad\quad{}
\otimes T_j(v_1 \wedge \cdots \wedge v_k).\end{gathered}$$ We compare this expression to $D_u({\underline{\omega}}) \beta(a')$, i.e., $$\begin{gathered}
\sum_j \partial_j\big(a e^{-(n/2 - k)h u} \big) e^{(n/2 - (k+1))h u} e^{h u} a' e^{-hu} \otimes T_j( v_1 \wedge \cdots \wedge v_k) \\
\qquad{}= \sum_j \partial_j(a) a' e^{-h u} \otimes T_j( v_1 \wedge \cdots \wedge v_k) \\
\qquad\quad{} + \sum_{j} a \partial _j\big(e^{-(n/2 - k)h u} \big) e^{(n/2 - (k+1))h u} e^{h u} a' e^{-hu} \otimes T_j( v_1 \wedge \cdots \wedge v_k).\end{gathered}$$ In these two sums, the only terms that could lead to an unbounded contribution are those containing $\partial _j(a)$, but these two terms cancel. At this point, we must perform the same computation on Part (II) to make sure that the automorphism $\beta$ is also suitable for this case. A very similar computation proves that this it is indeed the case – the key property is that $e^{-(n/2 - k)h u} e^{(n/2 - (k+1))h u} = e^{- hu} = e^{(n/2 - k)h u} e^{-(n/2 - (k-1))h u}$ – and thus the operator (defined *a priori* only on ${C}$) $$\begin{gathered}
D_u \pi^\text{op}\big( (a')^\text{op} \big) - \pi^\text{op}( \beta(a')^\text{op}) D_u,\end{gathered}$$ where $\pi^\text{op}\big( (a')^\text{op} \big) = {\mathcal{R}}_{a'} \otimes \operatorname{Id}_{\bigwedge^\bullet {\mathfrak{g}}^*}$, extends to a bounded operator on ${\mathscr{H}}$.
Since the operator $D_u$ is odd with respect to the grading operator $\gamma$, we can write $D_u$ as combination of $D_u^+ \colon {\mathscr{H}}_\text{even} \to {\mathscr{H}}_\text{odd}$ and $D_u^- \colon {\mathscr{H}}_\text{odd} \to {\mathscr{H}}_\text{even}$. The odd Fredholm operator admits a (possibly) nontrivial index defined as $$\begin{gathered}
\label{Eqn:DefOddIndex}
\operatorname{Index}_\text{odd}(D_u) = \dim \ker(D_u^+) - \dim \ker(D_u^-)\end{gathered}$$ (see, e.g., [@EltNCG equation (9.36), p. 397]).
Existence of a Chern–Gauss–Bonnet theorem\
for conformal perturbations of $\boldsymbol{C^*}$-dynamical systems {#Sec:CGBTheorem}
===================================================================
In this section we show that the Hodge decomposition theorem proved in Section \[Sec:HodgeOperator\] indicates the existence of an analog of the Chern–Gauss–Bonnet theorem for the $C^*$-dynamical systems studied in the present article. Let us explain the classical case before stating the statement for our setting. Indeed, because of the natural isomorphism between the space of harmonic differential forms and the de Rham cohomology groups, for a classical closed manifold $M$, the index of the operator $d+d^*\colon \Omega^\text{even} M \to \Omega^\text{odd} M$ is equal to the Euler characteristic of $M$. On the other hand the McKean–Singer index theorem asserts that the index is given by $$\begin{gathered}
\operatorname{Index}\big(d+d^*\colon \Omega^\text{even} \to \Omega^\text{odd} \big)=\sum_{i=0}^{\dim M} (-1)^i\operatorname{Tr}\big(e^{-t \triangle_i}\big),\end{gathered}$$ where $\triangle_i = d^* d + d d^*$ is the Laplacian on the space of $i$-differential forms on $M$, and $t$ is any positive number. This formula, furthermore, contains local geometric information as $t \to 0^+$, since there is a small time asymptotic expansion of the form $$\begin{gathered}
\operatorname{Tr}\big(e^{-t \triangle_i}\big)
\sim
t^{- \dim M/2} \sum_{j=0}^\infty a_{2j}(\triangle_i) t^j.\end{gathered}$$ The coefficients $a_{2j}(\triangle_i)$ are local geometric invariants, which depend on the high frequency behaviour of the eigenvalues of the Laplacian and are the integrals of some invariantly defined local functions $a_{2j}(x, \triangle_i)$ against the volume form of $M$. Independence of the index from $t$ implies that the alternating sum of the constant terms in the above asymptotic expansions for $\triangle_i$ gives the index. Hence, using the Hodge decomposition theorem, $$\begin{gathered}
\chi(M)=\operatorname{Index}\big(d+d^*\colon \Omega^\text{even} \to \Omega^\text{odd} \big)
=
\int_M \sum_{i=0}^{\dim M} (-1)^ia_{\dim M}(x,\triangle_i) \, d \text{vol}_g.\end{gathered}$$ In fact, the integrand in the latter coincides with the Pfaffian of the curvature form, which is a remarkable and difficult identification [@HeatEquationABP].
With notations and assumptions as in Section \[Sec:HodgeOperator\], we obtain the following result which indicates the existence of an analog of the Chern–Gauss–Bonnet theorem in the setting of $C^*$-dynamical systems studied in this article.
The Euler characteristic $\chi$ of the complex ${(d_{u}, {\mathscr{H}}_k)}$ is related to the odd index defined in $$\begin{gathered}
\chi
=
\sum_{k=0}^n (-1)^k \dim H^k {(d_{u}, {\mathscr{H}}_k)}
=
\sum_{k=0}^n (-1)^k \ker(\Delta_k )
=
\operatorname{Index}_\textnormal{odd} (D_u),\end{gathered}$$ and is independent of the conformal factor $e^{-h}$.
The first equality is actually the definition of the Euler characteristic $\chi$. The second equality is an immediate consequence of Corollary \[Cor:CohomGp\]. The third equality and the last statement can be justified by using Remark \[remarkfinitedimensionalityofcoh\] and Proposition \[Prop:Stability\]. That is, ${\underline{\omega}}\in \ker(\Delta_k)$ means in particular that ${\underline{\omega}}$ is in the domain of $\Delta_k$, which is included in the domain of $D_u$ (by definition). We then have $$\begin{gathered}
0 = \langle {\underline{\omega}}, \Delta_k {\underline{\omega}}\rangle = \langle D_u {\underline{\omega}}, D_u {\underline{\omega}}\rangle,\end{gathered}$$ which proves that ${\underline{\omega}}\in \ker(D_u)$. For a $k$-form ${\underline{\omega}}$, the converse is obvious. It follows that $\ker(D_u^+) = \bigoplus_{k \geqslant 0} \ker(\Delta_{2 k})$ and $\ker(D_u^-) = \bigoplus_{k \geqslant 0} \ker(\Delta_{2 k+1})$, which yields $$\begin{gathered}
\operatorname{Index}_\text{odd}(D_u) = \dim \ker(D_u^+) - \dim \ker(D_u^-)
= \bigoplus_{k \geqslant 0} \dim \ker(\Delta_{2 k}) - \bigoplus_{k \geqslant 0} \ker(\Delta_{2 k+1}) = \chi.\end{gathered}$$ The dimension of these groups are independent of the conformal factor $e^{-h}$ as a consequence of Proposition \[Prop:Stability\].
An alternative proof of the index property using only bounded operators can be obtained using Sobolev spaces. For a clear account of these spaces and their analytic properties in our setting, we refer the reader to the paper [@TrSpLieGpWahl]. Also, in order to have a complete analog of the Chern–Gauss–Bonnet theorem, one needs to find a local geometric formula for the index, which is proved above to be a conformal invariant.
The heat kernels of Laplacians of conformally perturbed metrics on certain noncommutative spaces such as the noncommutative $n$-tori $\mathbb{T}_{\Theta}^n$ admit asymptotic expansions of the form $$\begin{gathered}
\label{Eqn:AsymptExp0}
\operatorname{Tr}( e^{-t \Delta_k}) \sim \sum_{j=0}^\infty a_j( \Delta_k) t^{(j-n)/2}, \qquad t \to 0^+.\end{gathered}$$ In fact, for noncommutative tori, each Laplacian $\Delta_k$ is an elliptic selfadjoint differential operator of order 2, and asymptotic expansions of this form can be derived by using the heat kernel method explained in [@Gilkey] while employing Connes’ pseudodifferential calculus [@CAlgGeoDiff]. This method was indeed used in [@ModCurvCM; @GaussBonnet-ConnesTretkoff; @GaussBonnet-FK; @ScalarCurvNCtorusFK; @ScalarCurv4NCTFK], for calculating and studying the term in the expansion that is related to the scalar curvature of noncommutative two and four tori. Going through this process for noncommutative tori $\mathbb{T}_{\Theta}^n$, one can see that the odd coefficients in the latter asymptotic expansion will vanish, since in their explicit formula in terms of the pseudodifferential symbol of $\Delta_k$, there is an integration over the Euclidean space $\mathbb{R}^n$ of an odd function involved (see [@Gilkey p. 54 and Theorem 1.7.6, p. 58]). Thus in the case of the noncommutative torus $\mathbb{T}_{\Theta}^n$ we can write as $$\begin{gathered}
\label{Eqn:AsymptExp}
\operatorname{Tr}\big( e^{-t \Delta_k}\big) \sim t^{-n/2} \sum_{j=0}^\infty a_{2j} ( \Delta_k) t^j, \qquad t \to 0^+.\end{gathered}$$ Now, using the McKean–Singer index formula [@Gilkey Lemma 1.6.5, p. 47] and our analog of Hodge decomposition theorem, for any $t > 0$ we have $$\begin{gathered}
\label{Eqn:Chi}
\chi = \sum_{k=0}^n (-1)^k \operatorname{Tr}\big( e^{-t \Delta_k}\big).\end{gathered}$$ Thus from equations and , and using the independence of the Euler characteristic from $t$ which implies that only the constant term from contributes to the calculation of the Euler characteristic, we can write $$\begin{gathered}
\chi = \sum_{k=0}^n (-1)^k a_{n}(\Delta_k) = \sum_{k=0}^n (-1)^k \varphi_0({\mathcal{R}}_k),\end{gathered}$$ where the local geometric invariants ${\mathcal{R}}_k$ are derived from the pseudodifferential symbols of the Laplacian $\Delta_k$, by a heat kernel method. This method was used for example in [@ModCurvCM; @ScalarCurvNCtorusFK; @ScalarCurv4NCTFK] for computation of scalar curvature for noncommutative two and four tori. The alternating sum of the ${\mathcal{R}}_k$ gives a noncommutative analog of the local expression for the Euler class.
Summary and conclusions {#Sec:Conclusions}
=======================
The Chern–Gauss–Bonnet theorem is an important generalization of the Gauss–Bonnet theorem for surfaces, which states that the Euler characteristic of an even-dimensional Riemannian manifold can be computed as the integral of a characteristic class, namely the Pfaffian of the curvature form, which is a local invariant of the geometry. In particular, it shows that the integral of this geometric invariant is independent of the metric and depends only on the topology of the manifold. The results obtained in this paper show that the analog of this theorem holds for a general ergodic $C^*$-dynamical system, whose algebra and Lie group are not necessarily commutative. To be more precise, the family of metrics considered for a dynamical system is obtained by using an invertible positive element of the $C^*$-algebra to conformally perturb a fixed metric defined via the unique invariant trace, and our result is about the invariance of a quantity, which is a natural analog of the Euler characteristic, from the conformal factor.
This type of results were previously proved for the noncommutative two torus $\mathbb{T}_\theta^2$. That is, the analog of the Gauss–Bonnet theorem was proved in [@GaussBonnet-ConnesTretkoff] and extended to general translation invariant complex structures on these very important but particular $C^*$-algebras in [@GaussBonnet-FK], where a conformal factor varies the metric. The differential geometry of $C^*$-dynamical systems were developed and studied in [@CAlgGeoDiff], where the noncommutative two torus $\mathbb{T}_\theta^2$ played a crucial role. However the investigation of the analog of the Gauss–Bonnet theorem for $\mathbb{T}_\theta^2$, when the flat metric is conformally perturbed was pioneered in [@PreprintCC], where after heavy calculations, some noncommutative features seemingly indicated that the theorem does not hold. However, studying the spectral action in the presence of a dilaton [@SpActScaleChC], the development of the theory of twisted spectral triples [@TrSpTypeIII], and further studies of examples of complex structures on noncommutative manifolds [@LandietalComplex], led to convincing observations that the Gauss–Bonnet theorem holds for the noncommutative two torus. Then, by further analysis of the expressions and functions of a modular automorphism obtained in [@PreprintCC], Connes and Tretkoff proved the desired result in [@GaussBonnet-ConnesTretkoff] for the simplest translation invariant conformal structure, and the generalization of their result was established in [@GaussBonnet-FK] (where the use of a computer for the heavy computations was inevitable).
It is remarkable that, a non-computational proof of the Gauss–Bonnet theorem for the noncommutative two torus is given in [@ModCurvCM], which is based on the work [@BransonetalConInd], where the conformal index of a Riemannian manifold is defined using properties of conformally covariant operators and the variational properties of their spectral zeta functions. Therefore, since computations are enormously more involved in dimensions higher than two, it is of great importance to use spectral methods to show the existence of the analog of the Chern–Gauss–Bonnet theorem, which is presented in this article, not only for nocommutative tori, but for general $C^*$-dynamical systems. We have also paid special attention to the spectral properties of the analog of the Hodge–de Rham operator of the perturbed metric: we have proved its selfadjointness and shown that the spectral dimension is preserved. We have then shown that this operator gives rise to a spectral triple with the unitary left action of the algebra, and gives a twisted spectral triple with the unitary action of the opposite algebra on the right, generalizing the construction in [@GaussBonnet-ConnesTretkoff] on the noncommutative two torus and providing abstractly a large family of twisted spectral triples.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank the Hausdorff Research Institute for Mathematics (HIM) for their hospitality and support during the trimester program on Noncommutative Geometry and its Applications in 2014, where the present work was partially carried out. They also thank the anonymous referees for their constructive feedback. Parts of this article were obtained and written while the second author was working as a postdoc at the University of Glasgow. He would like to thank C. Voigt for enabling his stay in Scotland.
[99]{}
Albeverio S., H[ø]{}egh-Krohn R., Ergodic actions by compact groups on [$C^{\ast} $]{}-algebras, [*Math. Z.*](http://dx.doi.org/10.1007/BF01215076) **174** (1980), 1–17.
Atiyah M., Bott R., Patodi V.K., On the heat equation and the index theorem, [*Invent. Math.*](http://dx.doi.org/10.1007/BF01425417) **19** (1973), 279–330.
Bhuyain T.A., Marcolli M., The [R]{}icci flow on noncommutative two-tori, [*Lett. Math. Phys.*](http://dx.doi.org/10.1007/s11005-012-0550-0) **101** (2012), 173–194, [arXiv:1107.4788](http://arxiv.org/abs/1107.4788).
Branson T.P., [Ø]{}rsted B., Conformal indices of [R]{}iemannian manifolds, *Compositio Math.* **60** (1986), 261–293.
Bratteli O., Derivations, dissipations and group actions on [$C^*$]{}-algebras, [*Lecture Notes in Math.*](http://dx.doi.org/10.1007/BFb0098817), Vol. 1229, Springer-Verlag, Berlin, 1986.
Carey A.L., Neshveyev S., Nest R., Rennie A., Twisted cyclic theory, equivariant [$KK$]{}-theory and [KMS]{} states, [*J. Reine Angew. Math.*](http://dx.doi.org/10.1515/CRELLE.2011.007) **650** (2011), 161–191, [arXiv:0808.3029](http://arxiv.org/abs/0808.3029).
Carey A.L., Phillips J., Rennie A., Semifinite spectral triples associated with graph [$C^\ast$]{}-algebras, in Traces in Number Theory, Geometry and Quantum Fields, *Aspects Math.*, Vol. E38, Friedr. Vieweg, Wiesbaden, 2008, 35–56, [arXiv:0707.3853](http://arxiv.org/abs/0707.3853).
Carey A.L., Phillips J., Rennie A., Twisted cyclic theory and an index theory for the gauge invariant [KMS]{} state on the [C]{}untz algebra [$O_n$]{}, [*J. K-Theory*](http://dx.doi.org/10.1017/is009010003jkt092) **6** (2010), 339–380, [arXiv:0801.4605](http://arxiv.org/abs/0801.4605).
Chamseddine A.H., Connes A., The spectral action principle, [*Comm. Math. Phys.*](http://dx.doi.org/10.1007/s002200050126) **186** (1997), 731–750, [hep-th/9606001](http://arxiv.org/abs/hep-th/9606001).
Chamseddine A.H., Connes A., Scale invariance in the spectral action, [*J. Math. Phys.*](http://dx.doi.org/10.1063/1.2196748) **47** (2006), 063504, 19 pages, [hep-th/0512169](http://arxiv.org/abs/hep-th/0512169).
Cohen P.B., Connes A., Conformal geometry of the irrational rotation algebra, [P]{}reprint MPI, 1992.
Connes A., [$C^{\ast}$]{} algèbres et géométrie différentielle, *C. R. Acad. Sci. Paris Sér. A-B* **290** (1980), A599–A604, [hep-th/0101093](http://arxiv.org/abs/hep-th/0101093).
Connes A., Noncommutative differential geometry, *Inst. Hautes Études Sci. Publ. Math.* **62** (1985), 257–360.
Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
Connes A., Moscovici H., The local index formula in noncommutative geometry, [*Geom. Funct. Anal.*](http://dx.doi.org/10.1007/BF01895667) **5** (1995), 174–243.
Connes A., Moscovici H., Type [III]{} and spectral triples, in Traces in number theory, geometry and quantum fields, *Aspects Math.*, Vol. E38, Friedr. Vieweg, Wiesbaden, 2008, 57–71, [math.OA/0609703](http://arxiv.org/abs/math.OA/0609703).
Connes A., Moscovici H., Modular curvature for noncommutative two-tori, [*J. Amer. Math. Soc.*](http://dx.doi.org/10.1090/S0894-0347-2014-00793-1) **27** (2014), 639–684, [arXiv:1110.3500](http://arxiv.org/abs/1110.3500).
Connes A., Tretkoff P., The [G]{}auss–[B]{}onnet theorem for the noncommutative two torus, in Noncommutative Geometry, Arithmetic, and Related Topics, Johns Hopkins Univ. Press, Baltimore, MD, 2011, 141–158, [arXiv:0910.0188](http://arxiv.org/abs/0910.0188).
Cuntz J., Simple [$C^*$]{}-algebras generated by isometries, [*Comm. Math. Phys.*](http://dx.doi.org/10.1007/BF01625776) **57** (1977), 173–185.
D[a]{}browski L., Sitarz A., An asymmetric noncommutative torus, [*SIGMA*](http://dx.doi.org/10.3842/SIGMA.2015.075) **11** (2015), 075, 11 pages, [arXiv:1406.4645](http://arxiv.org/abs/1406.4645).
Devastato A., Martinetti P., Twisted spectral triple for the standard model and spontaneous breaking of the grand symmetry, [arXiv:1411.1320](http://arxiv.org/abs/1411.1320).
Fathizadeh F., On the scalar curvature for the noncommutative four torus, [*J. Math. Phys.*](http://dx.doi.org/10.1063/1.4922815) **56** (2015), 062303, 14 pages, [arXiv:1410.8705](http://arxiv.org/abs/1410.8705).
Fathizadeh F., Khalkhali M., The algebra of formal twisted pseudodifferential symbols and a noncommutative residue, [*Lett. Math. Phys.*](http://dx.doi.org/10.1007/s11005-010-0419-z) **94** (2010), 41–61, [arXiv:0810.0484](http://arxiv.org/abs/0810.0484).
Fathizadeh F., Khalkhali M., Twisted spectral triples and [C]{}onnes’ character formula, in Perspectives on Noncommutative Geometry, *Fields Inst. Commun.*, Vol. 61, Amer. Math. Soc., Providence, RI, 2011, 79–101, [arXiv:1106.6127](http://arxiv.org/abs/1106.6127).
Fathizadeh F., Khalkhali M., The [G]{}auss–[B]{}onnet theorem for noncommutative two tori with a general conformal structure, [*J. Noncommut. Geom.*](http://dx.doi.org/10.4171/JNCG/97) **6** (2012), 457–480, [arXiv:1005.4947](http://arxiv.org/abs/1005.4947).
Fathizadeh F., Khalkhali M., Scalar curvature for the noncommutative two torus, [*J. Noncommut. Geom.*](http://dx.doi.org/10.4171/JNCG/145) **7** (2013), 1145–1183, [arXiv:1110.3511](http://arxiv.org/abs/1110.3511).
Fathizadeh F., Khalkhali M., Weyl’s law and [C]{}onnes’ trace theorem for noncommutative two tori, [*Lett. Math. Phys.*](http://dx.doi.org/10.1007/s11005-012-0593-2) **103** (2013), 1–18, [arXiv:1111.1358](http://arxiv.org/abs/1111.1358).
Fathizadeh F., Khalkhali M., Scalar curvature for noncommutative four-tori, [*J. Noncommut. Geom.*](http://dx.doi.org/10.4171/JNCG/198) **9** (2015), 473–503, [arXiv:1301.6135](http://arxiv.org/abs/1301.6135).
Fathizadeh F., Wong M.W., Noncommutative residues for pseudo-differential operators on the noncommutative two-torus, [*J. Pseudo-Differ. Oper. Appl.*](http://dx.doi.org/10.1007/s11868-011-0030-9) **2** (2011), 289–302.
Gabriel O., Grensing M., Ergodic actions and spectral triples, [arXiv:1302.0426](http://arxiv.org/abs/1302.0426).
Gilkey P.B., Invariance theory, the heat equation, and the [A]{}tiyah–[S]{}inger index theorem, *Mathematics Lecture Series*, Vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984.
Gracia-Bond[í]{}a J.M., V[á]{}rilly J.C., Figueroa H., Elements of noncommutative geometry, [*Birkhäuser Advanced Texts: Basler Lehrbücher*](http://dx.doi.org/10.1007/978-1-4612-0005-5), Birkhäuser Boston, Inc., Boston, MA, 2001.
Greenfield M., Marcolli M., Teh K., Twisted spectral triples and quantum statistical mechanical systems, *p-Adic Numbers Ultrametric Anal. Appl.* **6** (2014), 81–104.
Higson N., The residue index theorem of [C]{}onnes and [M]{}oscovici, in Surveys in noncommutative geometry, *Clay Math. Proc.*, Vol. 6, Amer. Math. Soc., Providence, RI, 2006, 71–126.
H[ø]{}egh-Krohn R., Landstad M.B., St[ø]{}rmer E., Compact ergodic groups of automorphisms, [*Ann. of Math.*](http://dx.doi.org/10.2307/1971377) **114** (1981), 75–86.
Iochum B., Masson T., Crossed product extensions of spectral triples, *J. Noncommut. Geom.*, [t]{}o appear, [arXiv:1406.4642](http://arxiv.org/abs/1406.4642).
Kaad J., On modular semifinite index theory, [arXiv:1111.6546](http://arxiv.org/abs/1111.6546).
Kaad J., Senior R., A twisted spectral triple for quantum [${\rm SU}(2)$]{}, [*J. Geom. Phys.*](http://dx.doi.org/10.1016/j.geomphys.2011.12.019) **62** (2012), 731–739, [arXiv:1109.2326](http://arxiv.org/abs/1109.2326).
Kato T., Perturbation theory for linear operators, [*Classics in Mathematics*](http://dx.doi.org/10.1007/978-3-642-66282-9), Springer-Verlag, Berlin, 1995.
Khalkhali M., Landi G., van Suijlekom W.D., Holomorphic structures on the quantum projective line, [*Int. Math. Res. Not.*](http://dx.doi.org/10.1093/imrn/rnq097) **2011** (2011), 851–884, [arXiv:0907.0154](http://arxiv.org/abs/0907.0154).
Knapp A.W., Lie groups, [L]{}ie algebras, and cohomology, *Mathematical Notes*, Vol. 34, Princeton University Press, Princeton, NJ, 1988.
Kr[ý]{}sl S., Hodge theory for elliptic complexes over unital [$C^*$]{}-algebras, [*Ann. Global Anal. Geom.*](http://dx.doi.org/10.1007/s10455-013-9394-9) **45** (2014), 197–210, [arXiv:1309.4560](http://arxiv.org/abs/1309.4560).
Kr[ý]{}sl S., Hodge theory for complexes over [$C^*$]{}-algebras with an application to [$A$]{}-ellipticity, [*Ann. Global Anal. Geom.*](http://dx.doi.org/10.1007/s10455-015-9449-1) **47** (2015), 359–372, [arXiv:1309.4560](http://arxiv.org/abs/1309.4560).
Kr[ý]{}sl S., Elliptic complexes over [$C^*$]{}-algebras of compact operators, [*J. Geom. Phys.*](http://dx.doi.org/10.1016/j.geomphys.2015.12.001) **101** (2016), 27–37, [arXiv:1506.06244](http://arxiv.org/abs/1506.06244).
Lesch M., Moscovici H., Modular curvature and [Morita]{} equivalence, [arXiv:1505.00964](http://arxiv.org/abs/1505.00964).
Lord S., Sukochev F., Zanin D., Singular traces. Theory and applications, *De Gruyter Studies in Mathematics*, Vol. 46, De Gruyter, Berlin, 2013.
Moscovici H., Local index formula and twisted spectral triples, in Quanta of Maths, *Clay Math. Proc.*, Vol. 11, Amer. Math. Soc., Providence, RI, 2010, 465–500, [arXiv:0902.0835](http://arxiv.org/abs/0902.0835).
Paterson A.L.T., Contractive spectral triples for crossed products, *Math. Scand.* **114** (2014), 275–298, [arXiv:1204.4404](http://arxiv.org/abs/1204.4404).
Ponge R., Wang H., Noncommutative geometry and conformal geometry. [III]{}. [V]{}afa–[W]{}itten inequality and [P]{}oincaré duality, [*Adv. Math.*](http://dx.doi.org/10.1016/j.aim.2014.12.009) **272** (2015), 761–819, [arXiv:1310.6138](http://arxiv.org/abs/1310.6138).
Reed M., Simon B., Methods of modern mathematical physics. [I]{}. Functional analysis, Academic Press, Inc., New York, 1980.
Reed M., Simon B., Methods of modern mathematical physics. [II]{}. Fourier analysis, selfadjointness, Academic Press, Inc., New York, 1975.
Rieffel M.A., Metrics on states from actions of compact groups, *Doc. Math.* **3** (1998), 215–229, [math.OA/9807084](http://arxiv.org/abs/math.OA/9807084).
Simon B., Trace ideals and their applications, *Mathematical Surveys and Monographs*, Vol. 120, 2nd ed., Amer. Math. Soc., Providence, RI, 2005.
St[ø]{}rmer E., Spectra of ergodic transformations, [*J. Funct. Anal.*](http://dx.doi.org/10.1016/0022-1236(74)90019-6) **15** (1974), 202–215.
Wahl C., Index theory for actions of compact [L]{}ie groups on [$C^*$]{}-algebras, *J. Operator Theory* **63** (2010), 217–242, [arXiv:0707.3207](http://arxiv.org/abs/0707.3207).
Wassermann A., Ergodic actions of compact groups on operator algebras. [I]{}. [G]{}eneral theory, [*Ann. of Math.*](http://dx.doi.org/10.2307/1971422) **130** (1989), 273–319.
Wassermann A., Ergodic actions of compact groups on operator algebras. [II]{}. [C]{}lassification of full multiplicity ergodic actions, [*Canad. J. Math.*](http://dx.doi.org/10.4153/CJM-1988-068-4) **40** (1988), 1482–1527.
Wassermann A., Ergodic actions of compact groups on operator algebras. [III]{}. [C]{}lassification for [${\rm SU}(2)$]{}, [*Invent. Math.*](http://dx.doi.org/10.1007/BF01394336) **93** (1988), 309–354.
Wodzicki M., Local invariants of spectral asymmetry, [*Invent. Math.*](http://dx.doi.org/10.1007/BF01403095) **75** (1984), 143–177.
Wodzicki M., Noncommutative residue. [I]{}. [F]{}undamentals, in [$K$]{}-Theory, Arithmetic and Geometry ([M]{}oscow, 1984–1986), [*Lecture Notes in Math.*](http://dx.doi.org/10.1007/BFb0078372), Vol. 1289, Springer, Berlin, 1987, 320–399.
|
**Angular Momentum Imparted\
To Test Particles by\
Gravitational Waves\
**
**By**
**Muhammad Shoaib\
e-mail: safridi@gmail.com**
A thesis submitted in partial fulfilment\
of the Requirements of Quaid-I-Azam University Islamabad, Pakistan\
For the degree of Master of Philosophy
September 1999
*I would like to dedicate this thesis to\
**my parents**\
*
GRAVITATIONAL WAVES
====================
Introduction
-------------
Almost every student of science is familiar with the phenomenon of waves, such as water waves ( ripples rolling across the ocean), sound waves (vibration in the air), etc. The above mentioned kinds of waves are easily observable. Electromagnetic waves are comparatively difficult to understand but not as difficult as gravitational waves. The existence of gravitational waves was disputed for a long time but is generally accepted now.
What are gravitational waves? How do they propagate? and what is their energy content? These questions are addressed in the first two chapters. In the third chapter the pseudo-Newtonian formalism and its extension is reviewed in general and the formula for the momentum imparted to test particles in arbitrary spacetimes is reviewed in particular. In chapter four the analysis of a paper claiming to determine the spin for gravitational waves is given, and compared with the spin given by a geodesic analysis. It is demonstrated that the other claim is inconsistent. Finally in chapter five a summary of the work is given with the conclusion.
Idealization
-------------
The only way to come to grips with so complicated a subject as General Relativity is by idealization[ . ]{}Study one idealization after another. Build a catalogue of idealization, of their properties and of techniques for analyzing them and then arrive at a conclusion.
Let’s now see how can we idealize gravitational waves. As one idealizes “water waves” as small ripples of geometry rolling across the ocean so one gives the name “gravitational waves” to small ripples rolling across the space time. Both of the waves are idealizations. One cannot, with infinite accuracy, delineate at any moment which drops of water are in the waves and which are in the underlying ocean. Similarly one cannot tell precisely which parts of spacetime are in the ripples and which are in the cosmological backgrounds. One can almost do so; otherwise one would not speak of “waves”. Look at the ocean, the seascape is dominated by the waves. Changes occur at the surface of the ocean, which propagates obeying the following \[1\] wave equation:
$$(\frac{1}{g^{2}}\frac{\partial ^{4}}{\partial t^{4}}+\frac{\partial ^{2}}{\partial y^{2}}+\frac{\partial ^{2}}{\partial x^{2}})(\textrm{height of the
surface)}=0. \label{1.1}$$
Similarly gravitational waves are perturbations of spacetime. As I have already stated that gravitational waves are ripples of geometry. So let me support my statement. Take a massive body and disturb it violently, the near field adjusts rapidly, but the far field must wait for the signal that the mass has moved to propagate to it at a finite speed $``c"$. Thus there is a travelling kink which falls off in strength with distance. Hence the gravitational waves are small ripples rolling across the spacetime.
Now get more sophisticated. Notice from a space ship the large-scale curvature of the ocean’s surface-curvature because the Earth is round. Curvature because the Earth, Sun and Moon pull the water. As waves propagate long distances, this curvature bends their fronts and changes slightly their simple wave equation. Spacetime is similar. Propagating through the universe, according to Einstein’s theory must be a complex pattern of small-scale ripples in the spacetime curvature, ripples produced by binary stars, by gravitational collapse, by explosion in galactic nuclei etc.
Linear approximation for the investigation of gravitational waves
------------------------------------------------------------------
We are discussing gravitational waves in the frame work of General Relativity, a nonlinear field theory of gravity. Though many of the interesting consequences of General Relativity comes from its non linearity it is worthwhile to study its linear approximation. Linearization actually leads one to the gravitational waves.
What makes General Relativity nonlinear? As the field in General Relativity is the metric tensor, its appearance in the field equations non-linearly gives rise to the non-linearity of General Relativity. We cannot change the way the metric tensor enters into the curvature but we can write the curved spacetime metric as the flat spacetime metric tensor, $\eta _{\mu \nu }$, and an additional term, $h_{\mu \nu }.$ We then require that $h_{\mu \nu }$ and its derivative occur only once in the field equations and higher powers be neglected\[6\].
$$g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu }.$$
Let
$$g^{\mu \nu }=\eta ^{\mu \nu }+f^{\mu \nu },$$
where $\eta ^{\mu \nu }\;$is the inverse of $\eta _{\mu \nu }$. Then by definition,
$$g^{\mu \rho }g_{\rho \pi }=(\eta ^{\mu \rho }+f^{\mu \rho })(\eta _{\rho \pi
}+h_{\rho \pi }),$$
Which implies that
$$\delta _{\pi }^{\mu }=\delta _{\pi }^{\mu }+f^{\mu \rho }\eta _{\rho \pi
}+\eta ^{\mu \rho }h_{\rho \pi }+f^{\mu \rho }h_{\rho \pi }.$$
Cancelling $\delta _{\pi }^{\mu }$ on both sides and multiplying through by $\eta ^{\nu \pi }$ we get
$$f^{\mu \nu }+\eta ^{\mu \rho }\eta ^{\nu \pi }h_{\rho \pi }+\eta ^{\nu \pi
}f^{\mu \rho }h_{\rho \pi }=0.$$
The last term is clearly quadratic in the difference between the curved and flat spacetime metric tensor. Thus to first order,
$$f^{\mu \nu }=-\eta ^{\mu \rho }\eta ^{\nu \pi }h_{\rho \pi }+O(h^{2}).$$
Using the flat spacetime metric tensor to raise and lower indices we can write
$$f^{\mu \nu }=-h^{\mu \nu }+O(h^{2}).$$
Using equation (1.3), equation (1.8) becomes
$$\begin{aligned}
g^{\mu \nu } &=&\eta ^{\mu \nu }-h^{\mu \nu }+O(h^{2}) \\
&\approx &\eta ^{\mu \nu }-h^{\mu \nu }. \nonumber\end{aligned}$$
Using this linearization and for the moment taking Cartesian coordinates, so that there are no derivatives of $\eta ^{\mu \nu },\,$the Christoffel symbols linearize to:
$$\left\{
\begin{array}{c}
\rho \\
\mu \;\upsilon
\end{array}
\right\} \approx \frac{1}{2}\eta ^{\rho \pi }(h_{\mu \nu ,\pi }+h_{\nu \pi
,\mu }-h_{\mu \nu ,\pi }).$$
Clearly terms quadratic in the Christoffel symbols become quadratic in $h$ and can be neglected compared with linear terms. Thus the linearized Ricci tensor is
$$\begin{aligned}
R_{\mu \nu } &=&\left\{
\begin{array}{c}
\rho \\
\mu \;\upsilon
\end{array}
\right\} _{,\rho }-\left\{
\begin{array}{c}
\rho \\
\mu \;\rho
\end{array}
\right\} _{,\nu } \\
&\approx &\frac{1}{2}[\eta ^{\rho \pi }(h_{\mu \pi ,\nu }+h_{\nu \pi ,\mu
}-h_{\mu \nu ,\pi })]_{,\rho } \\
&&-\frac{1}{2}[\eta ^{\rho \pi }(h_{\mu \pi ,\rho }+h_{\rho \pi ,\mu
}-h_{\mu \rho ,\pi })]_{,\nu } \\
&=&\frac{1}{2}\eta ^{\rho \pi }(h_{\mu \pi ,\nu \rho }+h_{\nu \pi ,\mu \rho
}-h_{\mu \nu ,\rho \pi }-h_{\rho \pi ,\mu \nu })\end{aligned}$$
A choice of coordinates can be made to have the first two and the last term in the brackets disappear. To see this first we note that we can rewrite the Ricci tensor as:
$$\begin{aligned}
R_{\mu \nu } &\approx &\frac{1}{2}(h_{\mu }^{\rho }-\frac{1}{2}h\delta _{\mu
}^{\rho })_{,\rho \nu }+\frac{1}{2}(h_{\nu }^{\rho }-\frac{1}{2}h\delta
_{\nu }^{\rho })_{,\mu \rho } \nonumber \\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\frac{1}{2}\eta ^{\rho
\pi }h_{\mu \nu ,\rho \pi },\end{aligned}$$
where $h=h_{\mu }^{\mu }$ then we can break the first bracket in to two terms. Now consider an infinitesimal transformation
$$x^{\mu }\longrightarrow x^{^{\prime }\mu }=x^{\mu }+\xi ^{\mu }(x^{\rho }),$$
so that the terms quadratic in $h$ or $\xi $ can be neglected. Since this is only a coordinate transformation, it must leave the metric invariant and hence
$$\begin{aligned}
ds^{2} &=&g_{\mu \nu }(x^{\rho })dx^{\mu }dx^{\nu }=g_{\mu \nu }^{^{\prime
}}(x^{\prime \rho })dx^{^{\prime }\mu }dx^{\prime \nu } \nonumber \\
&=&g_{\mu \nu }^{\prime }(x^{\prime \rho })(dx^{\mu }+\xi _{,\alpha }^{\mu
}dx^{\alpha })(dx^{\nu }+\xi _{,\beta }^{\nu }dx^{\beta }).\end{aligned}$$
Using the linearization procedure it is easy to see that
$$h_{\mu \nu }^{^{\prime }}\approx h_{\mu \nu }-\xi _{\mu ,\nu }-\xi _{\nu
,\mu }.$$
Thus we have
$$\begin{aligned}
(h_{\mu }^{\rho }-\frac{1}{2}h\delta _{\mu }^{\rho }) &=&(h_{\mu }^{\prime
\rho }-\frac{1}{2}h^{\prime }\delta _{\mu }^{\rho })-\eta ^{\nu \rho }\xi
_{\mu ,\nu } \nonumber \\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\,\xi
_{,\nu }^{\rho }+\xi _{,\nu }^{\nu }\delta _{\mu }^{\rho }.\end{aligned}$$
Differentiating the above equation relative to $x^{\rho }\,$it is easy to see that the last two terms simply cancel and we have
$$(h_{\mu }^{\rho }-\frac{1}{2}h\delta _{\mu }^{\rho })_{,\rho }=(h_{\mu
}^{\prime \rho }-\frac{1}{2}h^{\prime }\delta _{\mu }^{\rho })_{,\rho }-\eta
^{\nu \rho }\xi _{\mu ,\nu \rho }. \label{eq14}$$
Taking the harmonic gauge condition $$\square \xi _{\mu }=-\phi _{\mu ,\rho }^{\rho },$$ we can make
$$(h_{\mu }^{\rho }-\frac{1}{2}h\delta _{\mu }^{\rho })_{,\rho }=\phi _{\mu
,\rho }^{\rho }=0. \label{eq15}$$
It should be pointed out here that the use of Cartesian coordinates was not crucial, but merely for convenience. With any other coordinates we could have to introduce the corresponding flat spacetime Christoffel symbols. This has been avoided so as not cause confusion of notation. We now drop the primes and so obtain $$R_{\mu \nu }\approx -\frac{1}{2}\eta ^{\rho \pi }h_{\mu \nu ,\rho \pi }=-\frac{1}{2}\square h_{\mu \nu }.$$
We can contract Equation (\[eq15\]) to obtain the Ricci scalar ($R=-\frac{1}{2}\square h)$ and hence combine them to obtain the Einstein tensor. Thus the Einstein field equation becomes $$\square \phi _{\mu \nu }=-2\kappa T_{\mu \nu }.$$ Here $T_{\mu \nu }$ is the stress energy tensor and $\kappa $ is the proportionality constant. In regions where $T_{\mu \nu }=0,$ $\phi _{\mu \nu
}$ then satisfies the wave equation, Thus $\phi _{\mu \nu }$ represent gravitational waves.
Plane wave solution in the linearized theory
---------------------------------------------
The simplest of all solutions to the linearized equation $h_{\mu \nu ,\alpha
}^{\alpha }=0$ is the monochromatic plane wave solution \[1\]
$$h_{\mu \nu }=\Re [A_{\mu \nu }\exp (ik_{\alpha }x^{\alpha })].$$
Here $\Re $ means that one must take the real part in the bracket. While $A_{\mu \nu }\,$is the amplitude and $k_{\mu }$ is the wave vector, satisfying
$$\begin{aligned}
k_{\alpha }k^{\alpha } &=&0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textrm{(}{\bf k}\textrm{\thinspace a\thinspace null\thinspace vector),} \\
A_{\mu \alpha }k^{\alpha } &=&0\,\,\,\,\,\,\,\,\,\,\,\,({\bf
A}\textrm{ orthogonal to }{\bf k}\textrm{).} \nonumber\end{aligned}$$
This solution describes a wave with the frequency
$$\omega =k^{0}=(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})^{1/2},$$ which propagate with the speed of light in the direction $\,(1/k^{0})(k_{x},k_{y},k_{z})$. At first sight the amplitude $A_{\mu \nu \textrm{ }}$ appears to have six independent components (ten, less the four orthogonality constraints $A_{\mu
\alpha }k^{\alpha }=0$). But this cannot be right. The gravitational field has two dynamic degrees of freedom (not six).
The plane wave vector
$$\xi ^{\mu }=-iC^{\mu }\exp (ik_{\alpha }x^{\alpha }),$$
with four arbitrary constants $C^{\mu },$ generates a gauge transformation that can change arbitrarily four of the six independent components of $A_{\mu \nu }.\,$One gets rid of this arbitrariness by choosing a specific gauge.
The transverse traceless (TT) gauge
------------------------------------
Consider the four velocity ${\bf u}$ through all the space time and impose the condition \[1\]
$$A_{\mu \nu }u^{\nu }=0.$$
These are only three constraints on $A_{\mu \nu }$. Why not four? Because ${\bf A}$ is orthogonal to ${\bf \,k}$ i.e. $k^{\mu }(A_{\mu \nu }u^{\nu })=0$, as already mentioned. As a fourth constraint use a gauge transformation to set$\,\,A_{\mu }^{\mu }=0.$ So there are now four constraints in all, $A_{\mu \alpha }u^{\alpha }=A_{\mu \alpha }k^{\alpha }=A_{\alpha }^{\alpha
}=0,$ on the ten components of amplitude. Thus the two remaining free components of $A_{\mu \nu }$ represents the two degrees of freedom in the plane gravitational wave.
It is useful to restate the four constraints,$\,A_{\mu \alpha }u^{\alpha
}=A_{\mu \alpha }k^{\alpha }=A_{\alpha }^{\alpha }=0$ in the Lorentz frame where $u^{0}=1,u^{j}=0$ and in a frame where $k^{\alpha }$ does not appear explicitly:
$$h_{\mu 0}=0\textrm{ , i.e only the spatial componemts
}h_{jk\textrm{ }}\textrm{are non-zero} \label{1}$$
$$h_{kj,j}=0\textrm{ i.e the spatial components are divergence free}
\label{2}$$
$$h_{kk}=h_{\mu }^{\mu }=0\textrm{ i.e the spatial components are
trace free.} \label{3}$$
The gauge conditions are all linear in $h_{\mu \nu }$ so an arbitrary wave will also satisfy the above gauge conditions. In this gauge only the $h_{jk}$ are non-zero. So we need only to impose the six wave equations $$\square h_{jk}=h_{jk,\alpha \alpha }=0$$ Now the definition. Any symmetric tensor satisfying the constraints (\[1\]), (\[2\]) and (\[3\]) is called a transverse traceless tensor. Why it is called transverse traceless? Because it is purely spatial ($h_{\mu 0}=0$) and, if thought of as a wave is transverse to the direction of propagation ($h_{ij,j}=h_{ij}k_{j}=0$) and traceless because $h_{kk}=0$.
The special gauge in which $h_{\mu \nu }$ reduces to its transverse traceless part is called the transverse traceless gauge . The conditions (\[1\]), (\[2\]) and (\[3\]) defining this gauge can be summarized as: $$h_{\mu \nu }=h_{\mu \nu }^{TT}.$$
Only pure waves can be reduced to TT gauge. In the TT-gauge the space components $$R_{j0ko}=R_{0j0k}=-R_{j00k}=-R_{0jk0}.$$ of the Riemannian curvature tensor have an especially simple form $$R_{j0k0}=-\frac{1}{2}h_{jk,00}^{TT}.$$ As the curvature tensor is gauge invariant therefore $h_{\mu \nu }$ cannot be reduced to still fewer components than it has in the TT-gauge.
Comparison with electromagnetic waves
--------------------------------------
A simple system consisting of two charges of equal magnitude but of opposite signs, each situated at a distance $r/2$ from the origin $O$ (see $Fig.1.1$), which is taken to lie on the line connecting the charges, is the simplest example of an electric dipole. The oscillation in the dipole generates electromagnetic waves.
Now we define the electric dipole moment of the pair of equal charges as the product of charge $q$ and the separation $r,$
$${\bf d}=qr{\bf e,} \label{3.30}$$
where ${\bf e}$ is the unit vector from negative to positive charge. Consider a system of charges $q_{a}$ and let $r_{a}$ be their radius vectors then
$${\bf d}={\sum_a }q_{a}r_{a}, \label{3.31}$$
is called the dipole moment of the system of charges.
Consider masses in an isolated, nearly Newtonian system, moving about each other. They emit radiations. For an order of magnitude estimate, one can apply the familiar radiation formula of electromagnetic theory, with the replacement $q^{2}\rightarrow -m^{2}$ which converts the static Coulomb’s law into Newton’s law of attraction. Although it introduces a moderate error in numerical factor and changes angular distribution, but it gives an estimate of the total power radiated. In electromagnetic theory, electric dipole radiation dominates, with power output or luminosity “$L$” given by\[1\], $${\bf L}_{elec\,\,dip}=\frac{2}{3}q^{2}a^{2},$$ where $a$ is the acceleration in the dipole. Then using equation (\[3.30\]) we can write
$${\bf L}_{elec\,dip}=\frac{2}{3}\stackrel{\cdot \cdot }{d}^{2}. \label{3.33}$$
The gravitational analogue of the electric dipole moment is the mass dipole moment,
$${\bf d\,}={\sum_{\,A} }m_{A}x_{A}, \label{3.34}$$
its first time-rate of change is the total momentum of the system,
$$\stackrel{.}{{\bf d\,}}=\sum m_{A}\stackrel{.}{x_{A}^{.}=\,{\bf P}},$$
where ${\bf P}$ is the momentum of the system. The time-rate of change of the mass dipole moment has to vanish because of the law of conservation of momentum, $\stackrel{..}{d\,\,}=\,\stackrel{.}{P\,}=\,0$ therefore there can be no mass dipole radiation in gravitation physics.
The next strongest type of electromagnetic radiation are magnetic dipole and electric quadrupole. Magnetic dipole radiation is generated by the second time derivative of the magnetic moment,$\,\stackrel{..}{\mu }.$ Here again the gravitational analogue is a constant of motion of the angular momentum i.e.,
$$\begin{aligned}
\mu &=&\sum (position\,of\,\,A)\times (current\,due\,to\,A) \nonumber \\
&=&\sum r_{A}\times (mV_{A})={\bf J}=\textrm{{\it constant}}\end{aligned}$$
Which shows that $\stackrel{..}{\mu }=0,$ so it can not radiate. Thus, there can be no gravitational dipole radiation of any sort. Physical example is a system of two masses attached by a spring, acts as an oscillating dipole.
### Comparison with plane electromagnetic waves
Consider the metric \[1\] (with signature $(+,-,-,-)$)
$$ds^{2}=L^{2}(u)(dx^{2}+dy^{2})-dudv. \label{A}$$
where $u=t-z\,$and$\,v=t+z\,\,\,$which is always flat. It satisfies the vacuum Einstein equations $R_{\mu \nu }=0\,\,$which implies that$\,\,L^{\prime \prime }=0\,\;$(see Appendix 1 for the proof). Now if $L^{\prime \prime }=0$ the spacetime is static therefore cannot represent gravitational waves. In this metric the electromagnetic potential$${\bf A}=A_{\mu }{\bf d}x^{\mu }=A(u)dx \label{B}$$ satisfies the Maxwell equation for arbitrary $A(u).$ It represents an electromagnetic plane wave analogous to the gravitational plane wave. The only non-zero components of this wave are
$$F_{ux}=A^{\prime }\;i.e\;\,\,\,\,\,\,\,\,\,F_{tx}=-F_{zx}=A^{\prime }.\,\,$$
$\,\,\,\,\,\,$
So the wave propagates in the $z$-direction, the magnetic vector oscillates in the $y$-direction and electric vector in the $x$-direction. The only non-zero vector of the stress energy tensor is $$T_{uu}=(4\pi L^{2})^{-1}(A^{\prime })^{2}$$ It can easily be verified that the Maxwell equation are satisfied by (\[B\]) in (\[A\]).
To make the metric acceptable, we need to impose the Einstein equations $R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R=\kappa T_{\mu \nu },\kappa $ is generally $\frac{8\pi G}{c^{4}},$but here we use gravitational units i.e. $G=c=1$ so it becomes $8\pi .$ As all the nonvanishing Ricci tensor components are the same and they read $\frac{-2L^{\prime \prime }}{L},$ so the Einstein field equations becomes $L^{\prime \prime }+(4\pi T_{\mu \mu
})L=0.$ This has exactly the form of the equation $L^{\prime \prime }+(\beta
^{\prime })^{2}L=0,$ (which will be discussed in detail in chapter 2) for gravitational plane wave.
REALITY OF GRAVITATIONAL WAVES
==============================
In the linearization procedure the stress energy tensor was taken to be zero, which creates a conceptual problem, namely the question of reality of gravitational waves. This will be discussed in detail in the first section of this chapter. For this purpose some exact gravitational wave solutions of the Einstein field equations are presented. Returning to the purpose of this chapter the energy contents of the waves are discussed. Finally as an additional but necessary, topic some sources of gravitational waves are discussed with in the scope of this work.
The conceptual problem
-----------------------
The exact gravitational waves being by definition, solutions of the vacuum field equations, have a zero stress energy tensor. This creates a conceptual problem. How can these solutions of the Einstein field equations represent waves if they carry no energy? Essentially, this problem arises because energy is not a well defined concept in General Relativity. For energy to be well defined in General Relativity the metric must have a time like isometry (Killing vector) so as to allow time translational invariance. This will not generally be true. Infact, spacetimes for which it is true are static whereas gravitational wave solutions must be non-static. Thus energy is not well defined for spacetimes containing gravitational waves. The question then is, how can we tell that these solutions really do behave as we would expect of the waves?
One way to answer this question: the very process of linearization provides the energy. The point is that if $\Box \phi _{\mu \nu }=0$ exactly then $R_{\mu \nu }\approx 0$ to order $h$ but $R_{\mu \nu }\neq 0$ to higher orders in $h$. Thus $T_{\mu \nu }\approx 0$ only to order $h$ but $T_{\mu
\nu }\neq 0$ generally. Conversely if $T_{\mu \nu }=0\,$andhence$\,R_{\mu \nu }=0$ exactly then $\Box \phi _{\mu
\nu }\approx 0$ only to order $h$. Thus we can expand $R_{\mu \nu }$ in powers of $h,\,$retaining linear terms on the left side of the equation and transposing all higher powers to the right side. These higher order terms become an effective stress energy tensor and the linearized equations give the gravitational waves.
Some exact solutions of gravitational waves
--------------------------------------------
So far we have obtained the wave equations for gravity by linearizing the Einstein field equations. In principle the solutions so obtained could be exact solutions of the vacuum Einstein field equations, of course there could be trivial static solutions which effectively satisfy the Laplace equation. But they do not represent moving waves so we are interested in the solutions which are non-static.
The first solution to be discovered was for cylindrical gravitational waves, by Einstein and Rosen in (1937) \[2\]. So first consider this:
### Cylindrical gravitational wave solution
A cylindrically symmetric metric depending on two arbitrary functions $\gamma $ and $\psi $ of the time $t$ and cylindrical radial coordinate $\rho
$, is
$$ds^{2}=e^{2(\gamma -\psi )}(dt^{2}-d\rho ^{2})-e^{-2\psi }\rho ^{2}d\varphi
^{2}-e^{2\psi }dz^{2}. \label{2.1}$$
The metric tensor is
$$g_{ab}=\left(
\begin{array}{cccc}
e^{2(\gamma -\psi )} & 0 & 0 & 0 \\
0 & -e^{2(\gamma -\psi )} & 0 & 0 \\
0 & 0 & -\rho ^{2}e^{-2\psi } & 0 \\
0 & 0 & 0 & -e^{-2\psi }
\end{array}
\right) .$$
Its inverse is
$$g^{ab}=\left(
\begin{array}{cccc}
e^{-2(\gamma -\psi )} & 0 & 0 & 0 \\
0 & -e^{-2(\gamma -\psi )} & 0 & 0 \\
0 & 0 & -\rho ^{-2}e^{-2\psi } & 0 \\
0 & 0 & 0 & -e^{2\psi }
\end{array}
\right)$$
The non-zero Christoffel symbols are:
$$\left\{
\begin{array}{c}
0 \\
0\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
1\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
1 \\
1\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
1 \\
0\,\,\,1
\end{array}
\right\} =\gamma ^{.}-\psi ^{.};$$
$$\left\{
\begin{array}{c}
0 \\
0\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
1\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
1 \\
0\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
1 \\
1\,\,\,1
\end{array}
\right\} =\gamma ^{\prime }-\psi ^{\prime };$$
$$\left\{
\begin{array}{c}
0 \\
2\,\,\,2
\end{array}
\right\} =-\rho ^{2}\psi ^{\cdot }e^{-2\gamma
};\,\,\,\,\,\,\,\,\,\,\,\,\left\{
\begin{array}{c}
0 \\
3\,\,\,3
\end{array}
\right\} =\psi ^{\cdot }e^{2(2\psi -\gamma
)};\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$\left\{
\begin{array}{c}
1 \\
2\,\,\,2
\end{array}
\right\} =\rho (\rho \psi ^{\prime }-1)e^{-2\gamma };\,\,\,\,\,\,\,\left\{
\begin{array}{c}
1 \\
3\,\,\,3
\end{array}
\right\} =-\psi ^{\prime }e^{2(2\psi -\gamma )}\,;\,\,\,\,\,\,\,\,\,$$
$$\,\,\,\left\{
\begin{array}{c}
2 \\
0\,\,\,2
\end{array}
\right\} =\,\,\,\left\{
\begin{array}{c}
2 \\
2\,\,\,0
\end{array}
\right\} =-\psi ^{\cdot };\left\{
\begin{array}{c}
2 \\
2\,\,\,1
\end{array}
\right\} =-(\psi ^{\prime }-1/\rho )\,;\,\,\,\,\,\,$$
$$\left\{
\begin{array}{c}
3 \\
0\,\,3
\end{array}
\right\} =\left\{
\begin{array}{c}
3 \\
3\,\,\,0
\end{array}
\right\} =\psi ^{\cdot };\,\,\,\,\,\,\,\,\left\{
\begin{array}{c}
3 \\
1\,\,\,\,3
\end{array}
\right\} =\psi ^{\prime
}\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$\left\{
\begin{array}{c}
\mu \\
\mu \,\,0
\end{array}
\right\} =(\ln \sqrt{g})_{,0}=2(\gamma ^{.}-\psi
^{.})\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$\left\{
\begin{array}{c}
\mu \\
\mu \,\,1
\end{array}
\right\} =(\ln \sqrt{g})_{,1}=2(\gamma ^{^{\prime }}-\psi ^{^{\prime
}})+1/\rho
\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
The “$\,^{\prime }\,$” refers to differentiation with respect to $\rho $ and the dot differentiation with respect $t.\,$The non vanishing components of the Ricci tensor are $R_{00},\,R_{11},\,R_{22}$ and $R_{33}$.Only four components are needed for the purpose. Using the Christoffel symbols listed above we obtain the following Ricci tensor components,
$$\begin{aligned}
R_{00} &=&\left\{
\begin{array}{c}
\mu \\
0\,\,\mu
\end{array}
\right\} _{,\mu }-(\ln \sqrt{g})_{,00}+(\ln \sqrt{g})_{,\mu }\left\{
\begin{array}{c}
\mu \\
0\,\,\,\,\,0
\end{array}
\right\} -\left\{
\begin{array}{c}
\mu \\
0\,\,\,\,\,\nu
\end{array}
\right\} \left\{
\begin{array}{c}
\nu \\
0\,\,\,\,\,\,\mu \,\,
\end{array}
\right\} \,\,\,\,\,\,\, \nonumber \\
&=&\left\{
\begin{array}{c}
0 \\
0\,\,0
\end{array}
\right\} _{,0}+\left\{
\begin{array}{c}
1 \\
0\,\,0
\end{array}
\right\} _{,1}-(\ln \sqrt{g})_{,00}+(\ln \sqrt{g})_{,0}\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,0
\end{array}
\right\} \nonumber \\
&&-\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,0
\end{array}
\right\} +(\ln \sqrt{g})_{,1}\,\left\{
\begin{array}{c}
1 \\
0\,\,\,\,\,\,\,0
\end{array}
\right\} \,-2\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1\,
\end{array}
\right\} \left\{
\begin{array}{c}
1 \\
0\,\,\,\,\,\,0\,\,
\end{array}
\right\} \nonumber \\
&&-\left\{
\begin{array}{c}
2 \\
0\,\,\,\,\,\,2\,\,
\end{array}
\right\} ^{2}-\left\{
\begin{array}{c}
3 \\
0\,\,\,\,\,\,3\,\,
\end{array}
\right\} ^{2}\end{aligned}$$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\,\,\,\,\,\,\,\,\,\,\,\,$$$\begin{aligned}
\,\,\,\,\,\,\,\,\,\,\,\, &=&\left\{
\begin{array}{c}
0 \\
0\,\,0
\end{array}
\right\} _{,0}+\left\{
\begin{array}{c}
1 \\
0\,\,0
\end{array}
\right\} _{,1}-(\ln \sqrt{g})_{,00}+(\ln \sqrt{g})_{,0}\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,0
\end{array}
\right\} -\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,0
\end{array}
\right\} \nonumber \\
&&+(\ln \sqrt{g})_{,1}\,\left\{
\begin{array}{c}
1 \\
0\,\,\,\,\,\,\,0
\end{array}
\right\} \,-2\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1\,
\end{array}
\right\} \left\{
\begin{array}{c}
1 \\
0\,\,\,\,\,\,0\,\,
\end{array}
\right\} -\left\{
\begin{array}{c}
2 \\
0\,\,\,\,\,\,2\,\,
\end{array}
\right\} ^{2}-\left\{
\begin{array}{c}
3 \\
0\,\,\,\,\,\,3\,\,
\end{array}
\right\} ^{2}\,\,\,\,\end{aligned}$$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
After simplification we get
$$\begin{aligned}
R_{00} &=&\gamma ^{\cdot \cdot }-\psi ^{\cdot \cdot }+\gamma ^{\prime \prime
}-\psi ^{\prime \prime }-2(\gamma ^{\cdot \cdot }-\psi ^{\cdot \cdot
})+(\gamma ^{\cdot }-\psi ^{\cdot })^{2}-2(\gamma ^{\prime }-\psi ^{\prime
})^{2} \nonumber \\
&&-(\gamma ^{\cdot }-\psi ^{\cdot })^{2}-2(\psi ^{\cdot })^{2}+[\frac{1}{\rho }+2(\gamma ^{\prime }-\psi ^{\prime })](\gamma ^{\prime }-\psi ^{\prime
})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \nonumber \\
&=&-(\gamma ^{\cdot \cdot }-\psi ^{\cdot \cdot })+(\gamma ^{\prime \prime
}-\psi ^{\prime \prime })+\frac{1}{\rho }(\gamma ^{\prime }-\psi ^{\prime
})-2(\psi ^{\cdot
})^{2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{aligned}$$
Similarly other non-vanishing components are,
$$R_{11}=(\gamma ^{\cdot \cdot }-\psi ^{\cdot \cdot })-(\gamma ^{\prime \prime
}-\psi ^{\prime \prime })+\frac{1}{\rho }(\gamma ^{\prime }+\psi ^{\prime
})-2(\psi ^{\prime \prime
})^{2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$R_{22}=\rho ^{2}e^{-\gamma }(-\psi ^{\cdot \cdot }+\psi ^{\prime \prime }+\frac{1}{\rho }\psi ^{\prime
})\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$R_{01}=\frac{1}{\rho }\gamma ^{\cdot }-2\psi ^{\cdot }\psi ^{\prime
}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
So the Einstein field equations becomes
$$(\gamma ^{\cdot \cdot }-\psi ^{\cdot \cdot })-(\gamma ^{\prime \prime }-\psi
^{\prime \prime })-\frac{1}{\rho }(\gamma ^{\prime }-\psi ^{\prime })+2(\psi
^{\cdot
})^{2}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{1a}$$
$\,\,\,$
$$(\gamma ^{..}-\psi ^{..})-(\gamma ^{\prime \prime }-\psi ^{\prime \prime })+\frac{1}{\rho }(\gamma ^{\prime }+\psi ^{\prime })-2(\psi ^{\prime \prime
})^{2}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{2a}$$
$$\rho ^{2}e^{-\gamma }(-\psi ^{\cdot \cdot }+\psi ^{\prime \prime }+\frac{1}{\rho }\psi ^{\prime
})\,=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{a}$$
$$\frac{1}{\rho }\gamma ^{\cdot }-2\psi ^{\cdot }\psi ^{\prime
}\,=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{3a}$$
Equation (\[a\]) is the usual cylindrical form of the wave$\,.$ As this is a second order linear differential equation, the solution has two arbitrary constants, one corresponding to the ingoing cylindrical wave and the other corresponding to the outgoing. Retaining only the outgoing waves with amplitude $A$ and frequency $\omega $ we have
$$\psi (t,\rho )=A[J_{0}(x)\cos (\omega t)+N_{0}(x)\sin (\omega t)]. \label{4}$$
where $x=\omega \rho \,,\,J_{0}(x)\,$and$\,\,N_{0}(x)\,$ are the zero order Bessel and Neuman functions respectively. Add equation (\[1a\]) and (\[2a\]) to obtain
$$\gamma ^{\prime }=\rho (\psi ^{\cdot 2}+\psi ^{\prime 2}) \label{5}$$
Equations (\[3a\]) and (\[5\]) gives the space and time derivative of $\gamma (t,\rho )$ in terms of functions that are now known through equation (\[4\]). The only thing required is the integration with respect to each variable. The time integration is relatively easy and the space integration, though tedious, is in principle easy (using the standard formulae for the integrals of the Bessel and Neumann function). The resulting solution of $\gamma $ is then
$$\left.
\begin{array}{c}
\gamma (t,\rho )=\frac{1}{2}A^{2}x\{J_{0}(x)J_{0}^{\prime
}(x)+N_{0}(x)N_{0}^{\prime }(x)+x[J_{0}^{\prime }(x)^{2}+N_{0}^{\prime
}(x)^{2}] \\
+[J_{0}(x)J_{0}^{\prime }(x)-N_{0}(x)N_{0}^{\prime }(x)]\cos (2\omega t) \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+[J_{0}(x)N_{0}^{\prime }(x)+J_{0}^{\prime }(x)N_{0}(x)]\sin (2\omega t)\}-\frac{2}{\pi }A^{2}\omega t
\end{array}
\right\} . \label{6*}$$
Where prime now refers to differentiation with respect to $x$ not $\rho $. Hence the metric
$$ds^{2}=e^{2(\gamma -\psi )}(dt^{2}-d\rho ^{2})-e^{-2\psi }\rho ^{2}d\varphi
^{2}-e^{2\psi }dz^{2}.$$
represents cylindrical gravitational wave with the above definition of $\gamma \,$and $\psi \,.$
### Plane gravitational wave solution
The solution which we are going to present here was discovered by Bondi and Robinson in 1957 \[3\]. We take a line element which incorporates the symmetries of a plane and represents a wave going in the x-direction.
$$ds^{2}=e^{2\Omega (u)}(dt^{2}-dx^{2})-u^{2}(e^{2\beta (u)}dy^{2}-e^{-2\beta
(u)}dz^{2}). \label{7}$$
$\,\,\,\,$Here all coefficients in the metric are functions of $(t-x)$ which are represented by $u.$ The$\,(x,y,z)$ are not the usual Cartesian coordinates but rectangular coordinates in a curved space-time. Thus the metric tensor of equation(\[7\]) is
$$g_{ab}=\left(
\begin{array}{cccc}
e^{2\Omega (u)} & 0 & 0 & 0 \\
0 & -e^{2\Omega (u)} & 0 & 0 \\
0 & 0 & -u^{2}e^{2\beta (u)} & 0 \\
0 & 0 & 0 & -u^{2}e^{-2\beta (u)}
\end{array}
\right) .$$
Its inverse can be written as
$$g^{ab}=\left(
\begin{array}{cccc}
e^{-2\Omega (u)} & 0 & 0 & 0 \\
0 & -e^{-2\Omega (u)} & 0 & 0 \\
0 & 0 & -u^{-2}e^{-2\beta (u)} & 0 \\
0 & 0 & 0 & -u^{-2}e^{2\beta (u)}
\end{array}
\right) .$$
Here it should be noted for convenience that
$\Omega _{,0}=\Omega ^{\prime }=-\Omega _{,1}$and$\,\,\beta
_{,0}=\beta ^{\prime }=-\beta _{,1}.$ Here “ $\prime $ ” represents derivative with respect to $u$.$\,$$$\begin{aligned}
&&^{\left\{
\begin{array}{c}
0 \\
0\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
1\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
1 \\
1\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
1 \\
0\,\,\,1
\end{array}
\right\} =-\left\{
\begin{array}{c}
0 \\
0\,\,\,1
\end{array}
\right\} =-\left\{
\begin{array}{c}
1 \\
0\,\,\,0
\end{array}
\right\} }\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
&=&-\left\{
\begin{array}{c}
1 \\
1\,\,\,1
\end{array}
\right\} =-\Omega ^{\prime }; \nonumber\end{aligned}$$
$$\begin{aligned}
\left\{
\begin{array}{c}
0 \\
2\,\,\,2
\end{array}
\right\} &=&\left\{
\begin{array}{c}
1 \\
2\,\,\,2
\end{array}
\right\} =u(u\beta ^{\prime }+1)e^{2(\beta -\Omega
)}\,\,\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
\left\{
\begin{array}{c}
0 \\
3\,\,\,3
\end{array}
\right\} &=&\left\{
\begin{array}{c}
1 \\
3\,\,\,3
\end{array}
\right\} =u(-u\beta ^{\prime }+1)e^{-2(\beta +\Omega )}; \nonumber\end{aligned}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{
\begin{array}{c}
2 \\
0\,\,\,2
\end{array}
\right\} =\,\,\,\left\{
\begin{array}{c}
2 \\
2\,\,\,0
\end{array}
\right\} =-\left\{
\begin{array}{c}
2 \\
2\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
2 \\
2\,\,\,1
\end{array}
\right\} =\beta ^{\prime }+\frac{1}{u};$$
$$\,\,\,\,\,\,\,\,\,\,\,\left\{
\begin{array}{c}
3 \\
0\,\,3
\end{array}
\right\} =\left\{
\begin{array}{c}
3 \\
3\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
3 \\
1\,\,\,\,3
\end{array}
\right\} =\left\{
\begin{array}{c}
3 \\
3\,\,\,\,1
\end{array}
\right\} =-\beta ^{\prime }+\frac{1}{u}. \label{8}$$
The non zero Ricci tensor components are $R_{00},R_{01}\,$and$\,R_{11}.$
$$R_{00}=\left\{
\begin{array}{c}
\mu \\
0\,\,\mu
\end{array}
\right\} _{,\mu }-(\ln \sqrt{g})_{,00}+(\ln \sqrt{g})_{,\mu }\left\{
\begin{array}{c}
\mu \\
0\,\,\,\,\,0
\end{array}
\right\} -\left\{
\begin{array}{c}
\mu \\
0\,\,\,\,\,\nu
\end{array}
\right\} \left\{
\begin{array}{c}
\nu \\
0\,\,\,\,\,\,\mu \,\,
\end{array}
\right\} .\,\,\,\,\,$$
For the given metric (\[7\]) we have
$$(\ln \sqrt{\mid g\mid })=2\ln u+2\Omega
.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$\ln \sqrt{\mid g\mid })^{\prime }=2(\Omega ^{\prime }+\frac{1}{u}$$
$$(\ln \sqrt{\mid g\mid })^{\prime \prime }=2(\Omega ^{\prime \prime }-\frac{1}{u^{2}})$$
Now using the Christoffel symbols listed in equation (\[8\]) we get.
$$\begin{aligned}
R_{00} &=&\left\{
\begin{array}{c}
0 \\
0\,\,0
\end{array}
\right\} _{,0}+\left\{
\begin{array}{c}
1 \\
0\,\,0
\end{array}
\right\} _{,1}-(\ln \sqrt{g})_{,00}+(\ln \sqrt{g})_{,0}\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,0
\end{array}
\right\} \\
&&-\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,0
\end{array}
\right\} +(\ln \sqrt{g})_{,1}\,\left\{
\begin{array}{c}
1 \\
0\,\,\,\,\,\,\,0
\end{array}
\right\} \,-2\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1\,
\end{array}
\right\} \left\{
\begin{array}{c}
1 \\
0\,\,\,\,\,\,0\,\,
\end{array}
\right\} \\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\left\{
\begin{array}{c}
2 \\
0\,\,\,\,\,\,2\,\,
\end{array}
\right\} ^{2}-\left\{
\begin{array}{c}
3 \\
0\,\,\,\,\,\,3\,\,
\end{array}
\right\} ^{2}\,.\,\,\,\end{aligned}$$
$$R_{00}=4\Omega ^{\prime }/u-2\beta ^{\prime
2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
Here the vacuum Einstein equation reduces to
$$4\frac{\Omega ^{\prime }}{u}-2\beta ^{\prime
2}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{9}$$
It is an exact plane gravitational wave solution.
$$R_{01}=-4\frac{\Omega ^{\prime }}{u}+2\beta ^{\prime
2}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{9a}$$
$$R_{11}=4\frac{\Omega ^{\prime }}{u}-2\beta ^{\prime
2}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{9b}$$
Similarly equations (\[9a\])and (\[9b\]) reduces to same value as equation (\[9\]), where other components of Ricci tensor are zero. Hence the only non-trivial exact solution of the plane gravitational wave is,
$$\Omega ^{\prime }(u)=\frac{1}{2}u\beta ^{\prime 2}.$$
Interaction of a particle with plane gravitational waves
---------------------------------------------------------
Here it will be shown that gravitational waves carry energy following the method of Weber and Wheeler \[3\], considering the plane gravitational wave (\[7\]) , which is discussed in the previous section.
As already mentioned, here we will show that plane gravitational waves carry energy. It can be shown by analyzing the motion of a particle which is initially at rest and interacts with the plane gravitational wave. Write the geodesic equation$.$
$$\stackrel{\cdot \cdot }{x}^{\mu }+\left\{
\begin{array}{c}
\mu \\
\nu \,\,\,\,\,\rho
\end{array}
\right\} \stackrel{\cdot }{x}^{\nu }\stackrel{\cdot }{x}^{\rho }=0.
\label{11}$$
Using the Christoffel symbols, listed in (\[8\]), and the usual summation convention over the repeated indices, equation (\[11\]) becomes:
$$\frac{d^{2}x}{ds^{2}}+\Omega ^{\prime }(\frac{dt}{ds})^{2}+2\Omega ^{\prime }\frac{dt}{ds}\frac{dx}{ds}-\Omega ^{\prime }(\frac{dx}{ds})^{2}+u(u\beta
^{\prime }+1)e^{2(\beta -\Omega )}(\frac{ds}{dy})^{2}$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+u(-u\beta
^{\prime }+1)e^{-2(\beta +\Omega )}(\frac{dz}{ds})^{2}=0. \label{12}$$ Here in the above equation (\[12\]) $\frac{dt}{ds}$ is unknown. It can easily be found as follows. Dividing both sides of (\[7\]) by $ds^{2}$ and simplifying, we get
$$\frac{dt}{ds}=[(1+(-2\Omega +2\Omega ^{2}-\frac{4}{3}\Omega ^{3}+...+(\frac{dx}{ds})^{2}]^{1/2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
Using the binomial expansion, we get the following after further simplification.
$$\frac{dt}{ds}=1-\Omega +\Omega ^{2}-\frac{2}{3}\Omega ^{3}+...+(\frac{dx}{ds})^{2}+...\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{13}$$
Consider $\Omega $ as a first order quantity and $\beta $ a second order quantity. Let $$x=x_{(0)}+x_{(1)}+x_{(2)}+...\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ Using these considerations and equation (\[13\]), equation(\[12\]) gives the following approximation equations.
$$\frac{d^{2}x_{(0)}}{ds^{2}}+2\Omega ^{\prime }\frac{dt}{ds}\frac{dx_{(0)}}{ds}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{14}$$
$$\frac{d^{2}x_{(1)}}{ds^{2}}+2\Omega ^{\prime }\frac{dt}{ds}\frac{dx_{(1)}}{ds}-2\Omega ^{\prime }\frac{dx_{(0)}}{ds}\frac{dx_{(1)}}{ds}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{15}$$
$$\frac{d^{2}x_{(2)}}{ds^{2}}+2\Omega ^{\prime }(\frac{dt}{ds}-\frac{dx_{(0)}}{ds})\frac{dx_{(2)}}{ds}-2\Omega ^{\prime }(\frac{dx_{(1)}}{ds})^{2}=0.
\label{16}$$
Equations (\[14\]), (\[15\]) and (\[16\]) are zero order, first order and second order approximation equations respectively. With the previously developed tools we can integrate the approximation equations. Here it should also be noted that as $\Omega $ is a first order quantity, its derivative will also be a first order quantity. The result is:
$i)$ The zero order approximation
$$\frac{dx_{(0)}}{ds}=-2\Omega ^{\prime }(1-\Omega
)x_{(0)}+c_{1},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{17}$$
$ii)$ The first order approximation
$$\frac{dx_{(1)}}{ds}=-2c_{2}\Omega ^{\prime
}x_{(1)}+c_{3};\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{18}$$
$iii)$ The second order approximation
$$\frac{dx_{(2)}}{ds}=-2c_{4}\Omega ^{\prime }x_{(2)}+2c_{5}\Omega ^{\prime
}s+c_{6}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{19}$$
Thus equations (\[17\]), (\[18\]) and (\[19\]) guarantee that a non-zero momentum is imparted to the particle after interacting with the plane gravitational wave. Hence gravitational waves carry energy.
Sources of gravitational waves
-------------------------------
### Spinning rod
A rod spinning about an axis perpendicular to its length is one of the first sources of gravitational radiation ever to have been considered \[13\].
Consider a steel beam of radius $r=1meter$ length $l=20\,meter,$ density $\rho =7.8gm/cm^{3}$, mass $M=4.9\times 10^{8}gm$ and tensile strength $t=40,000lbft/in^{2}$. Let the beam rotates about its middle, so it rotates end over end with an angular velocity $\omega \,$limited by the balance centrifugal force and tensile strength.
$$\omega =(\frac{8t}{\rho l^{2}})^{1/2}=28radians/\sec
\textrm{\thinspace
\thinspace }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
The internal power flow is
$$\begin{aligned}
L_{internal} &=&(\frac{1}{2}I\omega ^{2})\omega =\frac{1}{28}Ml^{2}\omega
^{3} \nonumber \\
&\approx &2\times 10^{8}erg/\sec \approx
10^{-41}L_{0}\,\,joule\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{aligned}$$
where $L_{0}=c^{5}/G.\,$The order of magnitude of the power radiated is only $L_{GW}\sim (10^{-41})^{2}L_{0}\sim 10^{-23}erg/\sec .$ Evidently the construction of a laboratory generator of gravitational wave is unattractive, as this is a very small quantity which cannot be detected without new engineering or new ideas. There are however a great variety of astrophysical sources of gravitational waves. We list some of them and then discuss them lightly without going into actual calculation.
### Astrophysical sources of gravitational waves
Astrophysical sources of gravitational waves are the following[ :]{}
1. 1. Pulsar
2. Double star system;
3. Gravitational collapse of a few solar mass star;
4. Formation of a large black hole;.
Now let me explain them.
$\left( {\bf a}\right) $[** **]{}[Gravitational radiation from a pulsar]{}
Consider a highly dynamic astrophysical system. In particular take it to be a wildly rotating pulsar. If its mass is $M$ and its size is $R$ then by virial theorem its kinetic energy is $\sim M^{2}/R^{2}$ . The characteristic time scale for mass to move from one side of the system to the other is
$$T\sim R/(mean\,velocity)\sim
R/(M/R)^{1/2}=(R^{3}/M)^{1/2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
The internal power flow is
$$L_{int}\sim \frac{kinetic\,energy}{T}\sim (\frac{M^{2}}{R})(\frac{M^{2}}{R^{3}})^{1/2}\sim
(M/R)^{5/2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
The gravitational wave output is the square of this quantity or $L_{GW}\sim (\frac{M}{R})^{5}L_{0}.$ Clearly the maximum power output occurs when the system is near its gravitational radius, and because nothing, not even gravitational waves can escape from inside the gravitational radius. The maximum value of the output is $\sim L_{0}=3.63\times 10^{59}erg/\sec $ regardless of the nature of the system.
$\left( {\bf b}\right) $ [Double star system]{}
It has been estimated that at least one-fifth of all the stars are binary systems. We will go into the details of how it happens so frequently because that is a topic of astrophysics and hydrodynamics.
Consider two stars of masses $m_{1\,\textrm{ }}$and $m_{2}$ revolving in a circular orbit about their common centre of gravity. For their circular frequency of revolution, we have the standard formula: $$\omega ^{2}=(m_{1+}m_{2)}/r^{3}\textrm{ (geometrical units of mass
and time)}$$ The calculated rate of loss of energy by radiation is \[13\].
$$-\frac{dE}{dt}=(\frac{32}{5})(m_{1}m_{2}/(m_{1}+m_{2}))^{2}r^{4}\omega
^{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
The following are important types of double star systems.
$
\begin{array}{lll}
\textrm{Type name} & -(\frac{dE}{dt})_{grav} & \textrm{on Earth} \\
\eta \,\cos & 5.2\times 10^{10}erg/\sec & 1.4\times 10^{-29}erg/cm^{2}\sec
\\
\xi \,\beta oo & 3.6\times 10^{12}erg/\sec & 6.7\times 10^{-28}erg/cm^{2}\sec
\\
Wuma & 4.7\times 10^{29}erg/\sec & 3.2\times 10^{-13}erg/cm^{2}\sec
\end{array}
$
$\left( {\bf c}\right) $
Collapse to form neutron stars or black holes in the mass range $1$ to $10\,M_{\odot }$ $(M_{\odot }=2\times 10^{33}gm$ is a solar mass$)$ will radiate waves in the frequency range 1 to 10 kHz with an amplitude that depends on how much symmetry there is in the collapse. These collapses, at least sometimes, result in supernova explosions. The rate at which supernova occurs is relatively well known, but the fraction of collapse events that produce strong enough gravitational waves is not well known. The characteristic period of the waves is proportional to the light-travel time around the collapsed object, the dominant frequency scale is $1/M$. For sufficiently large $M$, the source will produce low frequency waves detectable in space.
$\left( {\bf d}\right) $[** **]{}[Formation of a giant black holes]{}
Many astrophysicist believe that the most plausible explanation for quasars and active galactic nuclei is that they contain massive ( $10^{6}-10^{9}M_{\odot })\,$black holes that accrete gas and stars to fuel their activity. There is growing evidence that even so called normal galaxies, like our own and Andromeda, contains black holes of modest size $(10^{4}-10^{6}M_{\odot })\,$in their nuclei. It is not clear how such holes form, but if they form by the rapid collapse of a cluster of stars or of a single supermassive star, then with a modest degree of non-symmetry in collapse, they could produce amplitudes $h\approx 10^{-16}\,to\,\,10^{-18}$meters in the low frequency range observable from space. If a detector has spectral noise density of $10^{-20}H_{z}^{-\frac{1}{2}}$then such events could have signal to noise ratio ($\frac{S}{N})\,$of as much as 1000. This strong signal would permit a detailed study of the event. If every galaxy has one such black hole formed in this way , then there could be one event per year in a galaxy. If no such events are seen, then either giant black holes don’t exist or they form much more gradually or with too much spherical symmetry.
REVIEW OF THE PSEUDO-NEWTONIAN FORMALISM AND ITS EXTENSION
==========================================================
In the first section a review of the Pseudo-Newtonian $(\psi N)$ formalism is given. Section 2 provides a review of the extension of the $\psi N$ formalism. In section 3 the extended $\psi N\,\,(e\psi N)$ formalism is used to develop a formula for the momentum imparted to test particles in arbitrary spacetime. Finally this formula is applied to plane and cylindrical gravitational waves, both of which give very reasonable results.
[ The $\Psi N$ formalism]{}
---------------------------
The $\Psi N$ formalism \[7\] is based on the observation that, whereas the gravitational force is not detectable in a freely falling frame (FFF), that is so only at a point. It is detectable over a finite spatial extent as the tidal force. It could be measured by an accelerometer, as shown in fig.3.1.
This accelerometer has a spring of length $``l"$ which connects two masses. The spring ends in a needle which can move on the dial of the accelerometer to give a measure of the tension in the spring. Thus an observer in the FFF can observe the position of the spring by observing the moment of the needle on the dial. In Newtonian gravitation theory, the needle will show a zero position of the accelerometer in the absence of a central force. The force exerted by the source pulls the mass near it more than the mass further away. Thus the spring is stretched and the needle will move in the positive direction.
If the spring is compressed the needle will show a negative deflection otherwise it will show a positive deflection. This would occur if both discs and the source consisted of like electric charges. Hence the negative deflection corresponds to a repulsive source and the positive deflection corresponds to an attractive source. The strength of the source would be shown by the extent that the needle moved.
Mathematically the tidal acceleration is given by
$$A^{\mu }=R_{\nu \rho \pi }^{\mu }t^{\nu }l^{\rho }t^{\pi
}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\,\mu ,\nu
,...=0,1,2,3)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$\,\,\,\,$Here ${\bf R}\,$is the Riemann tensor, ${\bf t}$ a timelike killing vector, ${\bf l}$ is the spacelike separation vector representing accelerometer. Thus in geometrical terms tidal force is given by
$$F^{\mu }=-mR_{\nu \rho \pi }^{\mu }t^{\nu }l^{\rho }t^{\pi
}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.1}$$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$where $m\,$is the mass of the test particle, $t^{\nu \textrm{ }}$the timelike vector tangent to the particles path and $l^{\rho }$ the separation vector, which provides the observation of the tidal force. In the FFF $t^{\nu }=f\delta _{0}^{\nu
},\,\,\,\,f^{2}=\frac{1}{g_{00}}.$ Thus equation (\[00.1\]) becomes:
$$F^{\mu }=-mf^{2}R_{0\rho 0}^{\mu }l^{\rho
}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.2}$$
Regarding this as an eigenvalue equation, we get the maximum tidal force along the eigenvector of the matrix $R_{0\rho 0}^{\mu }$. Since $l^{\rho }$ is a purely spacelike vector in the free fall frame, the maximum tidal force will be a purely space like vector. Thus we get $$\stackrel{\_}{F}^{i}=\stackrel{\_}{l}^{j}\gamma
_{;j}^{i}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(i,j=1,2,3)\,\,\,\,\,\,$$ where “ ; ” stands for covariant derivative. Thus $\gamma ^{i}$ is the relativistic analogue of the Newtonian gravitational force. Further this force is the gradient of a scalar quantity \[7\] $$\gamma ^{i}=-\phi
_{,i}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ Where an appropriate expression for $\phi $ is
$$\phi =\frac{1}{2}(g_{00}-1).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
When expressed in this way a Lorentz factor has to be introduced by hand. $\gamma ^{i}\,$and$\,\phi \,$are called the $\psi N$ force and potential respectively.
The $e\protect\psi N$ force
----------------------------
The quantity whose directional derivative along the accelerometer, placed along the principal direction, gives the extremised tidal force, which is zero in Minkowski space. Thus the $e\psi N$ force, $F^{\mu }$, satisfies the equation \[9\]
$$F^{*\mu }=l^{\nu }F_{;\nu }^{\mu
}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{00.3}$$
Where $F^{*\mu }$ is the extremised tidal force. The fact that the zero component of the left side is zero does not guarantee that the zero component of $F^{\mu }$ is zero.
The space could be made look flat in a small neighborhood of any point $P$ in by a special choice of inertial coordinates. This locally inertial coordinate system has the metric of special relativity at some point $P$ and so the Christoffel symbols $\Gamma _{\nu \rho }^{\mu }$ are zero at $P.$ It is not, however, possible to make $R_{\mu \nu \rho \pi }=0$ at $P$ by any choice of coordinates, unless the space is flat in the neighborhood of $P$. This is because $R_{\mu \nu \rho \pi }$ is a tensor; if it vanishes in one coordinate system it will do so in any other because of the transformation laws. Here we will restrict ourselves to Riemann normal coordinates spatially, but not temporally.
Equation (\[00.3\]) can be written in the space and time break up as
$$l^{i}(F_{,i}^{0}+\Gamma
_{ij}^{0}F^{j})=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.4}$$
$$l^{j}(F_{,j}^{i}+\Gamma
_{0j}^{i}F^{0})=F^{*i}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{00.5}$$
A simultaneous solution of the above equations can be obtained by using Riemann normal coordinates for the spatial directions but not for the time coordinates. Thus the $e\psi N$ force four vector is (see Appendix 2 for the proof):
$$F^{0}=m[(\ln A)_{,0}-\Gamma _{00}^{0}+\Gamma _{0j}^{i}\Gamma
_{0i}^{j}/A]f^{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.5a}$$
$$F^{i}=\Gamma
_{00}^{i}f^{2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.5b}$$
where $A=(\ln \sqrt{-g})_{,0}\,\,\,\,\,\,\,\,g=\det (g_{ij})$ For these, block diagonalized metrics
$\Gamma _{00}^{0}=\frac{1}{2}g^{00}g_{00,0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\Gamma _{00}^{i}=-\frac{1}{2}g^{ij}g_{00,j}$
$\Gamma _{0j}^{i}=\frac{1}{2}g^{ij}g_{jk,0}$
Thus, writing the covariant form of the $e\psi N$ force
$$F_{0}=m[(\ln
Af)_{,0}-g_{,0}^{ij}g_{ij,0}/4A],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.5c}$$
$$F_{i}=m(\ln
f)_{,i}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
It is worth considering the significance of the zero component of the $e\psi
N$ force, which is its major difference from the $\psi N$ force. In special relativistic terms, which are relevant for discussing forces in a Minkowski space, the zero component of the four vector force corresponds to a proper rate of change of energy of the test particle. Further we know that, in general, an accelerated particle either radiates or absorbs energy according as $\frac{dE}{dt}$ is less than or greater than zero. Thus $F_{0}$, should also correspond to energy-emission or absorption by the background spacetime. This point will be discussed further in the next chapter.
General formula for the momentum imparted to test particles in arbitrary spacetime
-----------------------------------------------------------------------------------
There has been a debate whether gravitational waves really exist \[3,4\]. To demonstrate the reality of gravitational waves Ehler and Kundt \[4\] considered a sphere of test particles in the path of plane fronted gravitational waves and showed that a constant momentum was imparted to the test particles. This was latter extended by Weber and Wheeler \[3\] for cylindrical gravitational waves. The $\psi N$ force and $e\psi N$ force are based on an operational procedure embodying the same principle \[6\]. The proper time integral of the force four vector will be the momentum four vector. Here it will be verified that this procedure gives the Ehler-Kundt result for plane fronted gravitational waves. When it is applied to cylindrical gravitational waves it is found that the result so obtained is physically reasonable and gives an exact expression for the momentum imparted to test particles, corresponding to the approximation given by Weber and Wheeler.
From Equations (\[00.5a\]) and (\[00.5b\]) the momentum four vector,$\,P_{\mu }$, is obtained as:
$$P_{\mu }=\int F_{\mu
}dt.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.8}$$
Now we apply this formula to plane fronted gravitational waves and cylindrical gravitational waves.
### Plane-fronted gravitational waves
The metric for Plane-fronted gravitational waves is
$$ds^{2}=dt^{2}-dx^{2}-L^{2}(t,x)\{\exp (2\beta (t,x))dy^{2}+\exp (-2\beta
(t,x))dz^{2}\}.$$
Where $L\,$and $\beta $ are arbitrary functions.
The metric tensor is
$$g_{ab}=\left(
\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -L^{2}e^{2\beta } & 0 \\
0 & 0 & 0 & -L^{2}e^{-2\beta }
\end{array}
\right) .$$
Its inverse is
$$g^{ab}=\left(
\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -L^{-2}e^{-2\beta } & 0 \\
0 & 0 & 0 & -L^{-2}e^{2\beta }
\end{array}
\right) .$$
To find the momentum four vector we need the force four vector. We calculate the force four vector. Here in this special case:
$$g=\det (g_{ij})=-L^{4}(t,x),\,\,\,f=\frac{1}{\sqrt{g_{00}}}=1\,\,$$
$$\ln \sqrt{-g}=2\ln
L\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
which implies that
$$Af=(\ln \sqrt{-g}),_{0}=2\frac{\stackrel{.}{L}}{L}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.9}$$
$$(\ln (Af))_{,0}=\frac{\stackrel{..}{L}}{L}-\frac{\stackrel{.}{L}}{L}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{00.10}$$
$$g_{,0}^{ij}g_{ij,o}=-8(\stackrel{.}{\beta }^{2}+\stackrel{.}{L}^{2}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.11}$$
So from equations (\[00.9\]), (\[00.10\]), (\[00.11\]) and (\[00.5a\]) we get
$$F_{0}=m(\stackrel{..}{L}+\stackrel{.}{\beta }^{2}L)/L.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
From the vacuum Einstein equations we have (already discussed in chapter 2)
$$\stackrel{..}{L}+\stackrel{.}{\beta }^{2}L=0,$$
.which implies that $F_{0}=0$ that is the zero component of the force four-vector is zero. Also $F_{i}=0$ because ($\ln \sqrt{g_{00}})_{,i}=0.$
Thus the momentum four vector becomes $P_{\mu }=$[*constant*]{}$.$ Hence there is a constant energy and momentum imparted to the test particles. The constant here determines the strength of the wave. This exactly coincides with the Ehler-Kundt method in that they demonstrate that the test particles acquires a constant momentum and hence a constant energy from a plane gravitational wave.
### Cylindrical gravitational waves
Consider the cylindrically symmetric metric (\[2.1\]). We first calculate the force four vector. Here in this special case:
$$g=\det (g_{ij})=-e^{4(\gamma -\psi
)}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.12}$$
$$\ln \sqrt{-g}=2(\gamma -\psi
)\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.13}$$
$$A=2(\gamma ^{.}-\psi ^{.}),\,\,\,\,\,\,\,\,\,f=e^{\psi -\gamma
}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.14}$$
$$(\ln (Af))_{,0}=\frac{(\gamma ^{..}-\psi ^{..})\,\,\,\,\,}{(\gamma ^{.}-\psi
^{.})\,\,\,\,\,}-(\gamma ^{.}-\psi
^{.}).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.15}$$
$$g_{,0}^{ij}g_{ij,0}=-4(\gamma ^{.}-\psi ^{.})^{2}-8\psi
^{.2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{00.16}$$
where a dot denotes differentiation with respect to $t$.
The vacuum Einstein field equation gives \[3\].
$$\psi (t,\rho )=AJ_{0}(\omega \rho )\cos (\omega t)+BN_{0}(\omega \rho )\sin
(\omega t), \label{00.17}$$
where $\,J_{0}(\omega \rho )\,$and$\,\,N_{0}(\omega \rho )\,$ are the zero order Basel and Neuman functions respectively. $A$ and $B$ are arbitrary constants corresponding to the strength of gravitational wave.
$$\begin{aligned}
\gamma (t,\rho ) &=&\frac{1}{2}\omega \rho \{(A^{2}J_{0}(\omega \rho
)J_{0}^{\prime }(\omega \rho )-B^{2}N_{0}(x)N_{0}^{\prime }(x))\cos (2\omega
t) \nonumber \\
&&-AB[(J_{0}(\omega \rho )N_{0}^{\prime }(\omega \rho )+J_{0}^{\prime
}(x)N_{0}(x))\sin (2\omega t)- \nonumber \\
&&-2(J_{0}(\omega \rho )N_{0}^{\prime }(\omega \rho )-J_{0}^{\prime
}(x)N_{0}(x))\omega t]\} \label{00.19}\end{aligned}$$
Where prime now refers to differentiation with respect to $\omega \rho
,\,\omega $ being the angular frequency.
Using equations (\[00.15\]), (\[00.16\]), (\[00.17\]) and (\[00.19\]) we get the zero component of the force four vector.
$$\begin{aligned}
F_{0} &=&-m\omega \{[AJ_{0}\cos (\omega t)+BN_{0}\sin (\omega t)]-2\rho
\omega [(A^{2}J_{0}J_{0}^{\prime }-B^{2}N_{0}N_{0}^{\prime })\cos (2\omega t)
\nonumber \\
&&-AB(J_{0}N_{0}^{\prime }+N_{0}J_{0}^{\prime })\sin (2\omega t)]+\frac{2[AJ_{0}\sin (\omega t)-BN_{0}\cos (\omega t)]^{2}}{AJ_{0}\sin (\omega t)}
\nonumber \\
&&-BN_{0}\cos (\omega t)-2\omega \rho [AJ_{0}\sin (\omega t)-BN_{0}\cos
(\omega t)] \nonumber \\
&&[AJ_{0}^{\prime }\cos (\omega t)+BN_{0}^{\prime }\sin (\omega t)]\}.
\label{00.20}\end{aligned}$$
As $\gamma $ and $\psi $ are functions of $t$ and $\rho $ only so $F_{2}\,$and$F_{3}$ are zero.
$$\begin{aligned}
F_{1} &=&m(\ln f)_{,1}=-m\omega \{[AJ_{0}^{\prime }\cos (\omega
t)-BN_{0}^{\prime }\sin (\omega t)-\frac{1}{2}[(A^{2}J_{0}J_{0}^{\prime
}-B^{2}N_{0}N_{0}^{\prime }) \nonumber \\
&&+\omega \rho (A^{2}J_{0}J_{0}^{\prime }-B^{2}N_{0}N_{0}^{\prime })^{\prime
}]\cos (2\omega t)-\frac{1}{2}AB[2(J_{0}N_{0}^{\prime }+J_{0}^{\prime
}N_{0})^{\prime }]\sin (2\omega t) \nonumber \\
&&-\frac{1}{2}AB[4(J_{0}N_{0}^{\prime }-J_{0}^{\prime }N_{0})+2\omega \rho
(J_{0}N_{0}^{\prime }-J_{0}^{\prime }N_{0})^{\prime }]\omega t\}.\end{aligned}$$
The corresponding $P_{0}$ and $P_{1}$ are
$$P_{0}=-m[\ln \mid AJ_{0}\sin (\omega t)-BN_{0}\cos (\omega t)\mid +\ln \mid
1-2\omega \rho [AJ_{0}^{\prime }\cos (\omega t)$$ $$+BN_{0}^{\prime }\sin (\omega t)]\,\mid \left( 1+\frac{A^{2}J_{0}J_{0}^{\prime }+B^{2}N_{0}N_{0}^{\prime }}{\omega \rho (AJ_{0}^{\prime
2}+BN_{0}^{\prime 2})}\right)
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$-\frac{AB(AJ_{0}N_{0}N_{0}^{\prime }+N_{0}J_{0}^{\prime }-A\omega \rho
J_{0}J_{0}^{\prime }N_{0}^{\prime }-N_{0}^{\prime }J_{0})}{\omega \rho
(A^{2}J_{0}^{\prime 2}+B^{2}N_{0}^{\prime 2})\sqrt{1-4\omega ^{2}\rho
^{2}(A^{2}J_{0}^{\prime 2}+B^{2}N_{0}^{\prime 2})}}$$ $\,\,\,\,\,\,\,\,\,$$$\begin{aligned}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\tan ^{-1} &\mid &\frac{(1+2A\omega \rho J_{0}^{\prime })\tan (\frac{1}{2}\omega t)-2B\omega \rho N_{0}^{\prime }}{\sqrt{1-4\omega ^{2}\rho
^{2}(A^{2}J_{0}^{\prime 2}+B^{2}N_{0}^{\prime 2})}}\mid +\frac{AB(N_{0}J_{0}^{\prime }-J_{0}N_{0}^{\prime }}{\rho (A^{2}J_{0}^{\prime
2}+B^{2}N_{0}^{\prime 2})}t \nonumber \\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+f_{1}(\omega \rho ).\,\,\, \label{00.21a}\end{aligned}$$ $\,\,\,\,\,$$$\begin{aligned}
P_{1} &=&-m\{[AJ_{0}^{\prime }\sin (\omega t)-BN_{0}^{\prime }\cos (\omega
t)]-\frac{1}{4}[(A^{2}J_{0}J_{0}^{\prime }-B^{2}N_{0}N_{0}^{\prime })
\nonumber \\
&&+\omega \rho (A^{2}J_{0}J_{0}^{\prime }-B^{2}N_{0}N_{0}^{\prime })^{\prime
}]\sin (2\omega t)-\frac{1}{2}AB[(J_{0}N_{0}^{\prime }+J_{0}^{\prime }N_{0})
\nonumber \\
&&\omega \rho (J_{0}N_{0}^{\prime }+J_{0}^{\prime }N_{0})^{\prime }]\cos
(2\omega t)-AB\omega ^{2}t^{2}[(J_{0}N_{0}^{\prime }+J_{0}^{\prime }N_{0})
\nonumber \\
&&-\omega \rho (J_{0}N_{0}^{\prime }-J_{0}^{\prime }N_{0})^{\prime
}]\}+f_{2}(\omega \rho ). \label{00.22}\end{aligned}$$ Where $f_{1}$ and $f_{2}$ are arbitrary constants of integration. Weber and Wheeler exclude solution that contain irregular Bessel function, $N_{0}(\omega \rho )$, as not well defined at the origin. Taking the Weber-Wheeler solutions equations (\[00.21a\]) and (\[00.22\]) reduces to
$$\begin{aligned}
P_{0} &=&-m[\ln \mid AJ_{0}\sin (\omega t)\mid +(1+AJ_{0}/\omega \rho
J_{0}^{\prime })\ln \mid 1-2\omega \rho AJ_{0}^{\prime }\cos (\omega t)\mid
\label{00.23} \\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+f_{1}(\omega \rho ). \nonumber\end{aligned}$$
$$P_{1}=-m\{AJ_{0}^{\prime }\sin (\omega t)-\frac{1}{4}A^{2}J_{0}J_{0}^{\prime
}+\rho \omega (J_{0}J_{0}^{\prime })^{\prime }\sin (2\omega t)+f_{2}(\omega
\rho )\}. \label{00.24}$$
We see that the quantity $P_{1}$ given by equation (\[00.24\]) can be made zero for the large and small $\rho $ limits by choosing $\,f_{2}$ equal to zero. This is physically reasonable expression for the momentum imparted to test particles by cylindrical gravitational waves. The quantity $P_{0}$ given by equation (\[00.23\]) remains finite for small $\rho $ and can also be made finite for large $\rho $ by choosing $f_{1}=-\ln (J_{0})$. However, there is a singularity at $\omega t=n\pi $. This problem does not arise in the general expression given by equation (\[00.21a\]). However, in that case there appears a term linear in time which creates interpretational problems. Also $P_{0}$ and $P_{1}$ becomes singular at $\rho =0$ if $B\neq 0$.
SPIN IMPARTED TO TEST PARTICLES BY GRAVITATIONAL WAVES
=======================================================
We can write the $e\psi N$ force four vector (discussed in section 3.2) as $\left[ 7\right] $
$$F_{0}=-U_{,0}\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F_{i}=-V_{,i}.$$
where $$U=m\left[ \ln \left( \frac{Af}{B}\right) -\int \frac{g_{,0}^{ij}g_{ij,0}}{4A}dt\right] ,$$ $$V=-m\ln
(f),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ where B is a constant with units of time inverse so as to make $\frac{A}{B}$ dimensionless. Here $V$ is the $e\psi N\,$ potential but there is no good interpretation of $U$. Thus there is a problem of interpretation of $F_{0}$. The $F_{i}$ was reinterpreted $\left[ 8\right] $ as the rate of change of the momentum imparted to test particles by the gravitational field, i.e. $F_{i}=\frac{dP_{i}}{d\tau }$ (where $\tau $ is the proper time). In this interpretation it would be natural to identify $F_{0}\,$as $\frac{dP_{0}}{d\tau }$. The problem now is to interpret $P_{0}$ since one would normally take $P_{0}=E=\sqrt{m^{2}+P^{i}P^{j}g_{ij}}$ where $m$ is the mass of the test particle . Hence $\int F_{0}dt$ cannot be this $P_{0}$. Sharif’s suggestion $[11]$ for the interpretation of $P_{0},$ that it gives the spin angular momentum imparted to test rods, is given in the first section of this chapter, but in the same section it turns out to be inconsistent. To find an alternative check on its validity the geodesic $\left[ 4\right] $ analysis for the angular momentum imparted to test particles by gravitational waves is undertaken in section 2. This formalism is applied to various cases in section 3.
Spin angular momentum imparted by gravitational waves
------------------------------------------------------
Sharif \[11\] considers a test rod of length $\lambda $ in the path of a gravitational wave whose preferred direction is given by $l^{i}$ in the preferred reference frame. He argues that the rod will acquire maximum angular momentum from the wave if it lies in the plane given by $e_{\rho jki}l^{\rho },$ where $e_{\mu \nu \rho \pi }$ is a totally skew symmetric fourth rank tensor. Thus the spin vector will be given by
$$S^{i}=\frac{1}{2}e^{ijk\nu }e_{jkl}l^{l}P_{\nu }.$$
For $i=1,$$$\begin{aligned}
S^{1} &=&\frac{1}{2}e^{1jk\nu }e_{jkl}l^{l}P_{\nu } \nonumber \\
&=&\frac{1}{2}[e^{1230}e_{23l}+e^{1320}e_{32l}]l^{l}P_{0} \nonumber \\
&=&P_{0}l^{1}.\end{aligned}$$ Similarly
$$S^{2}=P_{0}l^{2}\textrm{ and }S^{3}=P_{0}l^{3}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
So in the preferred direction the spin vector would be proportional to $l^{i} $ such that
$$S^{i}=P_{0}l^{i}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$
The angular momentum imparted would be the magnitude of the spin vector . Thus the maximum angular momentum imparted to a test rod when it lies in the plane perpendicular to the preferred direction is:
$$S=P_{0}\lambda =m\lambda \int [(\ln (Af)_{,0}-g_{,0}^{ij}g_{ij,0}/4A]dt.
\label{4.1}$$
Hence the physical significance of the zero component of the momentum four vector would be that it provides an expression for the spin imparted to a test rod in an arbitrary spacetime.
This formula was applied to plane and cylindrical gravitational waves to give the following results.
[**1. Plane gravitational waves.**]{}
Using metric (\[7\]) in equation (\[4.1\]) we get (for detailed calculations see section 3.2) $$P_{0}=\textrm{{\it constant.}}$$
and thus the spin would also be constant.
[**2. Cylindrical gravitational waves.**]{}
Following the same procedure for the metric (2.1) we get (for detailed calculations see section 3.2)
$$S=-m\lambda [(1+AJ_{0}/\omega \rho J_{0}^{\prime })\ln \mid
1-2\omega \rho AJ_{0}^{\prime }\cos (\omega t)\mid +\textrm{{\it
constant.}}$$
Notice that there can be no spin angular momentum imparted to test particles in a perfectly homogeneous and isotropic cosmological model \[1\]; its high degree of symmetry $-$ in particular, spherical symmetry is incompatible with spin being imparted to test particles. However when we use Sharif’s formula for cosmological models, it gives exactly this error.
We give examples which had already partly been constructed by M. Sharif \[14\].
[**3.The Friedman model**]{}:
Consider the Friedman model, which is isotropic and homogeneous,
$$ds^{2}=dt^{2}-a^{2}(t)[d\chi ^{2}+f_{k}^{2}(\chi )d\Omega
^{2}],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{4.2}$$
where
$k=1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f_{1}(\chi )=\sin (\chi
),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$k=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f_{0}(\chi )=\chi ,$
$k=-1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f_{-1}(\chi
)=\sinh (\chi ),$
$\chi $ being the hyperspherical angle and $a(t)$ the scale parameter given by:
$$\left.
\begin{array}{c}
a(t)=a_{0}(1-\cos \eta )/2,\,\,\,\,t=a_{0}(\eta -\sin \eta
),\,\,\,\,\,\,\,\,\,\,\,\,\,0\leq \eta \leq 2\pi ,\,\,k=1 \\
\,\,\,\,\,\,\,\,\,=a_{0}^{1/3}t^{2/3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k=0; \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=a_{0}(\cosh \eta
-1)/2,\,\,\,\,t=a_{0}(\sinh \eta -\eta )/2,\,\,0\leq \eta \leq \infty
,\,\,\,k=-1;
\end{array}
\right\} \,\,\,\,\,\, \label{a(t)}$$
Using the metric (\[4.2\]) the $e\psi N$ force for the Friedmann models will be $$F_{0}=-m\frac{a^{..}}{a^{.}},$$ and
$$P_{0}=\int F_{0}dt=-m\ln (a^{.}). \label{F0}$$
As $S=m\lambda \int F_{0}dt.$ We get the spin for Friedmann models $$S=-m\lambda \ln (a^{.}). \label{S}$$ [**a. Closed Friedmann model**]{}
Using the metric (\[4.2\]) with $k=1$ in equation (\[F0\]) and (\[S\]) we get $$P_{0}=m[\ln \sqrt{1-\cos \eta }-\frac{3}{8}\cos \eta +\frac{1}{16}\cos
^{2}\eta +\ln \sqrt{2}+\frac{5}{16}],\,\,\,\,\,\,\, \label{4.3a}$$ and thus $$S=m\lambda [\ln \sqrt{1-\cos \eta }-\frac{3}{8}\cos \eta +\frac{1}{16}\cos
^{2}\eta +\ln \sqrt{2}+\frac{5}{16}]. \label{4.3}$$
[**b. Flat Friedmann model.**]{}
Again Using the metric (\[4.2\]) with $k=0$ in equation (\[F0\]) and (\[S\]) we get$\,\,\,\,\,\,\,$ $$P_{0}=m\ln \left( \frac{\eta }{2}\right) . \label{4.4a}$$ And thus the spin for a flat Friedmann model is $$S=m\lambda \ln \left( \frac{\eta }{2}\right)
\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{4.4}$$ [**c. Open Friedmann model.**]{}
Finally we obtain the spin for the open Friedmann model by the same procedure as the following $$P_{0}=-m\ln \mid \frac{\sinh \eta }{\cosh \eta -1}\mid .\, \label{4.5**}$$ $$S=-m\lambda \ln \mid \frac{\sinh \eta }{\cosh \eta -1}\mid .\, \label{4.5*}$$ $.$
Equations (\[4.3\]), (\[4.4\]) and (\[4.5\*\]) tells us that a non-zero spin is imparted by flat, open, and closed Friedmann models. Since no spin can be imparted thus Sharif’s interpretation of $P_{0}$ cannot be correct.
[**4. The Kasner model**]{}
Consider the Kasner model for a homogeneous anisotropic universe (near the cosmological singularity) $$ds^{2}=dt^{2}-t^{2p_{1}}dx^{2}-t^{2p_{2}}dy^{2}-t^{2p_{3}}dz^{2} \label{k}$$ Here $p_{i}$ are numbers such that $p_{1}+p_{2}+p_{3}=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1.$
The metric tensor is $$g_{\mu \nu }=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -t^{2p_{1}} & 0 & 0 \\
0 & 0 & -t^{2p_{2}} & 0 \\
0 & 0 & 0 & -t^{2p3}
\end{array}
\right)$$ Its inverse is $$g^{\mu \nu }=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -t^{-2p_{1}} & 0 & 0 \\
0 & 0 & -t^{-2p_{2}} & 0 \\
0 & 0 & 0 & -t^{-2p3}
\end{array}
\right)$$
To find the momentum four vector we need the force four vector. We therefore calculate it. Now
$$g=\det (g_{ij})=-t^{2},\,\,\,f=\frac{1}{\sqrt{g_{00}}}=1,\,\,$$
$$\ln \sqrt{-g}=\ln
t,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
which implies that $$Af=(\ln \sqrt{-g})_{,0}=\frac{1}{t},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(\ln (Af))_{,0}=-\frac{1}{t},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{kas1}$$ $$g_{,0}^{ij}g_{ij,o}=\frac{-4}{t^{2}}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{kas2}$$ So from equations (\[kas1\]) and (\[kas2\]) we get $$F_{0}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ Hence from equation (\[4.1\]) we get $$S=\textrm{{\it constant.}}$$ By physical consideration we could set $S=0.$
[**5. The De Sitter universe (usual coordinates)**]{}
Consider the metric $$ds^{2}=(1-r^{2}/D^{2})dt^{2}-(1-r^{2}/D^{2})^{-1}dr^{2}-r^{2}(d\theta
^{2}+\sin ^{2}\theta d\varphi ^{2}), \label{des}$$
The metric tensor is $$g_{\mu \nu }=\left(
\begin{array}{cccc}
(1-r^{2}/D^{2}) & 0 & 0 & 0 \\
0 & -(1-r^{2}/D^{2})^{-1} & 0 & 0 \\
0 & 0 & -r^{2} & 0 \\
0 & 0 & 0 & -r^{2}\sin ^{2}\theta
\end{array}
\right) .$$ Its inverse is $$g^{\mu \nu }=\left(
\begin{array}{cccc}
-\frac{D^{2}}{-D^{2}+r^{2}} & 0 & 0 & 0 \\
0 & \frac{-D^{2}+r^{2}}{D^{2}} & 0 & 0 \\
0 & 0 & -\frac{1}{r^{2}} & 0 \\
0 & 0 & 0 & -\frac{1}{r^{2}\sin ^{2}\theta }
\end{array}
\right)$$ Again proceeding on the same lines for the momentum four vector, we have $$g=\det (g_{ij})=-r^{4}\sin ^{2}\theta ,\,\,\,f=\frac{1}{\sqrt{g_{00}}}=\frac{1}{\sqrt{(1-r^{2}/D^{2})}}\,,\,$$ $$\ln \sqrt{-g}=\ln (r^{2}\sin \theta
),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ which implies that
$$Af=(\ln \sqrt{-g})_{,0}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$(\ln
(Af))_{,0}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{des1}$$
$$g_{,0}^{ij}g_{ij,o}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{des2}$$
So from equations (\[des1\]), (\[des2\]) we get
$$F_{0}=0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
Hence from equation (\[4.1\]) we get $$S=\textrm{{\it constant.}}$$ Clearly, we would need to take $S=0$ here.
[**6. The Lemaitre form of the De Sitter universe**]{}
Consider the empty space solution of the Einstein field equations with cosmological constant, $$ds^{2}=dt^{2}-a_{0}^{2}e^{2(\Lambda /3)^{1/2}t}\left[ d\chi ^{2}+\chi
^{2}d\theta ^{2}+\chi ^{2}\sin ^{2}\theta d\varphi ^{2}\right] . \label{d2}$$ The metric tensor is $$g_{\mu \nu }=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -a_{0}^{2}e^{2(\Lambda /3)^{1/2}t} & 0 & 0 \\
0 & 0 & -a_{0}^{2}e^{2(\Lambda /3)^{1/2}t}\chi ^{2} & 0 \\
0 & 0 & 0 & -a_{0}^{2}e^{2(\Lambda /3)^{1/2}t}\chi ^{2}\sin ^{2}\theta
\end{array}
\right) .$$ Its inverse is $$g^{\mu \nu }=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -\frac{1}{a_{0}^{2}e^{\frac{2}{3}\sqrt{3}\sqrt{\Lambda }t}} & 0 & 0 \\
0 & 0 & -\frac{1}{a_{0}^{2}e^{\frac{2}{3}\sqrt{3}\sqrt{\Lambda }t}\chi ^{2}}
& 0 \\
0 & 0 & 0 & -\frac{1}{a_{0}^{2}e^{\frac{2}{3}\sqrt{3}\sqrt{\Lambda }t}\chi
^{2}\sin ^{2}\theta }
\end{array}
\right) .$$
To find the momentum four vector we need the force four vector. We therefore calculate it. Now
$$g=\det (g_{ij})=-a_{0}^{6}e^{2\sqrt{3}\sqrt{\Lambda }t)}\chi ^{4}\sin
^{2}\theta ,\,\,\,f=\frac{1}{\sqrt{g_{00}}}=1\,\,,$$
$$\ln \sqrt{-g}=3\ln a_{0}+2\ln \chi +\sqrt{3}\sqrt{\Lambda }t+\frac{1}{2}\ln
\left( -\cos \theta +1\right) +\frac{1}{2}\ln \left( \cos \theta +1\right) ,$$
which implies that
$$Af=(\ln \sqrt{-g})_{,0}=\sqrt{3}\sqrt{\Lambda },\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$(\ln
(Af))_{,0}=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{lem1}$$
$$g_{,0}^{ij}g_{ij,o}=-4\Lambda
.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{lem2}$$
So from equations (\[lem1\]), (\[lem2\]) and (\[00.5c\]) we get $$F_{0}=m\left( \frac{\Lambda }{3}\right)
^{1/2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ Hence from equation (\[4.1\]) we get $$S=m\lambda \left( \frac{\Lambda }{3}\right) ^{1/2}t+\textrm{{\it
constant.}} \label{Lp0}$$ This does not seem reasonable. Since the other form of the metric gives a different result, it is clear that the interpretation is not even internally consistent.
The geodesic analysis for angular momentum imparted to test particles by gravitational waves
---------------------------------------------------------------------------------------------
In section $4.1$ we have concluded that $S$ is not the spin angular momentum imparted to test particles by gravitational waves. Then the question arises, if $S$ is not the spin then what is the spin angular momentum. So in this section we use the geodesic analysis $\left[ 4\right] $ to find the angular momentum imparted to test particles by gravitational waves. This formula is further applied to various cases.
Consider a time like congruence of the world lines (not necessarily geodetic) with tangent vector $u^{a}.$ Decompose $u_{a;b}$ by means of the operator $h_{ab}$ projecting in to the infinitesimal 3-space orthogonal to $u^{a}$ $[4].$
$$u_{a;b}=-\omega _{ab}+\sigma _{ab}+\frac{1}{3}\theta h_{ab}-u_{a}^{.}u_{b}.$$
where $$\,-\omega _{ab}\equiv
u_{[a,b]}+u_{[a}^{.}u_{b]}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,u_{a}^{.}\equiv
u_{a;b}u^{b}. \label{4.7}$$ $$\theta \equiv
u_{;a}^{a}\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,h_{ab}\equiv g_{ab}+u_{a}u_{b}$$ $$\,\,\,\,\,\,\,\,\,\sigma _{ab}\equiv u_{(a;b)}+u_{(a}^{.}u_{b)}-\frac{1}{3}\theta h_{ab}\,\,\,\,\,\,\,\,\,\,\,\,\,(\sigma _{a}^{a}=0).$$ For an observer along one of the world lines and using Fermi propagated axes, $\omega _{ab}$ describe velocity of rotation, $\sigma _{ab\textrm{ }}$shear and $\theta $ describe expansion of the neighboring free particles. Since the $\psi N-$formalism uses the fermi-walker frame, it could be expected that the results of this analysis should be consistent with it. For our purpose only $\omega _{ab}$ is needed. Choose the coordinates so that the tangent vector is $u^{a}=f\delta _{0}^{a}.$ In the case $g_{00}=1,\,u^{a}=\delta _{0}^{a}.$ If $g_{00}\neq 1,\,u^{a}=\frac{1}{\sqrt{g_{00}}}\delta _{0}^{a}\,$Thus, from equation (\[4.7\]) we have $$-\omega _{ab}=(u_{a,b}-u_{b,a})/2+(u_{a}^{.}u_{b}-u_{b}^{.}u_{a})/2.$$ The first and second component on the right hand side of the above equation vanishes. Thus we have$\,\,$$$-\omega _{ab}=(u_{a}^{.}u_{b}-u_{b}^{.}u_{a})/2.$$ Further using equation (\[4.7\]) we have
$$\begin{aligned}
-2\omega _{ab} &=&\left\{
\begin{array}{c}
d \\
c\,\,\,\,\,a
\end{array}
\right\} u_{d}u^{c}u_{b}-\left\{
\begin{array}{c}
d \\
c\,\,\,\,\,b
\end{array}
\right\} u_{d}u^{c}u_{a} \\
&=&\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,a
\end{array}
\right\} \delta _{b}^{0}-\left\{
\begin{array}{c}
0 \\
0\,\,\,\,b
\end{array}
\right\} \delta _{a}^{0}. \nonumber\end{aligned}$$
Taking $a\,$to be $i$ and $b$ to be zero, we finally obtain the components of the spin vector: $$\omega _{i0}=-\frac{1}{2}\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,\,\,i
\end{array}
\right\} .\,\,\,\,\,\,\,(i=1,2,3) \label{4.8}$$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
This simple formula appears because $u^{a}=\delta _{0}^{a}\,$only. This gives the angular momentum imparted to test particles by gravitational waves. We now apply this formula to different types of gravitational waves.
Applications
-------------
### Plane gravitational waves
For a metric $g_{\mu \nu }$ construct the new metric
$$\bar{g}_{\mu \nu }=\eta _{\mu \nu }+[g_{\mu \nu }-\eta _{\mu \nu }]u(s-s_{0})
\label{newdef}$$
where $\eta _{\mu \nu }$ is the Minkowski spacetime and $u(s-s_{0})$ is the step function defined as:
$u(s-s_{0})=0\,\,\,\,\,\,\,\,\,\,$when$\,\,\,\,\,\,\,\,\,s<s_{0}\,$
$u(s-s_{0})\,\,=1\,\,\,\,\,\,\,\,\,\,\,\,\,\,$when$\,\,\,\,\,\,\,s>s_{0}\,.$
$.$According to this definition the metric tensor becomes: $$\bar{g}_{\mu \nu }=\left(
\begin{array}{cccc}
\left[ (e^{2\Omega (\alpha )}-1)u+1\right] & 0 & 0 & 0 \\
0 & (1-e^{2\Omega (\alpha )})u-1 & 0 & 0 \\
0 & 0 & (-\alpha ^{2}e^{2\beta (\alpha )}+1)u-1 & 0 \\
0 & 0 & 0 & (-\alpha ^{2}e^{-2\beta (\alpha )}+1)u-1
\end{array}
\right) .$$ where $\alpha =t-x$. The inverse of this metric is
$$\bar{g}_{\mu \nu }=\left(
\begin{array}{cccc}
\frac{1}{(e^{2\Omega (\alpha )}-1)u+1} & 0 & 0 & 0 \\
0 & \frac{1}{-1+(-e^{2\Omega (\alpha )}+1)u} & 0 & 0 \\
0 & 0 & \frac{1}{-1+(-\alpha ^{2}e^{2\beta (\alpha )}+1)u} & 0 \\
0 & 0 & 0 & \frac{1}{-1+(-\alpha ^{2}e^{-2\beta (\alpha )}+1)u}
\end{array}
\right) .$$
Now
$$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,\,1
\end{array}
\right\} =\frac{1}{2}g^{00}g_{00,1},\,\,\,\,\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,\,2
\end{array}
\right\} =\frac{1}{2}g^{00}g_{00,2},\,\,\,\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,\,3
\end{array}
\right\} =\frac{1}{2}g^{00}g_{00,3}. \label{4.11}$$
If we find $g_{00,1},\,g_{00,2}\,$and $g_{00,3}$ then we are done. Since $g_{00}=(e^{2\Omega (\alpha )}-1)u+1$. $$\begin{aligned}
g_{00,1} &=&[(e^{2\Omega (\alpha )}-1)u+1]_{,1} \\
&=&2\Omega ^{\prime }(\alpha )e^{2\Omega (\alpha )}u-(e^{2\Omega (\alpha
)}-1)\frac{\partial s}{\partial x}\delta (s-s_{0}). \nonumber\end{aligned}$$ Dividing equation (\[7\]) by $ds^{2}$ and keeping $t,y$ and $z$ fixed, we get: $$\frac{\partial s}{\partial x}=e^{\Omega (\alpha
)}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ Consequently we have
$$g_{00,1}=-2\Omega ^{\prime }(\alpha )e^{2\Omega (\alpha
)}u(s-s_{0})+e^{\Omega (\alpha )}(e^{2\Omega (\alpha )}-1)\delta (s-s_{0}).
\label{4.14}$$
Further $$g_{00,2}=-(e^{2\Omega (\alpha )}-1)\frac{\partial s}{\partial y}\delta
(s-s_{0}). \label{4.15}$$ Again dividing equation (\[7\]) by $ds^{2}$ and keeping $t,x$ and $z$ fixed, we get $$\frac{\partial s}{\partial y}=\alpha e^{\beta (\alpha
)}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ Putting this value of$\frac{\partial s}{\partial y}\,$in to equation (\[4.15\]) we get
$$g_{00,2}=-e^{\beta (\alpha )}(e^{2\Omega (\alpha )}-1)\delta (s-s_{0}).
\label{4.16}$$
Also, $$g_{00,3}=-\alpha (e^{2\Omega (\alpha )}-1)\frac{\partial s}{\partial z}\delta (s-s_{0}). \label{4.17}$$ As in the previous cases, dividing equation (\[7\]) by $ds^{2}$ and keeping $t,x$ and $y$ fixed, we get $$\frac{\partial s}{\partial z}=\alpha e^{-\beta (\alpha
)}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ Putting $d\frac{\partial s}{\partial z}$ in equation (\[4.17\]) we get:
$$g_{00,3}=-\alpha e^{-\beta (\alpha )}(e^{2\Omega (\alpha )}-1)\delta
(s-s_{0}). \label{4.18}$$
Using equations (\[4.14\]), (\[4.16\]) and (\[4.18\]) in equation (\[4.11\]) we get:
$$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1
\end{array}
\right\} =\frac{1}{2(e^{2\Omega (\alpha )}-1)u+1}\left[ -2\Omega ^{\prime
}(\alpha )e^{2\Omega (\alpha )}u(s-s_{0})+e^{\Omega (\alpha )}(e^{2\Omega
(\alpha )}-1)\delta (s-s_{0})\right] \label{4.19}$$
$$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,2
\end{array}
\right\} =\frac{1}{2(e^{2\Omega (\alpha )}-1)u+1}\left[ \alpha e^{\beta
(\alpha )}(e^{2\Omega (\alpha )}-1)\delta (s-s_{0})\right]
,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{4.20}$$
$$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,3
\end{array}
\right\} =\frac{1}{2(e^{2\Omega (\alpha )}-1)u+1}\left[ \alpha e^{-\beta
(\alpha )}(e^{2\Omega (\alpha )}-1)\delta (s-s_{0})\right]
.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{4.21}$$
Now using equations (\[4.19\]), (\[4.20\]) and (\[4.21\]) in equation (\[4.8\]) we get components of the spin vector
$$\omega _{10}=-\frac{1}{4(e^{2\Omega (\alpha )}-1)u+1}\left[ -2\Omega
^{\prime }(\alpha )e^{2\Omega (\alpha )}u(s-s_{0})+e^{\Omega (\alpha
)}(e^{2\Omega (\alpha )}-1)\delta (s-s_{0})\right] , \label{4.86}$$
$$\omega _{20}=-\frac{1}{4(e^{2\Omega (\alpha )}-1)u+1}\left[ \alpha e^{\beta
(\alpha )}(e^{2\Omega (\alpha )}-1)\delta (s-s_{0})\right]
,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$\omega _{30}=-\frac{1}{4(e^{2\Omega (\alpha )}-1)u+1}\left[ \alpha e^{-\beta
(\alpha )}(e^{2\Omega (\alpha )}-1)\delta (s-s_{0})\right]
.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
In equation (\[4.86\]) the first part in the brackets gives the spin imparted to test particles by the plane gravitational waves while the second part gives the spin of the wave itself.
### Cylindrical gravitational waves
Now use the same construction for the metric (\[2.1\]) to get
$$\begin{aligned}
g_{00} &=&(e^{2(\gamma -\psi )}-1)u(s-s_{0})+1, \nonumber \\
g_{11} &=&-(e^{2(\gamma -\psi )}-1)u(s-s_{0})-1, \nonumber \\
g_{22} &=&-\rho ^{2}\left[ (e^{-2\psi }-1)u(s-s_{0})+1\right] , \nonumber \\
g_{33} &=&-(e^{2\psi }-1)u(s-s_{0})+1.\end{aligned}$$
The inverse of this metric is $$\bar{g}^{\mu \nu }=\left(
\begin{array}{llll}
g^{00} & 0 & 0 & 0 \\
0 & g^{11} & 0 & 0 \\
0 & 0 & g^{22} & 0 \\
0 & 0 & 0 & g^{33}
\end{array}
\right) ,$$ where
$g^{00}=\frac{1}{(e^{2(\gamma -\psi )}-1)u(s-s_{0})+1},$
$g^{11}=\frac{1}{-(e^{2(\gamma -\psi )}-1)u(s-s_{0})-1},$
$g^{22}=\frac{1}{-\rho ^{2}\left[ (e^{-2\psi }-1)u(s-s_{0})+1\right] },$
$g^{33}=\frac{1}{-(e^{2\psi }-1)u(s-s_{0})-1}.$
In this case
$$g_{00,1}=2e^{2\left( \gamma -\psi \right) }(\gamma ^{\prime }-\psi ^{\prime
})u(s-s_{0})+(e^{2\left( \gamma -\psi \right) }-1)\frac{\partial s}{\partial
\rho }\delta (s-s_{0)}. \label{c1}$$
Dividing equation (\[2.1\]) by $ds^{2}$ and keeping $t,\varphi ,z$ fixed, we get
$$1=e^{2\left( \gamma -\psi \right) }\left( \frac{\partial \rho }{\partial s}\right)
^{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
which implies that
$$\frac{\partial s}{\partial \rho }=e^{\left( \gamma -\psi \right)
}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{qwe}$$
Now putting $\frac{\partial s}{\partial \rho }$ in equation (\[c1\]) we get
$$g_{00,1}=+2e^{2\left( \gamma -\psi \right) }\left( \gamma ^{^{\prime }}-\psi
^{^{\prime }}\right) u\left( s-s_{0}\right) +e^{\gamma -\psi }\left(
e^{2\left( \gamma -\psi \right) }-1\right) \delta \left( s-s_{0}\right) ,
\label{C2}$$
$$\begin{aligned}
g_{00,2} &=&-\left[ \left( e^{2\left( \gamma -\psi \right) }-1\right) u+1\right] _{,2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \nonumber
\\
&=&-(e^{2\left( \gamma -\psi \right) }-1)\frac{\partial s}{\partial \varphi }\delta \left( s-s_{0}\right) , \label{c3}\end{aligned}$$
$$g_{00,3}=-\left( e^{2\left( \gamma -\psi \right) }-1\right) \frac{\partial s}{\partial z}\delta \left( s-s_{0}\right)
.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{c4}$$
Dividing equation (\[2.1\]) by $ds^{2}$ and keeping $t$,$\rho ,z$ fixed, we get
$$1=e^{2\psi }\rho ^{2}\left( \frac{\partial y}{\partial s}\right)
^{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
which implies that
$$\frac{\partial s}{\partial y}=\rho e^{-\psi
}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{c5}$$
Now dividing equation (\[2.1\]) by $ds^{2}$and keeping $t,\rho ,y$ fixed, we get $$1=e^{2\psi }\left( \frac{\partial z}{\partial s}\right)
^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\frac{\partial s}{\partial z}=e^{\psi
}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{c6}$$ Putting equation (\[c5\]) in to equation (\[c3\]) we get, $$g_{00,2}=-\left( e^{2\left( \gamma -\psi \right) }-1\right) \rho e^{-\psi
}\delta \left( s-s_{0}\right)
.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{c7}$$ Putting equation (\[c6\]) in to equation (\[c4\]) we get, $$g_{00,3}=-\left( e^{2\left( \gamma -\psi \right) }-1\right) e^{\psi }\delta
\left( s-s_{0}\right)
.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{c8}$$ Using equations (\[C2\]), (\[c7\]) and (\[c8\]) in equation (\[4.11\]) we get: $$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1
\end{array}
\right\} =-\frac{1}{2(e^{2(\gamma -\psi )}-1)u+1}\left[ 2e^{2(\gamma -\psi
)}(\gamma ^{\prime }-\psi ^{\prime })u+e^{\gamma -\psi }(e^{2(\gamma -\psi
)}-1)\delta (s-s_{0})\right] , \label{c9}$$ $$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,2
\end{array}
\right\} =-\frac{1}{2(c^{2}e^{2(\gamma -\psi )}-1)u+1}\left[ (e^{2(\gamma
-\psi )}-1)\rho e^{-\psi }\delta (s-s_{0})\right] ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{c10}$$ $$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,3
\end{array}
\right\} =-\frac{1}{2(e^{2(\gamma -\psi )}-1)u+1}\left[ (c^{2}e^{2(\gamma
-\psi )}-1)e^{\psi }\delta (s-s_{0})\right] .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{c11}$$ Now using equations (\[c9\]), (\[c10\]) and (\[c11\]) in equation (\[4.8\]) we get the components of the spin vector
$$\omega _{10}=\frac{1}{4(e^{2(\gamma -\psi )}-1)u+1}\left[ 2e^{2(\gamma -\psi
)}(\gamma ^{\prime }-\psi ^{\prime })u+e^{\gamma -\psi }(e^{2(\gamma -\psi
)}-1)\delta (s-s_{0})\right] ,$$
$$\omega _{20}=\frac{1}{4(e^{2(\gamma -\psi )}-1)u+1}\left[ (e^{2(\gamma -\psi
)}-1)\rho e^{-\psi }\delta (s-s_{0})\right] ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
$$\omega _{30}=\frac{1}{4(e^{2(\gamma -\psi )}-1)u+1}\left[ (e^{2(\gamma -\psi
)}-1)e^{\psi }\delta (s-s_{0})\right] .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
### The Friedmann model
Consider the Friedmann model. The metric for this model is given by equation (\[4.2\]) and the metric tensor is $$g_{\mu \nu }=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -a^{2}(t) & 0 & 0 \\
0 & 0 & -a^{2}(t)f_{k}^{2}(\chi ) & 0 \\
0 & 0 & 0 & -a^{2}(t)f_{k}^{2}(\chi )\sin ^{2}\theta
\end{array}
\right) .$$ According to equation (\[4.8\]) the only thing we need is $g_{00,1},\,g_{00,2},\,$and $\,g_{00,3}.$ As there are no wave fronts. Therefore we have $g_{00}=1.$ Here obviously all it’s derivatives will vanish. Hence $$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,2
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,3
\end{array}
\right\} =0.$$ Thus all components of spin vector are zero. i.e. $$\omega _{10}=\omega _{20}=\omega _{30}=0.$$ As we have stated earlier, there is no spin in an isotropic and homogeneous universe model, this analysis gives zero spin for this model as required.
### The Kasner model
Consider the Kasner model. The metric for the Kasner model is given by equation (\[k\]) and the metric tensor is $$g_{\mu \nu }=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -t^{2p_{1}} & 0 & 0 \\
0 & 0 & -t^{2p_{2}} & 0 \\
0 & 0 & 0 & -t^{2p3}
\end{array}
\right) . \label{K}$$ According to equation (\[4.8\]) the only thing we need is $g_{00,1},\,g_{00,2},\,$and $\,g_{00,3}.$ As there is no “wave-front” so the original metric stands. From (\[K\]) we have $g_{00}=1.$ Here obviously all it’s derivatives will vanish. Hence $$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,2
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,3
\end{array}
\right\} =0.$$ Thus all components of spin vector are zero. i.e. $$\omega _{10}=\omega _{20}=\omega _{30}=0.$$
### The De Sitter universe (usual coordinates)
The metric for this model is given by equation (\[des\]) and the metric tensor is $$g_{\mu \nu }=\left(
\begin{array}{cccc}
(1-r^{2}/D^{2}) & 0 & 0 & 0 \\
0 & -(1-r^{2}/D^{2})^{-1} & 0 & 0 \\
0 & 0 & -r^{2} & 0 \\
0 & 0 & 0 & -r^{2}\sin ^{2}\theta
\end{array}
\right) . \label{4.117}$$ As $g_{00}\neq 1$ so $\omega _{i0}$ calculated in section $4.2$ cannot work. Let $u^{a}=f\delta _{0}^{a}.$ Here by definition $$u^{a}u^{b}g_{ab}=-1\,=(u^{0})^{2}g_{00}\textrm{\thinspace or }\,u^{0}=\frac{1}{\sqrt{g_{00}}}$$
consequently we have $$u^{a}=\frac{1}{\sqrt{g_{00}}}\delta _{0}^{a}.$$ Now we only need to calculate $u_{[a,b]}\,$and$\,u_{[a}^{.}u_{b]}$$$\begin{aligned}
u_{a,b}\, &=&\left( \sqrt{g_{00}}\delta _{a}^{0}\right) _{,b} \nonumber \\
&=&(\ln \sqrt{g_{00}})_{,1}\delta _{a}^{0}\delta _{b}^{1}.\end{aligned}$$
Hence we have $$u_{\left[ a,b\right] }=(\ln \sqrt{g_{00}})_{,1}\delta _{[a}^{0}\delta
_{b]}^{1} \label{4.65}$$ Similarly we get $$u_{[a}^{.}u_{b]}=-\frac{1}{2}\left( \delta _{b}^{0}\delta _{a}^{1}-\delta
_{a}^{0}\delta _{b}^{1}\right) \sqrt{g_{00}}\left\{
\begin{array}{c}
0 \\
0\,\,\,\,1
\end{array}
\right\} \label{4.66}$$ From equation $\left( \ref{4.65}\right) $ and $\left( \ref{4.66}\right) $ the sum $u_{\left[ a,b\right] }+u_{[a}^{.}u_{b]}$ for this metric vanishes and thus equation $\left( \ref{4.7}\right) $ implies that $\omega _{ab}$ vanishes.
### The Lemaitre form of the De Sitter universe
The metric for this model is given by (\[d2\]). The metric tensor is $$g_{\mu \nu }=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -a_{0}^{2}e^{2(\Lambda /3)^{1/2}t} & 0 & 0 \\
0 & 0 & -a_{0}^{2}e^{2(\Lambda /3)^{1/2}t}\chi ^{2} & 0 \\
0 & 0 & 0 & -a_{0}^{2}e^{2(\Lambda /3)^{1/2}t}\chi ^{2}\sin ^{2}\theta
\end{array}
\right) .$$ By the definition (\[newdef\]) of the metric tensor $g_{00}=1.\,$Here obviously all it’s derivatives will vanish. Hence $$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,2
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,3
\end{array}
\right\} =0.$$ Thus all components of spin vector are zero. i.e. $$\omega _{10}=\omega _{20}=\omega _{30}=0.$$ Thus we have got the same result for both the forms of the De Sitter universe which proves that the formalism is consistent.
### The G $\stackrel{..}{o}$del universe model
Consider the G$\stackrel{..}{o}$del (spinning) universe model $$ds^{2}=dt^{2}-dx^{2}-dy^{2}+2m(x)dtdz-l(x)dz^{2}, \label{G}$$ where $m(x)=Ae^{ax}$ and $l(x)=A^{2}(\frac{a^{2}}{b^{2}}-1)e^{2ax}.\,$The metric tensor is $$g_{\mu \nu }=\left(
\begin{array}{cccc}
1 & 0 & 0 & m(x) \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
m(x) & 0 & 0 & -l(x)
\end{array}
\right) ,$$ Its inverse is $$g^{\mu \nu }=\left(
\begin{array}{cccc}
\frac{l(x)}{l(x)+m^{2}(x)} & 0 & 0 & \frac{m(x)}{l(x)+m^{2}(x)} \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
\frac{m(x)}{l(x)+m^{2}(x)} & 0 & 0 & -\frac{1}{l(x)+m^{2}(x)}
\end{array}
\right)$$ According to equation (\[4.8\]) the only thing we need are the following three Christoffel symbols$\,\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1
\end{array}
\right\} $, $\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,2
\end{array}
\right\} $ and $\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,3
\end{array}
\right\} .$ Now
$$\begin{aligned}
\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,1
\end{array}
\right\} &=&\frac{1}{2}g^{03}g_{03,1} \nonumber \\
&=&\frac{1}{2}\frac{m(x)}{l(x)+m^{2}(x)}(Ae^{ax})_{,1} \nonumber \\
&=&\frac{1}{2}\frac{am^{2}(x)}{l(x)+m^{2}(x)},\end{aligned}$$
$$\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,2
\end{array}
\right\} =\left\{
\begin{array}{c}
0 \\
0\,\,\,\,\,3
\end{array}
\right\} =0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$
Thus all components of the spin vector are $$\begin{aligned}
\omega _{10} &=&-\frac{1}{4}\frac{am^{2}(x)}{l(x)+m^{2}(x)} \nonumber \\
\omega _{20} &=&\omega _{30}=0.\end{aligned}$$
SUMMARY AND CONCLUSION
======================
Linearized General Relativity predicts gravitational waves. These waves are analogous to electromagnetic waves. There may be different types of gravitational waves i.e. plane and cylindrical gravitational waves etc. This background was reviewed in chapter 1. Work on plane and cylindrical gravitational waves $\left[
3\right] $ has been helpful in understanding them further. The question of the reality of gravitational waves was discussed in chapter 2. There we used Weber-Wheeler $\left[ 3\right] $ method for a particle in the path of a plane gravitational wave and obtained the standard constant momentum imparted to the particle. Some astrophysical sources of gravitational waves were also discussed. In chapter 3 the work of Qadir and Sharif $\left[
8\right] $ in which a general formula is developed for the momentum imparted to test particles by gravitational waves in arbitrary spacetime, was reviewed. In this paper the problem of the identification of their zero component of the momentum four vector was mentioned. Sharif $\left[ 11\right] $ gave a suggestion that it could be interpreted as the spin imparted to a test rod in an arbitrary spacetime. This suggestion was reviewed in chapter 4. Further analysis by Sharif $\left[ 14\right] $ was shown here to provide a counterexample for this suggestion i.e. it gives a non zero spin for an isotropic and homogeneous universe model. Further, when the example of the De Sitter universe was considered, the interpretation gave different results for the static and Lemaitre forms. This proved that the interpretation is not even internally consistent. As such $P_{0}$ cannot be interpreted as the spin imparted to test rods. A geodesic analysis $\left[ 4\right] $ was used in section $4.2$ to evaluate the spin imparted to test particles in various cases. In the cases of plane and cylindrical gravitational waves we got very reasonable results. As required , the Friedmann, the Kasner and De Sitter models do not impart spin to test particles according to this analysis. Further we got the same results for both forms of the De Sitter universe, which confirms the validity of our analysis. Finally, the G$\stackrel{..}{o}$del universe model gave a non-zero $\left( \textrm{non-constant}\right) $ angular momentum, precisely as it should.
If $P_{0}$ does not provide an expression for the spin imparted to test rods in an arbitrary spacetime then what is its correct interpretation?. Consider $E=\sqrt{P^{i}P^{j}g_{ij}+m^{2}}$ and $P_{0}=\int F_{0}dt.$ Define the difference between the two as $\Delta E=P^{0}-E$ for some of the spacetimes considered earlier. By definition the usual energy $E$ associated with a momentum $P^{i}$ is given by $$P^{\mu }P^{\nu }g_{\mu \nu }=m^{2}=\left( E^{2}\right)
g_{00}+P^{i}P^{j}g_{ij}.$$ Hence $$E=\frac{1}{\sqrt{g_{00}}}\left( m^{2}-P^{i}P^{j}g_{ij}\right) ^{1/2}.$$ Now, we want to compare this with $$P^{0}=g^{00}\int F_{0}dt.$$ The energy difference, $\Delta E,$ is then $$\Delta E=g^{00}\int F_{0}dt-\frac{1}{\sqrt{g_{00}}}\left(
m^{2}-P^{i}P^{j}g_{ij}\right) ^{1/2}.$$ When $F_{0}$ is zero we get no further understanding from $\Delta E$. As such we will not compute these cases. Again for G$\stackrel{..}{o}$del universe we have not calculated $F_{0}$ as the metric is not in the block diagonalized form $\left( g_{0i}=0\right) \,$and a gauge transformation would need to be made before it could be applied. Therefore we only work out $\Delta E$ for the cases $\left( a\right) $ cylindrical gravitational waves; $\left( b\right) $ the Friedmann models and $\left( c\right) $ the Lemaitre form of the De Sitter universe. Finally we will discuss the consequences of these results.
$\left( {\bf a}\right) $ [**Cylindrical gravitational waves**]{}
Consider the metric given by equation $\left( \ref{2.1}\right) $ and use the non-zero components of momentum four vector given by equations $\left( \ref{00.23}\right) $ and $\left( \ref{00.24}\right) $ to get $$E=me^{-(\gamma -\psi )}\left( e^{-2\left( \gamma -\psi \right) }\left[
AJ_{0}^{\prime }\sin (\omega t)-\frac{1}{4}A^{2}J_{0}J_{0}^{\prime }+\rho
\omega (J_{0}J_{0}^{\prime })^{\prime }\sin (2\omega t)\right] ^{2}+1\right)
^{1/2}$$ Consequently we have
$$\Delta E=-me^{-(\gamma -\psi )}[(1-e^{-2\left( \gamma -\psi \right)
}(AJ_{0}^{\prime }\sin (\omega t)-\frac{1}{4}A^{2}J_{0}J_{0}^{\prime }+\rho
\omega (J_{0}J_{0}^{\prime })^{\prime }\sin (2\omega t))^{2})^{1/2}$$ $$\;\;\;\;\;\;\;\;\;\;\;+e^{-\left( \gamma -\psi \right) }(\ln \mid AJ_{0}\sin
(\omega t)\mid +(1+AJ_{0}/\omega \rho J_{0}^{\prime })\ln \mid 1-2\omega
\rho AJ_{0}^{\prime }\cos (\omega t]\mid )].$$
$\left( {\bf b}\right) $[**The Friedmann models**]{}
Since $P^{1},P^{2}$ and $P^{3}$ are zero for the Friedmann models, from equations $\left( \ref{4.2}\right) ,$ $\left( \ref{4.3}\right) $, $\left(
\ref{4.4a}\right) $ and $\left( \ref{4.5**}\right) $ we get the energy difference for the three models: $$\Delta E_{k=1}=m[\ln \sqrt{1-\cos \eta }-\frac{3}{8}\cos \eta +\frac{1}{16}\cos ^{2}\eta +\ln \sqrt{2}+\frac{5}{16}];$$ $$\Delta E_{k=0}=m\ln (\frac{\eta }{2});\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\Delta E_{k=-1}=m\ln \frac{\cosh \eta -1}{\sinh \eta }.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $.$
These are plotted in Fig. $\left( 5.1\right) ,\,\left( 5.2\right) \,$and $\left( 5.3\right) \,$and discussed shortly.
$\left( {\bf c}\right) $[** The Lemaitre form of the De Sitter universe**]{}
Again $P^{1},\,P^{2}\,$and $P^{3}$ are zero for the metric given by equation $\left( \ref{d2}\right) .$ Thus using equation $\left( \ref{Lp0}\right) $ we get $$\Delta E=m\sqrt{\Lambda /3}t.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ as shown in Fig $\left( 5.4\right) .$
The three expressions for the Friedmann models have the same asymptotic behavior for sufficiently small values of $\eta $, namely $\Delta E\sim m\ln
\eta $. For $k=\pm 1$ there is a correction term $\sim \pm \frac{\eta ^{2}}{12}$. We have inserted a constant term for $k=+1$ so that the expressions for the three should match up to the zero order terms. At the phase of maximum expansion of the closed model we get $\Delta E=m(\ln 2+\frac{3}{4}).$ We could equally well, have set $\Delta E=0$ at the phase of maximum expansion and had a difference for it from the other two cases for small $\eta $ (in the constant term). Note that for all the three models $\Delta E$ diverges as $\eta \rightarrow 0$ and it also diverges as $\eta \rightarrow
2\pi $ for the closed model. For $k=+1$ we have chosen to display the constant term so that $\Delta E=0$ at $\eta =\pi .$ The proposal does not seem inconsistent, but still needs further discussion.
$\smallskip $
[**APPENDIX 1**]{}
Consider the metric \[1\] (with signature $(+,-,-,-)$)
$$ds^{2}=L^{2}(u)(dx^{2}-dy^{2})-dudv \label{A1.1}$$
where $u=t-z\,$and$\,v=t+z\,\,\,$which is always flat.
The metric tensor is.
$$g_{ab}=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -L^{2}(u) & 0 & 0 \\
0 & 0 & -L^{2}(u) & 0 \\
0 & 0 & 0 & -1
\end{array}
\right) \label{A1.2}$$
Its inverse is.
$$g^{ab}=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & -L^{-2}(u) & 0 & 0 \\
0 & 0 & -L^{-2}(u) & 0 \\
0 & 0 & 0 & -1
\end{array}
\right) \label{A1.3}$$
The non vanishing Christoffel symbols are:
$$\left\{
\begin{array}{c}
1 \\
1\,\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
2 \\
2\,\,\,\,0
\end{array}
\right\} =\left\{
\begin{array}{c}
1 \\
0\,\,\,\,1
\end{array}
\right\} =\frac{L^{\prime }(u)}{L(u)\,}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A1.4}$$
$$\left\{
\begin{array}{c}
0 \\
1\,\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
3 \\
1\,\,\,\,1
\end{array}
\right\} =L(u)L^{\prime
}(u).\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A1.5}$$
$$\,\,\left\{
\begin{array}{c}
1 \\
3\,\,\,\,1
\end{array}
\right\} =\left\{
\begin{array}{c}
2 \\
0\,\,\,\,2
\end{array}
\right\} =\frac{L^{\prime }(u)}{L(u)\,}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A1.6}$$
$$\,\left\{
\begin{array}{c}
1 \\
2\,\,\,\,2
\end{array}
\right\} =\left\{
\begin{array}{c}
3 \\
2\,\,\,\,2
\end{array}
\right\} =L(u)L^{\prime
}(u)\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A1.7}$$
$$\,\left\{
\begin{array}{c}
2 \\
3\,\,\,\,2
\end{array}
\right\} =\left\{
\begin{array}{c}
1 \\
1\,\,\,\,3
\end{array}
\right\} =\left\{
\begin{array}{c}
2 \\
2\,\,\,\,3
\end{array}
\right\} =\frac{L^{\prime }(u)}{L(u)\,}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A1.8}$$
The primes refers to differentiation with respect to $u$. The nonvanishing components of the Ricci tensor are $R_{00},\,R_{03},\,R_{30}\,$and$\,\,R_{33}.$ Using Christoffel symbols listed above we obtain the following Ricci tensor components.
$$R_{00}=R_{33}=\frac{-2L^{\prime \prime }(u)}{L(u)}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{A1.9}$$
$$R_{30}=R_{03}=\frac{2L^{\prime \prime }(u)}{L(u)}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{A1.10}$$
From equations $\left( A1.9\right) $ and $\left( A1.10\right) $ the required result follows.
[**APPENDIX 2**]{}
We shall prove here that $$F^{0}=m[(\ln A)_{,0}-\Gamma _{00}^{0}+\Gamma _{0j}^{i}\Gamma
_{0i}^{j}/A]f^{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A2.1}$$ and
$$F^{i}=\Gamma
_{00}^{i}f^{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A2.2}$$
are the solutions of the following equations. $$l^{i}(F_{,i}^{0}+\Gamma
_{ij}^{0}F^{j})=0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A2.3}$$
$$l^{j}(F_{,j}^{i}+\Gamma
_{0j}^{i}F^{0})=F^{*i}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \label{A2.4}$$
where $$F^{*i}=m(\Gamma _{00,j}^{i}-\Gamma _{0j,0}^{i}+\Gamma
_{0j}^{i}\Gamma _{00}^{0}-\Gamma _{0k}^{i}\Gamma
_{0}^{k})f^{2}l^{j}. \label{A2.5}$$ For the verification of equation $\left( A2.3\right) $ we note that $$A=\Gamma
_{0i}^{i},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f^{2}=\frac{1}{g_{00}}. \label{A2.6}$$ Thus $$(\ln A)_{,0}=g_{,0}^{ij}/g^{ij}+g_{ij,00}/g_{ij,0}. \label{A2.7}$$ Here we will make use of the Riemann normal coordinates (RNCs) for spatial directions. We have $$g_{0i}=0 \label{A2.8}$$ $$g_{\mu \nu ,0}\neq 0,\,\,\,g_{\mu \nu ,i}=0=g_{\mu \nu ,0i}.
\label{A2.9}$$ Using this approximation we can write $$(\ln A)_{,0i}=0=\Gamma _{00,i}^{0}=\Gamma _{0j}^{i}\Gamma
_{0i,i}^{j}. \label{A2.10}$$ Now $$\begin{aligned}
F_{,i}^{0} &=&m[(\ln A)_{,0}i-\Gamma _{00,i}^{0}+(\Gamma _{0j}^{i}\Gamma
_{0i}^{j}/A)_{,i}]f^{2} \nonumber \\
&&+m[(\ln A)_{,0}-\Gamma _{00}^{0}+\Gamma _{0j}^{i}\Gamma
_{0i}^{j}/A](f^{2})_{,i}. \label{A2.11}\end{aligned}$$ Using RNCs the first, second, third and the last term on the right hand side vanishes. Therefore $$F_{,i}^{0}=0 \label{A2.12}$$ Thus equation $\left( A2.3\right) $ is satisfied. Equation $\left(
A2.4\right) $ can directly be obtained just by replacing the values of $F^{0} $ and $F^{i}$ from equations $\left( A2.1\right) $ and $\left(
A2.2\right) $.
Hence $F^{0}$ and $F^{i}$ given by equations $\left( A2.1\right) $ and $\left( A2.2\right) $ are the solutions of equations $\left( A2.3\right) $ and $\left( A2.4\right) $.
**References**
1. C. W. Misner, K. S. Thorne and J. A. Wheeler, [*Gravitation,*]{} W. H. Freeman,San Francisco, (1973).
2. D. Kramer,H. Stephanie, E. Herlt and McCallum, [*Exact solutions of Einstein’s Field Equations,* ]{}(Cambridge university, Press, Cambridge, 1979).
3. J. Weber, [*General Relativity and Gravitational Waves,* ]{}(Interscience, NewYork, 1961).
4. J. Ehlers and W. Kundt, ed. L. Witten (Wiley, New York, 1962).
5. J. Weber and J. A. Wheeler, Rev. Mod. Phy. [**29**]{} (1957) 509.
6. S. M. Mahajan, A. Qadir, P. M. Valanju, Nuovo Cimento [**B65**]{} (1981) 404; J. Quamar, Ph.D thesis, Quaid-i-Azam University (1984); A. Qadir and J. Quamar,[* Proceedings of the Third Marcel Grossman Meeting on General Relativity,* ]{}ed. Hu Ning[* (*]{}Science Press and North Holland Publishing Co. 1983) 189.
7. M. Sharif, Ph.D thesis, Quaid-i-Azam University (1991).
8. A. Qadir and M Sharif, Physics LettersA[**, 167**]{} (1992) 331.
9. A. Qadir and M Sharif, Nuovo Cimento B, [**107** ]{} (1992) 1071.
10. A. Qadir, NuovoCimento [**112B**]{} (1997) 485.
11. M Sharif, Astrophysics and Space Science [**253**]{} (1997)
12. Martin Rees, R Ruffini, J. A. Wheeler [*Black Holes Gravitational Waves and Cosmology,* ]{}(Gordon and Breech, Science Publishers Inc.)1976.
13. A. Qadir. [*Einstein’s General Theory of Relativity,*]{} (preprint).
14. M. Sharif, to appear in Astrophysics and Space Science.
15. R. Penrose and W. Rindlers,[* Spinors and Spacetime,* ]{}Vol 1, Cambridge University Press 1984.
|
---
abstract: 'We analyse a version of the policy iteration algorithm for the discounted infinite-horizon problem for controlled multidimensional diffusion processes, where both the drift and the diffusion coefficient can be controlled. We prove that, under assumptions on the problem data, the payoffs generated by the algorithm converge monotonically to the value function and an accumulation point of the sequence of policies is an optimal policy. The algorithm is stated and analysed in continuous time and state, with discretisation featuring neither in theorems nor the proofs. A key technical tool used to show that the algorithm is well-defined is the mirror coupling of Lindvall and Rogers.'
address:
- |
Department of Statistics, University of Warwick, and\
The Alan Turing Institute, UK
- |
Department of Mathematics, King’s College London, and\
The Alan Turing Institute, UK
- 'Department of Statistics, University of Warwick, UK'
author:
- 'Saul D. Jacka'
- Aleksandar Mijatović
- Dejan Širaj
date:
-
-
title: Coupling and a generalised Policy Iteration Algorithm in continuous time
---
[^1]
Introduction {#ch3INT}
============
Howard’s policy iteration algorithm (PIA) [@Howard] is a well-known tool for solving control problems for Markov decision processes (see e.g. [@Bertsekas] for a recent survey of approximate policy iteration methods for finite state, discrete time, stochastic dynamic programming problems). The algorithm can be recast for general state-spaces continuous-time control problems, where allowed actions can be chosen from a Polish space. In this paper we investigate the convergence of the PIA for an infinite horizon discounted cost problem in the context of controlled diffusion processes in ${\mathbb{R}}^d$, where the control takes place in an arbitrary compact metric space. The main aim of the paper is to analyse the convergence of a sequence of policies and the corresponding payoff functions produced by the PIA under assumptions that are at least in principle verifiable in terms of the model data.
Our control setting is similar to that of [@Ari], where an ergodic cost criterion was considered. The main differences, beyond the cost criterion, are as follows: (1) we allow the controller to modulate the drift as well as the diffusion coefficient; (2) we consider a generalised version of the PIA where an arbitrary scaling function can be used to simplify the algorithm; (3) we investigate the convergence not only of payoffs but also of policies produced by the PIA; (4) we work with Markov policies that are defined for every $x\in{\mathbb{R}}^d$, not almost everywhere, and obtain a locally uniform convergence of a subsequence to the optimal policy.
This last point in this list is particularly important in our setting, as our aim is to design and analyse an algorithm that can in principle at least be used in to construct an optimal policy. This requirement forces us to work in the context of classical solutions of PDEs, rather than relying on generalised solutions of the Poisson equation in appropriate Sobolev spaces. The latter approach, followed in [@Ari], is based on the fact that there exists a canonical solution to the Poisson equation in $W^{2,p}_{\textrm{loc}}({\mathbb{R}}^d)$, see [@Ari_Borkar]. The analysis of the PIA can than be performed using Krylov’s extension [@Krylov] of Itô’s formula to functions in the Sobolev space $W^{2,p}_{\textrm{loc}}({\mathbb{R}}^d)$.
Our method for solving the Poisson equation in the classical sense is based on the coupling of Lindvall and Rogers [@article6]. This coupling plays a crucial role in the proof of Proposition \[operatorIHM\], guaranteeing that a payoff function for a locally Lipschitz Markov policy is the classical solution of the corresponding Poisson equation. The main technical contribution of the paper is the result in Lemma \[mdc\], which shows that the mirror coupling from [@article6] is successful with very high probability, when the diffusion processes are started sufficiently close together. Interestingly, the condition in [@article6] for the coupling to be successful is not satisfied in our setting in general. The proof of Lemma \[mdc\] is based on a local path-wise comparison of a time-change of the distance between the coupled diffusions and an appropriately chosen Bessel process.
The convergence of the policies and payoffs in the PIA is obtained in several steps. First we show that the PIA always improves the payoff. Then we prove, using a “diagonalisation” argument and an Arzela-Ascoli type result, that a subsequence of the policies produced by the PIA converges locally uniformly to a locally Lipschitz limiting policy. The final stage of the argument shows that this limiting policy is indeed an optimal policy with payoff equal to the pointwise limit of the payoffs produced by the PIA. These steps are detailed in Section \[ch3IHM\] and proved in Section \[sec:Proofs\].
The literature on the PIA for Markov decision processes in various settings is extensive (see e.g. [@Doshi], [@Lerma], [@Hordi], [@Lasser], [@Meyn], [@Parr], [@Mahadevan], [@Rust], [@Santos] and [@Zhu] and the references therein). Our approach is in some sense closest to the continuous time analysis in general state spaces in [@Doshi], where the convergence of the subsequence of the policies is established in the case of finite action space. In [@Zhu] this restriction is removed, but the controlled processes considered do not include diffusions. A recent application of the PIA to impulse control in continuous time is given in [@Erhan].
Finally, as mentioned above, we observe that the PIA can be generalised by multiplying the expression to be minimised by an arbitrary positive scaling function that can depend both on the state and the control action (see below). A choice of the scaling function clearly influences the sequence of policies produced by the algorithm. In particular, in the one-dimensional case, the scaling function can be used to eliminate the second derivative of the payoff from the algorithm. This idea is described in Section \[ch3IHO\]. A numerical example of the PIA is reported in Section \[ch3EXA\]. In this examples at least, the convergence of the PIA is very fast as the algorithm finds an optimal policy in fewer than six iterations.
Multidimensional controlled diffusion processes {#ch3IHM}
===============================================
Let $(A,d_A)$ be a compact metric space of available control actions and, for some $d,m\in{\mathbb{N}}$, let ${\sigma}: {\mathbb{R}}^d \times A \to {\mathbb{R}}^{d \times m}$ and $\mu : {\mathbb{R}}^d \times A \to {\mathbb{R}}^d$ be measurable functions. Let the set ${\mathcal{A}}(x)$ of *admissible policies* at $x\in{\mathbb{R}}^d$ consist of processes $\Pi = (\Pi_t)_{t \geq 0}$ satisfying the following: $\Pi$ is $A$-valued, adapted to a filtration $({\mathcal{F}}_t)_{t\geq0}$ and there exists an $({\mathcal{F}}_t)$-adapted process $X^{\Pi,x} = \big( X^{\Pi,x}_t \big)_{t \geq 0}$ satisfying the SDE $$\begin{aligned}
\label{sdeIHM}
X_t^{\Pi,x} = x + \int_0^t {\sigma}\left( X_s^{\Pi,x}, \Pi_s \right) \mathrm{d}B_s +
\int_0^t \mu \left( X_s^{\Pi,x},\Pi_s \right) \mathrm{d}s, \quad t \geq 0,\end{aligned}$$ where $B = (B_t)_{t \geq 0}$ is an $m$-dimensional $({\mathcal{F}}_t)$-Brownian motion. Note that the filtration $({\mathcal{F}}_t)_{t\geq0}$ (and indeed the entire filtered probability space) depends on the policy $\Pi$ in ${\mathcal{A}}(x)$. Pick measurable functions $\alpha : {\mathbb{R}}^d \times A \to {\mathbb{R}}$ and $f : {\mathbb{R}}^d \times A \to {\mathbb{R}}$. For any $x \in {\mathbb{R}}^d$ and $\Pi \in {\mathcal{A}}(x)$, define the *payoff* of the policy $\Pi$ by $$\begin{aligned}
V_\Pi(x) := {\mathbb{E}}\left( \int_0^{\infty} \mathrm{e}^{-\int_0^t \alpha \left( X^{\Pi,x}_s, \Pi_s \right) \mathrm{d}s} f \left( X^{\Pi,x}_t, \Pi_t \right) \mathrm{d}t \right).\end{aligned}$$ **Control problem.** Construct the *value function* $V$, defined by $$V(x) := \inf_{\Pi \in {\mathcal{A}}(x)} V_\Pi(x), \quad x \in {\mathbb{R}}^d,$$ and, if it exists, an optimal control $\{\Pi^x\in{\mathcal{A}}(x):x\in{\mathbb{R}}^d\}$, satisfying $V(x)=V_{\Pi^x}(x)$.
Note first that the problem is specified completely by the deterministic data ${\sigma}$, $\mu$, $\alpha$ and $f$. In order to define an algorithm that solves this problem, the functions ${\sigma}$, $\mu$, $\alpha$, $f$ are assumed to satisfy Assumption \[ass1IHM\] below throughout this section.
\[ass1IHM\] The functions ${\sigma}$, $\mu$, $\alpha$ and $f$ are bounded, and Lipschitz on compacts in ${\mathbb{R}}^d \times A$, i.e. for every compact set $K \subseteq {\mathbb{R}}^d\times A$ there exists a constant $C_K > 0$ such that $$\label{eq:h_bound_LLiP}
\|h(x,p) - h(y,r)\| \leq C_K \left( \|x-y\|^2 + d_A(p,r)^2 \right)^{\frac{1}{2}}$$ holds for every $(x,p),(y,r) \in K$, and $h \in \{ {\sigma}, \mu, \alpha, f \}$. In addition, $\alpha(x,p)>\epsilon_0>0$ for all $(x,p)\in{\mathbb{R}}^d\times A$, and there exists $\lambda > 0$ such that $$\label{lambda}
\langle {\sigma}(x,p) {\sigma}(x,p)^Tv, v\rangle
\geq \lambda \|v\|^2
\qquad \text{for all} \quad x \in {\mathbb{R}}^d, p \in A, v \in {\mathbb{R}}^d.$$
\[rem:matrix\_norm\] Here, and throughout the paper, $\|\cdot\|$ and $\langle\cdot,\cdot\rangle$ denote the Euclidean norm and inner product respectively. The norm $\|M\| = \sup \left\{ \|Mv\|/\|v\|: v \in {\mathbb{R}}^m \backslash \{0\} \right\}=\sqrt{\lambda_{\max}(M M^T)}$, for a matrix $M \in {\mathbb{R}}^{d \times m}$, is used in for $h=\sigma$, where $\lambda_{\max}(M M^T)$ is the largest eigenvalue of the non-negative definite matrix $MM^T\in{\mathbb{R}}^{d\times d}$ and $M^T\in{\mathbb{R}}^{m \times d}$ denotes the transpose of $M$.
\[rem:UElip\] The uniform ellipticity condition in is the multidimensional analogue of the volatility being bounded away from $0$. Hence, for all $x\in{\mathbb{R}}^d$ and $p\in A$, the smallest eigenvalue of ${\sigma}(x,p) {\sigma}(x,p)^T\in{\mathbb{R}}^{d\times d}$ is at least of size $\lambda$ and, in particular, $m\geq d$ (cf. Remark \[rem:d\_and\_m\] below).
A measurable function $\pi : {\mathbb{R}}^d \to A$ is a *Markov policy* (or synonymously *Markov control*) if for $x \in {\mathbb{R}}^d$ there exists an ${\mathbb{R}}^d$-valued process $X^{\pi,x} = \left( X^{\pi,x}_t \right)_{t \geq 0}$ that satisfies the following SDE: $$\label{sde2IHM}
X_t^{\pi,x} = x + \int_0^t {\sigma}\left( X_s^{\pi,x}, \pi \left( X_s^{\pi,x} \right) \right) \mathrm{d}B_s +
\int_0^t \mu \left( X_s^{\pi,x}, \pi \left( X_s^{\pi,x} \right) \right) \mathrm{d}s, \quad t \geq 0.$$ Let $({\mathcal{F}}_t)_{t\geq0}$ be a filtration with respect to which $(X^{\pi,x},B)$ is $({\mathcal{F}}_t)$-adapted and $B$ is an $({\mathcal{F}}_t)$-Brownian motion. Such a filtration $({\mathcal{F}}_t)_{t\geq0}$ exists by the definition of a solution of SDE , see e.g. [@Karatzas Def. 5.3.1, p. 300]. Then $({\mathcal{F}}_t)_{t\geq0}$ can be taken to be the filtration in the definition of the policy $\pi(X^{\pi,x})\in{\mathcal{A}}(x)$. Moreover, without loss of generality, we may assume that $({\mathcal{F}}_t)_{t\geq0}$ satisfies the usual conditions.
For any function $\pi : {\mathbb{R}}^d \to A$ that is Lipschitz on compacts (i.e. locally Lipschitz) in ${\mathbb{R}}^d$, Assumption \[ass1IHM\] implies (see e.g. [@Borodin p. 45] and the references therein) that the SDE in has a unique, strong non-exploding solution, thus making $\pi$ a Markov policy. The payoff function of a locally Lipschitz Markov policy is a classical solution of a linear PDE, a fact key for to work. To state this formally, recall that $h:{\mathbb{R}}^d\to{\mathbb{R}}^k$ (for any $k\in{\mathbb{N}}$) is $(1/2)$-Hölder continuous on a compact $D\subset {\mathbb{R}}^d$ if $\|h(x')-h(x'')\|\leq K_D \|x'-x''\|^{1/2}$ holds for some constant $K_D>0$ and all $x',x''\in D$. Streamline the notation for Markov policies as follows: $$\label{eq:Markov_payoff}
\begin{array}{l}
V_\pi(\cdot) := V_{\pi(X^{\pi,\cdot})}(\cdot) \quad \text{and}
\quad L_\pi h := \frac{1}{2} \operatorname{Tr}\left( {\sigma}_\pi^T {\mathrm{H}}h \,{\sigma}_\pi \right) + \mu_\pi^T {\nabla}h
\quad \text{for} \quad h \in {\mathcal{C}}^2({\mathbb{R}}^d),\\
{\sigma}_\pi(\cdot) := {\sigma}(\cdot,\pi(\cdot)), \quad \mu_\pi(\cdot) := \mu(\cdot,\pi(\cdot)),
\quad \alpha_\pi(\cdot) := \alpha(\cdot,\pi(\cdot)), \quad f_\pi(\cdot) := f(\cdot,\pi(\cdot)),
\end{array}$$ where ${\mathrm{H}}h$ and ${\nabla}h$ are the Hessian and gradient of $h$, respectively, and $\operatorname{Tr}(M)$ denotes the trace of any matrix $M\in{\mathbb{R}}^{m\times m}$.
\[operatorIHM\] Let Assumption \[ass1IHM\] hold. For a locally Lipschitz Markov policy $\pi$ we have: $V_\pi\in{\mathcal{C}}^2({\mathbb{R}}^d)$ is the unique bounded solution of the Poisson equation $L_\pi V_\pi - \alpha_\pi V_\pi + f_\pi = 0$ and ${\mathrm{H}}V_\pi$ is $(1/2)$-Hölder on compacts in ${\mathbb{R}}^d$.
\[rem:d\_and\_m\] The proof of Proposition \[operatorIHM\], given in Section \[subsec:Proofs\_Multidim\], depends crucially on the coupling in Lindvall and Rogers [@article6] of two $d$-dimensional diffusions via a coupling of the corresponding $d$-dimensional driving Brownian motions (see Lemma \[mdc\] below). Moreover, it is easy to see that the probability of coupling could in general be zero if the dimension $m$ of the Brownian noise is strictly less than $d$. In this case the controlled diffusions in ${\mathbb{R}}^d$, started at distinct points, could remain forever on disjoint submanifolds of positive co-dimension in ${\mathbb{R}}^d$ for any choice of control. In particular, Proposition \[operatorIHM\] fails in the case $m<d$, as demonstrated by Example \[ex:Prof\_fails\] below.
\[ex:Prof\_fails\] Let $m=1$, $d=2$, $A=[-1,1]$, $f(x_1,x_2,a):=|x_1+x_2|+|x_1-x_2|+a^2/2$, $\alpha(x_1,x_2,a)\equiv 1$, ${\sigma}(x_1,x_2,a)\equiv(1, 1)^T$ and $\mu(x_1,x_2,a)=a (1 ,-1)^T$. Then, for a constant policy $\pi_a\equiv a\in A$, the controlled process started at $x\in{\mathbb{R}}^2$ is given by $X^{\pi,x}_t= x+(1, 1)^T B_t + a (1 ,-1)^T t$, for $t\geq0$. In particular, for $a=0$ and any $x=(x_1,x_2)^T$, we have $V_{\pi_0}(x)=|x_1-x_2|+g(x_1+x_2)$, where $g(y):={\mathbb{E}}|y+2 B_{e_1}|$, $y\in{\mathbb{R}}$, and $e_1$ is an exponential random variable with mean $1$, independent of $B$. Since $B_{e_1}$ has a smooth density, it follows that $g$ is also smooth, implying that $V_{\pi_0}$ cannot satisfy the conclusion of Proposition \[operatorIHM\].
\[rem:d\_is\_m\] As we are using the standard weak formulation of the control problem (i.e. the filtered probability space is *not* specified in advance), all that matters for a Markov policy $\pi$ is the law of the controlled process $X^{\pi,\cdot}$, which solves the martingale problem in . Moreover, this law is uniquely determined by the symmetric-matrix valued function $(x,a)\mapsto {\sigma}(x,a){\sigma}(x,a)^T\in{\mathbb{R}}^{d\times d}$ (and of course the drift $(x,a)\mapsto\mu(x,a)$). Since the symmetric square root of ${\sigma}(x,a){\sigma}(x,a)^T$ in ${\mathbb{R}}^{d\times d}$ satisfies Assumption \[ass1IHM\], in the remainder of the paper we assume, without loss of generality, that the noise and the controlled process are of the same dimension, i.e. $d=m$ (cf. Remark \[rem:UElip\]).
For any locally Lipschitz Markov policy $\pi$, the process $X^{\pi,\cdot}$ is strong Markov. Hence [@Dynkin Thm. 1.7] implies that $V_\pi$ is in the domain $\mathcal{D}(A_\pi)$ of the generator $A_\pi$ of $X^{\pi,\cdot}$ and that the Poisson equation $A_\pi V_\pi - \alpha_\pi V_\pi + f_\pi = 0$ holds. Recall that for a bounded continuous $g:{\mathbb{R}}^d\to{\mathbb{R}}$ in $\mathcal{D}(A_\pi)$ the limit $(A_\pi g)(x):= \lim_{t\to0} ({\mathbb{E}}g(X^{\pi,x}_t) - g(x))/t$ exists for all $x\in{\mathbb{R}}^d$. Furthermore, if $g$ is also in ${\mathcal{C}}^2({\mathbb{R}}^d)$, it is known that $A_\pi g = L_\pi g$. However, [@Dynkin Thm. 1.7] does not imply that $V_\pi$ is in ${\mathcal{C}}^2({\mathbb{R}}^d)$. The PDE in Proposition \[operatorIHM\], key for to work, is established via the coupling argument in Section \[subsec:Proofs\_Multidim\].
If a policy $\pi:{\mathbb{R}}^d\to A$ is constant (i.e. $\pi\equiv p\in A$), write ${\sigma}_p$, $\mu_p$, $\alpha_p$, $f_p$, $L_p$ and $V_p$ instead of ${\sigma}_{\pi}$, $\mu_{\pi}$, $\alpha_{\pi}$, $f_{\pi}$, $L_{\pi}$ and $V_\pi$, respectively. Let $S:{\mathbb{R}}^d\times A\to(0,\infty)$ be a continuous function and, for any $p\in A$, denote $S_p(x):=S(x,p)$. Under Assumption \[ass1IHM\], the function $p \mapsto S_p(x)(L_p h(x) - \alpha_p(x) h(x) + f_p(x))$, $p \in A$, is continuous for any $x \in {\mathbb{R}}^d$ and $h\in {\mathcal{C}}^2({\mathbb{R}}^d)$. Since $A$ is compact, there exists $I_h(x)\in A$, which minimises this function.
\[ass2\_and\_a\_half\_IHM\] For any $h\in {\mathcal{C}}^2({\mathbb{R}}^d)$, the function $I_h:{\mathbb{R}}^d\to A$ can be chosen to be locally Lipschtiz on ${\mathbb{R}}^d$. The continuous scaling function $S$ satisfies $\epsilon_S<S<M_S$ for some $\epsilon_S,M_S\in (0,\infty)$.
**Generalised Policy Iteration Algorithm**
\[rem:Well\_defined\_alg\] By Assumption \[ass2\_and\_a\_half\_IHM\], the policy $\pi_{n+1}$ defined in is locally Lipschitz for all $0\leq n\leq N-1$. Hence, by Proposition \[operatorIHM\], the algorithm is well defined.
In the classical case of we take $S\equiv1$. A non-trivial scaling function $S$, which plays an important role in the one-dimensional context (see Section \[ch3IHO\] below), makes the algorithm into a generalised Policy Iteration Algorithm. Thm \[decreasingIHM\] requires only the positivity and continuity of $S$. The uniform bounds on $S$ in Assumption \[ass2\_and\_a\_half\_IHM\] are used only in the proof of Thm \[verificationIHM\].
always leads to an improved payoff (Theorem \[decreasingIHM\] is proved in Section \[subsec:Proofs\_Multidim\] below):
\[decreasingIHM\] Under Assumptions \[ass1IHM\]–\[ass2\_and\_a\_half\_IHM\], the inequality $V_{\pi_{n+1}} \leq V_{\pi_{n}}$ holds on ${\mathbb{R}}^d$ for all $n \in \{0,\ldots,N-1\}$.
The sequence $\{ V_{\pi_N} \}_{N \in {\mathbb{N}}}$, obtained by running from a given policy $\pi_0$ for each $N\in{\mathbb{N}}$, is non-increasing and bounded. Hence we may define $$\label{eq:V_lim_def}
V_{\lim}(x) := \lim_{N \to \infty}V_{\pi_N}(x), \quad x \in {\mathbb{R}}^d.$$ However, the sequence of the corresponding Markov policies $\{\pi_N\}_{N\in{\mathbb{N}}}$ need not converge and, even if it did, the limit may be discontinuous and hence not necessarily a Markov policy.
If the algorithm stops before $N$, i.e.$\pi_{n+1} = \pi_n$ for some $n<N$, then clearly $V_{\pi_N} = V_{\pi_n}$ and $\pi_N = \pi_n$. As this holds for any $N>n$, with started at a given $\pi_0$, we may proceed directly to the verification lemma (Theorem \[verificationIHM\] below) to conclude that $V_{\pi_n}$ is the value function with an optimal policy $\pi_n$.
Controlling the convergence of the policies requires the following additional assumptions. Introduce the set ${\mathcal{S}}_{B,K}:=\{ h\in {\mathcal{C}}^2({\mathbb{R}}^d): \|{\nabla}h(x)\|<B^1, \|{\mathrm{H}}h(x)\|<B^2\text{ for }x\in D_K\}$, where $D_K:=\{x\in{\mathbb{R}}^d:\|x\|\leq K\}$ is a ball of radius $K>0$ and $B:=(B^1,B^2)\in(0,\infty)^2$ are constants.
\[ass2IHM\] For any $K>0$, there exist constants $B_K$ and $C_K$ satisfying the following: if $h\in{\mathcal{S}}_{B_K,K}$, then $I_h$ (defined in Assumption \[ass2\_and\_a\_half\_IHM\]) satisfies $d(I_h(x),I_h(y))\leq C_K \|x-y\|$ for all $x,y\in D_K$.
\[ass3IHM\] For any $K>0$, let $B_K,C_K$ be as in Assumption \[ass2IHM\]. Then for a locally Lipschitz Markov policy $\pi:{\mathbb{R}}^d\to A$, such that $d(\pi(x),\pi(y))\leq C_K \|x-y\|$, $x,y\in D_K$, the following holds: $V_\pi\in{\mathcal{S}}_{B_K,K}$.
Non-trivial problem data that satisfy Assumptions \[ass1IHM\]–\[ass3IHM\] are described in Section \[ch3EXA\] below. It is precisely these types of examples that motivated the form Assumptions \[ass2\_and\_a\_half\_IHM\]–\[ass3IHM\] take. Assumption \[ass1IHM\] is standard and Assumptions \[ass2\_and\_a\_half\_IHM\]–\[ass2IHM\] concern only the deterministic data specifying the problem. Assumption \[ass3IHM\] essentially states that $\|{\mathrm{H}}V_\pi\|$ has a prescribed bound on the ball $D_K$ if the coefficients of the PDE in Proposition \[operatorIHM\] have a prescribed Lipschitz constant. Schauder’s boundary estimates for elliptic PDEs [@FriedmanParabolic p. 86] suggest that this requirement is both natural and feasible. In fact, Assumption \[ass3IHM\] may follow from assumptions of the type \[ass1IHM\]–\[ass2IHM\] on the problem data. This is left for future research.
Proposition \[subsequenceIHM\] and Theorems \[limitsIHM\] and \[verificationIHM\], proved in Section \[subsec:Proofs\_Multidim\] below, show that converges.
\[subsequenceIHM\] Let Assumptions \[ass1IHM\]–\[ass3IHM\] hold. Then there exists a subsequence of $\{ \pi_{N} \}_{N \in {\mathbb{N}}}$ that converges uniformly on every compact subset of ${\mathbb{R}}^d$ to a locally Lipschitz Markov policy.
Let $\pi_{\lim}:{\mathbb{R}}^d\to A$ denote a locally Lipschitz Markov policy that is a locally uniform limit of a subsequence of $\{ \pi_{N} \}_{N \in {\mathbb{N}}}$. By Propositions \[operatorIHM\], $V_{\pi_{\lim}}$ is a well-defined function in ${\mathcal{C}}^2({\mathbb{R}}^d)$ that solves the corresponding Poisson equation. However, since $\pi_{\lim}$ clearly depends on its defining subsequence, so may $V_{\pi_{\lim}}$. Furthermore, $V_{\lim}$ may depend on the choice of $\pi_0$ in . But this is not so, since $V_{\lim}$ equals both the value function $V$ and the payoff for the policy $\pi_{\lim}$.
\[limitsIHM\] Under Assumptions \[ass1IHM\]–\[ass3IHM\], the equality $V_{\lim} = V_{\pi_{\lim}}$ holds on ${\mathbb{R}}^d$ for a policy $\pi_{\lim}$.
\[verificationIHM\] Let Assumptions \[ass1IHM\]–\[ass3IHM\] hold. Then for every $x \in {\mathbb{R}}^d$ and $\Pi \in {\mathcal{A}}(x)$ the inequality $V_{\lim}(x) \leq V_{\Pi}(x)$ holds. Hence $V_{\lim}$ equals the value function $V$, does not depend on the choice of $\pi_0$ in and $\pi_{\lim}$ is an optimal locally Lipschitz Markov policy for the control problem.
The key technical issue in the proof of Theorem \[verificationIHM\] is that the policies in the convergent subsequence constructed in the proof of Proposition \[subsequenceIHM\] are not improvements of their predecessors (cf. ). The idea of the proof is to work with a convergent subsequence of the pairs of policies $\{(\pi_N,\pi_{N+1})\}_{N\in{\mathbb{N}}}$, where $\{\pi_N\}_{N\in{\mathbb{N}}}$ is produced by the (see Section \[subsec:Proofs\_Multidim\] for details).
The one-dimensional case {#ch3IHO}
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There are two reason for considering the one-dimensional control problem in its own right.\
(A) The canonical choice for the scaling function $S:=1/{\sigma}^2$ simplifies the to $$\label{piaIHO}
\pi_{n+1}(x) \in {\mathop{\rm argmin}\limits}_{p \in A}
\left\{ (\mu(x,p) V_{\pi_{n}}'(x) - \alpha(x,p)V_{\pi_{n}}(x) + f(x,p))/{\sigma}^2(x,p) \right\},
$$ by removing the second derivative of the payoff function $V_{\pi_{n}}$ from the minimisation procedure. This reduction appears to make the numerical implementation of the converge extremely fast: in the example in Section \[subsec:Numerics\] below the optimal payoff and policy are obtained in fewer than half a dozen iterations.\
(B) It is natural to control the process $X^{\Pi,x}$ only up to its first exit from an interval $(a,b)$, where $a,b \in [-\infty,\infty]$, and generalise the payoff as follows: $$\begin{aligned}
&
V_\Pi(x) := {\mathbb{E}}\left( \int_0^{\tau_a^b( X^{\Pi,x})} \mathrm{e}^{-\int_0^t \alpha_{\Pi_s}(X^{\Pi,x}_s)
\mathrm{d}s} f_{\Pi_s}(X^{\Pi,x}_t) \mathrm{d}t
+ \mathrm{e}^{-\int_0^{\tau_a^b( X^{\Pi,x})} \alpha_{\Pi_t}(X^{\Pi,x}_t) \mathrm{d}t} g(X^{\Pi,x}_{\tau_a^b(X^{\Pi,x})})
\right).\end{aligned}$$ Here $\mu,{\sigma},\alpha,f : (a,b) \times A \to {\mathbb{R}}$ are measurable with the same notational convention as in Section \[ch3IHM\] ($f_p(x)=f(x,p)$ *etc.*). Furthermore, $\Pi \in {\mathcal{A}}(x)$ if $X^{\Pi,x}$ follows SDE on the stochastic interval $[0,\tau_a^b(X^{\Pi,x}))$, where $\tau_a^b(X^{\Pi,x}):= \inf \{ t \geq 0;\; X^{\Pi,x}_t \in\{a,b\} \}$ ($\inf \emptyset = \infty$), and $X^{\Pi,x}_{\tau_a^b(X^{\Pi,x})}=X^{\Pi,x}_t$ for $\tau_a^b(X^{\Pi,x})\leq t$ (i.e. $\Pi_t$, $t\in[\tau_a^b(X^{\Pi,x}),\infty)$, are irrelevant for $V_\Pi(x)$). Pick an arbitrary function $g : \{a,b\} \cap {\mathbb{R}}\to {\mathbb{R}}$ and set the control problem as in Section \[ch3IHM\] with ${\mathbb{R}}^d$ substituted by $(a,b)$.
\[rem:Improtant\] In Assumptions \[ass1IHM\]–\[ass3IHM\] we substitute ${\mathbb{R}}^d$ with $(a,b)$. In particular, inequality in Assumption \[ass1IHM\] takes the form $
{\sigma}^2(x,p)\geq \lambda
$ for all $x \in(a,b)$, $p \in A$. Assumption \[ass1IHM\] hence implies the requirement on the scaling function $S=1/{\sigma}^2$ in Assumption \[ass2\_and\_a\_half\_IHM\]. In Assumptions \[ass2IHM\]–\[ass3IHM\], the family of closed balls $(D_K)_{K>0}$ in ${\mathbb{R}}^d$ is substituted with a family of compact intervals $(D_K)_{K>0}$ in $(a,b)$, such that $\cup_{K>0} D_K = (a,b)$ and, if $K'<K$, $D_{K'}$ is contained in the interior of $D_K$.
On the event $\{\tau_a^b(X^{\Pi,x})=\infty\}$ we take $g(X^{\Pi,x}_{\tau_a^b(X^{\Pi,x})})/\exp(\int_0^{\tau_a^b( X^{\Pi,x})} \alpha_{\Pi_t}(X^{\Pi,x}_t)\mathrm{d}t)=0$, since by Assumption \[ass1IHM\] the integral is infinite. Note also that on this event $X^{\Pi,x}_{\tau_a^b(X^{\Pi,x})}$ may not be defined. Moreover, if $\{a,b\} \cap {\mathbb{R}}=\emptyset$, then by Assumption \[ass1IHM\] we have $\tau_a^b(X^{\Pi,x})=\infty$ a.s.
A measurable function $\pi : (a,b) \to A$ is a *Markov policy* if for every $x \in (a,b)$ there exists a process $X^{\pi,x} = \left( X^{\pi,x}_t \right)_{t \geq 0}$ satisfying SDE on the stochastic interval $[0,\tau_a^b( X^{\pi,x}))$ and $X_t^{\pi,x} = X_{\tau_a^b( X^{\pi,x})}^{\pi,x}$ if $\tau_a^b( X^{\pi,x} ) \leq t < \infty$. If $\pi$ is a Markov policy, then $\pi(X^{\pi,x}) := \left( \pi ( X^{\pi,x}_t ) \right)_{t \geq 0} \in {\mathcal{A}}(x)$ for every $x \in (a,b)$, where we pick arbitrary elements in $A$ for the values $\pi(a)$ and $\pi(b)$ if $a > -\infty$ and $b < \infty$, respectively. We use analogous notation to that in , e.g. $L_\pi h := \frac{1}{2}{\sigma}_\pi^2 h'' + \mu_\pi h'$ for any $h\in {\mathcal{C}}^2((a,b))$.
Any locally Lipschitz function $\pi : (a,b) \to A$ is a Markov policy, since the Engelbert-Schmidt conditions for the existence and uniqueness of the solution of the corresponding SDE (see e.g. [@Karatzas Sec. 5.5]) are satisfied by Assumption \[ass1IHM\]. By substituting the state space ${\mathbb{R}}^d$ with the interval $(a,b)$ in Proposition \[operatorIHM\], Theorem \[decreasingIHM\], Proposition \[subsequenceIHM\] and Theorems \[limitsIHM\] and \[verificationIHM\] of Section \[ch3IHM\], we obtain the results of the present section, which thus solve the control problem in the one-dimensional case. In the interest of brevity, we omit their statement. We stress that the main difference lies in the fact that the proofs, in Sections \[subsec:aux\_one\_dim\] and \[subsec:Proofs\_OneDim\] below, rely on the theory of ODEs and scalar SDEs. In particular, we need to prove that the payoff has a continuous extension to a finite boundary point of the state space.
Examples {#ch3EXA}
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Data satisfying Assumptions \[ass1IHM\]–\[ass3IHM\] {#subsec:Data_theory}
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We now describe a class of models that provably satisfies Assumptions \[ass1IHM\]–\[ass3IHM\]. The main aim in the present section is not to be exhaustive, but merely to demonstrate that the form (particularly) of Assumptions \[ass2IHM\]–\[ass3IHM\] is natural in the context of control problems considered here. The example we give is in dimension one. But it is clear from the construction below that it can easily be generalised.
Let $A:=[-a,a]$, for some constant $a>0$, and ${\sigma},\mu, f, \alpha:{\mathbb{R}}\times [-a,a]\to{\mathbb{R}}$ be given by $$\label{eq:Example_Class}
{\sigma}(x,p):={\sigma}_1(x),\quad \mu(x,p):=\mu_1(x)+p \mu_2,\quad f(x,p):= f_1(x) + f_2(p),\quad
\alpha(x,p)\equiv \alpha_0,$$ where ${\sigma}_1,\mu_1,f_1\in {\mathcal{C}}^1({\mathbb{R}})$, $f_2\in{\mathcal{C}}^2((-a,a))$ is convex and symmetric (i.e. $f_2(p)=f_2(-p)$ for all $p\in A$) and $\mu_2$ and $\alpha_0$ are constants. For any $h\in\{{\sigma}_1,\mu_1,f_1,f_2\}$ (resp. $h'\in\{{\sigma}_1',\mu_1',f_1',f_2'\}$) let the positive constant $C_h$ (resp. $C_h'$) satisfy $|h|\leq C_h$ (resp. $|h'|\leq C_h'$). In particular, we assume that the derivatives of ${\sigma}_1,\mu_1,f_1,f_2$ are bounded. Moreover we may (and do) take $C_{f_2}':=f_2'(a)$. Assume also that $\alpha_0>0$ and ${\sigma}_1^2>\lambda>0$ (so that As \[ass1IHM\] is satisfied) and the scaling function $S\equiv1$. Then the following proposition holds.
\[dataIHO\] Assume $\alpha_0> C_{\mu_1}'+ |\mu_2|(2+C_{f_1}'/C_{f_2}')$ and $|\mu_2|B^2_\infty<L_{f_2}:=\inf_{p\in A} f_2''(p)$, where $$B^2_\infty:=\left[2(C_{f_1}+C_{f_2})+(C_{\mu_1}+ a|\mu_2|)B^1_\infty\right]/\lambda\quad\text{and}\quad
B^1_\infty:=(C_{f_1}'+C_{f_2}')/(\alpha_0- C_{\mu_1}'- |\mu_2|),$$ then Assumptions \[ass1IHM\]–\[ass3IHM\] hold. Moreover, in Assumptions \[ass2IHM\]–\[ass3IHM\] we have $B_K=(B^1_\infty,B^2_\infty)$ and $C_K=1$ for any $K>0$.
It is clear that the assumptions in Proposition \[dataIHO\] define a non-empty subclass of models . Moreover, these assumptions are much stronger than what is required by our general Assumptions \[ass2IHM\]–\[ass3IHM\] since the proposition yields global (rather than local) bounds on the derivatives of the payoff functions and the Lipschitz coefficients of the policies arising in .
Pick $h\in{\mathcal{C}}^2({\mathbb{R}})$, such that $|h'(x)|<B_\infty^1$ and $|h''(x)|<B_\infty^2$ for all $x\in{\mathbb{R}}$. Then the function $I_h$ in As \[ass2IHM\] satisfies $I_h(x)={\mathop{\rm argmin}\limits}_{p\in A}\{p\mu_2 h'(x)+f_2(p)\}$. By assumption we have $$|\mu_2 h'(x)|\leq |\mu_2| B_\infty^1<C'_{f_2}=f_2'(a),\qquad\text{implying}\quad
I_h(x)=(f_2')^{-1}(-\mu_2 h'(x))\quad \forall x\in{\mathbb{R}}.$$ Differentiate $I_h$ to obtain $|I_h'(x)|\leq |\mu_2|B_\infty^2/L_{f_2}<1$, $x\in{\mathbb{R}}$, and note that Assumptions \[ass2\_and\_a\_half\_IHM\]–\[ass2IHM\] follow.
We now establish As \[ass3IHM\]. The idea is to start with any policy $\pi:{\mathbb{R}}\to A$ in ${\mathcal{C}}^2({\mathbb{R}})$, such that its derivative satisfies $|\pi'|\leq 1$ on all of ${\mathbb{R}}$ (e.g. a constant policy), and apply stochastic flow of diffeomeorphisms [@Protter Sec. V.10] to deduce the necessary regularity of the payoff function $V_\pi$. In the notation from , we have $|\mu_\pi'|\leq C_{\mu_1}'+ |\mu_2|$ and $|{\sigma}_\pi'|=|{\sigma}_1'|\leq C_{{\sigma}_1}'$. Hence, for each $x\in{\mathbb{R}}$, the stochastic exponential $Y=(Y_t)_{t\in{\mathbb{R}}_+}$, given by $$Y_t = 1+\int_0^t \mu_\pi'(X^{\pi,x}_s)Y_s\mathrm{d}s + \int_0^t{\sigma}_\pi'(X^{\pi,x}_s)Y_s\mathrm{d}W_s,$$ exists (we suppress the dependence on $x$ (and $\pi$) from the notation). Since the coefficients SDE are in ${\mathcal{C}}^1({\mathbb{R}})$ with bounded and locally Lipschitz first derivative, [@Protter Sec. V.10, Thm 49] implies that the flow of controlled processes $\{X^{\pi,x}\}_{x\in{\mathbb{R}}}$ may be constructed on the single probability space so that it is smooth in the initial condition $x$ with $\frac{\partial }{\partial x}X^{\pi,x}=Y$. The upshot here is that, by the argument in the proof of [@Fournie_et_al Prop. 3.2], we obtain a stochastic representation for the derivative $\frac{\partial}{\partial x} {\mathbb{E}}f_\pi(X^{\pi,x}_t)= {\mathbb{E}}[ Y_t f_\pi'(X^{\pi,x}_t)]$ for every $t\in{\mathbb{R}}_+$. Since $Y_t=M_t \exp \int_0^t\mu_\pi'(X^{\pi,x}_s)\mathrm{d}s$, where the stochastic exponential $M=\mathcal{E}(\int_0^\cdot{\sigma}_\pi'(X^{\pi,x}_s)\mathrm{d}W_s)$ is a true martingale by Novikov’s condition, the following inequality holds: ${\mathbb{E}}Y_t \leq \exp( t(C_{\mu_1}'+ |\mu_2|))$. Since $\alpha_0>C_{\mu_1}'+ |\mu_2|$ by assumption and the inequality $|f_\pi'|\leq C_{f_1}'+ C_{f_2}'$ holds, we have $|{\mathbb{E}}\int_0^\infty \mathrm{e}^{-\alpha_0 s} Y_s f_\pi'(X^{\pi,x}_s) \mathrm{d}s|< B_\infty^1$ for all $x\in{\mathbb{R}}$.
Recall that $V_\pi(x)={\mathbb{E}}\int_0^\infty \mathrm{e}^{-\alpha_0 s} f_\pi(X^{\pi,x}_s) \mathrm{d}s$. By [@Protter Sec. V.8, Thm 43], the family of random variables indexed by $\delta\in(0,1)$, $$\frac{1}{\delta} \int_0^\infty \mathrm{e}^{-\alpha_0 s} |f_\pi(X^{\pi,x+\delta}_s)-f_\pi(X^{\pi,x}_s)| \mathrm{d}s
\leq
\frac{1}{\delta} (C_{f_1}'+ C_{f_2}')\int_0^\infty \mathrm{e}^{-\alpha_0 s} |X^{\pi,x+\delta}_s-X^{\pi,x}_s| \mathrm{d}s,$$ is uniformly integrable. Hence $\lim_{\delta\to0} (V_\pi(x+\delta)-V_\pi(x))/\delta$ takes the form $$V_\pi'(x)={\mathbb{E}}\int_0^\infty \mathrm{e}^{-\alpha_0 s} Y_s f_\pi'(X^{\pi,x}_s) \mathrm{d}s,\quad\text{implying }\quad
|V_\pi'(x)|< B_\infty^1 \quad\text{for all $x\in{\mathbb{R}}$}.$$ This inequality, the fact ${\sigma}_1^2>\lambda$ and Proposition \[operatorIHM\] imply $|V_\pi''|<B_\infty^2$, concluding the proof.
The process $Y$ in the proof of Proposition \[dataIHO\] exists in the multidimensional setting, see [@Protter Sec. V.10, Thm 49]. Hence the same argument works in higher dimensions if we can deduce a bound on the Hessian of the payoff function from the PDE in Proposition \[operatorIHM\].
Numerical examples {#subsec:Numerics}
------------------
Consider the one-dimensional control problem: $A = [-1,1]$, $a = -10$, $b = 10$, $g(a) := a^2$, $g(b) := b^2$, ${\sigma}(x,p) := 1$, $\mu(x,p) := p$, $\alpha(x,p) := 1$ and $f(x,p) := x^2 + p^2$, which is in the class discussed in Section \[subsec:Data\_theory\]. Explicitly, we seek to compute $\inf_{\Pi \in {\mathcal{A}}(x)} V_\Pi(x)$ for every $x \in (-10,10)$, where the payoff $V_\Pi(x)$ of a policy $\Pi$ is defined in Section \[ch3IHO\].
We implemented , with the main step given by , in Matlab. The payoff function at each step is obtained as the solution to the differential equation from Proposition \[operatorIHM\] with the boundary conditions given by the function $g$. The new policy at each step can be calculated explicitly (cf. the proof of Proposition \[dataIHO\] above). Figures \[fig1\] and \[fig2\] graph the payoff functions and the policies (colour coded). The initial policy $\pi_0\equiv 1$ and its payoff correspond to the blue graphs.
The graphs suggest that convergence effectively occurs in just a few steps. Figures \[fig3\] and \[fig4\], containing the graphs of the differences of the consecutive payoffs and policies on the logarithmic scale, confirm this. In Figures \[fig1\] and \[fig2\] it seems that fewer graphs are presented than is stated in the caption. The reason for this is that the final few graphs coincide. Moreover, the policies only differ on a subinterval $(-2,2)$, because outside of it they coincide as it is optimal to chose one of the boundary points of $A=[-1,1]$. Finally, there is no numerical indication that the sequence of policies have more than one accumulation point as they appear to converge very fast indeed.
Proofs {#sec:Proofs}
======
Auxiliary results - the multidimensional case
---------------------------------------------
### The reflection coupling of Lindvall and Rogers [@article6] and the continuity of the payoff $V_\pi$
We now establish the continuity of the payoff function for a locally Lipschitz Markov policy $\pi$ under Assumption \[ass1IHM\]. The reflection coupling of Lindvall and Rogers [@article6] plays a crucial role in this. In fact, the continuity of $V_\pi$ hinges on the following property of the coupling in [@article6]: copies of $X^{\pi,x}$ and $X^{\pi,x'}$, started very close to each other, will meet with high probability before moving apart by a certain distance greater than $\|x-x'\|$ (see Lemma \[mdc\] below).
We first show that the coupling from [@article6] can be applied to the diffusion $X^{\pi,\cdot}$. As explained in Remark \[rem:d\_is\_m\], we may (and do) assume that the dimension of the noise and the controlled process are equal, i.e. $d=m$. By Assumption \[ass1IHM\] above ${\sigma}_\pi$ and $\mu_\pi$ are bounded and hence [@article6 As. (12)(ii)] holds. Inequality in Assumption \[ass1IHM\] implies that $\lambda_{\max}({\sigma}_\pi^{-1}{\sigma}_\pi^{-T})\leq 1/\lambda$. Hence, by Remark \[rem:matrix\_norm\], we have $\|{\sigma}_\pi^{-1}\|\leq 1/\sqrt{\lambda}$ and [@article6 As. (12)(ii)] also holds. The assumptions in [@article6 (12)(i)] requires that ${\sigma}_\pi$ and $\mu_\pi$ are globally Lipschitz. But this assumption is only used in [@article6] as a guarantee that the corresponding SDE has a unique strong solution, which is the case in our setting under the locally Lipschitz condition in Assumption \[ass1IHM\]. Hence, for any $x,x'\in{\mathbb{R}}^d$, the coupling from [@article6] can be applied to construct the process $(X^{\pi,x},X^{\pi,x'})$ so that $X^{\pi,x}$ follows SDE and $X^{\pi,x'}$ satisfies $$X^{\pi,x'}_t = x' + \int_0^t \mu_\pi \left( X^{\pi,x'}_s \right) \mathrm{d}s
+ \int_0^t {\sigma}_\pi \left( X^{\pi,x'}_s \right) H_s\mathrm{d}B_s,\qquad\text{for $t\in[0,\rho_0(Y))$,}$$ where $\rho_0(Y):=\inf\{t\geq 0: \|Y_t\|=0\}$ ($\inf \emptyset := \infty$) is the coupling time, $Y:=X^{\pi,x}-X^{\pi,x'}$, and $$\label{ref:EQn_H_u_t}
H_t := I - 2u_t u_t^T,\qquad\text{defined via}\quad
u_t := \frac{{\sigma}_\pi^{-1} \left( X^{\pi,x'}_t \right) Y_t}{\left\| {\sigma}_\pi^{-1} \left( X^{\pi,x'}_t \right) Y_t \right\|}
\qquad\text{for $t\in[0,\rho_0(Y))$,}$$ is the reflection in ${\mathbb{R}}^d$ about the hyperplane orthogonal to the unit vector $u_t$. Moreover, we have $ X_t^{\pi,x'}=X^{\pi,x}_t$ for all $t \in[\rho_0(Y),\infty)$. Note also that $H_t\in O(d)$ is an orthogonal matrix for $t\in[0,\rho_0(Y))$ and the process $B'=(B'_t)_{t\in{\mathbb{R}}_+}$, given by $B'_t:=\int_0^t ({\mathbb{I}}_{\left\{ s <\rho_0(Y) \right\}} H_t +{\mathbb{I}}_{\left\{ s\geq \rho_0(Y) \right\}} I) \mathrm{d}B_s$, is a Brownian motion by the Lévy characterisation theorem. Hence $X^{\pi,x'}$ satisfies the SDE $\mathrm{d}X^{\pi,x'}_t = {\sigma}_\pi(X^{\pi,x'}_t)\mathrm{d} B'_t + \mu_\pi(X^{\pi,x'}_t)\mathrm{d} t$ with $X^{\pi,x'}_0=x'$ (see [@article6 Sec. 3] for more details).
\[mdc\] Fix a locally Lipschitz Markov policy $\pi:{\mathbb{R}}^d\to A$ and $x\in{\mathbb{R}}^d$. Then for every $\epsilon\in(0,1)$ there exist $\bar \varphi\in(0,1]$ with the property: $\forall \varphi\in(0,\bar\varphi)$ $\exists \varphi'\in(0,\varphi)$ such that ${\mathbb{P}}(\rho_\varphi(Y)<\rho_0(Y))<\epsilon$ if $\|x-x'\|<\varphi'$, where $\rho_c(Y) := \inf \left\{ t \geq 0;\; \|Y_t\| = c \right\}$ ($\inf \emptyset = \infty$) for any $c>0$.
Note that the main assumption in [@article6 Thm. 1] is not satisfied in Lemma \[mdc\], as we have no assumption on the global variability of ${\sigma}_\pi$. Hence the coupling $(X^{\pi,x},X^{\pi,x'})$ is not necessarily successful even if the starting points $x$ and $x'$ are very close to each other, i.e. possibly ${\mathbb{P}}(\rho_0(Y)<\infty)<1$ even if $\|Y_0\|=\|x-x'\|$ is very close to zero. However, by Lemma \[mdc\], the coupling will occur with probability at least $1-\epsilon$ before the diffusions are more than $\bar \varphi=\bar \varphi(\epsilon)$ away from each other, implying the continuity of $V_\pi$ (cf. Lemma \[conIHM\] and Remark \[rem:No\_coupling\] below).
Let $\bar{S} := \|Y\|^2$, $\delta:={\sigma}_\pi ( X^{\pi,x})-{\sigma}_\pi (X^{\pi,x'})$ and $\beta := \mu_\pi ( X^{\pi,x} ) - \mu_\pi ( X^{\pi,x'})$. Define $$\label{eq:notation_alpha_beta}
\alpha_t := {\sigma}_\pi ( X^{\pi,x}_t)-{\sigma}_\pi (X^{\pi,x'}_t)H_t\qquad\text{and}\qquad
v_t := Y_t/\|Y_t\|\qquad\text{for $t\in[0,\rho_0(Y))$.}$$ In this proof $x\in{\mathbb{R}}^d$ is fixed and $x'\in{\mathbb{R}}^d$ is arbitrary in the ball of radius one centred at $x$. Recall that ${\nabla}h(z)=2z$ and ${\mathrm{H}}h(z) = 2 I$ for $h(z):=\|z\|^2$, $z\in{\mathbb{R}}^d$, and apply Itô’s lemma to $\bar S$: $$\begin{aligned}
\label{Sbar}
\bar{S}_t & = \|x-x'\|^2 + \int_0^t 2\sqrt{\bar{S}_s} v_s^T \alpha_s \mathrm{d}B_s
+ \int_0^t \left( 2\sqrt{\bar{S}_s} v_s^T \beta_s + \operatorname{Tr}\left(\alpha_s\alpha_s^T\right) \right) \mathrm{d}s,
\quad\text{$t\in[0,\rho_0(Y))$.}\end{aligned}$$
Our task is to study the behaviour of $\bar S$ when started very close to zero. To do this, we first establish the facts in and below, which in turn allow us to apply time-change and coupling techniques to prove the lemma. We start by proving the following: $$\label{eqtrace}
0\leq \operatorname{Tr}\left( \alpha_t \alpha_t^T \right) - \left\| v_t^T \alpha_t \right\|^2
= \operatorname{Tr}\left( \delta_t \delta_t^T \right) - \left\| v_t^T \delta_t \right\|^2 \leq M_x^2 \|Y_t\|^2 \quad\text{for $t \in [0,\rho_0(Y)\wedge \rho_1(Y))$,}$$ where $M_x>1$ is a Lipschitz constant for ${\sigma}_\pi$ and $\mu_\pi$ in the ball around $x$ of radius one. The first inequality in follows since the trace is the sum of the eigenvalues of $\alpha_t \alpha_t^T$, which are all non-negative, while $\left\| v_t^T \alpha_t \right\|^2$ is at most the largest eigenvalue. The second inequality follows since ${\sigma}_\pi$ is Lipschitz on any ball around $x$ and $\|Y_t\|<1$ for $t< \rho_1(Y)$. To establish the equality in note that, as $\| v_t^T A\|^2=\operatorname{Tr}(AA^Tv_tv_t^T)=\operatorname{Tr}(v_tv_t^TAA^T)$ for any $A\in {\mathbb{R}}^{d\times d}$, we have $$\operatorname{Tr}( \alpha_t \alpha_t^T ) - \| v_t^T \alpha_t \|^2 - (\operatorname{Tr}( \delta_t \delta_t^T ) - \| v_t^T \delta_t \|^2)=
\operatorname{Tr}((I-v_tv_t^T)(\alpha_t \alpha_t^T-\delta_t \delta_t^T)).$$ Recall that $H_t^{-1}=H_t^T=H_t$. We therefore find $$\begin{aligned}
\nonumber
\alpha_t \alpha_t^T-\delta_t \delta_t^T & = &
{\sigma}_\pi(X^{\pi,x'}_t)(I - H_t) {\sigma}_\pi( X^{\pi,x}_t)^T + {\sigma}_\pi( X^{\pi,x}_t) (I - H_t) {\sigma}_\pi (X^{\pi,x'}_t)^T \\
& = & 2 \left(v_t u_t^T{\sigma}_\pi( X^{\pi,x}_t)^T+ {\sigma}_\pi( X^{\pi,x}_t)u_t v_t^T\right)\|Y_t\|/ \| {\sigma}_\pi (X^{\pi,x'}_t)^{-1} Y_t \|,
\label{eq:diff_conj}\end{aligned}$$ where the second equality follows by definition and identity $v_t = Y_t/\|Y_t\|$. Hence follows.
Since $\operatorname{Tr}(v_t u_t^T{\sigma}_\pi( X^{\pi,x}_t)^T)=\langle{\sigma}_\pi( X^{\pi,x}_t){\sigma}_\pi(X^{\pi,x'}_t)^{-1}v_t,v_t\rangle\|Y_t\|/ \| {\sigma}_\pi (X^{\pi,x'}_t)^{-1} Y_t \|$ holds for times $t \in[0,\rho_0(Y))$, equalities and yield: $$\begin{aligned}
\left\| v_t^T \alpha_t \right\|^2 \geq \left\| v_t^T \alpha_t \right\|^2 - \left\| v_t^T \delta_t \right\|^2
= 4 \langle{\sigma}_\pi( X^{\pi,x}_t){\sigma}_\pi(X^{\pi,x'}_t)^{-1}v_t,v_t\rangle\|Y_t\|^2/ \| {\sigma}_\pi (X^{\pi,x'}_t)^{-1} Y_t \|^2.\end{aligned}$$ Inequality in Assumption \[ass1IHM\] implies $\|{\sigma}_\pi^{-1}\|\leq 1/\sqrt{\lambda}$ (cf. the second paragraph of this section). Hence $\|Y_t\|/ \| {\sigma}_\pi (X^{\pi,x'}_t)^{-1} Y_t \|\geq \sqrt{\lambda}$. By the definition of $\delta_t$ above we get $$\begin{aligned}
\left\| v_t^T \alpha_t \right\|^2
\geq 4 \lambda (1+\langle\delta_t{\sigma}_\pi(X^{\pi,x'}_t)^{-1}v_t,v_t\rangle)
\geq 4 \lambda (1-\|\delta_t\| \|{\sigma}_\pi(X^{\pi,x'}_t)^{-1}\|)
\geq 4 \lambda (1- \|\delta_t\|/\sqrt{\lambda}).\end{aligned}$$ For any $\epsilon\in(0,1)$, define $$\bar\varphi:=\min\{1,\epsilon\sqrt{\lambda}/M_x, \epsilon(1 - \epsilon)\lambda /M_x^2\},$$ where $M_x$ is as in above. Then, if $\|x-x'\|<\bar\varphi$ and $t\in[0,\rho_{\bar\varphi}(Y))$, we have $\|Y_t\|<\bar\varphi$ and hence $\|\delta_t\|\leq M_x \|Y_t\|\leq \epsilon \sqrt{\lambda}$. In particular, we get $$\label{eq:S_def}
\| v_t^T \alpha_t\|^2
\geq 4 \lambda (1 - \epsilon)>0 \quad \text{ for any $t\in[0,T)$, where $T:=\rho_0(Y)\wedge \rho_{\bar\varphi}(Y)$.}$$
Let $M>0$ denote a global upper bound on ${\sigma}_\pi$ and $\mu_\pi$, which exists by Assumption \[ass1IHM\]. Since the inequalities $\|v_t^T \alpha_t\|\leq\|\alpha_t\|\leq \|{\sigma}_\pi(X^{\pi,x}_t)\| + \|{\sigma}_\pi(X^{\pi,x'}_t)\|\leq 2M$ hold for all $t \in[0,\rho_0(Y))$, the increasing process $[N]=([N]_t)_{t\in{\mathbb{R}}_+}$, given by $[N]_t:=\int_0^t{\mathbb{I}}_{\left\{s<\rho_0(Y)\right\}} \|v_s^T \alpha_s\|^2 \mathrm{d}s$, is well-define and $[N]_t<\infty$ for every $t\in{\mathbb{R}}_+$. Hence $N=(N_t)_{t\in{\mathbb{R}}_+}$, given by $N_t := \int_0^t {\mathbb{I}}_{\{s<\rho_0(Y)\}}v_s^T \alpha_s \mathrm{d}B_s$, is a well-defined local martingale with a quadratic variation process given by $[N]$. Let $\tau=(\tau_s)_{s\in{\mathbb{R}}_+}$ and $W=(W_s)_{s\in{\mathbb{R}}^+}$ be the Dambis Dubins-Schwartz (DDS) time-change and Brownian motion, respectively, for the local martingale $N$ (see [@Karatzas Thm 3.4.6, p. 174]). More precisely, let $s\mapsto \tau_s := \inf \{ t \in{\mathbb{R}}_+: [N]_t > s \}$ (with $\inf \emptyset =\infty$) be the inverse of $t\mapsto[N]_t$. Then $W$ satisfies $W_{[N]_t} = N_t$ for all $t\in{\mathbb{R}}_+$. Moreover it holds that $\tau_s<\infty$ for $s<[N]_\infty:=\lim_{t\uparrow\infty} [N]_t$. If $[N]_\infty<\infty$ with positive probability, we have to extend the probability space to support $W$ (see e.g. [@Karatzas Prob. 3.4.7, p. 175]). This extension however has no bearing on the coupling $(X^{\pi,x},X^{\pi,x'})$.
Let $\hat \alpha_s:=\alpha_{\tau_{s}}$, $\hat \delta_s:=\delta_{\tau_{s}}$, $\hat \beta_s:=\beta_{\tau_{s}}$, $\hat v_s:=v_{\tau_s}$ and $\hat S_s:=\bar S_{\tau_{s}}$ for $s\in[0,[N]_{\rho_0(Y)})$, cf. above. Assume $\|x-x'\|<\bar\varphi$ and time-change the integrals in (see [@Karatzas Prop. 3.4.8, p. 176]) to get $$\begin{aligned}
\hat{S}_u = \|x-x'\|^2 + \int_0^u 2\sqrt{\hat{S}_s} \,\mathrm{d}W_s
+ \int_0^u \left( 1 + \nu_s\right) \mathrm{d}s,\qquad
\text{for any $u\in[0,[N]_T)$,}\end{aligned}$$ where $\nu_s:= \left(2\sqrt{\hat{S}_s} \hat{v}_s^T \hat{\beta}_s + \operatorname{Tr}( \hat{\delta}_s \hat{\delta}_s^T)
- \| \hat{v}_s^T \hat{\delta}_s \|^2\right)/
\| \hat{v}_s^T \hat{\alpha}_s \|^2$ and $T$ is defined in . By it holds that $\| \hat{v}_s^T \hat{\alpha}_s \|^2\geq 4 \lambda (1 - \epsilon)$ for all $s\in[0,[N]_T)$. Then and the definitions of $\hat \beta, \hat \delta$ and $\nu_s$ imply the inequalities $0\leq \nu_s< M_x^2\|Y_{\tau_s}\|^2/(\lambda(1-\epsilon))$ for all $s\in[0,[N]_T)$. Any $\varphi\in(0,\bar\varphi)$ satisfies $\varphi< \epsilon(1-\epsilon)\lambda/M_x^2$ and $R:=\rho_0(Y)\wedge\rho_\varphi(Y)\leq T$. Hence the Lipschitz property of ${\sigma}_\pi$ and $\nu_\pi$ on the ball of radius $\varphi$ around $x$ implies $$\label{eq:nu_bound}
\nu_s<\epsilon\qquad
\text{for all $s\in[0,[N]_R)$.}$$
The SDE $S_s = \|x-x'\| + \int_0^s 2\sqrt{S_r}\mathrm{d}W_r + (1 + \epsilon)s$, $s\in{\mathbb{R}}_+$, for the squared Bessel process of dimension $1+\epsilon$ has a pathwise unique (and hence strong) solution $S=(S_s)_{s\in{\mathbb{R}}_+}$, see [@Cerny_Engelbert App. A.3, p. 108]. Note that $S$ is driven by the DDS Brownian motion $W$ introduced above. Hence the coupling $(S,\hat S)$ on the stochastic interval $[0,[N]_R)$ allows us to compare the two processes pathwise. Assume $\|x-x'\|<\varphi$. Then the following equality holds: $$\sqrt{S_s} - \sqrt{\hat{S}_s} =\frac{1}{2} \int_0^s \left( \epsilon/\sqrt{S_r}- \nu_r/\sqrt{\hat{S}_r}\right) \mathrm{d}r \qquad \text{for any $s\in[0,[N]_R)$.}$$ Almost surely, the path of the process $(\sqrt{S_s} - \sqrt{\hat{S}_s})_{s\in[0,[N]_R)}$ is continuously differentiable and, by , its derivative is strictly positive at every zero of the path. Since the derivative is continuous, it must be strictly positive on a neighbourhood of each zero. This implies that the only zero is at $s=0$ (i.e. $S_0=\hat S_0$), and it holds that $$\label{eq:S_dominates_S_hat}
S_s\geq \hat S_s\qquad\text{for all $s\in[0,[N]_R)$.}$$
We now conclude the proof of the lemma. Assume as before that $\|x-x'\|<\varphi$ and define $\Upsilon_\varphi(\hat S):=\inf\{s\in[0,[N]_T): \hat S_s=\varphi\}$ (with $\inf \emptyset=\infty$). Note that the events $\{\Upsilon_\varphi(\hat S)<\infty\}$ and $\{[N]_{\rho_\varphi(Y)}<[N]_{\rho_0(Y)}\}$ coincide, since on either event we have $\Upsilon_\varphi(\hat S)=[N]_R=[N]_{\rho_\varphi(Y)}$. Hence, $$\label{eq:important_inclusion}
\{\rho_\varphi(Y)<\rho_0(Y)\}
=
\{\Upsilon_\varphi(\hat S)<\infty\} \subseteq \{\text{$S$ exits interval $(0,\varphi)$ at $\varphi$}\},$$ where the inclusion follows by . Recall that $s(z)=z^{(1-\epsilon)/2}$, $z\in{\mathbb{R}}_+$, is a scale function of the diffusion $S$. Hence ${\mathbb{P}}(\text{$S$ exits interval $(0,\varphi)$ at $\varphi$})=s(\|x-x'\|)/s(\varphi)$. Define $\varphi':=\epsilon\varphi$ and note that by we have: ${\mathbb{P}}(\rho_\varphi(Y)<\rho_0(Y))<\epsilon$ for any $x'\in{\mathbb{R}}^d$ satisfying $\|x-x'\|<\varphi'$.
\[conIHM\] Pick a locally Lipschitz Markov policy $\pi:{\mathbb{R}}^d\to A$ and let Assumption \[ass1IHM\] hold. Then the corresponding payoff function in , $V_\pi:{\mathbb{R}}^d\to{\mathbb{R}}_+$, is continuous.
Fix $x\in{\mathbb{R}}^d$ and pick arbitrary $\varepsilon \in(0,1)$. By Assumption \[ass1IHM\] there exists $\epsilon_0>0$, such that $\alpha_\pi\geq \epsilon_0$, and a constant $M>1$ that simultaneously bounds $\alpha_\pi,|f_\pi|<M$ and is a Lipschitz constant on the ball of radius one around $x$ for $\alpha_\pi$ and $f_\pi$. Apply Lemma \[mdc\] to $x,\epsilon:=\varepsilon\epsilon_0/(6M)$ and $\pi$ to obtain $\bar\varphi\in(0,1]$ such that $\forall\varphi\in(0,\bar\varphi)$ $\exists \varphi'\in(0,\varphi)$ such that ${\mathbb{P}}(\rho_\varphi<\rho_0)<\epsilon$ for every $x'\in{\mathbb{R}}^d$ satisfying $\|x-x'\|<\varphi'$ (here $\rho_\varphi,\rho_0$ stand for $\rho_\varphi(Y),\rho_0(Y)$, resp.). Specifically, define $$\label{eq:def_of_phi}
\varphi:=\min\{\bar\varphi/2, \varepsilon /(3(1+M/\epsilon_0)M/\epsilon_0),(\varepsilon\mathrm{e}\epsilon_0)/(3M^2)\}$$ and fix $\varphi'\in(0,\varphi)$ such that the conclusion of Lemma \[mdc\] holds. Throughout this proof we use the notation and notions from Lemma \[mdc\]. In particular, $(X^{\pi,x},X^{\pi,x'})$ denotes the coupling of two controlled processes started at $(x,x')$ and we assume that $\|x-x'\|<\varphi'$.
Recall that $V_\pi(x')={\mathbb{E}}F_\infty(X^{\pi,x'})$ for any $x'\in{\mathbb{R}}^d$, where $F_\infty(X^{\pi,x'})$ is given in . By decomposing the probability space into complementary events $\{\rho_\varphi>\rho_0\}$ and $\{\rho_\varphi<\rho_0\}$, we obtain the following inequality $
|V_\pi(x) - V_\pi(x')|\leq A + A'+ A''$, where $$\begin{aligned}
& A: = {\mathbb{E}}\left( {\mathbb{I}}_{\{\rho_\varphi>\rho_0\}} \int_0^{\rho_0} \left|
\mathrm{e}^{-\int_0^t \alpha_\pi \left( X^{\pi,x}_s \right) \mathrm{d}s} f_\pi \left( X^{\pi,x}_t \right)
- \mathrm{e}^{-\int_0^t \alpha_\pi \left( X^{\pi,x'}_s \right) \mathrm{d}s} f_\pi \left( X^{\pi, x'}_t \right)
\right| \mathrm{d}t \right), \\
& A':={\mathbb{E}}\left( {\mathbb{I}}_{\{\rho_\varphi>\rho_0\}}
\int_{\rho_0}^\infty \left|
\mathrm{e}^{-\int_0^t \alpha_\pi \left( X^{\pi,x}_s \right) \mathrm{d}s} f_\pi \left( X^{\pi,x}_t \right)
- \mathrm{e}^{-\int_0^t \alpha_\pi \left( X^{\pi,x'}_s \right) \mathrm{d}s} f_\pi \left( X^{\pi,x'}_t \right)
\right| \mathrm{d}t \right), \\
& A'':= {\mathbb{E}}\left( {\mathbb{I}}_{\{\rho_\varphi<\rho_0\}}
\int_0^\infty \left|
\mathrm{e}^{-\int_0^t \alpha_\pi \left( X^{\pi,x}_s \right) \mathrm{d}s} f_\pi \left( X^{\pi,x}_t \right)
- \mathrm{e}^{-\int_0^t \alpha_\pi \left( X^{\pi,x'}_s \right) \mathrm{d}s} f_\pi \left( X^{\pi,x'}_t \right)
\right| \mathrm{d}t \right).\end{aligned}$$ Hence, by Lemma \[mdc\], we have $A''\leq{\mathbb{P}}(\rho_\varphi<\rho_0) 2M/\epsilon_0<\epsilon2M/\epsilon_0=\varepsilon/3$.
Since in the summands $A$ and $A'$ the coupling succeeds before the components of $(X^{\pi,x},X^{\pi,x'})$ grow at least $\varphi$ apart, we can control these terms using the local regularity of $\alpha_\pi$ and $f_\pi$. Consider $A$. Add and subtract $\mathrm{e}^{-\int_0^t \alpha_\pi ( X^{\pi,x}_s ) \mathrm{d}s} f_\pi ( X^{\pi,x'}_t )$ to obtain the bound: $$\begin{aligned}
A \leq &
{\mathbb{E}}{\mathbb{I}}_{\{\rho_\varphi>\rho_0\}}
\int_0^{\rho_0} \left(
\mathrm{e}^{-\epsilon_0 t} \left|f_\pi \left( X^{\pi,x}_t \right)
- f_\pi\left( X^{\pi, x'}_t \right)
\right|+M \left|\mathrm{e}^{-\int_0^t \alpha_\pi \left( X^{\pi,x}_s \right)\mathrm{d}s}
- \mathrm{e}^{-\int_0^t \alpha_\pi \left( X^{\pi,x'}_s \right)\mathrm{d}s}\right|\right)\mathrm{d}t. \end{aligned}$$ On the event $\{\rho_\varphi>\rho_0\}$, for $t<\rho_\varphi$ it holds that $\|X^{\pi,x}_t-X^{\pi,x'}_t\|<\varphi$. Since $z\mapsto \mathrm{e}^{-z}$ has a positive derivative bounded above by one for $z\in{\mathbb{R}}_+$, $\alpha_\pi-\epsilon_0\geq0$ and both $f_\pi$ and $\alpha_\pi$ are Lipschitz with constant $M$ on the ball of radius $\varphi$ around $x$, we get $$\begin{aligned}
A \leq &
{\mathbb{E}}{\mathbb{I}}_{\{\rho_\varphi>\rho_0\}}
\int_0^{\rho_0} \left(M\varphi
\mathrm{e}^{-\epsilon_0 t} +M\mathrm{e}^{-\epsilon_0 t} \int_0^t\left|
\alpha_\pi ( X^{\pi,x}_s ) - \alpha_\pi ( X^{\pi,x'}_s)\right|\mathrm{d}s\right)\mathrm{d}t
< \varphi\left(\frac{M}{\epsilon_0} +\frac{M^2}{\epsilon_0^2}\right)\leq \frac{\varepsilon}{3},\end{aligned}$$ where the last inequality follows from . Furthermore, since $X^{\pi,x}_t=X^{\pi,x'}_t$ for all $t\geq \rho_0$, it holds that the following expectation equals $A'$: $${\mathbb{E}}{\mathbb{I}}_{\{\rho_\varphi>\rho_0\}}
\int_{\rho_0}^\infty \mathrm{e}^{-\rho_0\epsilon_0}\left|
\mathrm{e}^{-\int_0^{\rho_0} (\alpha_\pi ( X^{\pi,x}_s )-\epsilon_0) \mathrm{d}s}
- \mathrm{e}^{-\int_0^{\rho_0} (\alpha_\pi ( X^{\pi,x'}_s )-\epsilon_0) \mathrm{d}s}
\right|
\mathrm{e}^{-\int_{\rho_0}^t \alpha_\pi ( X^{\pi,x}_s ) \mathrm{d}s} \left|f_\pi ( X^{\pi,x}_t )\right|
\mathrm{d}t.$$ Since $|f_\pi|<M$, $\alpha_\pi\geq\epsilon_0$, $|\mathrm{e}^{-z}-\mathrm{e}^{-y}|\leq |z-y|$ for $z,y\in{\mathbb{R}}_+$ and $\mathrm{e}^{-t\epsilon_0}\leq\mathrm{e}^{-1}\epsilon_0$ for $t\in{\mathbb{R}}_+$ we find $$A'\leq
{\mathbb{E}}\left( {\mathbb{I}}_{\{\rho_\varphi>\rho_0\}} M
\mathrm{e}^{-\rho_0\epsilon_0}
\int_0^{\rho_0} \left|\alpha_\pi ( X^{\pi,x}_s ) -\alpha_\pi ( X^{\pi,x'}_s )\right| \mathrm{d}s
\right)
\leq M^2\varphi
{\mathbb{E}}\left( {\mathbb{I}}_{\{\rho_\varphi>\rho_0\}}
\mathrm{e}^{-\rho_0\epsilon_0} \rho_0\right)
\leq \varphi\frac{M^2}{\mathrm{e}\epsilon_0},$$ which is by less than $\varepsilon/3$. Hence, for any $\|x-x'\| \leq \varphi'$, we proved that $|V_\pi(x) - V_\pi(x')| < \epsilon$, which concludes the proof of the lemma.
\[rem:No\_coupling\] The proofs of Lemmas \[mdc\] and \[conIHM\] show that if the locally Lipschitz property in Assumption \[ass1IHM\] is substituted by the globally Lipschitz requirement, we can conclude that the payoff function $V_\pi$ is in fact uniformly continuous. However, the coupling from [@article6] may still not be successful, since the global Lipschitz condition controls globally the local variability of the coefficients. The coupling may fail because the assumptions in [@article6 Thm. 1] constrain the global variability of ${\sigma}_\pi$. In fact, the idea of the proof of Lemma \[mdc\] can be used to construct an example where ${\mathbb{P}}(\rho_0(Y)<\infty)<1$ by bounding the norm of $\|Y\|^2$ from below by a squared Bessel process of dimension greater than two on an event of positive probability.
### A version of the Ascoli-Arzela Theorem
The following fact is key for proving the existence of the optimal strategy and showing that a subsequence of $\{\pi_N\}_{N\in{\mathbb{N}}}$ in converges to it.
\[AA\] Let $(M_1,d_1)$ and $(M_2,d_2)$ be compact metric spaces, and for every $n \in {\mathbb{N}}$ let $f_n : M_1 \to M_2$. If the sequence $\{ f_n \}_{n \in {\mathbb{N}}}$ is equicontinuous, i.e. $$\forall \epsilon > 0 \quad \exists \delta > 0 \quad \forall x,y \in M_1 \quad \forall n \in {\mathbb{N}}: \quad
d_1(x,y) < \delta \implies d_2(f_n(x),f_n(y)) < \epsilon,$$ then there exists a uniformly convergent subsequence $\{f_{n_k}\}_{k\in{\mathbb{N}}}$, i.e. $\exists f:M_1\to M_2$ such that for every $\epsilon>0$ there exists $N\in{\mathbb{N}}$ such that $\sup_{x\in M_1}d_2(f_{n_k}(x),f(x))<\epsilon$ for all $k\geq N$.
Let $B( x, 1/m) := \{ y \in M_1: d_1(x,y) < 1/m \}$ be a ball of radius $1/m$, $m\in{\mathbb{N}}$, centred at $x\in M_1$. Since $M_1$ is compact and metric, it is totally bounded: $\exists S_m\subseteq M_1$ finite satisfying $M_1=\cup_{x\in S_m}B(x,1/m)$. Then $S:=\cup_{m\in{\mathbb{N}}} S_m=\{ x_n \in M_1;\ n \in {\mathbb{N}}\}$ is countable and dense in $M_1$. We now apply the standard diagonalisation argument to find the subsequence in the lemma.
Let $\iota_1:{\mathbb{N}}\to{\mathbb{N}}$ be an increasing function defining a subsequence $\{ f_{\iota_1(n)} \}_{n \in {\mathbb{N}}}$ that converges at $x_1$, i.e. $\lim_{n\to\infty}f_{\iota_1(n)}(x_1)$ exists in $M_2$. Such a function $\iota_1$ exists since $M_2$ is compact. Assume now that we have constructed an increasing $\iota_k:{\mathbb{N}}\to{\mathbb{N}}$ such that $\{ f_{\iota_k(n)} \}_{n \in {\mathbb{N}}}$ converges on the set $\{ x_1,\ldots,x_k\}$ for some $k\in{\mathbb{N}}$. Then there exists an increasing $\iota:{\mathbb{N}}\to{\mathbb{N}}$ such that the sequence of functions $\{ f_{\iota_{k+1}(n)} \}_{n \in {\mathbb{N}}}$, where $\iota_{k+1}:=\iota_k\circ\iota$, converges at $x_{k+1}$ as well as on the set $\{ x_1,\ldots,x_k\}$, as it is a subsequence of $\{ f_{\iota_k(n)} \}_{n \in {\mathbb{N}}}$. Since $k\in {\mathbb{N}}$ was arbitrary, we have defined a sequence of subsequences of $\{ f_n \}_{n \in {\mathbb{N}}}$, such that the $k$-th subsequence converges on $\{ x_1,\ldots,x_k\}$.
Consider the “diagonal” subsequence $\{ f_{n_k} \}_{k \in {\mathbb{N}}}$, $f_{n_k}:= f_{\iota_k(k)}$ for any $k \in {\mathbb{N}}$. By construction it converges on $S$. We now prove that it is uniformly Cauchy, which implies uniform convergence since $M_2$ is complete. Pick any $\epsilon>0$. By equicontinuity $\exists m \in {\mathbb{N}}$ such that for any $k\in{\mathbb{N}}$ and $x,y\in M_1$ satisfying $d_1(x,y) < 1/m$, it holds that $d_2(f_{n_k}(x),f_{n_k}(y)) < \epsilon/3$. Furthermore, since $S_m$ is finite, $\exists N \in {\mathbb{N}}$ such that for all natural numbers $k_1,k_2\geq N$ we have $d_2(f_{n_{k_1}}(y),f_{n_{k_2}}(y)) < \epsilon/3$ for all $y\in S_m$. Finally, for any $x \in M_1$ there exists $y\in S_m$ such that $d_1(x,y)<1/m$. Hence, for any $k_1,k_2\geq N$ it holds that $$d_2(f_{n_{k_1}}(x),f_{n_{k_2}}(x)) \leq
d_2(f_{n_{k_1}}(x),f_{n_{k_1}}(y)) +
d_2(f_{n_{k_1}}(y),f_{n_{k_2}}(y)) +
d_2(f_{n_{k_2}}(y),f_{n_{k_2}}(x))
< \epsilon.$$ Since $x\in M_1$ was arbitrary, the lemma follows.
### A uniformly integrable martingale
If the process $X^{\pi,x}$ in , controlled by a Markov policy $\pi$, exists for all $x\in{\mathbb{R}}^d$, then $X^{\pi,\cdot}$ is a strong Markov process [@Karatzas Thm 4.30, p. 322], since ${\sigma}$ and $\mu$ are bounded by Assumption \[ass1IHM\]. Define the additive functional $F(X^{\pi,x})=(F_t(X^{\pi,x}))_{t\in[0,\infty]}$, $$\label{eq:F_gains}
F_t(X^{\pi,x}):=\int_0^t\mathrm{e}^{-\int_0^u \alpha_\pi \left( X^{\pi,x}_s \right)\mathrm{d}s}
f_\pi \left( X^{\pi,x}_u \right) \mathrm{d}u \qquad \text{for $t\in[0,\infty]$. }$$
\[rem:V\_F\_bounded\] Note that $V_\pi(x)={\mathbb{E}}F_\infty(X^{\pi,x})$ and, by Assumption \[ass1IHM\], the process $|F(X^{\pi,x})|$ is bounded by some constant $C_0>0$. Hence $|F_\infty(X^{\pi,x})|<C_0$ and $|V_\pi(x)|<C_0$.
\[martingaleIHM\] The following holds for every Markov policy $\pi$, $x \in {\mathbb{R}}^d$ and $({\mathcal{F}}_t)$-stopping time $T$: $$\begin{aligned}
{\mathbb{E}}\big( F_\infty(X^{\pi,x})\vert {\mathcal{F}}_T \big) = F_T(X^{\pi,x}) + {\mathbb{I}}_{\{ T < \infty\}} \mathrm{e}^{-\int_0^{T} \alpha_\pi \left( X^{\pi,x}_s \right) \mathrm{d}s}\,
V_\pi\left( X^{\pi,x}_{T} \right).\end{aligned}$$ In particular, the process $M=(M_r)_{r\in[0,\infty]}$ is a uniformly integrable martingale, where $$\begin{aligned}
M_r := F_r(X^{\pi,x}) + {\mathbb{I}}_{\{ r < \infty\}} \mathrm{e}^{-\int_0^{r} \alpha_\pi \left( X^{\pi,x}_s \right) \mathrm{d}s} \,V_\pi\left( X^{\pi,x}_{r} \right). \end{aligned}$$
The following calculations imply the lemma: $$\begin{aligned}
& {\mathbb{E}}\big( F_\infty(X^{\pi,x})\vert {\mathcal{F}}_T \big) = F_T(X^{\pi,x})
+ {\mathbb{E}}\left( {\mathbb{I}}_{\{ T < \infty \}} \int_0^{\infty}
\mathrm{e}^{-\int_0^{t + T} \alpha_\pi \left( X^{\pi,x}_s \right) \mathrm{d}s}
f_\pi \left( X^{\pi,x}_{t + T} \right) \mathrm{d}t \,\middle\vert\, {\mathcal{F}}_{T} \right) \\
& = F_T(X^{\pi,x}) + {\mathbb{I}}_{\{ T < \infty \}} \mathrm{e}^{-\int_0^{T} \alpha_\pi \left( X^{\pi,x}_t \right) \mathrm{d}t}
\,{\mathbb{E}}\left( \int_0^{\infty} \mathrm{e}^{-\int_0^{t} \alpha_\pi \left( X^{\pi,x}_{s + T} \right) \mathrm{d}s}
f_\pi \left( X^{\pi,x}_{t + T} \right) \mathrm{d}t \,\middle\vert\, {\mathcal{F}}_{T} \right) \\
& = F_T(X^{\pi,x}) + {\mathbb{I}}_{\{ T < \infty \}} \mathrm{e}^{-\int_0^{T} \alpha_\pi \left( X^{\pi,x}_t \right) \mathrm{d}t}
\,V_\pi(X^{\pi,x}_{T}),\end{aligned}$$ where we applied the strong Markov property of $X^{\pi,\cdot}$ in the last step.
Proofs of results in Section \[ch3IHM\] {#subsec:Proofs_Multidim}
---------------------------------------
Assume $m=d$, cf. Remark \[rem:d\_is\_m\]. It suffices to prove that the PDE holds on the ball $D:=\{y\in{\mathbb{R}}^d:\|y-x'\|<1\}$ for any $x'\in{\mathbb{R}}^d$. Fix $x \in D$ and define $\tau:=\inf\{t\in{\mathbb{R}}_+:X^{\pi,x}_t\in\partial D\}$ (with $\inf\emptyset =\infty$) to be the first time the process $X^{\pi,x}$ hits the boundary $\partial D:=\{y\in{\mathbb{R}}^d:\|y-x'\|=1\}$ of $D$. Note that, by Assumption \[ass1IHM\], we have $\tau<\infty$.
Let $v \in {\mathcal{C}}^2(D) \cap {\mathcal{C}}(\bar{D})$, where $\bar D:=D\cup \partial D$, denote a solution of the boundary value problem $$L_\pi v - \alpha_\pi v + f_\pi = 0\quad \text{in} \quad D, \qquad \text{where $v = V_\pi$ on $\partial D$.}$$ Since $\pi$ is locally Lipschitz and in Assumption \[ass1IHM\] holds, the coefficients ${\sigma}_\pi,\mu_\pi,f_\pi,\alpha_\pi$ are $(1/2)$-Hölder (in fact Lipschitz) on $\bar D$. The boundary data $V_\pi|_{\partial D}$ is continuous by Lemma \[conIHM\], $\alpha_\pi \geq 0$ and ${\sigma}_\pi$ satisfies . Hence, by [@FriedmanParabolic Thm 19, p. 87], the function $v$ exists, is unique and ${\mathrm{H}}v$ is $(1/2)$-Hölder.
Note that, for all $t\in[0,\infty]$, we have $X^{\pi,x}_{t \wedge \tau}\in \bar D$. Hence we can define $$\label{eq:def_of_proc_S}
Y_t := F_{t \wedge \tau}(X^{\pi,x}) +
\mathrm{e}^{-\int_0^{t \wedge \tau} \alpha_{\pi} \left( X^{\pi,x}_r \right) \mathrm{d}r}
v \left( X^{\pi,x}_{t \wedge \tau} \right),\qquad \text{for $t\in[0,\infty]$,}$$ where the process $F_\cdot(X^{\pi,x})$ is given in above. The process $Y = (Y_t)_{t \in[0,\infty]}$ is bounded by a constant and by definition converges almost sure $\lim_{t\to \infty}Y_t= Y_\infty$. Since $v$ solves the boundary value problem above and $X^{\pi,x}$ satisfies SDE , Itô’s formula on the stochastic interval $[0,\tau]\subset{\mathbb{R}}_+$ yields $$\begin{aligned}
Y_t
& = v(x) + \int_0^{t \wedge \tau} \mathrm{e}^{-\int_0^s \alpha_{\pi} \left( X^{\pi,x}_r \right) \mathrm{d}r}
{\nabla}v\left( X^{\pi,x}_s \right)^T {\sigma}_\pi \left( X^{\pi,x}_s \right) \mathrm{d}B_s, \quad t \in[0,\infty],\end{aligned}$$ making $Y$ into a local martingale. Since $Y$ is bounded, it is a uniformly integrable martingale satisfying $v(x)=Y_0={\mathbb{E}}[ Y_\infty]$. Since $v=V_\pi$ on $\partial D$ and $X^{\pi,x}_\tau\in \partial D$, the definition of $Y$ in and Lemma \[martingaleIHM\] (applied to the stopping time $T:=\tau$) yield $$\begin{aligned}
Y_\infty
& = F_\tau(X^{\pi,x}) +
\mathrm{e}^{-\int_0^{\tau} \alpha_{\pi} \left( X^{\pi,x}_r \right) \mathrm{d}r}
V_\pi \left( X^{\pi,x}_{\tau} \right) = {\mathbb{E}}\left( F_\infty(X^{\pi,x}) \middle\vert {\mathcal{F}}_{\tau} \right),\end{aligned}$$ implying $v(x)={\mathbb{E}}( F_\infty(X^{\pi,x}))=V_\pi(x)$.
The uniqueness follows similarly: let $v$ be another bounded solution of the Poisson equation on ${\mathbb{R}}^d$. Define the process $Y$ as in with $\tau\equiv\infty$ and $t<\infty$. As above we have $v(x)={\mathbb{E}}Y_t$ for all $t\in{\mathbb{R}}_+$. Then the DCT, applicable since $v$ is bounded, yields $v(x)=\lim_{t\uparrow\infty} {\mathbb{E}}Y_t = V_\pi(x)$.
Let $\pi_n$ and $\pi_{n+1}$ be as in , $n\in{\mathbb{N}}\cup\{0\}$. Define $Y=(Y_t)_{t\in{\mathbb{R}}_+}$ by $$\begin{aligned}
\label{eq:S_defin}
& Y_t := F_t( X^{\pi_{n+1},x}) + \mathrm{e}^{-\int_0^t \alpha_{\pi_{n+1}} \left( X^{\pi_{n+1},x}_r \right) \mathrm{d}r}
\,V_{\pi_n} \left( X^{\pi_{n+1},x}_{t} \right), \quad t \in {\mathbb{R}}_+.\end{aligned}$$ where $F_\cdot(X^{\pi_{n+1},x})$ is given in above. Define $\tau_m:=\inf\{t\geq0:\|X^{\pi_{n+1},x}_t\|=m\}$ for any fixed $m>\|x\|$ and note that $\tau_m<\infty$ by Assumption \[ass1IHM\]. Itô’s formula, applicable by Proposition \[operatorIHM\], yields $$Y_{\cdot \wedge \tau_m} = V_{\pi_n}(x)
+ M + \int_0^{\cdot \wedge \tau_m} \mathrm{e}^{-\int_0^s \alpha_{\pi_{n+1}} \left( X^{\pi_{n+1},x}_r \right) \mathrm{d}r}
\left( f_{\pi_{n+1}} + L_{\pi_{n+1}} V_{\pi_n} - \alpha_{\pi_{n+1}} V_{\pi_n} \right)
\left( X^{\pi_{n+1},x}_s \right) \mathrm{d}s,$$ where $M=(M_t)_{t\in{\mathbb{R}}_+}$, $M_t:= \int_0^{t \wedge \tau_m} \mathrm{e}^{-\int_0^s \alpha_{\pi_{n+1}}
( X^{\pi_{n+1},x}_r) \mathrm{d}r}\,
( {\nabla}V_{\pi_n}^T {\sigma}_{\pi_{n+1}}) ( X^{\pi_{n+1},x}_s ) \mathrm{d}B_s$, is a local martingale. Since the functions ${\sigma}_{\pi_{n+1}}$ and ${\nabla}V_{\pi_n}$ are bounded on the ball $\{y\in{\mathbb{R}}^d:\|y\|\leq m\}$ by Assumption \[ass1IHM\] and Proposition \[operatorIHM\], respectively, and $\alpha_{\pi_{n+1}}>\epsilon_0>0$, the quadratic variation of $M$ is bounded above by a constant. Hence $M$ is a uniformly integrable martingale. In particular, ${\mathbb{E}}M_t=0$ for all $t\in{\mathbb{R}}_+$. By and Proposition \[operatorIHM\], we have $$\left(
f_{\pi_{n+1}} + L_{\pi_{n+1}} V_{\pi_n} - \alpha_{\pi_{n+1}} V_{\pi_n}\right)
S_{\pi_{n+1}}
\> \leq\>
\left(
f_{\pi_n} + L_{\pi_n} V_{\pi_n} - \alpha_{\pi_n} V_{\pi_n}\right) S_{\pi_n}= 0
\qquad \text{on ${\mathbb{R}}^d$.}$$ Since $S_{\pi_{n+1}}>0$, we have $E \left( Y_{t \wedge \tau_m} \right) \leq V_{\pi_n}(x)$. Hence , Assumption \[ass1IHM\] and the DCT, as $t\uparrow\infty$, yield $$\begin{aligned}
V_{\pi_n}(x) & \geq
{\mathbb{E}}F_{\tau_m}( X^{\pi_{n+1},x}) +
{\mathbb{E}}V_{\pi_n} \left( X^{\pi_{n+1},x}_{\tau_m} \right)
\mathrm{e}^{-\int_0^{\tau_m}
\alpha_{\pi_{n+1}} \left( X^{\pi_{n+1},x}_r \right) \mathrm{d}r}.
$$ Hence, by Remark \[rem:V\_F\_bounded\], we have $V_{\pi_n}(x)\geq {\mathbb{E}}F_{\tau_m}( X^{\pi_{n+1},x}) -
C_0 {\mathbb{E}}\mathrm{e}^{-\epsilon_0\tau_m}$. Since $X^{\pi_{n+1},x}$ satisfies SDE for all $t\in{\mathbb{R}}_+$, we have $\lim_{m\uparrow\infty}\tau_m=\infty$. The DCT and Remark \[rem:V\_F\_bounded\] yield $V_{\pi_{n+1}}(x)= {\mathbb{E}}F_\infty( X^{\pi_{n+1},x})=\lim_{m\uparrow\infty}{\mathbb{E}}F_{\tau_m}( X^{\pi_{n+1},x})- C_0 {\mathbb{E}}\mathrm{e}^{-\epsilon_0\tau_m}\leq V_{\pi_n}(x)$, which concludes the proof.
Run to produce a sequence of policies $\{\pi_N\}_{N\in{\mathbb{N}}}$, starting from a constant policy $\pi_0$. Fix an arbitrary $K_0>0$ and consider the restriction of this sequence onto the closed ball $D_{K_0}$. Since the Lipschitz constant of $\pi_0$ is equal to zero and hence smaller than $C_{K_0}$, Assumption \[ass3IHM\] implies $V_{\pi_0}\in{\mathcal{S}}_{B_{K_0},K_0}$. Assumption \[ass2IHM\] implies that the Lipschitz constant of $\pi_1$ is also at most $C_{K_0}$. Iterating this argument implies that all the policies in the sequence $\{\pi_N\}_{N\in{\mathbb{N}}}$ have the same Lipschitz constant on $D_{K_0}$, making it equicontinuous on $D_{K_0}$. By Lemma \[AA\] above, there exists a subsequence that converges uniformly on $D_{K_0}$ to a function $\pi_\infty^0:D_{K_0}\to A$. Moreover, $\pi_\infty^0$ is also Lipschitz with a constant bounded above by $C_{K_0}$.
Let $K_1:=2K_0$ and repeat the argument above for $K_1$ and the subsequence of $\{\pi_N\}_{N\in{\mathbb{N}}}$ constructed in the previous paragraph. This yields a further subsequence of the policies that converges uniformly to a Lipschitz function $\pi_\infty^1:D_{K_1}\to A$ with the Lipschitz constant bounded above by $C_{K_1}$. Since the sequence we started with converges pointwise to $\pi_\infty^0$ on $D_{K_0}\subset D_{K_1}$, so must its every subsequence. Hence it holds that $\pi_\infty^1(x)=\pi_\infty^0(x)$ for all $x\in D_{K_0}$.
For $k\in{\mathbb{N}}$, let $K_k:=2K_{k-1}$ and construct inductively $\pi_\infty^k:D_{K_k}\to A$ as above. Then the function $\pi_{\lim}:{\mathbb{R}}^d\to A$, given by $\pi_{\lim}(x):=\pi_\infty^n(x)$ for any $n\in{\mathbb{N}}$ such that $x\in D_{K_n}$, is well-defined and locally Lipschitz. Let the policy $\pi_{n_k}:{\mathbb{R}}^d\to A$ be the $k$-th element of the convergent subsequence used to define $\pi_\infty^k:D_{K_k}\to A$. Then, by construction, the “diagonal” subsequence $\{\pi_{n_k}\}_{k\in {\mathbb{N}}}$ of $\{\pi_N\}_{N\in{\mathbb{N}}}$ converges uniformly to $\pi_{\lim}$ on $D_K$ for any $K>0$.
Let $\{ \pi_{n_k} \}_{k \in {\mathbb{N}}}$ be a subsequence of the output of , $\{ \pi_N \}_{N \in {\mathbb{N}}}$, that converges locally uniformly to a policy $\pi_{\lim}=\lim_{k\uparrow\infty}\pi_{n_k}$. By and Theorem \[decreasingIHM\], $V_{\pi_{n_k}}\searrow V_{\lim}$ as $k\to\infty$. Fix $K>0$ and let $\tau_K:=\inf\{t\in{\mathbb{R}}_+:X^{\pi_{\lim},x}_t-x\in\partial D_K\}$ be the first time $X^{\pi_{\lim},x}$ hits the boundary of the closed ball $x+D_K$ with radius $K$, centred at an arbitrary $x\in{\mathbb{R}}^d$.
Pick $k \in {\mathbb{N}}$, $t\in{\mathbb{R}}_+$ and define $$S^k_t := \int_0^{t} \mathrm{e}^{-\int_0^s \alpha_{\pi_{n_k}} \left( X^{\pi_{\lim},x}_r \right) \mathrm{d}r}
f_{\pi_{n_k}} \left( X^{\pi_{\lim},x}_s \right) \mathrm{d}s +
\mathrm{e}^{-\int_0^t \alpha_{\pi_{n_k}} \left( X^{\pi_{\lim},x}_r \right) \mathrm{d}r}\,
V_{\pi_{n_k}} \left( X^{\pi_{\lim},x}_{t} \right).$$ Apply Itô’s formula to the process $S^k = (S^k_t)_{t \geq 0}$ on the stochastic interval $[0,\tau_K)$ to get $$\begin{aligned}
& S^k_{t \wedge \tau_K} = V_{\pi_{n_k}}(x)
+ \int_0^{t \wedge \tau_K} \mathrm{e}^{-\int_0^s \alpha_{\pi_{n_k}} \left( X^{\pi_{\lim},x}_r \right) \mathrm{d}r}\,
\left( {\nabla}V_{\pi_{n_k}} \right)^T {\sigma}_{\pi_{\lim}} \left( X^{\pi_{\lim},x}_s \right) \mathrm{d}B_s \\
& \qquad \qquad + \int_0^{t \wedge \tau_K} \mathrm{e}^{-\int_0^s \alpha_{\pi_{n_k}} \left( X^{\pi_{\lim},x}_r \right) \mathrm{d}r}
\left( f_{\pi_{n_k}} + L_{\pi_{\lim}} V_{\pi_{n_k}} - \alpha_{\pi_{n_k}} V_{\pi_{n_k}} \right)
\left( X^{\pi_{\lim},x}_s \right) \mathrm{d}s.\end{aligned}$$ Note that ${\sigma}_{\pi_{\lim}}$ and ${\nabla}V_{\pi_{n_k}}$ are bounded on $D_K$ by Assumption \[ass1IHM\] and Proposition \[operatorIHM\], respectively, and $\alpha_{\pi_{n_k}}>\epsilon_0>0$. Hence the quadratic variation of the stochastic integral is bounded, making it into a true martingale. This fact and the equality $\alpha_{\pi_{n_k}} V_{\pi_{n_k}}-f_{\pi_{n_k}}= L_{\pi_{n_k}} V_{\pi_{n_k}}$ (Prop. \[operatorIHM\]) yield $$\begin{aligned}
\nonumber
{\mathbb{E}}S^k_{t \wedge \tau_K}
& = & V_{\pi_{n_k}}(x) + {\mathbb{E}}\int_0^{t \wedge \tau_K} \mathrm{e}^{-\int_0^s
\alpha_{\pi_{n_k}} \left( X^{\pi_{\lim},x}_r \right) \mathrm{d}r} \left(
f_{\pi_{n_k}} + L_{\pi_{\lim}} V_{\pi_{n_k}} - \alpha_{\pi_{n_k}} V_{\pi_{n_k}}
\right) \left( X^{\pi_{\lim},x}_s \right) \mathrm{d}s \\
& = & V_{\pi_{n_k}}(x) +
{\mathbb{E}}\int_0^{t \wedge \tau_K} \mathrm{e}^{-\int_0^s \alpha_{\pi_{n_k}}
\left( X^{\pi_{\lim},x}_r \right) \mathrm{d}r} \left( L_{\pi_{\lim}} V_{\pi_{n_k}} - L_{\pi_{n_k}} V_{\pi_{n_k}} \right) \left( X^{\pi_{\lim},x}_s
\right) \mathrm{d}s.
\label{eq:pi_lim_equals_pi_k}\end{aligned}$$
Note $[L_{\pi_{\lim}} - L_{\pi_{n_k}}] V_{\pi_{n_k}} =
( \mu_{\pi_{\lim}} - \mu_{\pi_{n_k}} )^T {\nabla}V_{\pi_{n_k}}+
\frac{1}{2} \operatorname{Tr}( ( {\sigma}_{\pi_{\lim}} + {\sigma}_{\pi_{n_k}})^T {\mathrm{H}}V_{\pi_{n_k}} ( {\sigma}_{\pi_{\lim}} - {\sigma}_{\pi_{n_k}}))$. Since, for every $k$, $V_{\pi_{n_k}}$ solves the corresponding Poisson equation in Proposition \[operatorIHM\] and, by Assumption \[ass1IHM\] and Remark \[rem:V\_F\_bounded\], the family of functions $\{ {\sigma}_{\pi_{n_k}}, \mu_{\pi_{n_k}},\alpha_{\pi_{n_k}}, f_{\pi_{n_k}}, V_{\pi_{n_k}} :k \in {\mathbb{N}}\}$ is uniformly bounded on the ball $x+D_K$, Schauder’s boundary estimate for elliptic PDEs [@FriedmanParabolic p. 86] implies that the sequences $\{ {\nabla}V_{\pi_{n_k}} \}_{k \in {\mathbb{N}}}$ and $\{ {\mathrm{H}}V_{\pi_{n_k}} \}_{k \in {\mathbb{N}}}$ are also uniformly bounded on $x+D_K$. Since $\alpha_{\pi_{n_k}}>\epsilon_0>0$ for all $k\in{\mathbb{N}}$ and the limits $\lim_{k\uparrow\infty}\mu_{\pi_{n_k}}=\mu_{\pi_{\lim}}$ and $\lim_{k\uparrow\infty}{\sigma}_{\pi_{n_k}}={\sigma}_{\pi_{\lim}}$ are uniform on $x+D_K$, the DCT and the equality in imply $\lim_{k\uparrow\infty} {\mathbb{E}}S^k_{t \wedge \tau_K} = V_{\lim}(x)$. Hence, the definition of $S^k$ above, Assumption \[ass1IHM\], Remark \[rem:V\_F\_bounded\] and a further application of the DCT yield $$\label{eq:V_lim_final_expression}
V_{\lim}(x) =
{\mathbb{E}}\int_0^{t \wedge \tau_K} \mathrm{e}^{-\int_0^s \alpha_{\pi_{\lim}} \left( X^{\pi_{\lim},x}_r \right) \mathrm{d}r} f_{\pi_{\lim}} \left( X^{\pi_{\lim},x}_s \right) \mathrm{d}s
+ E_{t \wedge \tau_K},$$ where $E_{t \wedge \tau_K}:= {\mathbb{E}}\mathrm{e}^{-\int_0^{t \wedge \tau_K} \alpha_{\pi_{\lim}} \left( X^{\pi_{\lim},x}_r \right) \mathrm{d}r}\,
V_{\lim} \left( X^{\pi_{\lim},x}_{t \wedge \tau_K} \right)$.
By and Remark \[rem:V\_F\_bounded\], the inequality $|V_{\lim}(y)|\leq C_0$ holds for all $y\in{\mathbb{R}}^d$. By Assumption \[ass1IHM\] we hence get $$0\leq \limsup_{t\wedge K\to\infty}\left| E_{t\wedge K}\right| \leq C_0 \limsup_{t\wedge K\to\infty}{\mathbb{E}}\mathrm{e}^{-\epsilon_0(t \wedge \tau_K)}= 0,$$ since $\tau_K\uparrow\infty$ as $K\uparrow\infty$. The DCT applied to the first summand in , as $t\wedge K\to\infty$, yields the equality $V_{\lim}(x) = V_{\pi_{\lim}}(x)$. Since $x\in{\mathbb{R}}^d$ was arbitrary, the theorem follows.
The second assertion in the theorem follows from the first one and Theorem \[limitsIHM\]. We now establish the first assertion of Theorem \[verificationIHM\]. Equip $A\times A$ with a product metric, e.g. $d_\infty((p_1,p_2),(a_1,a_2)) := \max\{d_A(a_1,p_1), d_A(a_2,p_2)\}$, and let $\{ \pi_N\}_{N \in {\mathbb{N}}}$ be constructed by the . As in the proof of Proposition \[subsequenceIHM\], $\{ (\pi_{N+1}, \pi_N) : {\mathbb{R}}^d \to A \times A \}_{N \in {\mathbb{N}}}$ are Lipschitz on a closed ball $D_K$ of radius $K>0$ with the Lipschitz constant $C_K$, independent of $N$. Hence as in the proof of Proposition \[subsequenceIHM\], there exists a subsequence $\{ (\pi_{1+n_k}, \pi_{n_k}) \}_{k \in {\mathbb{N}}}$ that converges uniformly on every compact subset of ${\mathbb{R}}^d$ to a locally Lipschitz function $(\tilde \pi_{\lim}, \pi_{\lim}) : {\mathbb{R}}^d \to A \times A$.
Pick any $x\in{\mathbb{R}}^d$, a policy $\Pi\in{\mathcal{A}}(x)$, $K>0$ and let $\tau_K:=\inf\{t\in{\mathbb{R}}_+:X^{\Pi,x}_t-x\in\partial D_K\}$ be the first time the controlled process $X^{\Pi,x}$ hits the boundary of the closed ball $x+D_K$ with radius $K$ (centred at $x$). Since $\Pi_s\in A$ for all $s\in{\mathbb{R}}_+$, the implies the inequality $$\label{eq:Scaled_Pia_estimate}
S_{\Pi_s} (f_{\Pi_s} + L_{\Pi_s} V_{\pi_{n_k}} - \alpha_{\Pi_{s}} V_{\pi_{n_k}}) \geq
S_{\pi_{n_k + 1}} (f_{\pi_{n_k + 1}} + L_{\pi_{n_k + 1}} V_{\pi_{n_k}} - \alpha_{\pi_{n_k+1}} V_{\pi_{n_k}}) \quad \text{on ${\mathbb{R}}^d$.}$$ Denote $\mathcal{L}_\pi h := L_\pi h - \alpha_\pi h + f_\pi$ for any policy $\pi$ and $h \in {\mathcal{C}}^2({\mathbb{R}}^d)$. Then, for $k \in {\mathbb{N}}$, we find that $$\begin{aligned}
\nonumber
& {\mathbb{E}}\left( \int_0^{t \wedge \tau_K} \mathrm{e}^{-\int_0^s \alpha_{\Pi_r} \left( X^{\Pi,x}_r \right) \mathrm{d}r} f_{\Pi_s} \left( X^{\Pi,x}_s \right) \mathrm{d}s +
\mathrm{e}^{-\int_0^{t \wedge \tau_K} \alpha_{\Pi_r} \left( X^{\Pi,x}_r \right) \mathrm{d}r}\,
V_{\pi_{n_k}} \left( X^{\Pi,x}_{t \wedge \tau_K} \right) \right) \\
\nonumber
& \stackrel{\text{It\^{o}}}{=} V_{\pi_{n_k}}(x) + {\mathbb{E}}\int_0^{t \wedge \tau_K}
\mathrm{e}^{-\int_0^s \alpha_{\Pi_{r}} \left( X^{\Pi,x}_r \right) \mathrm{d}r}
\left( f_{\Pi_s} + L_{\Pi_s} V_{\pi_{n_k}} - \alpha_{\Pi_{s}} V_{\pi_{n_k}}
\right) \left( X^{\Pi,x}_s \right) \mathrm{d}s \\
& \geq V_{\pi_{n_k}}(x) +\frac{\epsilon_S}{M_S} {\mathbb{E}}\int_0^{t \wedge \tau_K} \mathrm{e}^{-\int_0^s \alpha_{\Pi_{r}} \left( X^{\Pi,x}_r \right) \mathrm{d}r}
\left( \mathcal{L}_{\pi_{n_k + 1}} V_{\pi_{n_k}} \right) \left( X^{\Pi,x}_s \right) \mathrm{d}s,
\label{eq:final_verification}\end{aligned}$$ where the last inequality follows from Assumption \[ass2\_and\_a\_half\_IHM\] and inequality .
The next task is to take the limit as $k\to\infty$ on both sides of inequality . Since the sequence $\{ \pi_{1+n_k}\}_{k \in {\mathbb{N}}}$ converges locally uniformly to the locally Lipschitz policy $\tilde\pi_{\lim}$ (resp. $\pi_{\lim}$), Theorem \[limitsIHM\] implies $V_{\tilde\pi_{\lim}} = V_{\lim}$ (resp. $V_{\pi_{\lim}}= V_{\lim}$). Proposition \[operatorIHM\] implies $\mathcal{L}_{\tilde\pi_{\lim}} V_{\lim} =0=\mathcal{L}_{\pi_{\lim}} V_{\lim}$. Hence we can express $\mathcal{L}_{\pi_{n_k + 1}} V_{\pi_{n_k}}= \mathcal{L}_{\pi_{n_k + 1}}V_{\pi_{n_k}}-\mathcal{L}_{\tilde\pi_{\lim}}V_{\pi_{n_k}}+
\mathcal{L}_{\tilde\pi_{\lim}} V_{\pi_{n_k}}-\mathcal{L}_{\tilde\pi_{\lim}} V_{\lim}$. By Schauder’s boundary estimate for elliptic PDEs [@FriedmanParabolic p. 86], the sequences $\{ {\nabla}V_{\pi_{n_k}} \}_{k \in {\mathbb{N}}}$ and $\{ {\mathrm{H}}V_{\pi_{n_k}} \}_{k \in {\mathbb{N}}}$ are uniformly bounded on $x+D_K$. By Assumption \[ass1IHM\] and Remark \[rem:V\_F\_bounded\], the bounded sequence $\{({\sigma}_{\pi_{n_k+1}}, \mu_{\pi_{n_k+1}},\alpha_{\pi_{n_k+1}}, f_{\pi_{n_k+1}}, V_{\pi_{n_k+1}}) \}_{k\in{\mathbb{N}}}$ tends to the limit $({\sigma}_{\tilde\pi_{\lim}}, \mu_{\tilde\pi_{\lim}},\alpha_{\tilde\pi_{\lim}}, f_{\tilde\pi_{\lim}}, V_{\tilde \pi_{\lim}})$ uniformly on $x+D_K$ as $k\uparrow\infty$. Hence, so does $$\label{eq:Gen_Conv}
\mathcal{L}_{\pi_{n_k + 1}}V_{\pi_{n_k}}-\mathcal{L}_{\tilde\pi_{\lim}}V_{\pi_{n_k}}=
[ L_{\pi_{n_k+1}}-L_{\tilde\pi_{\lim}}] V_{\pi_{n_k}} - (\alpha_{\pi_{n_k+1}}- \alpha_{\tilde \pi_{\lim}})V_{\pi_{n_k}} +
(f_{\pi_{n_k+1}}- f_{\tilde \pi_{\lim}})\to0. $$
By the elliptic version of Theorem 15 in [@FriedmanParabolic p. 80] applied to the family of PDEs $\mathcal{L}_{\pi_{n_k}}V_{\pi_{n_k}}=0$, $k\in{\mathbb{N}}$, there exists a subsequence of $\{V_{\pi_{n_k}}\}_{k\in{\mathbb{N}}}$ (again denoted by $\{V_{\pi_{n_k}}\}_{k\in{\mathbb{N}}}$), such that the corresponding sequence $\{( V_{\pi_{n_k}}, {\nabla}V_{\pi_{n_k}}, {\mathrm{H}}V_{\pi_{n_k}})\}_{k\in{\mathbb{N}}}$ converges uniformly on the closed ball $x+D_K$ to $(V_{\pi_{\lim}}, {\nabla}V_{\pi_{\lim}}, {\mathrm{H}}V_{\pi_{\lim}})=(V_{\lim}, {\nabla}V_{\lim}, {\mathrm{H}}V_{\lim})$. Hence it follows that $$\label{eq:Gen_Conv_1}
\mathcal{L}_{\tilde\pi_{\lim}} V_{\pi_{n_k}}-\mathcal{L}_{\tilde\pi_{\lim}} V_{\lim}=
L_{\tilde\pi_{\lim}} (V_{\pi_{n_k}} - V_{\lim})-\alpha_{\tilde\pi_{\lim}}(V_{\pi_{n_k}} - V_{\lim})\to0,
\qquad\text{as $k\to\infty$.}$$ Equations and imply that $\mathcal{L}_{\pi_{n_k + 1}} V_{\pi_{n_k}}\to0$ as $k\to\infty$ uniformly on the ball $x+D_K$.
Apply the DCT to the right-hand side of and Assumption \[ass1IHM\] and Remark \[rem:V\_F\_bounded\] to its left-hand side: $$\begin{aligned}
V_{\lim}(x) \leq
{\mathbb{E}}\int_0^{t \wedge \tau_K} \mathrm{e}^{-\int_0^s \alpha_{\Pi_r} \left( X^{\Pi,x}_r \right) \mathrm{d}r} f_{\Pi_s} \left( X^{\Pi,x}_s \right) \mathrm{d}s +
C_0 {\mathbb{E}}\mathrm{e}^{-\epsilon_0 (t \wedge \tau_K) }.\end{aligned}$$ Since this inequality holds for all $K,t>0$ and $\tau_K\uparrow\infty$ as $K\uparrow\infty$, the inequality $V_{\lim}(x)\leq V_\Pi(x)$ follows by the DCT as $t\wedge K\to\infty$ (cf. the last paragraph of the proof of Theorem \[limitsIHM\]).
Auxiliary results - the one-dimensional case {#subsec:aux_one_dim}
--------------------------------------------
Throughout Sections \[subsec:aux\_one\_dim\] and \[subsec:Proofs\_OneDim\], define $\tau_c^d(Z):= \inf \{ t \geq 0;\;Z_t \in\{c,d\} \}$ ($\inf \emptyset = \infty$) for any continuous stochastic process $(Z_t)_{t\in{\mathbb{R}}_+}$ in ${\mathbb{R}}$ and $-\infty\leq c<d\leq\infty$.
\[boundaryIHP\] For any Markov policy $\pi:(a,b)\to A$, the payoff function $V_\pi : (a,b) \to {\mathbb{R}}$ can be continuously extended by defining $V_\pi(a) := g(a)$ if $a > -\infty$ and $V_\pi(b) := g(b)$ if $b < \infty$.
Let $\{ x_n \}_{n \in {\mathbb{N}}}$ be a decreasing sequence in $(a,b)$ that converges to $a > -\infty$. We now prove that $\lim_{n \to \infty} V_\pi(x_n) = g(a)$ (the argument for $b$ is analogous).
Pick arbitrary $\epsilon > 0$. Since $\mu$ is bounded and ${\sigma}^2$ bounded and bounded away from $0$, a simple coupling argument yields that the process $X^{\pi,x_n}$ can be bounded by a Brownian motion with drift so that ${\mathbb{P}}\left( \tau_a^{\infty} \left( X^{\pi,x_n} \right) > \epsilon \right) < \epsilon$ and ${\mathbb{P}}\left( \tau_a^{\infty} \left( X^{\pi,x_n} \right) > \tau_{-\infty}^{b} \left( X^{\pi,x_n} \right) \right) < \epsilon$ hold for large $n \in {\mathbb{N}}$. Hence, there exists $n_0\in{\mathbb{N}}$ such that $$\begin{aligned}
{\mathbb{P}}\left( \tau_a^{\infty} \left( X^{\pi,x_n} \right) > \epsilon \wedge \tau_{-\infty}^{b} \left( X^{\pi,x_n} \right) \right)
\leq {\mathbb{P}}\left( \tau_a^{\infty} \left( X^{\pi,x_n} \right) > \epsilon \right)
+ {\mathbb{P}}\left( \tau_a^{\infty} \left( X^{\pi,x_n} \right) > \tau_{-\infty}^{b} \left( X^{\pi,x_n} \right) \right)
< 2 \epsilon\end{aligned}$$ for all $n\geq n_0$. Define the quantities $B_a^b:= | \mathrm{e}^{-\int_0^{\tau_a^b ( X^{\pi,x_n} )} \alpha_{\pi} ( X^{\pi,x_n}_s ) \mathrm{d}t}
g ( X^{\pi,x_n}_{\tau_a^b(X^{\pi,x_n})} ) - g(a) |$ and $A_a^b:=
\int_0^{\tau_a^b ( X^{\pi,x_n} )} \mathrm{e}^{-\int_0^t \alpha_{\pi} ( X^{\pi,x_n}_s )
\mathrm{d}s} | f_{\pi} ( X^{\pi,x_n}_t ) | \mathrm{d}t$ and the event $C:=\{ \tau_a^{\infty} \left( X^{\pi,x_n} \right) \leq \epsilon \wedge \tau_{-\infty}^{b} \left( X^{\pi,x_n} \right) \}$. Then we have $$\begin{aligned}
|V_\pi(x_n) - g(a)| \leq
{\mathbb{E}}\left( (A_a^b+B_a^b) {\mathbb{I}}_{\Omega\setminus C}+
(A_a^b+B_a^b) {\mathbb{I}}_{C} \right). \end{aligned}$$
We now show that there exists $M > 0$, which does not depend on $\epsilon$, such that $|V_\pi(x_n) - g(a)|$ is bounded above by $4M \epsilon$ for all $n \geq n_0$. The expectation on the event $\Omega\setminus C$, which has probability less than $2 \epsilon$, is smaller than $2M \epsilon$ since $f,g$ are bounded and $\alpha\geq\epsilon_0>0$. On the event $C$ we have $\tau_a^b ( X^{\pi,x_n} )\leq \epsilon$, which implies ${\mathbb{E}}A_a^b {\mathbb{I}}_C<M\epsilon$. On $C$ it holds that $X^{\pi,x_n}_{\tau_a^b(X^{\pi,x_n})}=a$. Hence, the elementary inequality $1 - \mathrm{e}^{-x} \leq x$ for $x \geq 0$, yields an upper bound on ${\mathbb{E}}B_a^b {\mathbb{I}}_C$ of the form $ | g(a)| {\mathbb{E}}\left( \int_0^{\epsilon} \alpha_{\pi} \left( X^{\pi,x_n}_t \right) \mathrm{d}t \right)$. This concludes the proof.
Lemma \[martingaleIHO\] is the analogue of Lemma \[martingaleIHM\] with an analogous proof, which we omit for brevity.
\[martingaleIHO\] The following holds for every Markov policy $\pi$, $x \in (a,b)$ and stopping time $\rho$: $$\begin{aligned}
& {\mathbb{E}}\Bigg( F_{\tau_a^b( X^{\pi,x} )}(X^{\pi,x})
+ \mathrm{e}^{-\int_0^{\tau_a^b(X^{\pi,x})} \alpha_\pi \left( X^{\pi,x}_t \right) \mathrm{d}t}\, g \left( X^{\pi,x}_{\tau_a^b(X^{\pi,x})} \right)
{\mathbb{I}}_{\{ \tau_a^b(X^{\pi,x}) < \infty \}} \Bigg\vert {\mathcal{F}}_\rho \Bigg) = M_\rho, \end{aligned}$$ where $M_r :=
F_{r \wedge \tau_a^b( X^{\pi,x} )}(X^{\pi,x})
+ {\mathbb{I}}_{\{ r < \infty\}} \mathrm{e}^{-\int_0^{r \wedge \tau_a^b ( X^{\pi,x} )} \alpha_\pi
( X^{\pi,x}_s ) \mathrm{d}s}\,
V_\pi( X^{\pi,x}_{r \wedge \tau_a^b ( X^{\pi,x})})$, for $r\in[0,\infty]$. In particular, the process $M=(M_r)_{r\in[0,\infty]}$ is a uniformly integrable martingale.
Proofs of results in Section \[ch3IHO\] {#subsec:Proofs_OneDim}
---------------------------------------
Recall that Assumption \[ass1IHM\] holds. We need to show that for any locally Lipschitz Markov policy $\pi:(a,b)\to A$ we have $V_\pi \in {\mathcal{C}}^2((a,b))$ and $L_\pi V_\pi - \alpha_\pi V_\pi + f_\pi = 0$.
Let $a < a' < a'' < x < b'' < b' < b$, and for any $c < d$ denote $\tau_{c}^{d} := \tau_{c}^{d}(X^{\pi,x})$. Let $v \in {\mathcal{C}}^2((a',b')) \cap {\mathcal{C}}([a',b'])$ be the unique solution of the boundary value problem $L_\pi v - \alpha_\pi v + f_\pi = 0$, $v(a') = V_\pi(a')$, $v(b') = V_\pi(b')$, guaranteed to exist by Theorem 19 in [@FriedmanParabolic p. 87], which is applicable by Assumption \[ass1IHM\]. Let $ S_t^{a'',b''} := F_{t \wedge \tau_{a''}^{b''}}(X^{\pi,x}) +
\mathrm{e}^{-\int_0^{t \wedge \tau_{a''}^{b''}} \alpha_{\pi} ( X^{\pi,x}_r ) \mathrm{d}r}
v ( X^{\pi,x}_{t \wedge \tau_{a''}^{b''}})$. Then, by Itô’s formula on $[0,\tau_{a''}^{b''}]$ and the definition of $v$, the process $S^{a'',b''} = (S_t^{a'',b''})_{t \geq 0}$ satisfies $$\begin{aligned}
S_t^{a'',b''} & = &
v(x) + \int_0^{t \wedge \tau_{a''}^{b''}} \mathrm{e}^{-\int_0^s \alpha_{\pi} \left( X^{\pi,x}_r \right) \mathrm{d}r}
{\sigma}_\pi v' \left( X^{\pi,x}_s \right) \mathrm{d}B_s.\end{aligned}$$ Hence $S^{a'',b''}$ is clearly a uniformly integrable martingale and the following equalities hold: $\lim_{t\uparrow\infty}{\mathbb{E}}| S_t^{a'',b''}- S_\infty^{a'',b''}|=0$ and $v(x)={\mathbb{E}}S_\infty^{a'',b''}$. Define $S^{a',b'}$ by substituting $\tau_{a''}^{b''}$ in the definition of $S^{a'',b''}$ with $\tau_{a'}^{b'}$. Since $X^{\pi,x}$ is continuous, we have $\lim_{a''\downarrow a',b''\uparrow b'}\tau_{a''}^{b''}=\tau_{a'}^{b'}$ a.s. Hence, by the DCT, $v(x) = \lim_{a''\downarrow a',b''\uparrow b'} {\mathbb{E}}S_\infty^{a'',b''} = {\mathbb{E}}S_\infty^{a',b'}$.
Note that the boundary conditions for $v$, the fact $X^{\pi,x}_{\tau_{a'}^{b'}}\in\{a',b'\}$ and Lemma \[martingaleIHO\] (with $\rho=\tau_{a'}^{b'}$) imply $ S_\infty^{a',b'}
= {\mathbb{E}}( F_{\tau_{a}^{b}}(X^{\pi,x})
+ \mathrm{e}^{-\int_0^{\tau_{a}^{b}}
\alpha_{\pi} ( X^{\pi,x}_r ) \mathrm{d}r}
g ( X^{\pi,x}_{\tau_{a}^{b}} ) {\mathbb{I}}_{\{ \tau_{a}^{b} < \infty \}} \vert {\mathcal{F}}_{\tau_{a'}^{b'}} )$. Taking expectations on both sides of this equality yields $v(x)
= V_\pi(x)$.
We claim that under Assumptions \[ass1IHM\]–\[ass2\_and\_a\_half\_IHM\], the inequality $V_{\pi_{n+1}}(x) \leq V_{\pi_{n}}(x)$ holds for all $x\in(a,b)$ and $n\in{\mathbb{N}}$, where $\pi_{n+1}$ is defined in .
Define the process $Y$ as in in Section \[subsec:Proofs\_Multidim\] and consider the stopped process $Y_{\cdot\wedge \tau_{a'}^{b'}}$, where $a < a' < x < b' < b$ and $\tau_{c}^{d} := \tau_{c}^{d}(X^{\pi_{n+1},x})$ for any $c < d$. Then the proof follows the same steps as the proof of Theorem \[decreasingIHM\] in Section \[subsec:Proofs\_Multidim\]. The only difference is that in the penultimate line of the proof of Section \[subsec:Proofs\_Multidim\] we apply the DCT and Lemma \[boundaryIHP\] (instead of the DCT only) to obtain $V_{\pi_n}(x)\geq V_{\pi_{n+1}}(x)$.
The proof of the one-dimensional case of Proposition \[subsequenceIHM\] is completely analogous to the multi-dimensional one and is hence omitted.
We need to show that $V_{\lim}(x) = V_{\pi_{\lim}}(x)$ holds for all $x\in(a,b)$. The proof follows along the same lines as in the multi-dimensional case of Section \[subsec:Proofs\_Multidim\]. The only difference lies in the fact that we stop the process $X^{\pi_{\lim},x}$ at $\tau_{a'}^{b'}(X^{\pi_{\lim},x})$, where $a<a'<x<b'<b$, and take the limit as $(a',b',t)\to(a,b,\infty)$.
The verification lemma in the one-dimensional case is established exactly as in the proof of Theorem \[verificationIHM\]. The details are omitted.
[10]{}
A. Arapostathis, On the Policy Iteration Algorithm for Nondegenerate Controlled Diffusions Under the Ergodic Criterion. *Optimization, Control, and Applications of Stochastic Systems*, 1–12, 2012.
A. Arapostathis and V.S. Borkar, Uniform recurrence properties of controlled diffusions and applications to optimal control. *SIAM J. Control Optim.*, 48(7), 152–160, 2010.
P. Azimzadeh, E. Bayraktar and G. Labahn, Convergence of approximation schemes for weakly nonlocal second order equations. *arXiv:1705.02922v2*, 2017.
P. Bertsekas, Approximate Policy Iteration: a Survey and Some New Methods. *Journal of Control Theory and Applications*, 9(3):310–335, 2011.
A. N. Borodin, P. Salminen, *Handbook of Brownian Motion – Facts and Formulae*. Birkhauser, Basel, second edition, 2002.
A. S. Cherny and H.-J. Engelbert, *Singular Stochastic Differential Equations*. Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg, 2005.
B. T. Doshi, Continuous Time Control of Markov Processes on an Arbitrary State Space: Discounted Rewards. *The Annals of Statistics*, 4(6):1219–1235, 1976.
E.B. Dynkin, *Markov Processes, vol I*. Springer-Verlag, Berlin, 1965.
E. Fournié, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, N. Touzi, Applications of Malliavin calculus to Monte Carlo methods in finance. *Finance and Stochastics*, 3:391–412, 1999.
A. Friedman, *Partial Differential Equations of Parabolic Type*. Prentice-Hall, Englewood Cliffs, N.J., 1964.
O. Hernández-Lerma and J. B. Lasserre, Policy iteration for average cost Markov control processes on Borel spaces. *Acta Applicandae Mathematicae*, 47(2):125–154, 1997.
A. Hordijk and M. L. Puterman, On the convergence of policy iteration in finite state undiscounted Markov decision processes: the unichain case. *Mathematics of Operations Research*, 12(1):163–176, 1987.
R. A. Howard, *Dynamic Programming and Markov Processes*. MIT Press, Cambridge, 1960.
I. Karatzas and S. E. Shreve, *Brownian Motion and Stochastic Calculus*. Springer-Verlag, New York, second edition, 1991.
N. V. Krylov, *Controlled Diffusion Processes*. Springer-Verlag, New York, 1980.
M. G. Lagoudakis and R. Parr, Least-Squares Policy Iteration. *Journal of Machine Learning Research*, 4:1107-1149, 2003.
J. B. Lasserre, A new policy iteration scheme for Markov decision processes using Schweitzer’s formula. *Journal of Applied Probability*, 31(1):268–273, 1994.
T. Lindvall and L. C. G. Rogers, Coupling of Multidimensional Diffusions by Reflection. *The Annals of Probability*, 14(3):860–872, 1986.
S. Mahadevan, Representation Policy Iteration. ArXiv:1207.1408 \[cs.AI\], 2012.
S. P. Meyn, The policy iteration algorithm for average reward Markov decision processes with general state space. *IEEE Transactions on Automatic Control*, 42(12):1663–1680, 1997.
P. E. Protter, *Stochastic Integration and Differential Equations*. Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin Heidelberg, 2004.
J. P. Rust, A Comparison of Policy Iteration Methods for Solving Continuous-State, Infinite-Horizon Markovian Decision Problems Using Random, Quasi-Random, and Deterministic Discretizations. Available at http://ssrn.com/abstract=37768, 1997.
M. S. Santos and J. Rust, Convergence properties of policy iteration. *SIAM Journal on Control and Optimization*, 42(6):2094–2115, 2004.
Q. X. Zhu, X. S. Yang and C. X. Huang, Policy iteration for continuous-time average reward Markov decision processes in Polish spaces. *Abstract and Applied Analysis*, 2009, Article ID103723, 17 pages.
[^1]: SD supported by the EPSRC grant EP/P00377X/1; AM supported by the EPSRC grant EP/P003818/1 and the Programme on Data-Centric Engineering funded by Lloyd’s Register Foundation; DŠ supported by the Slovene Human Resources Development and Scholarship Fund.
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abstract: 'Generalized complex geometry [@hit] and, more generally, Dirac geometry [@cou], [@cow], unify several familiar geometric structures into one uniform viewpoint. We introduce two new notions of morphisms between manifolds equipped with Dirac structures, giving two different Dirac categories. The first generalizes holomorphic and Poisson maps, while the second dual notion generalizes both symplectic and holomorphic maps. As an application, we consider Dirac groups (i.e. Lie groups with Dirac structure such that group multiplication is a Dirac map). We explain the conditions under which a group with Dirac structure is a Dirac group. More precisely, we explain the data and conditions for a Dirac group. Dirac groups turn out to be a generalization of Poisson groups.'
author:
- Brett Milburn
date:
title: Two Categories of Dirac Manifolds
---
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Introduction
============
Complex Dirac structures on manifolds generalize several basic concepts of differential geometry:Poisson, presymplectic, and complex structures as well as integrable distributions. Dirac structures are originally due to Courant and Weinstein [@cow], [@cou]. Formally, this is a generalization of integrable distributions where the tangent bundle $TM$ is replaced by $ {{\mathcal V}}_M=TM\oplus T^*M$ and the Lie algebroid structure on $TM$ is replaced by the Courant algebroid structure which is a bracket operation $[-,-]$ on $ {{\mathcal V}}_M$. Hitchin’s notion of a generalized complex structure [@hit] is a specific type of Dirac structure on which we will focus much of our attention here. Generalized complex geometry contains both complex and symplectic geometry and has found a number of applications in mathematics and physics [@kap], [@hit], [@gua].\
Although Dirac geometry is a common viewpoint from which to consider these more traditional structures, there has not yet been a definition of morphisms between manfiolds with Dirac structure which has been widely used or accepted. Several possible approaches have been put forth and are described in section \[comp\]. In this paper we define two categories of Dirac manifolds (i.e. manifolds with complex Dirac structures). The first notion of maps we call *Dirac maps*, and the corresponding category is seen to contain the categories of Poisson and complex manifolds as full subcategories. We call this category simply the *Dirac category*. Actually, the Dirac category seems to be exactly a unification or common setting for Poisson and complex manifolds. With Poisson and complex maps at two opposite ends of the spectrum, the general case is an interpolation between or mixture of the two.\
Our only application of Dirac maps at present is the study of groups in the category of Dirac manifolds. These *Dirac groups* are a combination of Poisson groups and complex groups, though it is useful to think of them as a generalization of Poisson groups. A particularly interesting class of Dirac groups are generalized complex groups. These turn out to be equivalent to holomorphic Poisson groups, so they provide a (non-holomorphic) point of view on quantization of holomorphic Poisson groups.\
The second notion, *dual-Dirac maps*, defines a *dual-Dirac category* which contains presymplectic and complex manifolds as full subcategories. Generalized complex structures can be viewed as an interpolation between symplectic and complex structures, especially in the hyperkähler setting [@gua]. The dual-dirac maps allow for a uniform point of view on symplectic and complex maps so that in any setting involving a variation between the two extremes, it makes sense to talk about maps in a consistent way. The notion of dual-Dirac maps has an additional property that it is stable under B-transforms.\
Dirac maps and dual-Dirac maps provide two new structures of a category on Hitchin’s generalized complex manifolds, i.e. two reasonable notions of generalized complex maps. We establish conditions for which multiplication in a Lie group is a dual-Dirac map and partially classify group objects in the dual-Dirac category.\
Finally, we generalize further by also considering categories of Dirac manifolds for which Dirac structures lie in arbitrary exact Courant algebroids (i.e. Courant algebroids ${{\mathcal E}}$ on $M$ which are extensions $0 {{\longrightarrow}}T^*M {{\longrightarrow}}{{\mathcal E}}{{\longrightarrow}}TM {{\longrightarrow}}0$ of the tangent Lie algebroid by the cotangent bundle). This generality is crucial for expected applications to Representation Theory of affine Lie algebras. We mostly consider the dual-Dirac category, which extends readily and naturally to the case of arbitrary exact Courant algebroids because of stability under B-transforms. We also find an extension of the Dirac category to exact Courant algebroids. However, this extension is less natural since it requires keeping track of additional data. The problem of a natural theory of Dirac maps for exact Courant algebroids seems to have real content. It may be related to the notion of *vertex algebroids*, which are a quantization of the notion of Courant algebroids.\
Recollections on Dirac and Generalized Complex Geometry {#expo}
=======================================================
This section is a brief expostion on the essential ideas in generalized complex geometry. Here we introduce the basic definitions and notational conventions used in this paper. For a systematic development of generalized complex structures as well as some of their applications, we refer the reader to [@gua].\
For a manifold $M$, generalized geometry is concerned with the bundle $\mathcal V _M := TM \oplus
T^*M$. There is a natural bilinear form on $\mathcal V _M$, given by the obvious pairing $\langle X + \xi , Y + \eta \rangle = X(\eta) + Y (\xi)$ for sections $X,Y$ of $TM$ and $\xi $, $\eta$ of $T^*M$. Furthermore, $\mathcal V _M$ is equipped with the *Courant bracket* defined by $$[X + \xi , Y + \eta ] = [X,Y] + \io _X d \eta +\frac{1}{2} d( \io _X \eta) - \io _Y d \xi - \frac{1}{2} d(\io _Y \xi),$$ where $\io$ denotes contraction in the first variable ($ \io _x \phi = \phi (x,-,...)$). The Courant bracket $[ \; , \; ]$ and the bilinear form $\langle \; , \; \rangle$ extend ${{\mathbb C}}$-bilinearly to $(\mathcal V _M )_{{\mathbb C}}= \mathcal V _M
\otimes {{\mathbb C}}$.
A *generalized almost complex structure* on $M$ is a map ${{\mathcal J}}: {{\mathcal V}}_M {{\longrightarrow}}{{\mathcal V}}_M$ such that ${{\mathcal J}}$ is orthogonal with respect to the inner product $\langle \; , \; \rangle$ and ${{\mathcal J}}^2 = -1$. Just as with complex structures, one may consider the $i$-eigenbundle, $D$, of ${{\mathcal J}}$ in $({{\mathcal V}}_M)_{{\mathbb C}}$. The Courant bracket defines an integrability condition ($[D , D] \subset D$) for ${{\mathcal J}}$ to be called a *generalized complex structure*. This follows the analogy with almost complex stuctures; an almost complex structure is a complex structure precisely when its $i$-eigenbundle is integrable with respect to the Lie bracket.
The two canonical examples of generalized complex structures come from complex and symplectic structures. Since ${{\mathcal V}}_M = TM \oplus T^*M$, we can express any map ${{\mathcal V}}_M {{\longrightarrow}}{{\mathcal V}}_M$ as a block matrix in terms of this decomposition, and we will follow this convention throughout the text. If $J$ is a complex structure,
$$\begin{bmatrix}
J & 0 \\
0 & -J^*
\end{bmatrix}$$
is a generalized complex structure. The $i $-eigenbundle of ${{\mathcal J}}$ is $E \oplus Ann(E)$, where $E$ is the $i$-eigenbundle of $J$, and $Ann(E)$ is the annihilator of $E$ in $T^*M$.\
For a symplectic structure $\om$, we get a generalized complex structure $$\begin{bmatrix}
0 & -\om _\sharp {^{-1}}\\
\om _\sharp & 0
\end{bmatrix}$$ where $\om _\sharp (x) := \om (x , -)$. The $i$-eigenbundle is the graph of $i\om _\sharp $ in $({{\mathcal V}}_M)_{{\mathbb C}}$. The fact that the symplectic form $\om$ is closed implies that this generalized almost complex structure is integrable, hence a generalized complex structure.\
The $i$-eigenbundle $D$ of a generalized complex structure ${{\mathcal J}}$ turns out to be an integrable maximal isotropic subbundle of $({{\mathcal V}}_M)_{{\mathbb C}}$, also known as a *complex Dirac structure*. Thus, the study of generalized geometry now lies in the framework of Dirac structures. With this in mind, we recall the following working definitions for our paper.
For any manifold, $M$,
1. A *real almost Dirac structure* on $M$ is a maximal isotropic subbundle $D$ of $\mathcal V _M$. A real almost Dirac structure is called a *real Dirac structure* if it is integrable with respect to the Courant bracket. Similarly, a *complex almost Dirac structure* is a maximal isotropic subbundle $D \subset (\mathcal V _M )_{{\mathbb C}}$, and a *complex Dirac structure* is an integrable complex almost Dirac structure.
2. A complex Dirac structure $D$ is said to be of *constant rank* if the projection map $pr : D {{\longrightarrow}}TM$ is of constant rank.
A generalized (almost) complex structure ${{\mathcal J}}$ is equivalent to is a complex (almost) Dirac structure $D$ such that $D \cap \overline D = 0$. Note that the integrability is a closed condition and that being generalized complex is an open condition. Henceforth we will think of generalized complex structures as complex Dirac structures.\
Since the complexification of any Dirac structure is a complex Dirac structure, both generalized complex structures and Dirac structures are complex Dirac structures. Thus, the set of complex Dirac structures contains real Dirac structures and generalized complex structures. Henceforth (almost) *Dirac structure* will always mean complex (almost) Dirac structure, and we will specify whether it is also real Dirac (i.e. if $\overline {\mathcal D } = \mathcal D$) if there is any ambiguity.\
Most of the complex Dirac structures considered in this paper will be of constant rank. It is checked in [@gua] that any complex Dirac structure of constant rank is of the form $$L(E , {\varepsilon}) := \{ X + \xi \in (\mathcal V _M)_{{\mathbb C}}\; | \; X \in E \; and \; \iota _X {\varepsilon}= \xi _{|E} \},$$ where $E$ is a subbundle of $TM$ and ${\varepsilon}\in \Ga (M , \wedge ^2 E^*)$. Complex and symplectic structures, for example, are of this form. For a subbundle $E$ of $TM$, we define the differential\
$d_E : \Ga (M, \wedge ^2 E^*) {{\longrightarrow}}\Ga (M, \wedge ^3 E^*)$ by the following forumula. For sections $X,Y,Z$ of $E$ and ${\varepsilon}\in \Ga (M , \wedge ^2 E^*)$, $$d_E {\varepsilon}(X,Y,Z) = {\varepsilon}(X,[Y,Z]) + {\varepsilon}(Y,[Z,X]) + {\varepsilon}(Z,[X,Y]) + X{\varepsilon}(Y,Z) - Y{\varepsilon}(X,Z) + Z{\varepsilon}(X,Y).$$ In other words, $d_E {\varepsilon}$ is the restriction to $\wedge ^3 E $ of the ordinary De Rham differential of any extension $\tilde {\varepsilon}\in \wedge ^2 T^*M $ of ${\varepsilon}$. Gualtieri [@gua] proves the following useful lemma.
\[21sept1\] A complex almost Dirac structure of constant rank $L(E ,
{\varepsilon})$ is integrable if and only if $E$ is integrable and $d_E {\varepsilon}=0$.
Dirac groups (§\[13Dec4\]) provide interesting examples of Dirac structures which are not of constant rank. In particular, generalized complex groups provide examples of generalized complex structures not of constant rank.
We recall from [@gua] that the following procedures create Dirac structures from geometric structures on $M$. The first three are real, and (4) and (5) are special cases of generalized complex structures (as we have seen).
1. To an integrable distribution $\mathcal{D}\subset TM$, assign $[\mathcal{D}\oplus Ann(\mathcal{D})]_{{\mathbb C}}$.
2. To a Poisson structure $\pi \in \Ga (M, \wedge ^2 TM)$, assign $L(\pi , T^*M)$.
3. To a presymplectic structure $\om \in \Om ^2 (M)$, assign $L(TM , \om)$.
4. To a complex structure $J$, assign $T^{(1,0)}M \oplus T^{* ,(0,1)}M$, where $T^{(1,0)} M$ and $T^{(0,1)}M$ denote the holomorphic and antiholomorphic tangent bundles respectively with respect to $J$.
5. To a symplectic structure $\om \in \Om ^2 (M)$, assign $L(TM_{{\mathbb C}}, i \om)$.
A symplectic form $\omega$ on $M$ determines a complex Dirac structure in one of two ways: $L(TM, \omega) $ and $L(T_{{\mathbb C}}M, -i\omega)$. The former is a real Dirac structure, and the latter is a generalized complex structure.
This way of representing a Dirac structure $L$ as $L(E,{\varepsilon})$ turns out to be extremely useful for our purposes. For any vector bundle $E$ and ${\varepsilon}\in \bigwedge ^2 E^*$, the convention used in this paper is for $\varepsilon _\sharp$ to denote the map $E {{\longrightarrow}}E^*$ determined by ${\varepsilon}$. That is, for $X,Y \in E$, $({\varepsilon}_\sharp X)(Y) = {\varepsilon}(X,Y)$.\
For a Dirac structure $D$, if the projection $pr : D {{\longrightarrow}}T^*M$ has constant rank, then there is some subbundle $U \subset T^*M$ and some $\pi \in \Ga (M, \bigwedge ^2 U^*)$ such that $D$ is of the form $$L(\pi , U) := \{ X + \xi \st X_{|U} = \io _\xi \pi \}.$$ If $U = T^*M$, then for $\pi \in \Ga (M , \wedge ^2 TM)$, $L(\pi , T^*M)$ is a Dirac structure if and only if $\pi$ is a Poisson bi-vector [@gua], [@vai]. Now presymplectic structures, complex structures, and Poisson structures can all be considered Dirac structures.\
We recall the notions of pullback and pushforward of linear Dirac structures [@gua]. For a map $F: V {{\longrightarrow}}W$ of vector spaces and a subspace $D \subset V \oplus V^*$, define $$F_\star D = \{ FX + \xi \in W \oplus W^* \st X + F^* \xi \in D \}$$ and for a subspace $D \subset W \oplus W^*$, define $$F^\star D = \{ X + F^* \xi \in V \oplus V \st FX + \xi \in D \} = (F^*)_\star D.$$
Now let $f : M {{\longrightarrow}}N$ be any map of manifolds. For a Dirac structure, $D$, on $N$, the pullback $f^\star D$ is defined pointwise by $(f^\star D)_p = (df_p)^\star D_{f(p)} $. It is not necessarily itself a Dirac structure.\
Twisted Courant Bracket and Automorphisms
-----------------------------------------
In addition to the standard Courant bracket on ${{\mathcal V}}_M$, Ševera and Weinstein noticed a twisted Courant bracket $[ \; , \; ] _H$ for each closed 3-form $H$ on $M$, defined as $$[X + \xi , Y + \eta ] _H = [X + \xi , Y + \eta ] + H(X,Y,-)$$ and ${{\mathcal V}}_{M,H} := (TM \oplus T^*M , [ \; , \; ] _H , \langle \; , \; \rangle )$ so that ${{\mathcal V}}_M$ is just ${{\mathcal V}}_{M,0}$. For any 2-form $B$ on $M$, there is an automorphism of the vector bundle $TM \oplus T^*M$,
$$\begin{bmatrix}
1 & 0 \\
B_\sharp & 1
\end{bmatrix},$$ denoted $e^B$. Indeed, $e^B$ is an isomorphism ${{\mathcal V}}_{M,H+dB} {{\longrightarrow}}{{\mathcal V}}_{M, H}$. In other words, $e^B$ is orthogonal with respect to $ \langle \; , \; \rangle $, and $[e^B u , e^B v]_H = e^B [u,v]_{H +dB}$ for all sections $u,v$ of $TM \oplus T^*M$. When $B$ is closed, then $e^B$ is an automorphism of ${{\mathcal V}}_{M,H}$. This is what we call a *B-transform*, i.e. an automorphism of ${{\mathcal V}}_{M,H}$ of the form $e^B$ for a closed 2-form $B$. In fact, the automorphism group of ${{\mathcal V}}_{M,H}$ is the semidirect product of the group of diffeomorphisms $M {{\longrightarrow}}M$ and closed 2-forms $Z^2(M)$ [@gua]. B-transforms are thought of as the symmetries of the Courant bracket.\
An *H-twisted Dirac structure* $D \subset {{\mathcal V}}_{M,H}$ is simply a maximal isotropic subundle which is integrable with respect to the $H$-twisted Courant bracket. We discuss examples of these in §\[twisted\] , some of which are generalizations of $H$-twisted Poisson structures [@swe].
Linear Dirac Maps {#13Dec1}
=================
A *linear Dirac structure* is a maximal isotropic subspace $D$ of $V \oplus V^*$ for some vector space $V$.
In order to define morphisms between manifolds with Dirac structures, we first develop this notion for linear maps between vector spaces equipped with linear Dirac structures.\
For a vector space V, we will denote by $p_V : V\oplus V^* \longrightarrow V $ and $p_{V^*} : V\oplus V^* \longrightarrow V^* $ the natural projection maps. There are natural notions of pullbacks and pushforwards of Dirac structures, first introduced by Weinstein. For a linear map $f : V {{\longrightarrow}}W$ the pullback of a linear Dirac structure $L_W \subset W\oplus W^*$ is defined as $f^\star L_W := \{ X + f^*\xi \in V \oplus V^* \st fX + \xi \in L \}$, and the pushforward of a linear Dirac structure $L_V$ on $V$ is $f_\star L_V := (f^*)^\star L_V$. These are again linear Dirac structures.\
\[a1\] Let $L_{1},L_{2}$ be linear Dirac structures on vector spaces ${V_1},{V_2}$. A linear map $f: {V_1} \longrightarrow {V_2} $ is said to be a *linear Dirac map* if:\
$$f(L_{{1}} \cap {V_1} ) \subset L_{{2}} \cap {V_2}, \; and
\tag{M1}$$ $$p_{{V_1} ^*} {^{-1}}(f^* (p_{{V_2} ^*} L_{{2}})) \cap L_{{1}} \subset f^\star L_{{2}}.
\tag{M2}$$
Such a map will also be written as $f: ({V_1},L_{V_1} ) \longrightarrow
({V_2},L_{V_2}) $. Notice that a linear Dirac structure on ${V_1}$ is also a linear Dirac structure on ${V_1}^*$, however Dirac maps are not invariant under duality. We will say that $f$ is a *linear dual-Dirac map* if $f^*:\ ({V_2}^*,L_{V_2})
{{\longrightarrow}}({V_1}^*,L_{V_1})$ is a linear Dirac map.
For a map $F: (V_1,L_1) \longrightarrow (V_2,L_2)$, condition (M2) of Definition \[a1\] is equivalent to either of the two following conditions: $$Y+\xi \in L_2 \; \; \mbox{and} \; \; X + f^*\xi \in L_1 \;
\Longrightarrow \; fX + \xi \in L_2 ,\tag{M2$^{\prime}$}$$ $$f^\star L_2 \subset L_1 + f{^{-1}}(L_2 \cap V_2) . \tag{M2$^{\prime\prime}$}$$
(M$2^\prime$) is simply a rewriting of (M2) in terms of elements. Now, (M$2^{\prime \prime}$) is equivalent to the following statements: $$(L_1 + f{^{-1}}(L_2 \cap V_2))^\perp \subset (f^\star L_2 )^\perp = f^\star L_2,$$ $$L_1 ^\perp \cap (f{^{-1}}(L_2 \cap V_2))^\perp \subset f^\star L_2 ,$$ $$L_1 \cap (f{^{-1}}(L_2 \cap V_2))^\perp \subset f^\star L_2 .$$ So to show that (M2) and (M$2^{\prime \prime}$) are equivalent, it suffices to show that $(f{^{-1}}(L_2 \cap V_2))^\perp = V_1 \oplus (f^* (p_{{V_2} ^*} L_{{2}}))$ or equivalently that $Ann(f{^{-1}}U) = f^* (p_{V_2 ^*} L_2)$. It is a general fact that for a map $V_1 \stackrel{f}{{{\longrightarrow}}} V_2 $ of vector spaces, and a subspace $U \subset V_2$, $Ann(f{^{-1}}U)) = f^* Ann(U)$. Thus, to show that $Ann(f{^{-1}}L_2 \cap V_2) = f^* (p_{{V_2} ^*} L_2)$, we need only show that $Ann(L_2 \cap V_2 ) = p_{V_2 ^*} L_{2}$. We want, therefore, that $(L_2 \cap V_2 )^\perp \subset V \oplus p_{V_2 ^*} L_2$, but we can rewrite this as $L_2 + V_2 = L_2^\perp + V_2 ^\perp \subset V_2 \oplus p_{V_2 ^*} L_2$, which is apparent. $\square$
Recall that for any linear Dirac structure $L_V$ there is a unique data of a subspace $E \subset
V$ and ${\varepsilon}\in \wedge ^2 E^* $ such that $L_V$ is of the form $
L(E, {\varepsilon}) := \{ x + \xi \in E \oplus V^* \st
\xi_{|_E} =
{\varepsilon}(x, -) \}$ ( [@gua]). Dually, there is a subspace $U \subset
V^*$ and $\pi \in \wedge ^2 U^* $ such that $L_V = L(\pi , U ):=
\{x + \xi \in V \oplus U | x_{|_U} = \pi(\xi, -) \} $. We denote the inclusion maps by $j_E : E \hookrightarrow V $ and $i_U : U
\hookrightarrow V^*$.\
For Dirac structures $L_k = L(\pi _k , U_k ) $ on $V_k$ ($k=1,2$), consider maps $f:V_1{{\longrightarrow}}V_2$ such that $f^* (U_2 ) \subset U_1 $. Denote the restriction of $f^*$ to $U_2$ by $\phi = \phi _f : U_2 {{\longrightarrow}}U_1$.\
The next two propositions \[a2\] and \[26nov1\] state conditions (M1) and (M2) of Definition \[a1\] in terms of presentations of linear Dirac structures as $L(\pi , U)$ or $L(E,{\varepsilon})$.
\[a2\] Let $L_k = L(\pi _k ,
U_k )$ be Dirac structures on $V_k,\ k=1,2$. A map $f: V_1 {{\longrightarrow}}V_2
$ is a Dirac map if and only if\
(D1) $f^* (U_2 ) \subset U_1 $ and the corresponding map $\phi=\phi_f$ satisfies\
(D2) $\phi^* \circ {{ (\pi_ 1)_\sharp}}\circ \phi= {{(\pi _2 )_\sharp}}$ (or equivalently, $\phi^* \pi_1 = \pi _2 $).
(D1) and (D2) can be expressed by the requirement that the following diagram commutes. $$\begin{CD}
U_1 @<\phi _f << U_2 \\
@VV (\pi _1)_\sharp V @VV (\pi _2 )_\sharp V\\
U_1 ^* @> \phi _f ^* >> U_2 ^*
\end{CD}$$
Conditions (D1) and (D2) of this proposition correspond to the conditions (M1) and (M2) of Definition \[a1\].
1. We show that $f^* (U_2 ) \subset U_1 $ if and only if $f(V_1
\cap L_1 ) \subset V_2 \cap L_2 $. But since $f^* (U_2) \subset U_1
$ if and only if $f(Ann(U_1)) \subset Ann(U_2) $, it suffices to show that $Ann(U_i ) = L_i \cap V_i $.\
Clearly $Ann(U_i) \subset L_i \cap V_i $ by definition of $L(\pi_i , U_i ) $. On the other hand, $L_i \cap V_i \subset L_i =
L_i ^\perp $. Hence $\langle L_i \cap V_i , L_i \rangle = 0 $ and so $\langle L_i \cap V_i
, U_i \rangle = 0 $. This implies that $L_i \cap V_i \subset Ann(U_i) $ and therefore $L_i \cap V_i = Ann(U_i) $.
2. First suppose that $\phi _f ^* \circ {{ (\pi_ 1)_\sharp}}\circ \phi _f = {{(\pi _2 )_\sharp}}$. By definition, $Y+\xi \in L_2$ if and only if ${{(\pi _2 )_\sharp}}\xi =
i_2 ^* Y $. Let $Y + \xi \in L_2 $ and $X+f^*\xi \in L_1$. Then $\i_1 ^* X = {{ (\pi_ 1)_\sharp}}(f^* \xi )= {{ (\pi_ 1)_\sharp}}(\phi( \xi) ) $. We have $${{(\pi _2 )_\sharp}}\xi = \phi ^* \circ {{ (\pi_ 1)_\sharp}}\circ \phi (\xi) =\phi ^* \circ i_1 ^* X =
i_2 ^* \circ f (X).$$ But $i_2 ^* \circ f (X) = {{(\pi _2 )_\sharp}}(\xi )$ means exactly that $fX + \xi
\in L_2$. Therefore condition (M2$^\prime$) is satisfied.\
Now suppose that conditions (M2$^\prime$) and (M1) are satisfied. Let $\xi \in
U_2$. Since $f^* \xi \in U_1 = p_{V_1 ^*}L_1$, there is some $X$ such that $X + f^*
\xi \in L_1 $. This means that $i_1 ^* X = {{ (\pi_ 1)_\sharp}}(\phi \xi ) $. But by condition (M2$^\prime$), $fX + \xi \in L_2 $, so $\i_2 ^* \circ f X =
{{(\pi _2 )_\sharp}}\xi $. However, $i_2 ^* \circ f X = \phi ^* \circ i_1 ^* X =
\phi^* \circ {{ (\pi_ 1)_\sharp}}\circ \phi \xi $. Therefore, $\phi^* \circ {{ (\pi_ 1)_\sharp}}\circ \phi \xi = {{(\pi _2 )_\sharp}}\xi $. But this is true for arbitrary $\xi \in U_2$, which means $\phi^* \circ {{ (\pi_ 1)_\sharp}}\circ \phi =
{{(\pi _2 )_\sharp}}$.$\square$
\[26nov1\] A linear map $f : (V_1 , L_1 = L(E_1 ,{\varepsilon}_1 ) ) {{\longrightarrow}}(V_2 , L_2 = L(E_2 ,{\varepsilon}_2 ) )$ is a linear Dirac map if and only if:\
1. $f(Ker(({\varepsilon}_1)_\sharp) ) \subset Ker(({\varepsilon}_ 2 )_\sharp ) $, and\
2. for $X_k \in E_k$, and $\xi_2 \in V_2 ^* $ such that $j_2 ^*
\xi_2 = ({\varepsilon}_ 2 )_\sharp (X_2) $ and $({\varepsilon}_1 )_\sharp (X_1) = j_1 ^* f^* \xi$, one has $$f(X_1) \in E_2\ \ \text{and}\ \ f(X_1)-X_2\in\ Ker(({\varepsilon}_ 2)_\sharp).$$
Upon observing that $Ker(({\varepsilon}_ i)_\sharp )=L_{V_i} \cap V_i$, (1) and (M1) are apparently equivalent. Condition (2) is a direct restatement of (M$2^\prime$) in terms of $E$ and ${\varepsilon}$. $\square$
\[a3\] Pairs $(V,L) $ of vector spaces with linear Dirac structures form a category in two ways by taking morphisms to be either the linear Dirac maps (this category will be denoted $\mathcal {LD} $) or the linear dual-Dirac maps (category $\mathcal {LD}^* $). The assignment $(V,L) \mapsto (V^*,L)$ gives an equivalence of $\mathcal {LD}^{opp}$ with $\mathcal {LD}^*$
We must show that the composition of two linear Dirac maps $(V_1 , L_1 ) \stackrel{f_1}{{{\longrightarrow}}}
(V_2 , L_2 ) \stackrel{f_2}{{{\longrightarrow}}} (V_3 , L_3 ) $ is a linear Dirac map. Let $L_j = L(\pi _j , U_j )$, $i_j : U_j \hookrightarrow V_j $, and $\phi _j : U_{j+1} {{\longrightarrow}}U_j $. Here we use the criteria of Proposition \[a2\].\
Because $f_1$ and $f_2$ satisfy (D1) so does the composition: $$(f_2 \circ f_1 )^* (U_3 ) = f_1 ^* \circ f_2 ^* (U_3 )
\subset f_1 ^* (U_2 ) \subset U_1$$ Again, because $f_1$ and $f_2$ also satisfy (D2), i.e., $
\phi _1 ^*
\circ {{ (\pi_ 1)_\sharp}}\circ \phi _1 = {{(\pi _2 )_\sharp}}$ and $\phi _2 ^* \circ {{(\pi _2 )_\sharp}}\circ \phi _2 = (\pi_ 3)_\sharp$, we get $$(\pi_ 3)_\sharp = \phi _2 ^* \circ \phi _1 ^*
\circ {{ (\pi_ 1)_\sharp}}\circ \phi _1 \circ \phi _2 = (\phi _1 \circ
\phi _2 ) \circ {{ (\pi_ 1)_\sharp}}\circ (\phi _1 \circ \phi _2 )
.$$ Finally, associativity and the existence of identity morphisms is obvious since linear Dirac maps are functions.$\square$
Dirac and Dual-Dirac Maps {#13Dec2}
=========================
Now linear Dirac and linear dual-Dirac maps are used to define maps between manifolds with Dirac structures.
Dirac Maps {#10feb2}
----------
\[1feb1\] Let $f : M{{\longrightarrow}}N $ be a $\mathcal C ^\infty $ map of manifolds, and let $L_M$ be an (almost) Dirac structure on M and $L_N $ an (almost) Dirac structure on N. The map $f$ is said to be *Dirac* or a *Dirac map* if at each point $p \in
M $, $df_p : (T_p M _{{\mathbb C}}, (L_M ) _p ) {{\longrightarrow}}(T_{f(p)} N _{{\mathbb C}}, (L_N )_{f(p)} )
$ is a linear Dirac map. If $L_M$ and $L_N$ are both generalized complex structures, we also say that $f$ is a *generalized complex map*. A Dirac map $f$ between manifolds with Dirac structures will also be denoted by $f : (M, L_M ) {{\longrightarrow}}(N, L_N ) $.
\[24nov1\] We can define a category $\mathcal D$ of Dirac manifolds by taking its objects to be all pairs $(M,L)$, where M is a $\mathcal C ^\infty$ manifold and L is a Dirac structure; the morphisms of $\mathcal D$ are the Dirac maps.
This follows from Proposition \[a3\]. For instance if $(M_1,L_1)\stackrel{f}{\longrightarrow} (M_2,L_2)
\stackrel{g}{\longrightarrow} (M_3, L_3) $ are Dirac maps, then for any $p \in M_1$ the differential $d(g\circ
f)_p = dg_{f(p)} \circ df_p$ is composition of linear Dirac maps, so it is itself a linear Dirac map.$\square$
Category $\mathcal D$ contains complex manifolds and Poisson manifolds as full subcategories. The category ${{\mathcal D}}$ also contains the category of manifolds as a full subcategory. For two symplectic manifolds $(M,\omega_M)$ and $(N,\omega_N)$ a map $f:M\to N$ which is a local isomorphism (i.e., $df_p $ is an isomorphism for all $p \in M$) is a symplectomorphism if and only if f is Dirac.
Consider Dirac manifolds $(M_1,L_1)$, $(M_2,L_2)$ and a map $f :M_1 \longrightarrow M_2 $. Since the property of being holomorphic, Poisson, or Dirac map is determined pointwise, we fix a point $p \in M_1 $ and $q= f(p)\in M_2$. We use the notation of Proposition \[a2\].\
If $L_1$ and $L_2$ are complex strucutres, we need to check that f is holomorphic if and only if f is Dirac. Let $E_1$ and $E_2$ be the holomorphic tangent bundle for complex structures so that $L_k = E_k\oplus Ann(E_k) =
L(0,Ann(E_k))$. Because $\pi_i = 0 $ for $i=1,2$, condition (M2) is trivially true. The map $f$ is holomorphic if and only if $df_p (E_1)_p \subset (E_2)_q $. Also, $f$ is Dirac if and only if $df^*(Ann(E_1)) \subset Ann(E_1) $. It is clear now that $f$ is Dirac exactly when $f$ is holomorphic.\
We check that if $L_1$ and $L_2$ are Poisson structures, then $f$ is a Poisson map if and only if f is Dirac. Now let $\pi_1 \in \wedge ^2 T_pM_1$, $\pi_2 \in \wedge^2 T_q M_2$ be two Poisson bivectors with Dirac structures $L_k = L(\pi _k ,T^*M_k ) $. Since $U_1 = T^* _p M_1$, (M1) is trivially satisfied. The map $f$ is Poisson if and only if $df _p (\pi_1 )_p
= (\pi _2 )_q $. However, one may easily verify that $df _p (\pi _1 )_p = (\pi _2)_q $ if and only if $((\pi _2)_q)_\sharp = df_p \circ ((\pi _1)_p)_\sharp \circ df_p ^*$. Therefore, by Proposition \[a2\], $f$ is Poisson if and only if $f$ is Dirac.\
Sending a manifold $M$ to $(M, TM) \in {{\mathcal D}}$ gives a full embedding of the category of manifolds into ${{\mathcal D}}$ since any $(X, L) {{\longrightarrow}}(M, TM)$ is Dirac for any map of manifolds $f: X {{\longrightarrow}}M$ and any Dirac structure $L$ on $X$.\
Symplectic structures $\omega _i$ determine Poisson structures $\pi _i$ by $(\pi _i) _\sharp = (\omega _i ) _\sharp {^{-1}}$ for $i =1,2$. The claim follows from the observation that, since $df$ is everywhere an isomorphism, $f$ is symplectic precisely if it is Poisson, i.e., if it is Dirac. $\square$
We note that for a Dirac map $f : (M , L_M ) {{\longrightarrow}}(N, L_N) $, even if it is an immersion or a submersion, it is not true that either of the Dirac structures $L_M$ or $L_N$ determines the other. For example, if $L _N = TN$ and $L_M = ( 0, U_1)$ for any subbundle $U_1$ of $TM$, then any $f$ is a Dirac map. Similarly, any map $f$ is Dirac if $L_M = T^*M$ and $L_N = L(0,U_2)$ for any subbundle $U_2$ of $T^*N$.
Dual-Dirac Maps {#13Dec3}
---------------
\[4feb1\] A map $f: M {{\longrightarrow}}N$ of manifolds with (almost) Dirac structures $ L_M, L_N$ is said to be a *dual-Dirac map* if $df_p$ is a linear dual-Dirac map for each $p \in M$.
The dual statement of Proposition \[a2\] is as follows.
\[24mar\] Let $V_1$, $V_2$ be vector spaces and $L_1 = L(E_1 ,
{\varepsilon}_1 ) \subset V_1 \oplus V_1 ^* $, $L_2 = L(E _2 , {\varepsilon}_2 ) \subset
V_2 \oplus V_2 ^* $ be linear Dirac structures. If $f: V_1 {{\longrightarrow}}V_2
$ is any linear map, then f is a linear dual-Dirac map if and only if\
$(D1^* )$ $f (E_1 ) \subset E_2 $ and\
$(D2^* )$ $f \circ ({\varepsilon}_2)_\sharp \circ f^* = ({\varepsilon}_1)_\sharp $ (or equivalently, $f^* {\varepsilon}_2 = {\varepsilon}_1 $).
These two conditions are equivalent to the existence and commutivity of the diagram $$\begin{CD}
E_1 @>> f> E_2 \\
@VV ({\varepsilon}_1)_\sharp V @VV ({\varepsilon}_2)_\sharp V \\
E_1 ^* @<< f^* < E_2 ^*
\end{CD}$$
Dirac manifolds with dual-Dirac maps form a category $\mathcal{D}^*$ of dual-Dirac manifolds.
If $U \stackrel{f}{{{\longrightarrow}}} V \stackrel {g}{{{\longrightarrow}}} W$ are two linear dual-Dirac maps, then $(g \circ f)^* = f^* \circ g^*$ is the composition of two linear dual-Dirac maps, so $g \circ f$ is linear dual-Dirac. Now the conclusion follows in the same way as for Proposition \[24nov1\].$\square$
The category ${{\mathcal D}}^*$ of dual-Dirac manifolds contains symplectic and complex manifolds as full subcategories.
If $(M_1,L_1) , (M_2,D_2) \in {{\mathcal D}}^*$ and $L_1$ and $L_2$ are complex structures, then $E_1$ and $E_2 $ are the holomorphic tangent bundles, and ${\varepsilon}_1 = {\varepsilon}_2 =0$. In light of Proposition \[24mar\], a map $f : M_1 {{\longrightarrow}}M_2$ is holomorphic if and only if it is dual-Dirac. If $L_j = L(TM_j , i{\varepsilon}_j)$ ($j = 1,2$) are symplectic structures, then Proposition \[24mar\] states that $f :M_1 {{\longrightarrow}}M_2$ is dual-Dirac if and only if $f$ is a symplectomorphism. $\square$
We say that a Dirac manifold $(M,L_1)$ is a *(dual)Dirac submanifold* of $(N,L_2)$ if $M $ is a submanifold of $N$ and the inclusion map $i: (M, L_1) \hookrightarrow (N , L_2)$ is a (dual)Dirac map. If both Dirac structures are generalized complex structures this gives two notions of generalized complex submanifolds. Gualtieri [@gua] offers a very different definition of generalized complex submanifold which is related to the notion of *branes*. The terminology “Dirac submanifolds” follows the standard use of the notion of a subobject in a category (that the inclusion is a morphism in the category). Gualtieri’s “generalized complex submanifolds” are submanifolds with extra structure which formalize mathematically certain subclasses of branes, objects from physics which are viewed as boundary conditions for Quantum Field Theories. These are objects of a different nature than the ambient manifold and are in particular not categorical subobjects. The inclusion maps of Ben-Bassat and Boyarchenko [@bbb] can also be used to define a type of generalized complex submanifolds.
B-transforms
------------
For a vector space $V$ and $B \in \wedge ^2 V^*$, we extend the composition $ V \stackrel{B_\sharp}{{{\longrightarrow}}} V^* \hookrightarrow V \oplus V^*$ by $0$ on $V$ to give a map $\tilde B _\sharp : V \oplus V^* {{\longrightarrow}}V \oplus V^*$ and a map $e ^{\tilde B _\sharp}$, which by abuse of notation, we denote by $e ^B$. This is the linear version of a B-transform, but we also refer to it as a B-transform. For a linear Dirac structure $L \subset V \oplus V^*$, we say that $e^BL$ is the *B-transform of L* by $B$. If $L = L(E , {\varepsilon})$, then $e ^B L = L(E, {\varepsilon}+ B_{|E\times E})$.
Dual-Dirac maps are stable under B-transforms in the sense that if $B \in \Omega ^2 (N)$, then $f : (M, L_M) {{\longrightarrow}}(N ,
L_N)$ being dual-Dirac implies $f : (M, e^{f^*B}L_M) {{\longrightarrow}}(N ,
e^BL_N)$ is dual-Dirac.
For dual-Dirac maps, it is enough to prove this statement pointwise for the derivative $df$ of $f$, where one may represent $L_M$ as $L_M = L(E_1 , {\varepsilon}_1)$ and $L_N$ as $L_N = L(E_2 , {\varepsilon}_2)$. Clearly $f^* {\varepsilon}_2 = {\varepsilon}_1$ implies $(f_{|E_1})^*({\varepsilon}_2 + B_{|E_2}
) = {\varepsilon}_1 + f^*B _{|E_1}$.$\square$
Crainic’s notion [@cra] of generalized complex maps is also stable under $B$-transforms. However, Dirac maps are not stable under B-transforms, as the following example shows. Let $\io : M \hookrightarrow M\times N$ be any inclusion $x \mapsto (x,c)$, let $L_1 = TM$, and let $L_2 = T(M\times N)$. For any closed non-zero $B \in T^*M \wedge T^*N \subset \wedge ^2 T^* (M\times N)$, $\io ^* B = 0$, and $(M , e^0L_1) {{\longrightarrow}}(M\times N , L( T(M\times N) , B))$ is not Dirac.
Comparison with Other Categories of Dirac Manifolds {#comp}
---------------------------------------------------
In addition to the Dirac maps and dual-Dirac maps described in this paper, there are several other distinct concepts of Dirac maps presented in the existing literature, some of them only defined for a subclass of Dirac manifolds.\
Crainic [@cra] defines a category of generalized complex manifolds which also contains complex manifolds as a full subcategory but which is only defined for generalized complex structures, not for Dirac structures in general. He defines “generalized holomorphic maps," which visibly form a category. The requirement to be a *Crainic-Dirac* map is very strong. For instance, if $L_M$ and $L_N$ are symplectic structures, then $f : (M, L_M) {{\longrightarrow}}(N, L_N)$ is Crainic-Dirac if and only if it is both symplectic and Poisson. Recall from §\[13Dec1\] that there are pushforwards and pullbacks of linear Dirac structures. Both pushforwards and pullbacks are useful operations in their own right, but they also define maps between manifolds with Dirac structures [@bur]. One asks of $f: (M,L_M) {{\longrightarrow}}(N, L_N)$ that $L_N = f_\star L_M$ or $L_M = f^\star L_N$. However, when considered as maps of manifolds with Dirac structures, they do not generalize holomorphic maps. Pullbacks and pushforwards only generalize holmorphic maps when $f$ is an immersion or submersion respectively.\
Alekseev, Bursztyn, and Meinrenken [@abm] have introduced a notion of Dirac maps, which consists of a pair $(f,B)$ of a map of manifolds $f : M {{\longrightarrow}}N$ and a 2-form $B \in \Om ^2 (M)$ such that $(f,B): (M,D) {{\longrightarrow}}(N, L)$ satisfies $f_\star (e^B D) = L$, where $f_\star L$ is the pushforward of $L$. This fits naturally into the picture when one considers generalized complex structures from the point of view of pure spinors. It is also useful for dealing with Courant algebroids other than ${{\mathcal V}}_M$. Our notions (defined below) of maps of Courant algebroids and “Courant-Dirac maps” use the same data $(f,B)$ of a map and a 2-form as in [@abm].\
Additionally, Ben-Bassat and Boyarchenko [@bbb] define Dirac inclusions and quotients which combine to give a notion of Dirac maps distinct from ours. Most recently, Ornea and Pantilie have defined “generalized holomorphic maps” [@op], which they have used to study generalized Kähler structures and which are compatible with Poisson and complex maps as well as being stable under B-transforms.
Cateogries of Courant Algebroids and Dirac Manifolds {#10feb1}
====================================================
Here we extend the definitions of Dirac and dual-Dirac maps to include Dirac structures contained in arbitrary exact Courant algebroids.
Exact Courant Algebroids
------------------------
A Courant algebroid on $M$ (defined in [@lwx]) is a quadruple $({{\mathcal E}}, \pi : {{\mathcal E}}{{\longrightarrow}}TM , \langle \, , \, \rangle , [ \, , \, ])$, where $\langle \, , \, \rangle$ is bilinear form and $[ \, , \, ]$ is a bracket operation on sections of ${{\mathcal E}}$. If we define $D : {{\mathcal C}}^\infty (M) {{\longrightarrow}}\Ga (M, {{\mathcal E}})$ by $\langle Df , A \rangle = \frac {1}{2} \pi (A) f$. We say that $({{\mathcal E}}, \pi : {{\mathcal E}}{{\longrightarrow}}TM , \langle \, , \, \rangle , [ \, , \, ])$ is a *Courant algebroid* if for all sections $A,B,C$ of ${{\mathcal E}}$ and functions $f, g$ on $M$, the following conditions hold:\
1. $\pi ([A,B])= [\pi A , \pi B]$\
2. $Jac (A,B,C) = D (Nij(A,B,C))$\
3. $[A,fB] = f[A,B] + (2\langle Df ,A \rangle )B - \langle A , B \rangle Df $\
4. $\pi \circ D = 0$\
5. $\pi (A) \langle B , C \rangle = \langle [A,B] + D\langle A , B \rangle , C \rangle + \langle B , [A,C] + D \langle A , C \rangle \rangle $.\
$Jac$ denotes the Jacobiator of the Courant bracket, and $Nij (A,B,C) := \tfrac{1}{3}(\langle [A,B],C \rangle + \langle [B,C],A \rangle + \langle [C,A], B \rangle ) $.\
An *exact Courant algebroid* is a Courant algebroid ${{\mathcal E}}{{\longrightarrow}}M$ for which $0 {{\longrightarrow}}T^* M \stackrel{\pi ^* }{{{\longrightarrow}}} {{\mathcal E}}\stackrel{\pi}{{{\longrightarrow}}} TM {{\longrightarrow}}0$ is exact. A map of Courant algebroids on $M$ is simply a map of vector bundles which preserves all of the defining structures of the Courant algebroids. Such a map is necessarily an isomorphism if the Courant algebroids are exact. For a closed 3-form $H$, there is an exact Courant algebroid ${{\mathcal V}}_{M,H}$, where $({{\mathcal V}}_{M,H} , \pi , \langle \, , \, \rangle ) = ({{\mathcal V}}_{M} , \pi , \langle \, , \, \rangle )$, but ${{\mathcal V}}_{M,H} $ has bracket given by $[X + \xi , Y + \eta ]_H = [X+\xi , Y + \eta ] + \io _Y \io _X H$ [@swe]. Ševera’s classification (as described in [@bcg]) is that any exact Courant algebroid is isomorphic to some ${{\mathcal V}}_{M,H}$. Further, it is known that any isomorphism ${{\mathcal V}}_{M,H} {{\longrightarrow}}{{\mathcal V}}_{M,K}$ is a B-transformation $e^B$, where $H= K + dB$ [@gua].\
We now define the pullback of an exact Courant algebroid. For a submanifold $\io : S \hookrightarrow M$ and an exact Courant algebroid ${{\mathcal E}}$ over $M$, ${{\mathcal E}}$ can be restricted to an exact Courant algebroid ${{\mathcal E}}_{|S} = \pi {^{-1}}(TS) / Ann(TS)$ on $S$ [@bcg], and when ${{\mathcal E}}= {{\mathcal V}}_{M,H}$, there is a canonical identification $({{\mathcal V}}_{M,H})_{|S} = {{\mathcal V}}_{S, \io ^* H}$. For a map of manifolds $f: M {{\longrightarrow}}N$ and a Courant algebroid ${{\mathcal E}}$ over $N$, we define the pullback $f^\star {{\mathcal E}}$ as follows. The restriction of ${{\mathcal V}}_M \oplus {{\mathcal E}}$ to the graph $\Ga _f$ of $f$ is an exact Courant algebroid on $\Ga _f$. The diffeomorphism $F = 1 \times f$ between $M$ and $\Ga _f$ now gives an exact Courant algebroid on $M$, which will be denoted by $f^\star{{\mathcal E}}$. Clearly, $f^\star {{\mathcal E}}= ({{\mathcal V}}_M \oplus _{T_M} f^* {{\mathcal E}})/F$, where $f^* {{\mathcal E}}$ is the pullback of vector bundles and $F$ is the subbundle of ${{\mathcal V}}_M \oplus _{TM} f^*{{\mathcal E}}$ which is the graph of $-f^* : f^* T^*N {{\longrightarrow}}T^*M$. Concretely, at a point $x \in M$, the fiber of $f^\star {{\mathcal E}}$ is
$$(f^\star {{\mathcal E}})_x = \{ (A,B) \in ({{\mathcal V}}_M )_x \oplus {{\mathcal E}}_{f(x)} \st \pi B = d_x f \pi A \} / \{ (-f^*\be , \be) \st \be \in T^* _x N\}.$$ When ${{\mathcal E}}= {{\mathcal V}}_N$, there is an isomorphism ${{\mathcal V}}_M {{\longrightarrow}}f^\star {{\mathcal V}}_N$ given by $X + \al \mapsto [X + \al , df X ]$, and more generally when ${{\mathcal E}}= {{\mathcal V}}_{N,H}$, $f^\star {{\mathcal E}}$ is identified with ${{\mathcal V}}_{M, f^*H} $ in the same way because ${{\mathcal E}}_{|\Ga _f} = {{\mathcal V}}_{M\times N , 0+H}$.\
Let ${{\mathcal E}}$ be an exact Courant algebroid on $M$. When $\io : S \hookrightarrow M$ is an embedding, $\io ^\star {{\mathcal E}}\simeq {{\mathcal E}}_{|S}$. For any manifold $N$, let $p : M \times N {{\longrightarrow}}M$ be the first projection map. Then $p ^\star {{\mathcal E}}\simeq {{\mathcal E}}\oplus {{\mathcal V}}_N$.
The pullback $\io ^\star {{\mathcal E}}$ is the exact Courant algebroid $({{\mathcal V}}_S \oplus {{\mathcal E}})_{|\Delta (S)} {{\longrightarrow}}\Delta S \tilde {{\longrightarrow}}S$, where $\Delta : S {{\longrightarrow}}S \times S$ is the diagonal map. There is an isomorphism ${{\mathcal E}}_{|S} {{\longrightarrow}}({{\mathcal V}}_S \oplus {{\mathcal E}})_{\Delta S}$ given by $[v] \mapsto [\pi v , v]$. For the projection map $p$, $p ^\star {{\mathcal E}}= ({{\mathcal V}}_M \oplus {{\mathcal V}}_N \oplus {{\mathcal E}})_{|N \times \Delta M} = {{\mathcal V}}_N \oplus ({{\mathcal V}}_M \oplus {{\mathcal E}})_{|\Delta M} = {{\mathcal V}}_N \oplus id_M ^\star {{\mathcal E}}$. But ${{\mathcal V}}_N \oplus id_M ^\star {{\mathcal E}}= {{\mathcal V}}_N \oplus {{\mathcal E}}$ by Lemma \[11feb1\] below. $\square$
Category of Spaces with Courant Algebroids
------------------------------------------
A *space with a Courant algebroid* is simply a Courant algebroid ${{\mathcal E}}{{\longrightarrow}}M$ over a manifold $M$, also denoted $(M, {{\mathcal E}})$. A morphism $(M, {{\mathcal E}}) {{\longrightarrow}}(N , {{\mathcal E}}^\prime)$ between two spaces with Courant algebroids is a pair $(f, \phi)$ of a map $f : M{{\longrightarrow}}N$ together with an isomorphism of Courant algebroids $\phi : {{\mathcal E}}{{\longrightarrow}}f^\star {{\mathcal E}}^\prime$.
\[11feb1\] The collection of spaces with Courant algebroids and the morphisms between them form a category ${{\mathcal C}}$.
The only thing to check is that morphisms compose. Let $(M_1 , {{\mathcal E}}_1 ) \stackrel {(f,\phi)}{{{\longrightarrow}}} (M_2 , {{\mathcal E}}_2 ) \stackrel {(g,\psi)}{{{\longrightarrow}}} (M_3 , {{\mathcal E}}_3)$. We first observe that for any isomorphism $\al : {{\mathcal E}}{{\longrightarrow}}{{\mathcal E}}^\prime$ of exact Courant algebroids on manifold $M$ and any submanifold $\io : S \hookrightarrow M$, $\al $ restricts to an isomorphism $\al _{|S} : {{\mathcal E}}_{|S} {{\longrightarrow}}{{\mathcal E}}^\prime _{|S}$. This implies that for any map $f: N {{\longrightarrow}}M$, there is a pullback $f^\star \al : f^\star {{\mathcal E}}{{\longrightarrow}}f^\star {{\mathcal E}}^\prime$. Since there are isomorphisms ${{\mathcal E}}_1 \stackrel {\phi}{{{\longrightarrow}}} f^\star {{\mathcal E}}_2$ and ${{\mathcal E}}_2 \stackrel {\psi}{{{\longrightarrow}}} g^\star {{\mathcal E}}_3$, there is also an isomorphism $ f^\star {{\mathcal E}}_2 \stackrel {f^\star \psi} {{{\longrightarrow}}} f^\star g^\star {{\mathcal E}}_3$. Therefore, there is an isomorphism $f^\star \psi \circ \phi : {{\mathcal E}}_1 {{\longrightarrow}}f^\star g^\star {{\mathcal E}}_3$. To complete the proof, we must show that $f^\star g^\star {{\mathcal E}}_3 = (g\circ f )^\star {{\mathcal E}}_3$.\
Let $\al : M_1 \tilde {{\longrightarrow}}\Ga _f$ and $\be : M_2 \tilde {{\longrightarrow}}\Ga _g$. There is a commutative diagram $$\begin{CD}
M_1 @> \al >> \Ga _f @>i>> M_1 \times M_2 @. \\
@| @VV\ga V @VV \delta V @. \\
M_1 @> \epsilon >> \Ga _{\be \circ f} @>j>> M_1 \times \Ga _g @>k>> M_1 \times M_2 \times M_3
\end{CD}$$
where $\al$, $\be$, $\delta$, $\ga$, $\epsilon$ are isomorphisms, and $i$, $j$, and $k$ are inclusions. Starting with ${{\mathcal V}}_{M_1} \oplus {{\mathcal V}}_{M_2} \oplus {{\mathcal E}}_3$ on $M_1 \times M_2 \times M_3$, we restrict the Courant algebroid for each inclusion and make identifications (pullback of vector bundles) for each isomorphism. Along the top row, we end up with $f^\star g^\star {{\mathcal E}}_3$ on $M_1$, and the result on the second row will therefore be the same.\
Next we observe that the inclusion $\Ga _{\be \circ f} \hookrightarrow M_1 \times \Ga _g \hookrightarrow M_1 \times M_2 \times M_3$ is the same as the inclusion $\Ga _{\be \circ f} \hookrightarrow \Ga _f \times M_3 \hookrightarrow M_1 \times M_2 \times M_3$. In general, for $Q \subset S \subset M$ and Courant algebroid ${{\mathcal E}}$ on $M$, ${{\mathcal E}}_{|Q} = ({{\mathcal E}}_{|S})_{|Q}$. Hence, restricting ${{\mathcal V}}_{M_1} \oplus {{\mathcal V}}_{M_2} \oplus {{\mathcal E}}_3$ to $\Ga _{\be \circ f}$ is the same as restricting ${{\mathcal V}}_{\Ga _f } \oplus {{\mathcal E}}_3$ on $\Ga _f \times M_3$ to $\Ga _{\be \circ f}$. There is now a commutative diagram $$\begin{CD}
M_1 @> \al >> \Ga _{\be \circ f} @>l>> \Ga _f \times M_3 \\
@| @VV\io V @VV \al {^{-1}}\times 1 V \\
M_1 @> \kappa >> \Ga _{g \circ f} @>m>> M_1 \times M_3
\end{CD}$$
where $\io$ and $\kappa$ are isomorphism and $l$ and $m$ are inclusions. Again restricting for each inclusion and making identifications for each isomorphism, the bottom row will yield $(g\circ f)^\star {{\mathcal E}}_3$ on $M_1$, whereas the previous diagram shows that the top row yields $f^\star g^\star {{\mathcal E}}_3$. $\square$
### Categories of Courant-Dirac manifolds {#twisted}
First consider Dirac structures $D_1 \subset {{\mathcal V}}_{M_1 , H_1}$, $D_2 \subset {{\mathcal V}}_{M_2 , H_2}$. Since ${{\mathcal V}}_{M_i , H_i} = {{\mathcal V}}_{M_i}$ as vector bundles, it still makes sense to say that a map $f : M_1 {{\longrightarrow}}M_2 $ is Dirac or dual-Dirac if it satisfies the conditions of Definitions \[1feb1\] and \[4feb1\] respectively.
\[4feb2\] We define *Courant dual-Dirac manifolds* to be triples $(M, {{\mathcal E}}, D)$ of a Dirac structure $D$ in an exact Courant algebroid ${{\mathcal E}}$ on manifold $M$. A morphism of Courant dual-Dirac manifolds $ (f, \phi): (M_1 , {{\mathcal E}}_1 , D_1 ) {{\longrightarrow}}(M_2 , {{\mathcal E}}_2 , D_2)$ is a map $(f,\phi) : (M_1 , {{\mathcal E}}_1 ) {{\longrightarrow}}(M_2 , {{\mathcal E}}_2 )$ of spaces with Courant algebroids such that:\
$(M1)^*$ : For any $[A,B] \in f^\star {{\mathcal E}}_2$, if $[A,B] \in \phi D_1$, then there exists $\be \in T^* _{f(x)} N$ such that $B + \be \in D_2$.\
$(M2)^*$ : If $[X + \al , B] \in \phi D_1$ with $B \in D_2$, then $[X, B] \in \phi D_1$.
As an example, a Courant dual-Dirac map $(M _1, {{\mathcal V}}_{M _1, H_1} , D_1 ) {{\longrightarrow}}(M_2 , {{\mathcal V}}_{M _2, H_2} , D_2 )$ consists of a pair $(f, B)$ such that $H_1 + dB = f^* H_2$ and $f: (M_1, e^B D_1 ) {{{\longrightarrow}}} (M_2 , D_2)$ is dual-Dirac in the sense of Definition \[4feb1\]. This follows from Ševera’s classification [@bcg].
1. Courant dual-Dirac manifolds form a category, which we denote by ${{\mathcal C}}{{\mathcal D}}^*$.
2. For the full subcategory ${{\mathcal T}}{{\mathcal D}}^*$ of all $(M, {{\mathcal V}}_{M , H}, D)$, the inclusion into ${{\mathcal C}}{{\mathcal D}}^*$ is an equivalence.
3. The category ${{\mathcal D}}^*$ is a subcategory of ${{\mathcal C}}{{\mathcal D}}^*$, obtained by allowing only ${{\mathcal E}}$’s of the form $ {{\mathcal V}}_M$ and only allowing maps $(f,B)$ with $B = 0$.
Since ${{\mathcal D}}^*$ is a category and morphisms in ${{\mathcal D}}^*$ are stable under B-transforms, it follows that ${{\mathcal T}}{{\mathcal D}}^*$ is a category. Morphisms in ${{\mathcal C}}{{\mathcal D}}^*$ have the property that if $\al _i : {{\mathcal E}}_i {{\longrightarrow}}{{\mathcal E}}_i ^\prime$ are isomorphisms of Courant algebroids on $M_i$, then for $(f , \phi ) \in Hom _{{\mathcal C}}( (M_1, {{\mathcal E}}_1 ), (M_2, {{\mathcal E}}_2)$, it is the case that $(f,\phi) : (M_1 , {{\mathcal E}}_1 , D_1 ) {{\longrightarrow}}(M_2 , {{\mathcal E}}_2 , D_2)$ is a morphism in ${{\mathcal C}}{{\mathcal D}}^*$ if and only if $(f, f^\star \al _2 \circ \phi \circ \al _1 {^{-1}}) : (M_1 , {{\mathcal E}}_1 ^\prime , \al _1 D_1 ) {{\longrightarrow}}(M_2 , {{\mathcal E}}_2 ^\prime, \al _2 D_2)$ is a morphism in ${{\mathcal C}}{{\mathcal D}}^*$. Hence, since every object in ${{\mathcal C}}{{\mathcal D}}^*$ is isomorphic to some object in ${{\mathcal T}}{{\mathcal D}}^*$ and ${{\mathcal T}}{{\mathcal D}}^*$ is a category, morphisms in ${{\mathcal C}}{{\mathcal D}}^* $ compose, making it a category equivalent to ${{\mathcal T}}{{\mathcal D}}^*$. Clearly ${{\mathcal D}}^*$ is a subcategory. $\square$
We define a collection ${{\mathcal C}}{{\mathcal D}}$ of *Courant Dirac manifolds* where $Ob({{\mathcal C}}{{\mathcal D}}) $ consists of quadruples $(M , {{\mathcal E}}, \al , D)$ of a manifold $M$ with exact Courant algebroid ${{\mathcal E}}$, an isomorphism $\al : {{\mathcal E}}{{\longrightarrow}}{{\mathcal V}}_{M,H}$ for some $H \in \Om ^3 (M)$, and a Dirac structure $D \subset {{\mathcal E}}$. A morphism $ (M_1 , {{\mathcal E}}_1 , \al _1 , D_1 ) {{\longrightarrow}}(M_2 , {{\mathcal E}}_2 , \al _2 , D_2 )$ is a map $(f,\phi) : (M_1 , {{\mathcal E}}_1 ) {{\longrightarrow}}(M_2 , {{\mathcal E}}_2)$ of spaces with Courant algebroids such that $f : (M_1 , \al _1 D_1 ) {{\longrightarrow}}(M_2 , \al _2 D _2)$ is a Dirac map in the sense of Definition \[1feb1\].
1. Since ${{\mathcal C}}$ and ${{\mathcal D}}$ are categories, ${{\mathcal C}}{{\mathcal D}}$ is a category.
2. There is no way to consistently state (M1) and (M2) (as in Definition \[4feb2\]) for Dirac structures contained in arbitrary exact Courant algebroids. This is possible for $(M1)^*$ and $(M2)^*$ because dual-Dirac maps are stable under B-transforms whereas Dirac maps are not. This failure forces us to choose isomorphisms $\al : {{\mathcal E}}_i {{\longrightarrow}}{{\mathcal V}}_{M _i , H _i}$, or equivalently isotropic sections $TM_i \stackrel{s_i}{ {{\longrightarrow}}} {{\mathcal E}}_i$ of $\pi _i$. In other words, different choices of isomorphisms $\al _1$, $\al _2$ will change whether $f : (M_1 , \al _1 D_1 ) {{\longrightarrow}}(M_2 , \al _2 D _2)$ is a dual-Dirac map in the sense of Definition \[1feb1\].
Dirac Groups {#13Dec4}
============
Here we will consider groups in the categories ${{\mathcal D}}$ and ${{\mathcal D}}^*$ but will also briefly mention groups in ${{\mathcal T}}{{\mathcal D}}$ and ${{\mathcal T}}{{\mathcal D}}^*$.
Almost Dirac Groups
-------------------
In this subsection we describe the data and conditions involved in requiring that a group with almost Dirac structure is a Dirac group (that is, group multiplication is a Dirac map).
If $(G,D)$ is a Lie group with (almost) Dirac structure $D$, we say that $(G,D)$ is a *(almost) Dirac group* if group multiplication $\mu :
G \times G {{\longrightarrow}}G$ is a Dirac map. If $D$ is a real Dirac structure and we say that $(G,D)$ is a *real Dirac group*, whereas if $D$ is a generalized complex structure, we say that $(G,D)$ is a *generalized complex group*.
For a Lie group $G$ and a bi-invariant subbundle $U \subset T^* G _{{\mathbb C}}$, a section $\be$ of $\wedge ^2 U^*$ is called *multiplicative* if $d\mu _{(g,h)} (\be _g + \be _h ) = \be _{gh}$ for all $g,h \in G$, i.e. $dL_g \be _h + d R_h \be _g = \be _{gh}$. For the remainder of this section we will, when convenient, use left translation to identify $TG$ and $T^*G$ with $G \times {{\mathfrak g}}$ and $G \times {{\mathfrak g}}^*$. This identifies $\Ga (G , TG) \simeq Map (G, {{\mathfrak g}})$, and with bi-invariant $U$ as above, $\Ga (G , \wedge ^2 U^*) \simeq Map (G , \wedge ^2 U_e ^*)\simeq Map (G , \wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ Ann(U_e))$.
Almost Dirac Groups
-------------------
\[24nov3\] Let $(G,D) $ be a group with almost Dirac structure. Then $(G,D)$ is an almost Dirac group if and only if there is a bi-invariant subbundle $U$ of $T^*G _{{\mathbb C}}$ and multiplicative section $\beta$ of $ \wedge ^2 U^*$ such that $D = L(\beta , U)$. In this case $\fk = Ann(U_e )$ is a $G$-invariant ideal of ${{\mathfrak{g} _{\mathbb{C}}}}$. Almost Dirac groups are thus parameterized by pairs $(\fk, \be)$ of a G-invariant ideal $\fk \subset {{\mathfrak{g} _{\mathbb{C}}}}$ and a multiplicative $\be : G {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$.
Let $\mu : G \times G {{\longrightarrow}}G$ denote group multiplication in G. The map $d\mu ^* _{(g,h)} : T^*_{gh} G {{\longrightarrow}}T_g ^* G \oplus
T_h ^* G $ is given by $d\mu ^* _{(g,h)} = (dR_h)_g ^* \oplus
(dL_g)_h ^*$. We prove this lemma for real Dirac structures, but the proof applies to arbitrary Dirac stuctures by complexifying ${{\mathcal V}}_G$.\
First we show that (M1) is equivalent to the requirement that there exists a bi-invariant subbundle $U$ for which $D$ is of the form $L(\be, U)$ for some $\be \in \Ga ( G, \wedge ^2 U^*)$. At each point $g \in G$, we can express $D_g $ as $D_g = L(\beta _g , U_g)$ for some $U _g \subset T_g ^*G$ and $\beta _g \in \wedge ^2 U_g ^*$. This defines some $U \subset T^*G$, and we must show that $U$ is a subbundle. The Dirac structure $\mathcal D$ on $G\times G$ is defined by $\mathcal D _{(g,h)} = D_g \oplus D_h = L(\beta _g + \beta
_h , U_g \oplus U_h )$. In order to satisfy (M1), $d\mu _{(g,h)} ^* (U_{gh} ) \subset U_g \oplus U_h $, which means $(dR_h)_g ^* U_{gh} \subset U_g $ and $(dL_g)_h ^* U_{gh}\subset U_h$.\
This is true for all $g, h \in G$. In particular, if $h= e$, $(dL_g)_e ^* U_g \subset U_e$, and if $g=e$, $(dR_h)_e ^* U_h \subset
U_e$. Hence, $ U_g \subset (dL_g)_e ^{-*} U_e $ and $U_g \subset
(dR_g)_e ^{-*} U_e$ for all $g \in U_g $.\
On the other hand, if $h = g{^{-1}}$, we get $(dR_{g{^{-1}}} )_g ^* U_e
\subset U_g$. This implies that dim$ U_g$ $\geq$ dim$U_e $. Therefore, $U_g = (dL_g)_e ^{-*} U_e = (dR_g)_e ^{-*} U_e$. We conclude that $U$ is a subbundle and in fact bi-invariant, which means that $U_e$ is $G$-invariant and $\mathfrak k :=
Ann(U_e) $ is $G$-invariant. Therefore $\mathfrak k$ is an ideal.
Condition (M2) states that $d\mu (\be _{g} + \be _{h} ) = \be _{gh } $, just as for Poisson groups. $\square$
It follows from Lemma \[24nov3\] that if $(G , L(\be , U))$ is a Dirac group, then $\be _e =0$ and $D_e = \mathfrak k \oplus Ann(\mathfrak k)$, where $\fk = Ann(U_e)$. With $TG \simeq G \times {{\mathfrak g}}$, multiplication $d\mu _{(g, h)} :
\mathfrak g \oplus \mathfrak g {{\longrightarrow}}\mathfrak g $ is given by $d\mu
_{(g, h)} (X,Y) = Ad(h{^{-1}}) X + Y$, and $d\mu _{(g,h)} ^* \xi = Ad(h)^{-*}\xi + \xi $. Hence $\be : G {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ is multiplicative if and only if $\be _{gh} = \be _h + Ad(h{^{-1}})\be _g$.
The following is a generalization of the Poisson case from [@luw], [@vai] of muliplicative bivectors in terms of cocycles. For an ideal $\fk \subset {{\mathfrak{g} _{\mathbb{C}}}}$, $\be : G {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ is a *cocycle* if $ad_x (d_e\be (y)) - ad_y (d_e \be (x)) - d_e \be ([x,y]) = 0 $ for all $ x, y \in {{\mathfrak g}}$.
\[5feb1\] Let $G$ be a Lie group, $\fk$ a G-invariant ideal and $\be \in \Ga (G , G \times \wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk)$ such that $\be (e) = 0$. If $\be$ is multiplicative, then $d_e \be :{{\mathfrak g}}{{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ is a $\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$-valued cocycle. Conversely, if $G$ is simply connected, then any cocycle $\epsilon : {{\mathfrak{g} _{\mathbb{C}}}}{{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ is the differential of some multiplicative $\be : G {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/\fk$.
Since $G\times \fk$ is an ideal in the sheaf of Lie algebras ${{\mathcal T}}G$, it makes sense to take the Lie derivative ${{\mathcal L}}_X \be$ for any left or right-invariant vector field $X$. The proof of (1) is now identical to the one in [@vai]. $\square$
If $(G,D)$ is an almost Dirac group, then inversion $\io : G {{\longrightarrow}}G$ ($g \mapsto g{^{-1}}$) and the identity map $\{e\} \hookrightarrow G$ are Dirac maps, so $(G, D)$ is a group object in the category of almost Dirac manifolds.
By Lemma \[24nov3\], $D = L(\be , U)$, and $\be _e = 0$. Now (D1) and (D2) are trivially satisfied for $\{e\} \hookrightarrow G$, so the identity map is Dirac. The derivative of inversion at $g\in G $ is given by $d\io _g = -Ad(g)$. Since $U$ is bi-invariant, by Lemma \[24nov3\], (D1) is satisfied. Since $\be$ is multiplicative, $\be _{gh} = \be _h + Ad(h{^{-1}})\be _g$. Letting $h = g{^{-1}}$, we get $\be _{g{^{-1}}} = -Ad(g)\be _g$, which is exactly what is needed for (D2) to be satisfied for $\io$. $\square$
Group multiplication $\mu : G \times G {{\longrightarrow}}G $ is Dirac if and only if $(\mu , 0)$ is an ABM-Dirac map (i.e. it is a Dirac map in the sense of [@abm]).
First we observe that a map $(f,0) : (M_1 , D_1 ) {{\longrightarrow}}(M_2 , D_2)$ is an ABM-Dirac map if and only if $X + f^* \xi \in D_1 \Longrightarrow fX + \xi \in D_2$. Let $(G,D)$ be a group in $ABM -{{\mathcal D}}$. Pointwise we can represent $D$ as $D_g = L(\be _g , U_g)$. Using left translation to get a trivialization ${{\mathcal V}}_G \simeq G\times ({{\mathfrak g}}\oplus {{\mathfrak g}}^*)$, we view $D _g \subset {{\mathfrak g}}\oplus {{\mathfrak g}}^*$. The condition for $\mu$ to be a map in $ABM-{{\mathcal D}}$ is: for all $g,h \in G$, $$X + Ad(h{^{-1}})^* \xi \in D_g \; \; and \; \; Y + \xi \in D_h \Longrightarrow Ad(h{^{-1}})X + Y + \xi \in D_{gh}.$$
In particular this implies that $Ad(h{^{-1}})^* Ann(U_g) + Ann(U_h) \subset D_{gh}$. But since $D_{gh} \cap {{\mathfrak g}}= Ann(U_{gh})$, we have $Ad(h{^{-1}})^* Ann(U_g) + Ann(U_h) \subset Ann(U_{gh})$. Then of course $Ann(U_h) \subset Ann(U_{gh})$ and $U_{gh} \subset U_h$ for all $g$, $h \in G$. Letting $h = e$ gives $U_g \subset U_e$, and letting $g = h{^{-1}}$ gives $U_e \subset U_h$ so that $U_g = U_e$ for all $g \in G$. But $Ad(h{^{-1}})^* Ann(U_g) + Ann(U_h) \subset Ann(U_{gh})$ also implies that $Ad(h)^*Ann(U_e) \subset Ann(U_e)$ for all $h \in G$, whence $U_e$ and $\fk = Ann(U_e)$ are $G$-invariant. Now the condition for $\mu $ being a morphism in $ABM-{{\mathcal D}}$ is that $(\be _{gh} )_\sharp = Ad(h{^{-1}}) (\be _g)_\sharp Ad (h{^{-1}})^* + (\be _g)_\sharp$, i.e. $\be _{gh} = Ad(h{^{-1}}) \be _g + \be _h$, which is the same as saying that $\be$ is multiplicative. Therefore, $(G, D)$ is a Dirac group. Conversely, let $(G , D = L(\be , U))$ be a Dirac group. Since $U$ is bi-invariant, $(M1)$ and $(M2 ^\prime)$ easily imply that $\mu$ is a morphism in $ABM-{{\mathcal D}}$. $\square$
Real Dirac Groups
-----------------
Here we consider Dirac groups $(G,D)$ for which $D$ is a real Dirac structure. We prove Proposition \[4auga\] which relates real Dirac group structures on $G$ to Poisson group structures on quotients of $G$.
\[4auga\] Let $(G, D)$ be a real Dirac group. If the connected subgroup $K \subset G$ with $Lie \, K = \mathfrak k$ is closed, then the data for the Dirac structure $D$ on $G$ is equivalent to the data for a Poisson group structure on $G/K$. If $G$ is semisimple, then Dirac group structures on $G$ are in bijection with Poisson group structures on quotients of $G$ by closed, normal, connected sugroups of $G$.
Suppose that there is a closed subgroup $K$ such that $Lie \, K = \mathfrak k$. Let $\pi : G {{\longrightarrow}}G/K $ denote the quotient map. When we view $G $ locally as a product of an open set $W$ of K and an open set $V$ of G/K, we see that $T_g K = dL_g \mathfrak k$ and so $T^*V = U$. One may check that $\pi _\star D$ is an almost Dirac structure, and $D = TK \oplus \pi ^* (\pi _\star
D) $. Then $\pi ^\star (\pi _\star D) = D $ implies that $\pi _\star
D$ is a Dirac structure. Since $pr_{T^* (G/K) } \pi _\star D = T^*(G/K)$, $\pi
_\star D = L(\beta , T^* (G/K))$ for some 2-form $\beta$ on $G/K$. But $\beta$ is a Poisson structure on $G/K$ since $\pi _\star D $ is involutive.\
To see that $(G/K , \beta ) $ is a Poisson group, we must must show that $\beta$ is multiplicative. Let $\mu : G\times G {{\longrightarrow}}G$ and $\overline \mu : G/K \times G/K {{\longrightarrow}}G/K $ denote group multiplication in $G$ and $G/K$ respectively. We know that $\overline \mu \circ \pi
\times \pi = \pi \circ \mu$. Recall that $\gamma$ is multiplicative, meaning $d\mu (\gamma _{g_1} + \gamma _{g_2} ) = \gamma _{g_1 g_2 } $ just as for Poisson groups. We also know that $\beta = (d\pi) \gamma $ because $\pi :(G,D) {{\longrightarrow}}(G/K , \pi _\star D ) $ is obviously a Dirac map. Therefore $\beta _{g_1K} + \beta _{g_2 K} = d(\pi \times \pi
)_{(g_1 , g_2 )}
(\gamma _{g_1} + \gamma _{g_2} ) $, so $$\begin{aligned}
d\overline \mu _{(\pi (g_1) , \pi (g_2
))} (\beta _{\pi (g_1)} + \beta _{\pi (g_2 )} ) & = & d\overline
\mu _{(\pi (g_1 ) , \pi(g_2
))} \circ d(\pi \times \pi
)_{g_1 , g_2 )}
(\gamma _{g_1} + \gamma _{g_2} ) \\ & = & d\pi _{g_1 g_2} \circ d\mu _{(g_1 ,
g_2 )} (\gamma _{g_1} + \gamma _{g_2} ) = d\pi _{g_1 g_2} (\gamma
_{g_1 g_2} ) \\ & = & \beta _{\pi (g_1 g_2 )} \; . \end{aligned}$$
Therefore $\beta$ is multiplicative and $(G/K , \beta ) $ is a Poisson group. This correspondence $D \mapsto \pi _\star D $ is injective because $\pi ^\star \pi _\star D = D $.\
To complete the proof, it suffices to show that for any Poisson group structure\
$L = L(\beta , T^* (G/K ))$ on $G/K$, $(G, \pi^\star L) $ is a Dirac group. From Lemma \[pdsprop2\], $\pi ^\star L $ is a Dirac structure on $G$ because $L$ is a Dirac structure on $G/K$. Note that $\pi ^\star L = L(\gamma , U ) $, where $U = Ann(\mathfrak k)$. Since $\mathfrak k$ is $G$-invariant, $U$ is bi-invariant. Thus, (M1) is satisfied.\
It remains to show that $\gamma$ is multiplicative. This follows in the same way as before. Since $\pi : (G, \pi ^\star L ) {{\longrightarrow}}(G/K ,
L) $ is Dirac, $d\pi \gamma = \beta$. Observe that $$\begin{aligned}
d\pi _{g_1 g_2} (\gamma _{g_1 g_2} ) & = &
\beta _{\pi(g_1 g_2 )} =
d\overline \mu _{(\pi (g_1 ) ,\pi ( g_2 )
)} (\beta _{\pi (g_1)} + \beta _{\pi (g_2 )} ) \\
& = & d\overline \mu
_{(\pi (g_1 ) ,\pi ( g_2 ) )} \circ (\pi \times \pi )_{(g_1 , g_2 )}
(\gamma _{g_1} + \gamma _{g_2} ) \\
& = & d\pi _{g_1 g_2} \circ d\mu _{(g_1 ,
g_2 )} (\gamma _{g_1} + \gamma _{g_2} ). \end{aligned}$$ Therefore $\gamma$ is multiplicative. This proves the first two statements.\
Now suppose that $G$ is semisimple. By the first part of this proposition, to prove our correspondence it suffices to show that every Ad-invariant $\mathfrak k \subset {{\mathfrak g}}$ is the Lie algebra of some normal closed subgroup. Any Ad-invariant $\mathfrak k \subset {{\mathfrak g}}$ is an ideal. Since ${{\mathfrak g}}$ is semisimple, $ {{\mathfrak g}}= \mathfrak k \oplus \mathfrak k ^\perp$, where $\mathfrak k ^\perp $ is determined by the Killing form on ${{\mathfrak g}}$. We know that $\mathfrak k ^\perp$ is also an ideal and semisimple. The identity component of the centralizer $Z_G(\mathfrak k ^\perp ) $ is a closed subgroup with Lie algebra $Z_{{\mathfrak g}}(\mathfrak k ^\perp) = \mathfrak k$.$\square$
Generalized Complex Groups
--------------------------
Here we classify generalized complex groups. In parallel with the work of Gualtieri, we hope that the following result may be helpful in a geometric construction of quantization of holomorphic Poisson groups.
\[27jan1\] A generalized complex group structure on a group $G$ is equivalent to a holomorphic Poisson group structure on $G$.
By Lemma \[24nov3\], $D = L(\beta , U ) $, where U is a bi-invariant subbundle of $G \times {{\mathfrak{g} _{\mathbb{C}}}}^* $. Then $\mathfrak k = Ann(U_e ) $ is an ideal in ${{\mathfrak{g} _{\mathbb{C}}}}$ and $G$-invariant. Also by Lemma \[24nov3\], $D_e = L(0,U_e)$, so $D_e
\oplus \overline{D_e} = {{\mathfrak{g} _{\mathbb C} \oplus \mathfrak{g} _\mathbb{C} ^* } }$ implies that $\fk \oplus \overline \fk = {{\mathfrak{g} _{\mathbb{C}}}}$. Therefore $\fk$ gives $G$ the structure of a complex group. We henceforth identify $TG \simeq G \times {{\mathfrak g}}$ by left translation. Let ${{\mathcal J}}: {{\mathcal V}}_G {{\longrightarrow}}{{\mathcal V}}_G$ be the map with i-eigenbundle $D$. Since $\fk \subset D_g$ for any $g$, ${{\mathcal J}}\fk = \fk$, and since $\fk \subset \overline D_g$, ${{\mathcal J}}\fk \subset \fk$ because $\overline D_g$ is the $-i$-eigenspace of ${{\mathcal J}}_g$. Therefore, since $\fk \oplus \overline \fk = {{\mathfrak{g} _{\mathbb{C}}}}$, ${{\mathcal J}}{{\mathfrak g}}\subset {{\mathfrak g}}$ and ${{\mathcal J}}$ is of the form $${{\mathcal J}}=
\left[
\begin{array}{cc}
J & Q \\
0 & -J^* \\
\end{array}
\right]$$ where $J$ is the complex structure with i-eigenbundle $G \times \fk$. Gualtieri shows that generalized complex structures of this form are equivalent to a complex structure $J$ with holomorphic Poisson structure $\rho _\sharp = JQ + iQ$ [@gua2] [@gua3]. One may check that the graph of $ -\tfrac{-i}{2} Q _{|((T^{0,1})^*}$ is contained in the i-eigenbundle of ${{\mathcal J}}$ so that $\be _\sharp = pr_{|T^{(0,1}} -\tfrac{-i}{2} Q _{|(T^{0,1})^*} = -\tfrac{-i}{2}(1+iJ) \circ Q _{|(T^{0,1})^*}$, from which one can compute that $\be _\sharp $ is a multiple of $Q + iJQ$. This implies that $\be$ is multiplicative if and only if $Q$ is multiplicative, whence the desired result follows. $\square$
We have noted that Crainic defines a category of generalized complex manifolds with morphisms which I will call here *Crainic-GC maps*. When $f$ is a submersion, as is the case for the group multiplication map, $f$ being Crainic-GC implies that $f$ is Dirac, but the converse is not true in general. However, we do notice the following result.
A Crainic-GC group is the same as a generalized complex group.
Vaisman [@vai2] shows that groups with generalized complex structure such that multiplication is a Crainic-GC map are equivalent to a complex Poisson groups such that the complex and Poisson structure form a Poisson-Nijenhuis structure. Laurent-Gengoux, Stiénon, and Xu [@lsx] describe how a Poisson-Nijenhuis structure is equivalent to a holomorphic Poisson structure. Thus, Vaisman’s result implies that a Crainic-GC group structure is equivalent to a holomorphic Poisson group structure. What Proposition \[27jan1\] shows is that Crainic-GC groups are the same as generalized complex groups.$\square$
Integrability and Dirac Groups
------------------------------
Here we generalize several formulations of integrability for Poisson groups treated in [@luw], [@vai] to the case of almost Dirac groups. We begin with an independent observation.
\[22jan4\] Let $(G , D = L(\be, U))$ be an almost Dirac group. If there is an ideal $V \subset {{\mathfrak{g} _{\mathbb{C}}}}$ complementary to $\fk = Ann(U_e)$ (for example, if ${{\mathfrak g}}$ is reductive), then $\be : G {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ can be lifted to $\hat \be : G \wedge ^2 V \subset \wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}$. The almost Dirac structure $D$ is integrable if and only if $L(\hat \be , T^*G _{{\mathbb C}})$ is integrable and $d_g \be (\fk ) = 0$ for all $g \in G$ (i.e. ${{\mathcal L}}_{\tilde X} \be = 0$ for all $X \in \fk$, where $\tilde X$ is left-invariant and $\tilde X _e = X$).
First suppose that $D$ is integrable. We first show that $\fk (\be ) = 0$, i.e. $d_x \be (\fk ) =0$. If $\eta$, $\si \in V^* \subset {{\mathfrak{g} _{\mathbb{C}}}}$. In order for $[ \be _\sharp \eta + \eta , \be _\sharp \si + \si ] $ to be a section of $D$, we need $\phi = [\be _\sharp \eta , \si ] - [\be _\sharp \si , \eta ] $ to be a section of $G \times V^* = Ann (G \times \fk)$. A simple computation yields that for $Y \in \fk$, $\phi (Y) = -Y(\be(\eta, \si))- \si [\be _\sharp \eta, Y] + \eta [\be _\sharp \si , Y]$. Another simple computation shows that $-\si[\be _\sharp \eta , Y] = Y (\be(\eta , \si))$ so that $\phi (Y) = Y(\be (\si , \eta))$. Integrability implies that $Y(\be (\eta , \si)) (g) = d_g \be (Y) (\eta , \si) = 0$.\
To check that $D^\prime$ is integrable, it is enough to check this on a frame, and in fact, it is enough to check that $[\hat \be _\sharp \eta + \eta , \al ]$ is a section of $D^\prime $ for constant sections $\al $ of $G \times \fk ^*$ and $\eta $ of $G \times V^*$. But $[\hat \be _\sharp \eta + \eta , \al ] = [\hat \be _\sharp \eta , \al] = \iota _{\hat \be _\sharp \eta} d \al + \tfrac{1}{2} d\iota_{\hat \be _\sharp \eta} \al = \iota _{\hat \be _\sharp \eta} d \al $. For a constant section $X$ of $G \times V$, $[\hat \be _\sharp \eta, \al ] (X) = d\al (\hat \be_\sharp \eta , x) = \hat \be _\sharp \eta (\al (X)) - x(\al(\hat \be_\sharp \eta)) + \al ([\hat \be _\sharp \eta, X]) = 0 -0+ \al ( [\hat \be _\sharp \eta , X] ) =0$ because $[\hat \be _\sharp \eta , X] $ is a section of $G\times V$ and $\al \in \fk ^*$. Therefore $[\hat \be _\sharp \eta + \eta , \al]$ is a section of $G \times \fk ^* \subset D^\prime$.\
Now suppose that $D^\prime$ is integrable and $\fk (\be ) =0$. The first paragraph of this proof shows that for constant sections $\eta $, $\si$ of $G \times V^*$, $[\be _\sharp \eta , \si ] - [\be _\sharp \si , \eta]$ is a section of $G \times V^*$ because $\fk (\be ) = 0$. Thus, $[\be_\sharp \eta + \eta , \be _\sharp \si + \si ] $ is a section of $D$. Now to show integrability of $D$, we simply must show that for constant sections $k$ of $G \times \fk$ and $\eta $ of $G \times V^*$, $[k ,\be _\sharp \eta + \eta ]$ is a section of $D$. Because $\fk$ is an ideal and $\fk (\be) =0$, one may easily check that $[k , \be _\sharp \eta] $ is a section of $G \times \fk$. Also $[k , \eta](X) =0$ for any constant section $X$ of $G \times {{\mathfrak{g} _{\mathbb{C}}}}$ so that $[k , \eta] = 0$. $\square$
When ${{\mathfrak g}}$ is semisimple, $\fk$ has a unique complementary ideal $\fk ^c$, so $D^\prime$ in Proposition \[22jan4\] is canonical.
Let $G$ be a semisimple Lie group. The Dirac group structures are parameterized by pairs ($\fk$, $\be$) of a G-invariant ideal $\fk $ of ${{\mathfrak{g} _{\mathbb{C}}}}$ and a complex Poisson group structures $\be : G {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}$ on $G$ such that $\be$ takes values in $\wedge ^2 \fk ^c $ and $d_g \be (\fk) = 0$ for all $g \in G$.
\[22jan3\] Given an ideal $\fk \subset {{\mathfrak{g} _{\mathbb{C}}}}$, there is a well defined Schouten bracket $[ \be , \be ] \in G {{\longrightarrow}}\wedge ^3 ({{\mathfrak{g} _{\mathbb{C}}}}/ \fk )^*)$. In fact, the Schouten bracket $[P,Q]$ makes sense for any $P ,Q : G {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk )$, as long as $d_g P (\fk ) = d_g Q (\fk) = 0$ for all $g \in G$.
Let $P$, $Q$ be as above. By choosing a splitting $\si$ of $\pi : {{\mathfrak{g} _{\mathbb{C}}}}{{\longrightarrow}}{{\mathfrak{g} _{\mathbb{C}}}}/\fk$, there are sections $\hat P , \hat Q $ of $ G \times \wedge ^2 \si ({{\mathfrak{g} _{\mathbb{C}}}}/ \fk ^*) \subset G \times \wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}})$, so we define the Schouten bracket $[P,Q] = \pi [\hat P , \hat Q]$. One may verify that since $\fk$ is an ideal, $[P,Q]$ does not depend on the choice of spliting. $\square$
Let $(G, D = L(\be, U))$ be an almost Dirac group. Let ${{\mathcal F}}\subset {{\mathcal C}}^\infty (G) \otimes {{\mathbb C}}$ denote the subsheaf consisting of all functions on which vectors in $G \times \fk \subset TG_{{\mathbb C}}$ vanish. Then $\be$ defines a bracket operation $\{ f, g \} = \be (df , dg)$ on ${{\mathcal F}}$ with values in ${{\mathcal C}}^\infty (G) _{{\mathbb C}}$. By considering $\hat \be$ as in Lemma \[22jan3\], this bracket can also be extended to one on $\wedge ^2 {{\mathcal C}}^\infty (G) \otimes {{\mathbb C}}$.
\[22jan1\] Let $(G,D)$ be an almost Dirac group as above. Then the following are equivalent:
1. $(G, D)$ is a Dirac group (i.e. $D$ is integrable)
2. The bracket operation $\{ \, , \, \} : {{\mathcal F}}\times {{\mathcal F}}{{\longrightarrow}}{{\mathcal C}}^\infty (G)_{{\mathbb C}}$ satisfies the Jacobi identity. In this case, $\{ \; , \}$ is a Lie bracket on ${{\mathcal F}}$.
3. $[\be , \be ] = 0$.
(1 $\iff$ 2) First we note that locally there exists a frame of $G \times U \subset G \times {{\mathfrak{g} _{\mathbb{C}}}}$ consisting of sections of the form $df$ for $f \in {{\mathcal F}}$. Gualtieri shows [@gua] that integrability of $D$ is equivalent to the Nijenhuis operator $Nij (A,B,C) = \tfrac{1}{3}(\langle [A,B],C \rangle + \langle [B,C],A \rangle + \langle [C,A], B \rangle ) $ vanishing on all sections $A,B,C $ of $D$. We may, choose sections of the form $X + df$, where $f \in {{\mathcal F}}(G)$ and $X_{|U} = \be (df, -)$. If $D$ is integrable, then $[X + df , Y + dg] = [X,Y] + d (\be(df,dg))$ is a section of $D$ so that $\be (df ,dg) \in {{\mathcal F}}(G)$. The integrability condition is $0 = Nij(X + df , Y + dg , Z + dh ) = \{ f , \{ g, h\} \} + \{ g , \{ h,f \} \} + \{ h , \{ f , g \} \}$. On the other hand, if $\{ \; , \; \}$ satisfies the Jacobi identity, then $Nij_{|D} = 0$, so $D$ is integrable. Hence $\be (df,dg) \in {{\mathcal F}}(G)$ for $f,g \in {{\mathcal F}}(G)$, which makes ${{\mathcal F}}$ a sheaf of Lie algebras.\
(2 $\iff$ 3) It follows directly from [@vai] equation 1.16, originally appearing in [@bhv], that with $\hat \be $ defined as in Lemma \[22jan3\], $f,g,h \in {{\mathcal F}}(G)$, $[\be , \be ] (df, dg,dh) = [\hat \be , \hat \be ] (df,dg,dh) = \{ f , \{ g, h\} \} + \{ g , \{ h,f \} \} + \{ h , \{ f , g \} \}$. $\square$
The classification [@vai] of Poisson group structures in terms of Lie bialgebras can be extended to Dirac group structures. Proposition \[22jan4\] shows that $d_e \be (\fk) = 0$, so $d_e \be: {{\mathfrak{g} _{\mathbb{C}}}}{{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ factors through ${{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ and can be thought of as $d_e \be : {{\mathfrak{g} _{\mathbb{C}}}}/ \fk {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}\fk$.
\[5feb2\] Let $G$ be a Lie group, $\fk$ a G-invariant ideal and $\be \in \Ga (G , G \times \wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk)$ such that $\be (e) = 0$. If the almost Dirac structure $L(\be , U = Ann(G \times \fk))$ is integrable, then $d_e \be (\fk ) = 0$, and $d_e \be ^* : \wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk {{\longrightarrow}}{{\mathfrak{g} _{\mathbb{C}}}}/\fk ^*$ defines a Lie bracket on ${{\mathfrak{g} _{\mathbb{C}}}}/ \fk ^*$. If $G$ is simply connected and $\be$ is multiplicative, then $L(\be , U = Ann(G \times \fk))$ is integrable if and only if $d_e \be ^*$ is a Lie bracket.
We only need to make a few modifications to the proof presented in [@vai]. First, as in Proposition \[22jan4\], $d_g \be (\fk) = 0$ is a necessary condition for integrability. As in [@vai], the bracket $[ \; , \; ] = d_e \be ^* : \wedge ^2 ({{\mathfrak{g} _{\mathbb{C}}}}/ \fk ) ^* {{\longrightarrow}}({{\mathfrak{g} _{\mathbb{C}}}}/ \fk )^*$ is given by $[\al , \be ] = d(\be(\tilde \al , \tilde \be )) _e$, where $\tilde \al $ and $\tilde \be$ are any sections of $G \times {{\mathfrak{g} _{\mathbb{C}}}}/ \fk ^*$ with $\tilde \al (e) = \al $, $ \tilde \be (e) = \be$. If $D$ is integrable, then by Proposition \[22jan1\], $\{ \; , \; \}$ is a Lie bracket on the sheaf ${{\mathcal F}}$. Let ${{\mathcal F}}_e ^i$ denote all germs of functions in the stalk ${{\mathcal F}}_e$ which vanish up to order $i$ at $e$. Then there is an isomorphism ${{\mathcal F}}^0 _e / {{\mathcal F}}^1 _e \tilde {{\longrightarrow}}U_e \subset {{\mathfrak{g} _{\mathbb{C}}}}^*$ sending $[f] \mapsto df_e$. Then $\{f,g\} = \be (df ,dg) \mapsto d(\be(df,dg))_e = [df_e , dg_e]$, so $d_e \be ^*$ satisfies the Jacobi identity. If, however, we assume that $d_e \be ^*$ satisfies the Jacobi identity, then the proof in [@vai] follows as written and $[\be, \be] = 0$, which by Proposition \[22jan1\] implies that $D$ is integrable. $\square$
If $G$ is simply connected, a Dirac group structure on $G$ is equivalent to an ideal $\fk$ of ${{\mathfrak{g} _{\mathbb{C}}}}$ and a Lie bialgebra structure on ${{\mathfrak{g} _{\mathbb{C}}}}/ \fk$.
First let $L(\be , U)$ be a Dirac group structure on $G$. For the ideal $\fk = Ann (U_e)$, a Lie bi-algebra structure on ${{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ is equivalent to a cocylce $\epsilon : {{\mathfrak{g} _{\mathbb{C}}}}/ \fk {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ such that $\epsilon ^*$ is a Lie bracket. Then by Proposition \[5feb1\], $d_e \be : {{\mathfrak g}}{{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ is a cocycle. Since $L(\be , U)$ is integrable, $d_e \be (\fk) = 0$ by Lemma \[22jan4\]. So $d_e \be$ factors to some cocycle $\epsilon : {{\mathfrak{g} _{\mathbb{C}}}}/ \fk {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$. By Lemma \[5feb2\], integrability of $L(\be , U)$ implies that ${\varepsilon}^*$ satisfies the Jacobi identity.\
Now let $\epsilon : {{\mathfrak{g} _{\mathbb{C}}}}/ \fk {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ define a Lie bi-algebra structure. Then letting $\epsilon ^\prime : {{\mathfrak g}}{{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ be the composition of $\epsilon $ with ${{\mathfrak g}}\hookrightarrow {{\mathfrak{g} _{\mathbb{C}}}}{{\longrightarrow}}{{\mathfrak{g} _{\mathbb{C}}}}/ \fk$. Then $\epsilon ^\prime$ is a cocycle for ${{\mathfrak g}}$. By Proposition \[5feb1\], there is a unique $\be : G {{\longrightarrow}}\wedge ^2 {{\mathfrak{g} _{\mathbb{C}}}}/ \fk$ such that $d _e \be = \epsilon ^\prime$. Since $\epsilon$ satisfies the Jacobi identity, $L(\be , Ann(\fk))$ is integrable (by Lemma \[5feb2\]).\
Clearly these two processes are inverses of each other. $\square$
Dual-Dirac Groups
-----------------
A *dual-Dirac group* is a group with Dirac structure such that group multiplication is a dual-Dirac map.
Dual-Dirac group structures on a Lie group $G$ are in bijection with pairs $(E, {\varepsilon})$ of a G-invariant ideal $E $ of ${{\mathfrak{g} _{\mathbb{C}}}}$ and ${\varepsilon}\in \wedge ^2 E^*$ such that ${\varepsilon}$ is G-invariant and ${\varepsilon}$ is a cocycle in the Lie algebra cohomology of $E$ with values in ${{\mathbb C}}$ (i.e. ${\varepsilon}(x, [y,z] ) + {\varepsilon}(y, [z,x]) + {\varepsilon}(z , [x,y]) = 0 $ for all $x,y,z \in E$). If $G$ is connected semisimple, dual-Dirac group structures on $G$ are given by ideals of ${{\mathfrak{g} _{\mathbb{C}}}}$.
Let $(G, D)$ be an almost dual-Dirac group. That is, multiplication is dual-Dirac and $D$ is an almost Dirac structure. Then poinwise, $D_g = L(E_g , {\varepsilon}_g )$. We again identify $TG _{{\mathbb C}}\simeq G \times {{\mathfrak{g} _{\mathbb{C}}}}$ by left-translation, and we think of $E $ as an assignment of a subspace of ${{\mathfrak{g} _{\mathbb{C}}}}$ for each $g \in G$. Condition $(M1)^*$ is that $Ad(h{^{-1}})E_g + E_h \subset E_{gh}$. Hence, $E_h \subset E_{gh}$ for all $g$, $h$. Clearly then $E_g = E_e$ for all $g$. Thus, $E$ is constant or left-invariant. But also $Ad(h{^{-1}}) E \subset E$ for all $h \in G$, so $E$ is a G-invariant ideal of ${{\mathfrak{g} _{\mathbb{C}}}}$. The requirement $(M2)^*$ is that $(Ad(h{^{-1}})^* {\varepsilon}_{gh} , {\varepsilon}_{gh} ) = ({\varepsilon}_g , {\varepsilon}_h )$, so ${\varepsilon}: G {{\longrightarrow}}\wedge ^2 E^*$ is also constant. Now $(M2 ^*) $ holds if and only if $Ad (G)^* {\varepsilon}= {\varepsilon}$.\
Gualtieri shows that almost Dirac structures of the form $L(E, {\varepsilon})$ are integrable if and only if $E$ is an integrable distribution and $d_E {\varepsilon}= 0$. Since $E$ is an ideal, $G \times E \subset G \times {{\mathfrak{g} _{\mathbb{C}}}}\simeq TG _{{\mathbb C}}$ is an integrable distribution.\
When $G$ is connected semisimple, $E$ is also semisimple, so $H^2 (E , {{\mathbb C}}) = 0$ [@wei]. Hence, ${\varepsilon}= \phi \circ [ \, , \, ]$ for some $\phi \in {{\mathfrak{g} _{\mathbb{C}}}}^*$. $G$-invariance of ${\varepsilon}$ implies that $\phi$ is $G$-invariant. Since $G$ is semisimple, this is only possible if $\phi = 0$. $\square$
Twisted Groups
==============
Here we define groups in ${{\mathcal T}}{{\mathcal D}}^*$ and ${{\mathcal T}}{{\mathcal D}}$ and describe the data for such groups.
Groups in ${{\mathcal T}}{{\mathcal D}}^*$
-------------------------------------------
We consider only Courant algebroids of the form ${{\mathcal V}}_{G , H}$, i.e. exact Courant algebroids in the category ${{\mathcal T}}{{\mathcal D}}^* \subset {{\mathcal C}}{{\mathcal D}}^*$. Products do not exist in ${{\mathcal C}}$ or ${{\mathcal C}}{{\mathcal D}}^*$, so we cannot have groups objects in these categories. However, we will say that a *group* in ${{\mathcal T}}{{\mathcal D}}^*$ consists of a group $G$, a Dirac structure $D \subset {{\mathcal V}}_{G, H}$, and a 2-form $B \in \Om ^2 (G\times G)$ such that $(\mu ,B) : (G \times G , D \oplus D ) {{\longrightarrow}}(G , D)$ is a morphism in ${{\mathcal T}}{{\mathcal D}}^*$. In Proposition \[4augb\] we trivialize the vector bundle ${{\mathcal V}}_{G, H}$ by left translation and let $\pi _1 \; , \pi _2 : G \times G {{\longrightarrow}}G$ be the first and second projection maps respectively.
\[4augb\] A group $(G,D,H,B)$ in ${{\mathcal T}}{{\mathcal D}}^*$ is given by the following data:
1. a G-invariant ideal $E \subset {{\mathfrak{g} _{\mathbb{C}}}}$,
2. a section ${\varepsilon}\in \Ga (G, \wedge ^2 \tilde E)$, where $\tilde E$ is the bi-invariant distribution defined by $E$,
3. a section $B = B_1 + B_2 + 0 $ of $\wedge ^2 T^*G =( \pi _1 ^* \wedge ^2 T^*G )\oplus (\pi _2 ^* \wedge ^2 T^*G )\oplus (\pi _1 ^* T^*G \otimes \pi _2 ^* T^*G )$ such that on $\tilde E$ one has $(B_1)_{(g,h)} = {\varepsilon}_g - Ad (h{^{-1}})^* {\varepsilon}_{gh} $, $(B_2)_{(g,h)} = {\varepsilon}_h - {\varepsilon}_{gh}$ for all $g,h \in G$. Similarly, we also require that $(dB_1)_{(g,h)} = H _g - Ad (h{^{-1}})^* H _{gh} $, $(dB_2)_{(g,h)} = H _h - H _{gh}$ for all $g,h \in G$,
4. $H_{|E} = -d_E {\varepsilon}$, where $d_E {\varepsilon}$ is the restriction to $\tilde E$ of the differential of any 2-form restricting to ${\varepsilon}$.
For such a $(E, B, {\varepsilon}, H)$, $D = L(\tilde E , {\varepsilon}) \subset {{\mathcal V}}_{G,H}$ is the corresponding Dirac structure on $G$. If we require that $B = \pi _1 ^* \om \oplus \pi _2 ^* \om $ for some $\om \in \Om ^2 (G)$, then $\om _{|E} = 0$ and ${\varepsilon}$ is constant (i.e. bi-invariant). In particular, if $G$ is semisimple, then ${\varepsilon}= 0$.
We use left translation to trivialize ${{\mathcal V}}_G$, and pointwise, there is $E_g \subset {{\mathfrak{g} _{\mathbb{C}}}}$ and ${\varepsilon}_g \in \wedge ^2 E_g ^*$ such that $D_g = L(E_g , {\varepsilon}_g)$. The condition $(M1)^*$ is that $Ad(h)\in E_g + E_h \subset E_{gh}$, which implies that $E := E_e = E_g $ for all $g \in G$ and $Ad(h {^{-1}})E \subset E$. Therefore, $E$ is a $G$-invariant ideal. Condition $(M2)^*$ is that $ (Ad(h{^{-1}})^*{\varepsilon}_{gh} , {\varepsilon}_{gh} ) = ({\varepsilon}_g , {\varepsilon}_h ) + B_{(g,h)}$, which is exactly condition (3) of this proposition. Therefore, for $(\mu , B)$ to be a morphism in ${{\mathcal T}}{{\mathcal D}}^*$, it is necessary and sufficient that $\mu ^* H = H \oplus H + dB$ as is explained in part (3) and that $E$ ,${\varepsilon}$, and $B$ be as in (1) and (2). For $D = L(E, {\varepsilon}) \subset {{\mathcal V}}_{G, H}$ to be integrable, $d_E {\varepsilon}= -H_{|E}$.\
Now if $B = \om + \om$, then (2) implies that $\om _{|E} = 0$, ${\varepsilon}$ is constant and $G$-invariant. When ${{\mathfrak{g} _{\mathbb{C}}}}$ is semisimple, so is $E$. The only possible ${\varepsilon}$ is ${\varepsilon}= 0$. $\square$
There is an example of a group in ${{\mathcal T}}{{\mathcal D}}^*$, called the Cartan-Dirac structure, described in detail in [@abm].
Groups in ${{\mathcal T}}{{\mathcal D}}$
----------------------------------------
Here we define twisted Dirac groups and show that they are a generalization of twisted Poisson groups.
We define the full subcategory ${{\mathcal T}}{{\mathcal D}}$ of ${{\mathcal C}}{{\mathcal D}}$ of all objects of the form $(M , {{\mathcal V}}_{M,H} , id , D) $. The data for an object in ${{\mathcal T}}{{\mathcal D}}$ is just a triple $(M,H,D)$. We say that $(G,H,D)$ is a *group in ${{\mathcal T}}{{\mathcal D}}$* if $G$ is a Lie group with multiplication $\mu$ such that $(\mu , B) : (G\times G , H \oplus H , D \oplus D ) {{\longrightarrow}}(G, H,D)$ is a map in ${{\mathcal T}}{{\mathcal D}}$ for some 2-form $B$ on $G \times G$.
The object $(G , D \subset {{\mathcal V}}_{M, H})$ in ${{\mathcal T}}{{\mathcal D}}$ is a group in ${{\mathcal T}}{{\mathcal D}}$ if and only if $(G,D)$ is an almost Dirac group and $H $ is a section of $\wedge ^2 U \subset \wedge ^2 T^*G$ such that $[\be , \be ] = \be _\sharp H$ and such that $\mu ^* (H \oplus H) -H$ is exact.
The condition for $\mu $ to be a map in ${{\mathcal T}}{{\mathcal D}}$ is simply for $\mu$ to be a Dirac map. This happens exactly when $(G,D)$ is an almost Dirac group. In this case, $D = L(\be , U)$. Integrability of $D \subset {{\mathcal V}}_{M,H}$ is equivalent to $Nij_H = 0$ on $D$, where $Nij_H (X + \xi , Y + \eta , Z + \al ) = Nij(X + \xi , Y + \eta , Z + \al ) + H(X,Y,Z)$. We recall [@gua] that $Nij_H (X + df ,Y + dg ,Z + dh) = Jac_{\{\; , \; \}} (f,g,h) + H(X, Y,Z)$. What this says is that when $X _{|U} = \be _\sharp (df)$, $Y _{|U} = \be _\sharp (dg)$, and $Z _{|U} = \be _\sharp (dh)$, $H(X,Y,Z) = -Jac_{\{\; , \; \}} (f,g,h) $. This means that $H(X,Y,Z)$ really only depends on the images $\overline X $, $\overline Y$, and $\overline Z$ in $U^*$, not on $X$, $Y$, $Z$ themselves. In other words, $H \in \wedge ^2 U \subset \wedge ^2 T^*G$. Now, $H(X,Y,Z) = H(\be _\sharp (df) , \be _\sharp (dg) , \be _\sharp (dh)) = \be _\sharp H (df , dg, dh)$. The integrability condition is that $\be _\sharp H = Jac _{\{ \, , \, \}} = [\be , \be ]$. The condition on $B$ and $H$ being related by the pullback $\mu ^*$ is unrelated to either of these conditions. $\square$
[10]{}
Alekseev, A.; Bursztyn, H.; Meinrenken, E.; Pure Spinors on Lie Groups (arXiv:0709.1452v1)
Ben-Bassat, O.; Boyarchenko, M.; Submanifolds of Generalized (Almost)Complex Manifolds (arXiv:math/ 0309013v1)
Bhaskara, B.H; Viswanath, K.; [*Poisson algebras and Poisson manifolds*]{}, Pitman Research Notes in Math. 174, Longman Sci., Harlow and New York, 1988
Bursztyn, H.; Cavalcanti, G; Gualtieri, M.; Reduction of Courant algebebroids and generalized complex structures (arXiv:math/0509640v3)
Bursztyn, H.; Radko, O.; Guage equivalence of Dirac structures and symplectic groupoids (arXiv:math/ 0202099v3)
Courant, T., Dirac Manifolds, [*Tans. Amer. Math. Soc.* ]{}, 319 pp. 631-661, 1990
Courant, T.; Weinstein, A.; Beyond Poisson Structures. [*Action hamiltoniennes de groupes. Troisieme theoreme de Lie* ]{}; vol. 27 of [*Travaux en Cours*]{}, pp. 39-49; Hermann, Paris, 1988
Crainic, M., Generalized complex stuctures and Lie brackets (DG/0412097)
Gualtieri, M. Branes on Poisson varieties (arxiv:0710.2719v1)
Gualtieri, M., Generalized Complex Geometry (math.DG/0401221 v1)
Gualtieri, Generalized Complex Geometry (arxiv:math/0703298v2)
Hitchin, N., Generalized Calabi-Yau manifolds (DG/0209099v1)
Kapustin, A., A-branes and Noncommutative Geometry (hep-th/0502212)
Laurent-Genoux, C.; Stiénon, M.; Xu, P.; [*Holomorphic Poisson Structures and Groupoids* ]{} (arxiv: 07074253 v1)
Liu, Z.-J.; Weinstein, A.; Xu, P.; Manin triples for Lie bialgebroids, [*J. Diff. Geom.*]{}, 45. pp. 547-574, 1997
Lu, J.H.; Weinstein, A., Poisson Lie groups, dressing transformations and Bruhat decompositions, Journal of Differential Geometry 31, pp. 501-526, 1990
Ornea, S; Pantilie, R.; Holomorphic Maps Between Generalized Complex Manifolds (arxiv: DG/0810.1865v4)
Ševera, P.; Weinstein, A. Poisson geometry with a 3-form backgroud, *Prog. Theor. Phys. Suppl.* , 144, pp. 145-154, 2001
Vaisman, I., [*Lectures on the geometry of Poisson manifolds*]{}, Birkhauser, 1994
Vaisman, I., Reduction and submanifolds of generalized complex manifolds (math.DG/0511013 v2)
Weibel, C.A., [*An introduction to homological algebra*]{}, Cambridge University Press, 1994
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The calculation of the magneto-crystalline anisotropy energy (MAE) [@vanVleck:37; @brooks; @fletcher; @sloncewskij; @asdente] of magnetic materials containing transition-metal elements from first principles electronic structure calculations is a long-standing problem. The MAE is defined as the difference of total energies with the orientations of magnetization pointing in different, e.g., (001) and (111), crystalline axis. The difference is not zero because of spin-orbit effect, which couples the magnetization to the lattice, and determines the direction of magnetization, called the easy axis.
Being a ground state property, the MAE should be accessible in principle via density functional theory (DFT) [@hk64; @ks65]. Despite the primary difficulty related to the smallness of MAE ($\sim 1\ \mu eV/$atom), great efforts to compute the quantity with advanced total energy methods based on local density approximation (LDA) combined with the development of faster computers, have seen success in predicting its correct orders of magnitudes [@efn; @halilov; @jansen; @tjew; @dks]. However, the correct easy axis of Ni has not been predicted by this method and the fundamental problem of understanding MAE is still open.
A great amount of work has been done to understand what is the difficulty in predicting the correct axis for Ni. Parameters within the LDA calculation have been varied to capture physical effects which might not be correctly described. These include (i) scaling spin-orbit coupling in order to enlarge its effect on the MAE [@halilov; @jansen], (ii) calculating torque to avoid comparing large numbers of energy [@jansen], (iii) studying the effects of the second Hunds rule in the orbital polarization theory [@tjew], (iv) analyzing possible changes in the position of the Fermi level by changing the number of valence electrons [@dks], (v) using the state tracking method [@freeman], and (iv) real space approach [@beiden].
In this paper we take a new view that the correlation effects within the $d$ shell are important for the magnetic anisotropy of $3d$ transition metals like Ni. These effects are not captured by the LDA but are described by Hubbard–like interactions presented in these systems and need to be treated by first principles methods[@anisimov].
Another effect which has not been investigated in the context of magnetic anisotropy calculations is the non-collinear nature of intra-atomic magnetization [@singh]. It is expected to be important when spin-orbit coupling and correlation effects come into play together. In this article we show that when we include these new ingredients into the calculation we solve the long-standing problem of predicting the correct easy axis of Ni.
We believe that the physics of transition metal compounds is intermediate between atomic limit where the localized $d$ electrons are treated in the real space and fully itinerant limit where the electrons are described by band theory in k space. A many–body method incorporating these two important limits is the dynamical mean–field theory (DMFT) [@gabi:rmp]. The DMFT approach has been extensively used to study model Hamiltonian of correlated electron systems in the weak, strong and intermediate coupling regimes. It has been very successful in describing the physics of realistic systems such as transition metal oxides and, therefore, is expected to treat properly the materials with $d$ or $f$ electrons.
Electron–electron correlation matrix $U_{\gamma _{1}\gamma _{2}\gamma
_{3}\gamma _{4}}=\left\langle m_{1}m_{3}\left| v_{C}\right|
m_{2}m_{4}\right\rangle \delta _{s_{1}s_{2}}\delta _{s_{3}s_{4}}$ for $d$ orbitals is the quantity which takes strong correlations into account. This matrix can be expressed via Slater integrals $F^{(i)} $, $i=0,2,4,6$ in the standard manner. The inclusion of this interaction generates self–energy $%
\Sigma_{\gamma_1\gamma_2}(i\omega_n,\vec{k})$ on top of the one–electron spectra. Within DMFT it is approximated by momentum independent self–energy $\Sigma_{\gamma_1\gamma_2}(i\omega_n)$.
A central quantity of the dynamical mean–field theory is the one–electron on–site Green function $$\begin{aligned}
G_{\gamma_1\gamma_2}(i\omega_n)=&\sum_{\vec{k}}&\left[(i\omega_n+\mu)
O_{\gamma_1\gamma_2}(\vec{k}) -H^0_{\gamma_1\gamma_2}(\vec{k})\right.
\nonumber \\
&+&\left. v_{dc}-\Sigma_{\gamma_1\gamma_2} (i\omega_n)\right]^{-1}.
\label{dmft}\end{aligned}$$ where $H^0_{\gamma_1\gamma_2}(\vec{k})$ is the one–electron Hamiltonian standardly treatable within the LDA. Since the latter already includes the electron-electron interactions in some averaged way, we subtract the double counting term $v_{dc}$ [@laz]. The use of realistic localized orbital representation such as linear muffin–tin orbitals [@OA75] leads us to include overlap matrix $O_{\gamma_1\gamma_2}(\vec{k})$ into the calculation.
The DMFT reduces the problem to solving effective impurity model where the correlated $d$ orbitals are treated as an impurity level hybridized with the bath of conduction electrons. The role of hybridization is played by the so–called bath Green function defined as follows: $$[{\cal G}_0^{-1}]_{\gamma_1\gamma_2}(i\omega_n)=
G_{\gamma_1\gamma_2}{}^{-1}(i\omega_n) +\Sigma_{\gamma_1\gamma_2}(i\omega_n).$$ Solving this impurity model gives access to the self–energy $%
\Sigma_{\gamma_1\gamma_2}(i\omega_n)$ for the correlated electrons. The one–electron Green function (\[dmft\]) is now modified with new $%
\Sigma_{\gamma_1\gamma_2}(i\omega_n)$, which generates a new bath Green function. Therefore, the whole problem requires self–consistency.
In this paper we confine ourselves to zero temperature and make an additional assumption on solving the impurity model using the Hartree–Fock approximation. In this approximation the self–energy reduces to $$\Sigma_{\gamma_1\gamma_2}=\sum_{\gamma_3\gamma_4} (U_{\gamma _{1}\gamma
_{2}\gamma _{3}\gamma _{4}}-U_{\gamma _{1}\gamma _{2}\gamma _{4}\gamma
_{3}}) \bar{n}_{\gamma _{3}\gamma _{4}} \label{HF}$$ where $\bar n_{\gamma_1\gamma_2}$ is the average occupation matrix for the correlated orbitals. The off-diagonal elements of the occupancy matrix are not zero when spin-orbit coupling is included [@sol]. The latter can be implemented following the prescription of Andersen [@OA75] or more recent one by Pederson [@pk].
In the Hartree–Fock limit the self–energy is frequency independent and real. The self–consistency condition of DMFT can be expressed in terms of the average occupation matrix: Having started from some $\bar n%
_{\gamma_1\gamma_2}$ we find $\Sigma_{\gamma_1\gamma_2}$ according to (\[HF\]). Fortunately, the computation of the on–site Green function (\[dmft\]) needs [*not*]{} to be performed. Since the self–energy is real, the new occupancies can be calculated from the eigenvectors of the one–electron Hamiltonians with $\Sigma_{\gamma_1\gamma_2}-v_{dc}$ added to its $dd$ block. The latter can be viewed as an orbital–dependent potential which has been introduced by the LDA+U method [@anisimov].
The LDA$+$U method has been very successful compared with experiments at zero temperature in ordered compounds. By establishing its equivalence to the static limit of the DMFT we see clearly that dynamical mean–field theory is a way of improving upon it, which is crucial for finite temperature properties.
In this work we study the effect of the Slater parameters $F_{0}$, $F_{2}$ and $F_{4}$ on the magnetic anisotropy energy. Slater integrals can be linked to intra–atomic repulsion $U$ and exchange $J$ obtained from LSDA supercell procedures via $U=F^{0}$ and $J=(F^{2}+F^{4})/14$. The ratio $%
F^{2}/F^{4}$ is to a good accuracy a constant $\sim 0.625$ for $d$ electrons [@afl]. The MAE is calculated by taking the difference of two total energies with different directions of magnetization (MAE=$E(111)-E(001)
$). The total energies are obtained via fully self consistent solutions. Since the total energy calculation requires high precision, full potential LMTO method [@Sav] has been employed. For the $\vec{k}$ space integration, we follow the analysis given by Trygg and co–workers [@tjew] and use the special point method [@froyen] with a Gaussian broadening [@mp] of $15\ mRy$. The validity and convergence of this procedure has been tested in their work [@tjew]. For convergence of the total energies within desired accuracy, about $15000\ k$-points are needed. We used $28000\ k$-points to reduce possible numerical noise, where the convergency is tested up to $84000k$-points. Our calculations include non-spherical terms of the charge density and potential both within the atomic spheres and in the interstitial region [@Sav]. All low-lying semi-core states are treated together with the valence states in a common Hamiltonian matrix in order to avoid unnecessary uncertainties. These calculations are spin polarized and assume the existence of long-range magnetic order. Spin-orbit coupling is implemented according to the suggestions by Andersen [@OA75]. We also treat magnetization as a general vector field, which realizes non-collinear intra-atomic nature of this quantity. Such general magnetization scheme has been recently discussed [@singh].
To incorporate the effects of intraatomic correlations on the magnetocrystalline anisotropy energy, we have to take into account the intra–atomic repulsion $U$ and the intraatomic exchange $J$. It is important to perform the calculations for fixed values of magnetic moments which themselves show a dependency on $U$ and $J$. Since the pure LSDA result ($U$=0, and $J$=0) reproduces the experimental values for magnetic moments in both Fe and Ni fairly well, we have scanned the $U-J$ parameter space and have obtained the path of $U$ and $J$ values which hold the theoretical moment constant, following the approach of Ref. [@Kudrnovsky].
We now discuss our calculated MAE. We first test our method in case of LDA ($%
U=J=0$). To compare with previous calculations, we turn off the non-collinearity of magnetization which makes it collinear with the quantization axis. The calculation gives correct orders of magnitude for both fcc Ni and bcc Fe but with the wrong easy axis for Ni, which is the same result as the previous result [@tjew]. Turning on the non-collinearity results in a a larger value of the absolute value of the MAE ($2.9\ \mu eV$) for Ni but the easy axis predicted to be (001) which is still wrong. The magnitude of the experimental MAE of Ni is $2.8\ \mu eV$ aligned along $(111)$ direction [@landolt].
We now describe the effect of correlations, which is crucial in predicting the correct axis of Ni (see Fig. \[mae\]). We walked along the path of parameters $U$ and $J$ which hold the magnetic moment to 0.6 $\mu _{B}$. The MAE first increases to $60\ \mu eV$ ($U=0.5\ eV$, $J=0.3\ eV$) and then decreases. While decreasing it makes a rather flat area from $U=1.4\ eV$, $%
J=0.9\ eV$ to $U=1.7\ eV$, $J=1.1\ eV$ where MAE is positive and around $10\
\mu eV$. After the flat area, the MAE changes from the wrong easy axis to the correct easy axis. The correct magnetic anisotropy is predicted at $U
=1.9\ eV$ and $J=1.2\ eV$. The change from the wrong easy axis to the correct easy axis occurs over the range of $\delta U\sim 0.2eV$, which is of the order of spin-orbit coupling constant ($\sim 0.1eV$).
For Fe, the MAE is calculated along the path of $U$ and $J$ values which fixes the magnetic moment to $2.2\ \mu _{B}$. At $U=0\ eV$ and $J=0\ eV$, the MAE is $0.5\ \mu eV$. It increases as we move along the contour in the direction of increasing $U$ and $J$. The correct MAE with the correct direction of magnetic moment is predicted at $U=1.2\ eV$ and $J=0.8\ eV$.
Notice that the trends in the values of $U$ and $J$ that are necessary to reproduce the correct magnetic anisotropy energy within LDA + U are similar to the values used to describe photoemission spectra of these materials [@kl99] within DMFT. The values of the parameters $U$ and $J$, are basis–set dependent, and method dependent, but the values of $U$ used in our LDA+ U calculation are within $1eV$ of those used in [@kl99]. Since DMFT contains the graphs which screen the on–site interaction which are ommitted in the LDA+ U functional, a larger value of $U$ is needed to produce the correct moment in DMFT.
We find direct correlation between the dependency of the MAE as a function of $U$-$J$ and the difference of magnetic moments ($\Delta m=-(m(111)-m(001)$) behaving similarly (see Fig. \[mae\]). For Ni the difference increases till $U=0.4\ eV$ and $J=0.2\ eV$, then decreases. While decreasing it makes a flat area from $U=0.9\ eV$ and $J=0.6$ to $U=1.7\ eV$ and $J=1.1\ eV$. After the flat area, the difference decreases rapidly. For Fe, the difference of magnetic moments slightly fluctuates till $U=0.7\ eV$ and $%
J=0.5\ eV$ and then decreases till $U=1.0\ eV$ and $J=0.7\ eV$ .
This concurrent change of MAE and the difference of magnetic moments suggests why some previous attempts based on force theorem [@dks] failed in predicting the correct easy axes. Force theorem replaces the difference of the total energies by the difference of one–electron energies. In this approach, the contribution from the slight difference in magnetic moments does not appear and, therefore, is not counted in properly. Unfortunately, we could not find any experimental data of magnetic moments with different orientations to the desired precision ($10^{-4}\mu_B$) to compare with.
We now present implications of our results on the calculated electronic structure for the case of Ni. One important feature which emerges from the calculation is the absence of the $X_2$ pocket (see Fig. \[fermi\]). This has been predicted by LDA but has not been found experimentally [@wc]. The band corresponding to the pocket is pushed down just below the Fermi level. This is expected since correlation effects are more important for slower electrons and the velocity near the pocket is rather small. It turns out that the whole band is submerged under the Fermi level. We also find that the removal of the $X_2$ point is near the point $U=1.9\ eV$ and $%
J=1.2\ eV$. For comparison, the corresponding band is just above the Fermi level at $U=1.9\ eV$ and $J=1.1\ eV$ forming a tiny pocket. This strengthens the connection between MAE and the absence of $X_2$ pocket.
There has been some suspicions that the incorrect position of the $X_2$ band within LDA was responsible for the incorrect prediction of the easy axis within this theory. Daalderop and coworkers [@dks] removed the $X_2$ pocket by increasing the number of valence electrons and found the correct easy axis. We therefore conclude that the absence of the pocket is one of the central elements in determining the magnetic anisotropy, and there is no need for any ad-hoc adjustment within a theory which takes into account the correlations.
We now describe the effects originated from (near) degenerate states close to the Fermi surface. These have been of primary interest in past analytic studies [@Kondorskii; @mori:74]. We will call such states [*degenerate Fermi surface crossing*]{} (DFSC) states. The contribution to MAE by non-DFSC states comes from the fourth order perturbation. Hence it is of the order of $\lambda^4$. The energy splitting between DFSC states due to spin-orbit coupling is of the order of $\lambda$ because the contribution comes from the first order perturbation. Using linear approximation of the dispersion relation $\epsilon(\vec{k}\lambda)$, the relevant volume in $k$-space was found of the order $\lambda^3$. Thus, these DFSC states make contribution of the order of $\lambda^4$. Moreover, there may be accidentally DFSC states appearing along a line on the Fermi surface, rather than at a point. We have found this case in our LDA calculation for Ni. Therefore the contribution of DFSC states is as important as the bulk non-DFSC states though the degeneracies occur only in small portion of the Brillouin zone.
The importance of the DFSC states leads us to comparative analysis of the LDA and LDA+U band structures near the Fermi level. In LDA, five bands are crossing the Fermi level at nearly the same points along the $\Gamma X$ direction. Two of the five bands are degenerate for the residual symmetry and the other three bands accidentally cross the Fermi surface at nearly the same points. There are two $sp$ bands with spin up and spin down, respectively. The other three bands are dominated by $d$ orbitals. In LDA$+$U, one of the $d$ bands is pushed down below the Fermi surface. The other four bands are divided into two degenerate pieces at the Fermi level (see Fig. \[fermi\]): Two symmetry related degenerate $d_\downarrow$ bands and two near degenerate $sp_\uparrow$ and $sp_\downarrow$ bands. In LDA, we found that two bands are accidentally near degenerate along the line on the Fermi surface within the plane $\Gamma X L$. One band is dominated by $%
d_\downarrow$ orbitals. The other is dominated by $d_\downarrow$ orbitals near $X$ and by $s_\downarrow$ orbitals off $X$. In LDA+U, these accidental DFSC states disappear(see Fig. \[fermi\]).
As we have seen, the on-site repulsion $U$ reduces the number of DFSC states along $\Gamma X$ direction. Based on the tight–binding model, the importance of DFSC states has been shown. We see that strong correlations reduce number of DFSC states in $\Gamma X$ direction and remove the near degenerate states on $\Gamma X L$ plane. We conclude that the change of DFSC states is another important element that determines the easy axis of Ni.
To conclude, we have demonstrated that it is possible to perform highly precise calculation of the total energy in order to obtain both the correct easy axes and the magnitudes of MAE for Fe and Ni. This has been accomplished by including the strong correlation effects via taking intra–atomic repulsion and exchange into account and incorporating the non–collinear magnetization. In both Fe and Ni, both $U$ and $J$ take physically acceptable values consistent with the values known from atomic physics. The calculations performed are state of the art in what can currently be achieved for realistic treatments of correlated solids. Further studies should be devoted to improving the quality of the solution of the impurity model within DMFT and extending the calculation to finite temperatures.
This research was supported by the ONR grant N0014-99-1-0653. GK would like to thank K. Hathaway for discussing the origin of magnetic anisotropy and G. Lonzarich for discussing dHvA data. We thank R. Chitra for stimulating discussion. We thank V. Oudovenko for setting up the computer cluster used to perform these calculations. We have also used the supercomputer at the Center for Advanced Information Processing, Rutgers. IY thanks K. H. Ahn for discussions.
J. H. van Vleck, Phys. Rev. [**52**]{}, 1178 (1937).
H. Brooks, Phys. Rev. [**58**]{}, 909 (1940).
G. C. Fletcher, Proc. R. Soc. London [**67A**]{}, 505 (1954).
J. C. Sloncewskij, J. Phys. Soc. Jpn. [**17**]{}, Suppl. B (1962).
M. Asdente and M. Delitala, Phys. Rev. [**163**]{}, 497 (1967).
P. Hohenberg and W. Kohn, Phys. Rev. [**136**]{}, 864 (1964).
W. Kohn and L. J. Sham, Phys. Rev. [**140**]{}, 1133 (1965).
H. Eckardt, L. Fritsche, and J. Noffke, J. Phys. F [**17**]{}, 943 (1987).
S. V. Halilov [*et al.*]{}, Phys. Rev. B [**57**]{}, 9557 (1998).
H. J. F. Jansen, J. Appl. Phys. [**67**]{}, 4555 (1990).
J. Trygg, B. Johansson, O. Eriksson, and J. M. Willis, Phys. Rev. Lett. [** 75**]{}, 2871 (1995).
G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, Phys. Rev. B [** 41**]{}, 11 919 (1990).
D. Wang, R. Wu, and A. J. Freeman, Phys. Rev. Lett. [**70**]{}, 867 (1993).
S. V. Beiden, W. M. Temmerman, Z. Szotek, G. A. Gehring, G. M. Stocks, Y. Wang, D. M. C Nicholson, W. A. Shelton, and H. Ebert, Phys. Rev. B [**57**]{}, 14 247, (1998).
, edited by V. I. Anisimov (Gordon and Breach Science Publishers, Amsterdam, 2000).
L. Nordstrom and D. Singh, Phys. Rev. Lett. [**76**]{}, 4420 (1996).
A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. [** 68**]{}, 13 (1996).
A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.Rev. B [**12**]{}, 3060 (1975).
O. K. Andersen, Phys. Rev. B [**12**]{}, 3060 (1975).
I. Y. Solovyev, A. I. Liechtenstein, and K. Terakura, Phys.Rev. Lett. [** 80**]{}, 5758 (1999).
M. R. Pederson and S. N. Khanna, Phys. Rev. B [**60**]{}, 9566 (1999).
V. I. Anisimov, F. Aryastiawan, and A. I. Lichtenstein, J.Phys.: Condensed Matter [**9**]{}, 767 (1997).
S. Y. Savrasov, Phys. Rev. B [**54**]{}, 16470 (1996).
S. Froyen, Phys. Rev. B [**39**]{}, 3168 (1989). A. P. Cracknell, J. Phys. C [**2**]{}, 1425 (1969).
R. J. Needs, R. M. Martin, and O. H. Nielsen, Phys. Rev. B [**33**]{}, 3778 (1986). K.-M. Ho, C. L. Fu, and B. N. Harmon, Phys. Rev.Lett. [**49**]{}, 673 (1982).
M. Pajda, J. Kudrnovsky, I. Turek, V. Drchal, P. Bruno, cond-mat/0007441;
M. B. Stearns, [in [*Magnetic Properties of $3d$, $4d$, and $5d$ Elements, Alloys and Compounds*]{}]{}, Vol. III of [*Landolt-Börnstein, New Series*]{}, edited by K.-H. Hellewege and O. Madelung, (Springer-Verlag, Berlin, 1987).
M. Katsenelson and A. Lichtenstein, J. Phys. Cond. Matt. [**11**]{}, 1037 (1999). M. Katsenelson and A. Lichtenstein, Phys. Rev. B. [**61**]{}, 8906 (2000).
C. S. Wang and J. Callaway, Phys. Rev. B [**9**]{}, 4897 (1973). F. Weling and J. Callaway, Phys. Rev. B [**26**]{}, 710 (1982).
E. I. Kondorskii and E. Straube, Sov. Phys.-JETP [**36**]{}, 188 (1973).
N. Mori, Y. Fukuda, and T. Ukai, J. Phys. Soc. Jpn. [**37**]{}, 1263 (1974).
|
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abstract: 'Nuclear structure observables usually most effectively probe the properties of nuclear matter at subsaturation densities rather than at saturation density. We demonstrate that the electric dipole polarizibility $\alpha _ {\text{D}}$ in $^{208}$Pb is sensitive to both the magnitude $E_{\text{sym}}(\rho_{\text{c}})$ and density slope $L(\rho_{\text{c}})$ of the symmetry energy at the subsaturation cross density $\rho_{\text{c}} = 0.11$ fm$^{-3}$. Using the experimental data of $\alpha _ {\text{D}}$ in $^{208}$Pb from RCNP and the recent accurate constraint of $E_{\text{sym}}(\rho_{\text{c}})$ from the binding energy difference of heavy isotope pairs, we extract a value of $L(\rho_{\text{c}}) = 47.3 \pm 7.8$ MeV. The implication of the present constraint of $L(\rho_{\text{c}})$ to the symmetry energy at saturation density, the neutron skin thickness of $^{208}$Pb and the core-crust transition density in neutron stars is discussed.'
author:
- Zhen Zhang
- 'Lie-Wen Chen[^1]'
title: 'Constraining the density slope of nuclear symmetry energy at subsaturation densities using electric dipole polarizability in $^{208}$Pb'
---
Introduction
============
Due to its multifaceted roles in nuclear physics and astrophysics [@Lat04; @Ste05; @Bar05; @LCK08] as well as new physics beyond the standard model [@Hor01b; @Sil05; @Wen09; @Zhe14], the symmetry energy has become a hot topic in current research frontiers of nuclear physics and astrophysics [@LiBAEPJA14]. During the last decade, a lot of experimental, observational and theoretical efforts have been devoted to constraining the magnitude $E_{\text{sym}}(\rho)$ and density slope $L(\rho)$ of the symmetry energy at nuclear saturation density $\rho_0$ ($\sim 0.16$ fm$^{-3}$), i.e., $E_{\text{sym}}(\rho_0)$ and $L(\rho_0)$. Although important progress has been made, large uncertainties on the values of $E_{\text{sym}}(\rho_0)$ and $L(\rho_0)$ still exist (See, e.g., Refs. [@Bar05; @Ste05; @LCK08; @LiBAEPJA14; @Tsa12; @Lat12; @ChenLW12; @LiBA12; @Hor14]). For instance, while the $E_{\text{sym}}(\rho_0)$ is determined to be around $32\pm 4$ MeV, the extracted $L(\rho_0)$ varies significantly from about $20$ to $115$ MeV, depending on the observables and analysis methods. To better understand the model dependence and narrow the uncertainties of the constraints is thus of extreme importance.
While many studies on heavy ion collisions and neutron stars have significantly improved our knowledge on the symmetry energy, more and more constraints on the symmetry energy have been obtained in recent years from analyzing the properties of finite nuclei, such as the nuclear binding energy [@Mye96; @Dan09; @Liu10; @Mol12; @LatX12], the neutron skin thickness [@War09; @Che10; @Vin14], and the resonances and excitations [@Sag07; @Kli07; @Car10; @Rei10; @Tam11; @Pie12; @Roc13; @Pie14; @Col14]. Furthermore, it has been realized that the properties of finite nuclei usually provide more precise constraints on $E_{\text{sym}}(\rho)$ and $L(\rho)$ at subsaturation densities rather than at saturation density $\rho_0$. This feature is understandable since the characteristic (average) density of finite nuclei is less than $\rho_0$. For example, the average density of heavy nuclei (e.g., $^{208}$Pb) is about $0.11$ fm$^{-3}$, and thus the properties of heavy nuclei most effectively probe the properties of nuclear matter around $0.11$ fm$^{-3}$ [@Tri08; @Cao08; @Cen09; @Che11; @Kha12; @Zha13; @Bro13; @Wan13; @Roc13a; @Dan14; @Fat14]. Indeed, a quite accurate constraint on the symmetry energy at the subsaturation cross density $\rho_{\text{c}} = 0.11$ fm$^{-3}$, i.e., $E_{\text{sym}}(\rho_{\text{c}})=26.65 \pm 0.20$MeV, has been recently obtained from analyzing the binding energy difference of heavy isotope pairs [@Zha13]. In contrast to the fact that many and precise constraints on the magnitude of $E_{\text{sym}}(\rho )$ around $\rho_{\text{c}}$ have been obtained, to the best of our knowledge, so far there is only one experimental constraint on the density slope $L(\rho_{\text{c}})$ which was obtained from analyzing the neutron skin data of Sn isotopes [@Zha13]. Knowledge on $L(\rho_{\text{c}})$ is not only important for understanding the density dependence of the symmetry energy itself, but also plays a central role in determining the neutron skin thickness of heavy nuclei and the core-crust transition density in neutron stars. Therefore, any new constraints on $L(\rho_{\text{c}})$ will be extremely useful.
In the present work, with the precise knowledge of $E_{\text{sym}}(\rho_{\text{c}})$, we demonstrate that the electric dipole polarizability $\alpha_{\text{D}}$ in $^{208}$Pb measured at the Research Center for Nuclear Physics (RCNP) via polarized proton inelastic scattering at forward angles, can put a strong limit on the $L(\rho_{\text{c}})$. We emphasize since at forward angles Coulomb excitation dominates, the extracted $\alpha_{\text{D}}$ at RCNP is expected to be a relatively clean isovector indicator with less uncertainties from strong interaction.
Model and method
================
The symmetry energy and Skyrme-Hartree-Fock approach
----------------------------------------------------
The equation of state (EOS) of asymmetric nuclear matter, given by its binding energy per nucleon, can be written as $$E(\rho ,\delta )=E_{0}(\rho )+E_{\mathrm{sym}}(\rho )\delta ^{2}+O(\delta
^{4}), \label{EOSANM}$$where $\rho $ is the baryon density, $\delta=(\rho _{n}-\rho _{p})/(\rho _{p}+\rho _{n})$ is isospin asymmetry, $E_{0}(\rho )=E(\rho ,\delta =0)$ is the EOS of symmetric nuclear matter, and the symmetry energy is expressed as $$E_{\mathrm{sym}}(\rho )=\frac{1}{2!}\frac{\partial ^{2}E(\rho ,\delta )}{\partial \delta ^{2}}|_{\delta =0}. \label{Esym}$$Around a reference density $\rho _{r}$, the $E_{\mathrm{sym}}(\rho )$ can be expanded in $\chi_r=(\rho -{\rho _{r}})/\rho _{r}$ as $$E_{\text{sym}}(\rho )=E_{\text{sym}}({\rho _{r}})+\frac{L(\rho _{r})}{3}\chi_r+O(\chi_r^2),$$ where $L(\rho _{r})=3{\rho _{r}}\frac{\partial E_{\mathrm{sym}}(\rho )}{\partial
\rho }|_{\rho ={\rho _{r}}}$ is the density slope parameter which characterizes the density dependence of the symmetry energy around $\rho _{r}$.
Our calculations in the present work are based on the Skyrme-Hartree-Fock (SHF) approach with the so-called standard Skyrme force (see, e.g., Ref. [@Cha97; @Fri86; @Klu09]) which includes $10$ parameters, i.e., the $9$ Skyrme force parameters $\sigma $, $t_{0}-t_{3}$, $x_{0}-x_{3}$, and the spin-orbit coupling constant $W_{0}$. Instead of directly using the $9$ Skyrme force parameters, we can express them explicitly in terms of $9$ macroscopic quantities, i.e., $\rho _{0}$, $E_{0}(\rho_{0})$, the incompressibility $K_{0}$, the isoscalar effective mass $m_{s,0}^{\ast }$, the isovector effective mass $m_{v,0}^{\ast }$, $E_{\text{sym}}({\rho _{r}})$, $L({\rho _{r}})$, the gradient coefficient $G_{S}$, and the symmetry-gradient coefficient $G_{V}$. In this case, we can examine the correlation of properties of finite nuclei with each individual macroscopic quantity by varying individually these macroscopic quantities within their empirical ranges. Recently, this correlation analysis method has been successfully applied to study nuclear matter properties from analyzing nuclear structure observables [@Che10; @Che11a; @Che11; @Che12; @Zha13], and will also be used in this work.
Random-phase approximation and electric dipole polarizability
-------------------------------------------------------------
The random-phase approximation (RPA) provides an important microscopic approach to calculate the electric dipole polarizability in finite nuclei. Within the framework of RPA theory, for a given excitation operator $\hat{F}_{JM}$, the reduced transition probability from RPA ground state $|\tilde{0}\rangle $ to RPA excitation state $|\nu \rangle $ is given by: $$\begin{split}
B(EJ:\tilde{0}\rightarrow|\nu\rangle)&=|\langle\nu||\hat{F}_{J}||\tilde{0}\rangle|^2\\
&=\left|\sum_{mi}\left( X_{mi}^{\nu}+Y_{mi}^{\nu}\right) |\langle m||\hat{F}_{J}||i\rangle\right| ^2
\end{split},$$ where $m (i)$ denotes the unoccupied (occupied) single nucleon state; $\langle m||\hat{F}_{J}||i\rangle$ is the reduced matrix element of $\hat{F}_{JM}$; and $X_{mi}^{\nu}$ and $Y_{mi}^{\nu}$ are the RPA amplitudes. The strength function then can be calculated as: $$S(E)=\sum_{\nu}|\langle\nu\Vert\hat{F}_J\Vert\tilde{0}\rangle|^2\delta(E-E_{\nu}),$$ where $E_{\nu}$ is the energy of RPA excitation state $|\nu\rangle$. Thus the moments of strength function can be obtained as: $$m_k=\int dE E^kS(E)=\sum_{\nu}|\langle\nu\Vert\hat{F}_J\Vert\tilde{0}\rangle|^2E_{\nu}^k.$$ In the case of electric dipole ($E1$) response, the excitation operator is defined as: $$\hat{F}_{1M} = \frac{eN}{A}\sum^Z_{i=1}r_iY_{\text{1M}}(\hat{r}_i)-\frac{eZ}{A}\sum^N_{i=1}r_iY_{\text{1M}}(\hat{r}_i),$$ where $Z$, $N$ and $A$ are proton, neutron and mass number, respectively; $r_i$ is the nucleon’s radial coordinate; $Y_{\text{1M}}(\hat{r_i})$ is the corresponding spherical harmonic function. For a given Skyrme interaction, we can calculate the inverse energy-weighted moment $m_{-1}$ using the HF-RPA method, and then obtain the electric dipole polarizability $\alpha_{D}$ as $$\alpha_{D}=\frac{8\pi}{9}e^2\int dEE^{-1}S(E)=\frac{8\pi}{9}e^2m_{-1}. \label{AlphaDMm1}$$
For the theoretical calculations of electric dipole polarizability in $^{208}$Pb in the present work, we employ the Skyrme-RPA program by Col$\grave{\text{o}}$ [*et al*]{} [@Colo13]. In this program, the SHF mean field and the RPA excitations are fully self-consistent. In particular, we calculate the isovector dipole strength in $^{208}$Pb with a spherical box extending up to $24$ fm, a radial mesh of $0.1$ fm and a cutoff energy of $E_{\text{C}} = 150$ MeV which denotes the maximum energy of the unoccupied single-particle states in the RPA model space. Then the inverse energy-weighted moment $m_{-1}$ is evaluated with an upper integration limit of $130$ MeV according to the experimental energy range [@Tam11], and thus the electric dipole polarizability $\alpha_{\text{D}}$ can be calculated invoking Eq. .
The symmetry energy and electric dipole polarizability
------------------------------------------------------
The electric dipole polarizability $\alpha_{\text{D}}$ has been shown to be a sensitive probe of the symmetry energy [@Rei10; @Pie12; @Roc13]. In particular, based on the droplet model, Roca-Maza [*et al.*]{} [@Roc13] obtained the following relation: $$\label{AlphaDDM}
\alpha_{\mathrm{D}}= \frac{\pi e^2}{54}
\frac{A\left\langle r^2 \right\rangle}{E_{\mathrm{sym}}(\rho_0)}
\left[1+\frac{5}{3}
\frac{E_{\mathrm{sym}}(\rho_0)-a_{\mathrm{sym}}(A)}{E_{\mathrm{sym}}(\rho_0)}
\right],$$ where $\langle r^2 \rangle$ is the mean-square radius and $a_{\mathrm{sym}}(A)$ is the symmetry energy coefficient of a finite nucleus of mass number $A$. Furthermore, using the empirical relation $a_{\mathrm{sym}}(A)\approx E_{\mathrm{sym}}(\rho_A)$ [@Cen09; @Che11; @Dan14] and expanding $E_{\mathrm{sym}}(\rho_A)$ as $$E_{\mathrm{sym}}(\rho_A)\approx E_{\mathrm{sym}}(\rho_0)-L(\rho_0)(\rho_0-\rho_A)/3\rho_0,
\label{EsymrA}$$ Roca Maza *et al.* demonstrated that $\alpha_{\mathrm{D}}$ is correlated with both $E_{\mathrm{sym}}(\rho_0)$ and $L(\rho_0)$. Particularly, based on a large and representative set of relativistic and nonrelativistic nuclear mean-field models, they found a strong linear correlation between $\alpha_{\mathrm{D}}E_{\mathrm{sym}}(\rho_0)$ and $L(\rho_0)$ and then extracted the constraint $L(\rho_0) = 43 \pm (6)_{\text{expt}}\pm(8)_{\text{theor}}\pm(12)_{\text{est}}$ MeV from the combination of the experimental determination of $\alpha_{\mathrm{D}}$ with the empirical estimate of $E_{\text{sym}}(\rho_0) = 31 \pm (2)_{\text{est}}$ MeV. One can see that the uncertainty of the estimated $E_{\mathrm{sym}}(\rho_0)$ leads to a large error of $12$ MeV for $L(\rho_0)$.
Instead of expressing $E_{\mathrm{sym}}(\rho_A)$ in terms of $E_{\mathrm{sym}}(\rho_0)$ and $L(\rho_0)$ as in Eq. (\[EsymrA\]), one can also express $E_{\mathrm{sym}}(\rho_0)$ in terms of $E_{\mathrm{sym}}(\rho_c)$ and $L(\rho_c)$ as $$E_{\mathrm{sym}}(\rho_0)\approx E_{\mathrm{sym}}(\rho_c)+L(\rho_c)(\rho_0-\rho_c)/3\rho_c.
\label{Esymr0rc}$$ Noting $\rho_{208}\approx \rho_c$ [@Cen09; @Che11; @Dan14], one can then see from Eqs. (\[AlphaDDM\]) and (\[Esymr0rc\]) that $\alpha_{\mathrm{D}}$ in $^{208}$Pb is also correlated with both $L(\rho_c)$ and $E_{\mathrm{sym}}(\rho_c)$. As we will see in the following, the microscopic RPA calculations indeed show that $\alpha_D$ is sensitive to $E_{\mathrm{sym}}(\rho_0)$ and $L(\rho_0)$ as well as to $L(\rho_c)$ and $E_{\mathrm{sym}}(\rho_c)$. Since $E_{\mathrm{sym}}(\rho_c)$ has been stringently constrained recently (see, e.g., $E_{\mathrm{sym}}(\rho_c) = 26.65\pm0.20$ MeV in Ref.[@Zha13]), the $\alpha_{\mathrm{D}}$ in $^{208}$Pb can thus be used to constrain the $L(\rho_c)$ parameter.
Results and discussions
=======================
To examine the correlation of the $\alpha_{\text{D}}$ in $^{208}$Pb with each macroscopic quantity, especially on $E_{\text{sym}}({\rho _r})$ and $L(\rho_r)$, we show in Fig. \[AlphaD\] the $\alpha_{\text{D}}$ in $^{208}$Pb from SHF with the Skyrme force MSL0 [@Che10] by varying individually $L(\rho_r)$, $G_{V}$, $G_{S}$, $E_{0}(\rho _{0})$, $E_{\text{sym}}(\rho _r)$, $K_{0}$, $m_{s,0}^{\ast }$, $m_{v,0}^{\ast }$, $\rho _{0}$, and $W_{0}$ within their empirical uncertain ranges, namely, varying one quantity at a time while keeping all others at their default values in MSL0, for $\rho_r=0.11$ and $0.16$ fm$^{-3}$, respectively. It is seen from Fig. \[AlphaD\] that, as Eq. (\[AlphaDDM\]) suggests, the $\alpha_{D}$ in $^{208}$Pb exhibits strong correlations with both $L(\rho_{\text{r}})$ and $E_{\text{sym}}(\rho _{\text{r}})$, while much weaker correlation with other macroscopic quantities. Particularly, the $\alpha_{\text{D}}$ decreases sensitively with $E_{\text{sym}}(\rho_r)$ while increases rapidly with $L({\rho_r})$, implying a fixed value of $\alpha_{\text{D}}$ will lead to a strong positive correlation between $E_{\text{sym}}(\rho_r)$ and $L({\rho_r})$. The results for $\rho_r = 0.16$ fm$^{-3}$ just confirm the correlations of $\alpha_{\mathrm{D}}$ with $E_{\mathrm{sym}}(\rho_0)$ and $L(\rho_0)$ reported in Ref.[@Roc13]. For $\rho_r=0.11$ fm$^{-3}$, given that the symmetry energy at $\rho_{\text{c}}=0.11$ fm$^{-3}$ has been well constrained as $E_{\text{sym}}(\rho_{\text{c}})=26.65\pm0.20$ MeV, one thus expects the $\alpha_{\text{D}}$ in $^{208}$Pb can constrain stringently the parameter $L({\rho_{\text{c}}})$.
Fixing the values of other $8$ macroscopic quantities, i.e., $G_{V}$, $G_{S}$, $E_{0}(\rho _{0})$, $K_{0}$, $m_{s,0}^{\ast }$, $m_{v,0}^{\ast }$, $\rho _{0}$ and $W_{0}$ at their default values in MSL0, we illustrate in Fig. \[AlphadLc\] by open up-triangles (down-triangles) the $\alpha_{\text{D}}$ in $^{208}$Pb as a function of $L(\rho_{\text{c}})$ for $E_{\text{sym}}(\rho_{\text{c}})=26.45~(26.85)$ MeV. As expected, it is seen from Fig. \[AlphadLc\] that the $\alpha_{\text{D}}$ in $^{208}$Pb increases (decreases) with $L({\rho_{\text{c}}})$ ($E_{\text{sym}}(\rho_{\text{c}})$) for a fixed $E_{\text{sym}} (\rho_{\text{c}})$ ($L({\rho_{\text{c}}})$). By comparing with the experimental data $\alpha_{\text{D}}=20.1\pm0.6$ fm$^3$, one can extract a strong constraint of $L(\rho_{\text{c}})=48.6\pm7.9$ MeV.
![(Color online) The electric dipole polarizability $\alpha_{\text{D}}$ in $^{208}$Pb as a function of $L(\rho_{\text{c}})$ for fixed $E_{\text{sym}}(\rho_{\text{c}})$. The open (solid) up- and down-triangles represent the results with $E_{\text{sym}}(\rho_{\text{c}}) = 26.45$ and $26.85$ MeV, respectively, from SHF-RPA calculations with the values of other parameters fixed in MSL0 (obtained in optimization). The band indicates the experimental value of $\alpha_D = 20.1\pm 0.6$ fm$^3$ from RCNP [@Tam11].[]{data-label="AlphadLc"}](AlphaDLc.eps)
The above constraint of $L(\rho_{\text{c}})=48.6\pm7.9$ MeV has been obtained by neglecting the weak correlations between the $\alpha_{\text{D}}$ in $^{208}$Pb and other $8$ macroscopic quantities. To test the robustness of this constraint and to obtain a more precise constraint, for fixed $E_{\text{sym}}(\rho_{\text{c}})$ and $L(\rho_{\text{c}})$, we optimize all other $8$ parameters instead of simply fixing them at their default values in MSL0, by minimizing the weighted sum of ${\chi}^2$ evaluated from the difference between SHF prediction and the experimental data for some selected observables using the simulated annealing technique [@Agr05]. In particular, in the optimization, we chose the following experimental data of spherical even-even nuclei, i.e., (i) the binding energy $E_B$ of $^{16}$O,$^{40,48}$Ca, $^{56,68}$Ni, $^{88}$Sr, $^{90}$Zr, $^{100,116,132}$Sn, $^{144}$Sm, $^{208}$Pb [@Wan12]; (ii) the charge rms radii $r_{\text{C}}$ of $^{16}$O, $^{40,48}$Ca, $^{56}$Ni, $^{88}$Sr, $^{90}$Zr, $^{116,132}$Sn, $^{144}$Sm, $^{208}$Pb [@Ang04; @Blanc05]; (iii) the breathing mode energy $E_{0}$ of $^{90}$Zr,$^{116}$Sn,$^{144}$Sm and $^{208}$Pb [@You99]. In the calculation of the breathing mode energy $E_{0}=\sqrt{m_1/m_{-1}}$, we evaluate the inverse energy-weighted sum rule $m_{-1}$ with the constrained Hartree-Fock (CHF) method and obtain the energy-weighted sum rule $m_{1}$ using the double commutator sum rule [@Boh79; @Col04; @Agr04; @Sil06]. In addition, in the optimization, we constrain the macroscopic parameters by requiring that (i) the neutron $3p_{1/2}-3p_{3/2}$ energy level splitting in $^{208}$Pb should lie in the range of $0.8-1.0$ MeV; (ii)$m_{s,0}^*$ should be greater than $m_{v,0}^*$ and here we set $m_{s,0}^*-m_{v,0}^* =0.1m$ ($m$ is nucleon mass in vacuum) to be consistent with the extraction from global nucleon optical potentials constrained by world data on nucleon-nucleus and (p,n) charge-exchange reactions [@XuC10]. As usual, in the optimization, we assign a theoretical error $1.2$ MeV to E$_B$, $0.025$ fm to $r_C$ while use the experimental error for breathing mode energy $E_0$ with a weight factor $0.08$, so that the respective $\chi^2$ evaluated from each sort of experimental data is roughly equal to the number of the corresponding data points [@Bev03].
Using the above optimization process, we evaluate the electric dipole polarizability $\alpha_{\text{D}}$ in $^{208}$Pb as a function of $L(\rho_{\text{c}})$ for a fixed $E_{\text{sym}}(\rho_{\text{c}})$, and the results are shown in Fig \[AlphadLc\] by solid up-triangles (down-triangles) for $E_{\text{sym}}(\rho_{\text{c}})=26.45~(26.85)$ MeV. It should be noted that for each pair of $E_{\text{sym}}(\rho_{\text{c}})$ and $L(\rho_{\text{c}})$ with fixed values, the other $8$ macroscopic quantities have been optimized accordingly as described above. It is interesting to see that the values of $\alpha_{\text{D}}$ with optimization are quite consistent with the results using the default values in MSL0 without optimization and only show a small upward shift compared with the latter. Comparing the results from optimization to the experimental data, one can obtain a constraint of $L(\rho_{\text{c}}) = 47.3 \pm 7.8$ MeV, which is again in good agreement with the constraint $L(\rho_{\text{c}})=48.6\pm7.9$ MeV extracted using the default values in MSL0. These features demonstrate the validity of neglecting the weak correlations between the $\alpha_{\text{D}}$ in $^{208}$Pb and other $8$ macroscopic quantities. The present constraint on $L(\rho_{\text{c}})$ further agrees very well with the constraint $L(\rho_{\text{c}}) = 46.0 \pm 4.5$ MeV extracted from analyzing the experimental data on the neutron skin thickness of Sn isotopes [@Zha13]. This is a very interesting finding since these two constraints are obtained from two completely independent experimental observables.
In addition, using the constrained $E_{\text{sym}}(\rho_{\text{c}})$ and $L(\rho_{\text{c}})$ together with the corresponding $8$ other optimized quantities, one can easily extract the $E_{\text{sym}}(\rho)$ and $L(\rho )$ at saturation density $\rho_0$, and the results are $E_{\text{sym}}(\rho_{\text{0}})=32.7\pm1.7$ MeV and $L(\rho_{\text{0}})=47.1\pm17.7$ MeV, which are essentially consistent with other constraints extracted from terrestrial experiments, astrophysical observations, and theoretical calculations with controlled uncertainties [@Tsa12; @Lat12; @ChenLW12; @LiBA12; @Tew13]. Especially, our present results agree surprisingly well with the constraint of $E_{\text{sym}}({\rho _{0}}) = 31.2$-$34.3$ MeV and $L({\rho _{0}}) = 36$-$55$ MeV (at 95% confidence level) obtained from analyzing the mass and radius of neutron stars [@Ste12] as well as that of $E_{\text{sym}}({\rho _{0}}) = 29.0$-$32.7$ MeV and $L({\rho _{0}}) = 40.5$-$61.9$ MeV extracted from the experimental, theoretical and observational analyses [@Lat12]. Our results are also in agreement with the constraint of $E_{\text{sym}}({\rho _{0}}) = 32.0\pm1.8$ MeV and $L({\rho _{0}}) = 43.1\pm15$ MeV from analyzing pygmy dipole resonances (PDR) of $^{130,132}$Sn [@Kli07] and that of $E_{\text{sym}}({\rho _{0}}) = 32.3\pm1.3$ MeV and $L({\rho _{0}}) = 64.8\pm15.7$ MeV from analyzing PDR of $^{68}$Ni and $^{132}$Sn [@Car10]. In addition, our results are further consistent with the constraint of $E_{\text{sym}}({\rho _{0}}) = 32.3\pm1.0$ MeV and $L(\rho_{\text{0}}) = 45.2 \pm 10.0$ MeV extracted from analyzing the experimental data of the binding energy difference of heavy isotope pairs and the neutron skins of Sn isotopes [@Zha13] as well as the constraint of $E_{\text{sym}}({\rho _{0}}) = 32.5\pm0.5$ MeV and $L({\rho _{0}}) = 70\pm15$ MeV from a new finite-range droplet model analysis of the nuclear mass [@Mol12].
Given that the neutron skin thickness ${\Delta}r_{np}$ of $^{208}$Pb is uniquely fixed by the slope parameter $L(\rho_{\text{c}})$ at $\rho_{\text{c}}=0.11$ fm$^{-3}$ [@Zha13], we can also extract a constraint ${\Delta}r_{np}=0.176\pm 0.027$ fm for $^{208}$Pb by using the optimized parameters together with $E_{\text{sym}}(\rho_{\text{c}})=26.65\pm0.20$ MeV and $L(\rho_{\text{c}}) = 47.3 \pm 7.8$ MeV. Our result is consistent with the estimated range ${\Delta}r_{np}=0.165\pm(0.009)_{\text{expt}}\pm(0.013)_{\text{theor}}
\pm(0.021)_{\text{est}}$ fm in Ref. [@Roc13] obtained by analyzing the experimental $\alpha_{\text{D}}$ in $^{208}$Pb with an empirical range of $E_{\text{sym}}(\rho_0)= 31\pm(2)_{\text{est}}$. One can see that our present constraint on ${\Delta}r_{np}$ of $^{208}$Pb has higher precision, indicating a more precise constraint on the symmetry energy at a subsaturation density is very helpful to extract ${\Delta}r_{np}$ of $^{208}$Pb from the electric dipole polarizability. Our result further agrees with the constraint $\Delta r_{np}=0.156^{+0.025}_{-0.021}$ fm obtained from the $^{208}$Pb dipole polarizability by using an empirical correlation between $\alpha_{\text{D}}$ and ${\Delta}r_{np}$ of $^{208}$Pb [@Tam11], the constraint ${\Delta}r_{np}=0.15\pm0.03({\text{stat.}})^{+0.01}_{-0.03}({\text{sys.}})$ fm extracted very recently from coherent pion photoproduction cross sections [@Tar14], and within the experimental error bar the constraint $\Delta r_{np}=0.33^{+0.16}_{-0.18}$ fm extracted from the PREX at JLab [@Abr12].
Furthermore, it has been well established that the core-crust transition density $\rho_{\text{t}}$ in neutron stars, which plays a crucial role in neutron star properties [@Lat04], is strongly correlated with the density slope $L(\rho_0)$ of the symmetry energy (see, e.g., Ref. [@XuJ09]). In particular, in Ref. [@Che10], the same correlation analysis method as in this work has been successfully applied to study the correlation between $\rho_{\text{t}}$ and the various macroscopic quantities, and indeed a strong correlation between $\rho_{\text{t}}$ and $L(\rho_0)$ has been found. As mentioned in Ref. [@Zha13], a similar strong correlation is also existed between $\rho_{\text{t}}$ and $L(\rho_{\text{c}})$, and this is demonstrated in Fig. \[Rhot\] which shows the same correlations as Fig. \[AlphaD\] but for the core-crust transition density $\rho_{\text{t}}$ in neutron stars. Here, the transition density $\rho_{\text{t}}$ is calculated by using a dynamical approach (see, e.g., Ref. [@XuJ09]). One can see from Fig. \[Rhot\] that, for both $\rho_{\text{r}} =0.11$ and $0.16$ fm$^{-3}$, $\rho_{\text{t}}$ exhibits a strong correlation with $L({\rho_r})$, a weak dependence on $E_{\mathrm{sym}}(\rho_r)$ and $K_0$, but almost no sensitivity to other macroscopic parameters. Employing the optimized values for other macroscopic parameters as well as $E_{\text{sym}}(\rho_{\text{c}})=26.65\pm0.20$ MeV and $L(\rho_{\text{c}}) = 47.3 \pm 7.8$ MeV, we then obtain a value of $\rho_{\text{t}}= 0.084\pm0.009$ fm$^{-3}$, which agrees well with the empirical values [@Lat04].
Summary and outlook
===================
In summary, we have demonstrated that the electric dipole polarizability $\alpha_{\text{D}}$ in $^{208}$Pb is sensitive to both the magnitude $E_{\text{sym}}(\rho_{\text{c}})$ and density slope $L(\rho_{\text{c}})$ of the symmetry energy at a subsaturation cross density $\rho_{\text{c}} = 0.11$ fm$^{-3}$, and it decreases (increases) with $E_{\text{sym}}(\rho_{\text{c}})$ ($L(\rho_{\text{c}})$), leading to a positive correlation between $L(\rho_{\text{c}})$ and $E_{\text{sym}}(\rho_{\text{c}})$ for a fixed value of $\alpha_{\text{D}}$ in $^{208}$Pb. Using the experimental value of $\alpha_{\text{D}}$ in $^{208}$Pb measured at RCNP and the very well-constrained range of $E_{\text{sym}}(\rho_{\text{c}})$, we have obtained a strong constraint on the slope parameter $L(\rho_{\text{c}}) = 47.3 \pm 7.8$ MeV. This constraint is in surprisingly good agreement with the previous solely existing constraint $L(\rho_{\text{c}}) = 46.0 \pm 4.5$ MeV from neutron skin data of Sn isotopes, demonstrating the robustness of these constraints on the value of the $L(\rho_{\text{c}})$ parameter.
The present constraint of $L(\rho_{\text{c}})$ further leads to $E_{\text{sym}}(\rho_{\text{0}})=32.7\pm1.7$ MeV and $L(\rho_{\text{0}}) = 47.1 \pm 17.7$ MeV for the symmetry energy at saturation density, the neutron skin thickness $\Delta r_{np} = 0.176 \pm 0.027$ fm for $^{208}$Pb, and $\rho_t = 0.084 \pm 0.009$ fm$^{-3}$ for the core-crust transition density of neutron stars. These results are nicely consistent with many other constraints extracted from terrestrial experiments, astrophysical observations, and theoretical calculations with controlled uncertainties.
Our present results are based on the standard SHF energy density functional. It will be interesting to see how the results change if different energy-density functionals, e.g., the relativistic mean field model or the extended non-standard SHF energy density functional, are applied. These works are in progress and will be reported elsewhere.
We are grateful to Li-Gang Cao for helpful discussions on the Skyrme-RPA code. This work was supported in part by the Major State Basic Research Development Program (973 Program) in China under Contract Nos. 2015CB856904 and 2013CB834405, the NNSF of China under Grant Nos. 11135011 and 11275125, the “Shu Guang" project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and the Science and Technology Commission of Shanghai Municipality (11DZ2260700).
[99]{}
J.M. Lattimer and M. Prakash, Science **304**, 536 (2004); Phys. Rep. **442**, 109 (2007).
A.W. Steiner, M. Prakash, J.M. Lattimer, and P.J. Ellis, Phys. Rep. **411**, 325 (2005).
V. Baran, M. Colonna, V. Greco, and M. Di Toro, Phys. Rep. **410**, 335 (2005).
B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. **464**, 113 (2008).
C.J. Horowitz, S.J. Pollock, P.A. Souder, and R. Michaels, Phys. Rev. C **63**, 025501 (2001).
T. Sil, M. Centelles, X. Vi$\tilde{\text{n}}$as, and J. Piekarewicz, Phys. Rev. C **71**, 045502 (2005).
D.H. Wen, B.A. Li, and L.W. Chen, Phys. Rev. Lett. **103**, 211102 (2009).
H. Zheng, Z. Zhang, and L.W. Chen, J. Cosmo. Astropart. Phys. **08**, 011 (2014) \[arXiv:1403.5134\].
Topical Issue Nuclear Symmetry Energy edited by B.A. Li, A. Ramos, G. Verde, I. Vidana, Eur. Phys. J. A **50**, (2014).
B.M. Tsang *et al*., Phys. Rev. C **86**, 015803 (2012).
J.M. Lattimer, Ann. Rev. Nucl. Part. Sci. **62**, 485 (2012).
L.W. Chen, Nuclear Structure in China 2012: Proceedings of the 14th National Conference on Nuclear Structure in China (NSC2012) (World Scientific, Singapore, 2012), pp. 43-54 \[arXiv:1212.0284\].
B.A. Li *et al*., J. Phys.: Conf. Series **413**, 012021 (2013) \[arXiv:1212.1178\].
C.J. Horowitz *et al*., J. Phys. G **41**, 093001 (2014).
W.D. Myers and W.J. Swiatecki, Nucl. Phys. **A601**, 141 (1996).
P. Danielewicz and J. Lee, Nucl. Phys. **A818**, 36 (2009).
M. Liu, N. Wang, Z. X. Li, and F. S. Zhang, Phys. Rev. C **82**, 064306 (2010).
P. M${\ddot{\text{o}}}$ller, W. D. Myers, H. Sagawa, and S. Yoshida, Phys. Rev. Lett. **108**, 052501 (2012).
J.M. Lattimer and Y. Lim, Astrophys. J. **771**, 51 (2013)
M. Warda, X. Vinas, X. Roca-Maza, and M. Centelles, Phys. Rev. C **80**, 024316 (2009).
L.W. Chen, C.M. Ko, B.A. Li, and J. Xu, Phys. Rev. C **82**, 024321 (2010).
X. Vinas, M. Centelles, X. Roca-Maza, and M. Warda, Eur. Phys. J. A **50**, 27 (2014).
H. Sagawa, S. Yoshida, X.-R. Zhou, K. Yako, and H. Sakai, Phys. Rev. C **76**, 024301 (2007).
A. Klimkiewicz *et al*., Phys. Rev. C **76**, 051603 (2007).
A. Carbone *et al*., Phys. Rev. C **81**, 041301(R) (2010).
P.–G. Reinhard and W. Nazarewicz, Phys. Rev. C **81**, 051303(R) (2010).
J. Piekarewicz *et al*., Phys. Rev. C **85**, 041302(R) (2012).
J. Piekarewicz, Eur. Phys. J. A **50**, 25 (2014).
G. Col$\grave{\text{o}}$, U. Garg, and H. Sagawa, Eur. Phys. J. A **50**, 26 (2014).
A. Tamii *et al*., Phys. Rev. Lett. **107**, 062502 (2011).
X. Roca-Maza *et al*., Phys. Rev. C **88**, 024316 (2013).
M. Centelles, X. Roca-Maza, X. Vin$\tilde{\text{a}}$s, and M. Warda, Phys. Rev. Lett. **102**, 122502 (2009)
L.W. Chen, Phys. Rev. C **83**, 044308 (2011).
E. Khan, J. Margueron, and I. Vidana, Phys. Rev. Lett. **109**, 092501 (2012).
F.J. Fattoyev, W.G. Newton, and B.A. Li, Phys. Rev. C **90**, 022801(R) (2014).
Z. Zhang and L.W. Chen, Phys. Lett. **B726**, 234 (2013).
P. Danielewicz and J. Lee, Nucl. Phys. **A922**, 1 (2014).
L. Trippa, G. Col$\grave{\text{o}}$, and E. Vigezzi, Phys. Rev. C **77**, 061304(R) (2008).
L.G. Cao and Z.Y. Ma, Chin. Phys. Lett. **25**, 1625 (2008).
X. Roca-Maza *et al*., Phys. Rev. C **87**, 034301 (2013).
B.A. Brown, Phys. Rev. Lett. **11**, 232502 (2013).
N. Wang, L. Ou, and M. Liu, Phys. Rev. C **87**, 034327 (2013).
E. Chabanat *et al*., Nucl. Phys. **A627**, 710 (1997).
J. Friedrich and P.-G. Reinhard, Phys. Rev. C **33**, 335 (1986).
P. Klüpfel *et al*., Phys. Rev. C **79**, 034310 (2009).
L.W. Chen and J.Z. Gu, J. Phys. G **39**, 035104 (2012).
L.W. Chen, Sci. China: Phys. Mech. Astro. **54** (Suppl. 1), s124 (2011).
G. Col$\grave{\text{o}}$, L.G. Cao, N. Van Giai, and L. Capelli, Comput. Phys. Commun. **184**, 142 (2013).
B.K. Agrawal, S. Shlomo, and V. Kim Au, Phys. Rev. C **72**, 014310 (2005).
M. Wang *et al*., Chin. Phys. C **36** (2012) 1287.
I. Angeli, At. Data Nucl. Data. Tab. **87** (2004) 185.
F. Le Blanc *et al*., Phys. Rev. C **72**, (2005) 034305.
D.H. Youngblood, H.L. Clark, and Y.W. Lui, Phys. Rev. Lett. **82**, 691 (1999).
O. Bohigas, A.M. Lane, and J. Martorell, Phys. Rep. **51**, 267 (1979).
G. Col$\grave{\text{o}}$, N. Van Giai, J. Meyer, K. Bennaceur, and P. Bonche, Phys. Rev. C **70**, 024307 (2004).
B.K. Agrawal and S. Shlomo, Phys. Rev. C **70**, 014308 (2004).
T. Sil, S. Shlomo, B.K. Agrawal, and P.-G. Reinhard, Phys. Rev. C **73**, 034316 (2006).
C. Xu, B.A. Li, and L.W. Chen, Phys. Rev. C **82**, 054607 (2010).
P.R. Bevington and D.K. Robinson, *Data reduction and error analysis for physical sciences* (3ed.) (McGraw-Hill, New York, 2003), p. 71.
I. Tews *et al*., Phys. Rev. Lett. **110**, 032504 (2013).
A.W. Steiner and S. Gandolfi, Phys. Rev. Lett. **108**, 081102 (2012).
C.M. Tarbert *et al*., Phys. Rev. Lett. **112**, 242502 (2014).
S. Abrahamyan *et al*., Phys. Rev. Lett **108**, 112502 (2012).
J. Xu, L.W. Chen, B.A. Li, and H.R. Ma, Phys. Rev. C **79**, 035802 (2009); Astrophys. J. **697**, 1549 (2009).
[^1]: Corresponding author (email: lwchen$@$sjtu.edu.cn)
|
---
abstract: 'Electron conductivity is an important material property that can provide a wealth of information about the underlying system. Especially, the response of the conductivity with respect to electromagnetic fields corresponds to various nonlinear charge currents, which have distinct symmetry requirements and hence can be used as efficient probes of different systems. To help the band-structure engineering of such nonlinear currents, a universal treatment of electron dynamics up to second order expressed in the basis of the unperturbed states are highly useful. In this work, we review the general semiclassical framework of the nonlinear charge currents.'
author:
- Yang Gao
title: Semiclassical Dynamics and Nonlinear Charge Current
---
Introduction
============
Electron transport has long been a central topic in solid state physics, not only due to its fascinating complexity but also because it can provide rich information about the underlying electronic structure and dynamics. In the most simple Drude theory of metals, the conductivity is determined by the density, effective mass of carriers, and the phenomenological relaxation time [@Mermin1976]. Such simple picture can be generalized using the dynamics of Bloch electrons, to take into account the real energy spectrum and the Fermi surface geometry, as well as the microscopic scattering process [@Mermin1976].
In recent years, this classic paradigm of electron transport receives profound modifications, due to a deeper understanding of the geometric structure in the momentum space induced by the wave functions instead of the energy spectrum. The most well-known examples are various Hall effects [@Nagaosa2010; @Xiao2010]. Central to the Hall effect is the concept of the Berry curvature. Its integration yields the Chern number, responsible for a topological classification of matter [@Qi2011]. Moreover, it directly modifies the electron motion and hence is indispensable in describing the static and optical transport phenomenon [@Xiao2010; @Morimoto2016; @Zhong2016; @Ma2015]. Berry curvature is also the analog of the magnetic field in the momentum space. As such, the source of Berry curvature is the momentum-space analog of magnetic monopoles, which can be realized by Weyl/Dirac points [@Son2013; @Armitage2018]. In three dimensions, such isolated monopoles in momentum space can response differently to electromagnetic fields and cause a characteristic strong negative longitudinal magnetoresistance [@Son2013; @Armitage2018].
On the other hand, the charge current can be expanded with respect to the electromagnetic fields as follows $$\label{eq_curexp}
J_i=\sigma_{ij}E_j+\sigma_{ijk}E_jE_k+\sigma_{(ij,k)}E_jB_k+\sigma_{(ij,kk)}E_jB_kB_k+\cdots\,.$$ Berry curvature offers a first order correction to the electron dynamics. As such, it can give an accurate description of $\sigma_{ij}$ whose antisymmetric part is the anomalous Hall effect. In addition to this linear current, there are various nonlinear currents with coefficients satisfying different symmetry requirements and hence can be used to probe distinct systems. To further demonstrate the symmetry requirement, we can expand the last three coefficients in terms of transport relaxation time $\tau$ [^1] $$\begin{aligned}
\label{eq_jee}\sigma_{ijk}&=\sigma_{ijk,0}+\sigma_{ijk,1}\tau+\sigma_{ijk,2}\tau^2\,,\\
\label{eq_jeb}\sigma_{(ij,k)}&=\sigma_{(ij,k),0}+\sigma_{(ij,k),1}\tau+\sigma_{(ij,k),2}\tau^2\,,\\
\label{eq_jebb}\sigma_{(ij,kk)}&=\sigma_{(ij,kk),0}+\sigma_{(ij,kk),1}\tau+\sigma_{(ij,kk),2}\tau^2+\sigma_{(ij,kk),3}\tau^3\,.\end{aligned}$$ The symmetry of different coefficients is summarized in Table. \[tb\_tb1\]. It is thus tempting to find how different coefficients depend on the geometric properties of Bloch electrons in the momentum space. This can provide experimental guide and help the band structure engineering of larger nonlinear charge currents. For this purpose, the electron dynamics should be extended to second order.
\[tb\_tb1\]
---------------------------------------------------------------------------------------
Even Odd
------ --------------------------------------------- ----------------------------------
Even $\begin{tabular}{c} $\sigma_{ijk,1}$
$\_[(ij,k),0]{}$\quad$\_[(ij,k),2]{}$\\
$\_[(ij,kk),1]{}$ \quad $\_[(ij,kk),3]{}$
\end{tabular}$
Odd $\begin{tabular}{c} $\sigma_{ijk,0}$$\sigma_{ijk,2}$
\quad$\_[(ij,k),1]{}$ \\
$ \_[(ij,kk),0]{}$\quad $\_[(ij,kk),2]{}$
\end{tabular}$
---------------------------------------------------------------------------------------
: Nonzero components of conductivity in crystals with different space inversion and time reversal symmetries. The requirement of the lower right entry should be understood as in crystals that simultaneously break time reversal and inversion symmetry but preserve the combined symmetry.
In this work, we will focus on the semiclassical treatment of the electron dynamics and review the effort of generalizing such theory up to second order and deriving the nonlinear currents in the Boltzmann transport framework. Our paper is organized as follows. In Sect. II, we discuss the generalization of the semiclassical equations of motion up to second order through an appropriately constructed wave packet up to first order. We then discuss the validity of the semiclassical theory though the magnetoelectric coefficient and magnetic susceptibility. With the correct semiclassical theory up to second order, in Sect. III, we extend the Boltzmann transport theory and derive its solution in steady-state up to third order. We then use this universal treatment to derive various nonlinear currents, including the electric-field-correction to the nonlinear Hall effect ($\sigma_{jii}$), linear magnetoresistance ($\sigma_{(ij,k)}$), and quadratic magnetoresistance ($\sigma_{(ij,kk)}$).
Semiclassical Theory of Bloch electrons in electromagnetic fields
=================================================================
To derive various nonlinear charge currents, one will need the correct electron dynamics beyond linear order under electromagnetic fields. For this purpose, certain perturbation technique should be employed. However, the periodic nature of crystals causes difficulties for such process. We note that the perturbation from electromagnetic fields reads $e\bm E\cdot \bm r+\frac{1}{2}e[\bm A(\bm r)\cdot \hat{\bm v}+\hat{\bm v}\cdot \bm A(\bm r)]$. It contains the position operator, which is unbounded in extended systems. Therefore, direct application of the time-independent perturbation is inappropriate.
To circumvent such difficulty, for magnetoconductivty one can first obtain the eigenstate of Bloch electrons under magnetic field, which are Landau levels for effective continuum models or Hofstadter spectrum for tight-binding models. Then the static magnetoconductivity is obtained using the standard Kubo formula under an appropriate limit. The dependence of the conductivity on magnetic field is encoded in the quantization of original Bloch states under magnetic field. Although this method is standard and has wide applicability, it cannot give universal understanding of the physics behind the magnetoconductivity, as the quantization into Landau levels or the Hofstadter spectrum varies for different systems.
Alternatively, one can construct a phase space using an appropriate position variable $\bm R$ and momentum variable $\bm P$ from the Bloch states. The electron dynamics is then contained in the equations of motion in the phase space. If $\bm R$ and $\bm P$ are canonical variables, their dynamics is solely determined by the energy. Therefore, the whole procedure amounts to deriving an effective energy in terms of $\bm R$ and $\bm P$ under external fields. If $\bm R$ and $\bm P$ are non-canonical, their Poisson brackets are also needed, which determine the structure of the equations of motion. The magnetoconductivity is then derived in combination with the Boltzmann equation for the distribution function. Although this method treats external fields as perturbations, and is only derived up to the second order currently, it has the benefit that it yields analytical results independent of the model detail. In fact, all terms in the equations of motion are expressed using the unperturbed Bloch functions, and hence can directly access both the spectrum and the geometric structure in the unperturbed Hilbert space.
In this section, we follow the second method by treating Bloch electrons as wave packets, whose center of mass position and momentum naturally span the phase space. We will first sketch the construction of the wave packet and show how its internal structure modifies the electron dynamics in phase space at first order of external fields. A detailed discussion of both aspects can be found in Ref. [@Xiao2010]. We then extend such theory up to second order and derive the second order correction to equations of motion. The validity of the semiclassical theory can be confirmed through the magnetoelectric effect and the quantization of effective energies into Landau levels under magnetic field. Since all elements in the semiclassical dynamics are expressed using unperturbed Bloch states, they can be evaluated using the output from the first-principles codes.
Properties of the wave packet
-----------------------------
The dynamics of Bloch electrons under uniform electromagnetic fields is governed by the following crystal Hamiltonian in non-relativistic limit $$\hat{H}_\text{f}=\hat{H}_0[\hat{\bm p}+e\bm A(\bm r);\bm r]+\frac{g\mu_B}{\hbar}\bm B\cdot \hat{\bm s}+e\bm E\cdot \bm r\,,$$ where $\hat{H}_0(\hat{\bm p};\bm r)$ is the unperturbed Hamiltonian for periodic crystals with $\hat{\bm p}$ being the momentum operator, $g$ is the gyromagnetic ratio, $\mu_B$ is the Bohr magneton, and $\hat{\bm s}$ is the spin operator. The magnetic field enters through both the minimal coupling and the Zeeman coupling. Generally speaking, $\hat{H}_\text{f}$ does not respect the translational symmetry, unless the magnetic flux through a unit cell is a rational number times the flux quantum such that the magnetic translational symmetry is recovered [@Zak1964a; @Zak1964b; @Hofstadter1976; @Chang1994].
Although $\hat{H}_\text{f}$ exactly determines the dynamics, to directly diagonalize it is a difficult task. Instead, we can simplify the problem by assuming that the solution to the Schr$\ddot{\rm o}$dinger equation $(i\hbar \partial_t-\hat{H}_\text{f})\psi=0$ takes the ansartz of a wave packet, which is the essence of the semiclassical theory. To construct the wave packet, we first evaluate the electromagnetic potential in the full Hamiltonian at the center of mass position $\bm r_c$ of the wave packet, and hence recover the translational symmetry of the original unperturbed crystal. The resulting local Hamiltonian reads: $$\hat{H}_c=\hat{H}_0[\hat{\bm p}+e\bm A(\bm r_c);\bm r]+\frac{g\mu_B}{\hbar}\bm B\cdot \hat{\bm s}+e\bm E\cdot \bm r_c\,,$$ Its eigenenergy forms the local Bloch bands $\varepsilon_n[\bm p+(e/\hbar)\bm A(\bm r_c)]+e\bm E\cdot \bm r_c$ with $n$ being the band index and $\hbar\bm p$ being crystal momentum. Its eigenfunction is the Bloch function $e^{i\bm p\cdot \bm r}|u_n[\bm p+e\bm A(\bm r_c)/\hbar]\rangle$.
In the following, we focus on a single band with index $0$ and assume that it is well separated from all the other bands. When electromagnetic fields are weak, the dynamics of a Bloch electron starting from some state in this band will still be confined in the same band. Therefore, the wave packet is constructed as the superposition of Bloch states from the band $0$ [@Sundaram1999] $$|W\rangle=\int d\bm p C_0(\bm p) e^{i\bm p\cdot \bm r}|u_0[\bm p+e\bm A(\bm r_c)/\hbar]\rangle\,.$$ $|W\rangle$ should be normalized, indicating $\int d\bm p |C_0|^2=1$.
There are two constraints on the coefficient $C_0$. First, the wave packet is assumed to be sharply localized in the momentum space. This requires that the magnitude of $C_0$ satisfies $|C_0|^2\approx \delta (\bm p-\bm p_c)$. Here $\hbar\bm p_c$ is the center of mass momentum of the wave packet, i.e. $$\bm p_c=\langle W|\bm p|W\rangle\,.$$ Secondly, the construction of the wave packet requires a local Hamiltonian around the center of mass position $\bm r_c$. This $\bm r_c$ has to be determined in a self-consistent manner, which exerts a constraint on the phase of $C_0$ [@Sundaram1999]: $$\label{eq_wrc}
\bm r_c=\langle W|\bm r|W\rangle=\left.\frac{\partial \gamma}{\partial \bm p}\right |_{\bm p=\bm p_c}+ \bm {\mathcal{A}}_0(\bm k_c)\,,$$ where $\gamma=-arg(C_0)$, $\bm {\mathcal{A}}_0(\bm p)=\langle u_0(\bm p)|i\bm \partial_{\bm p}|u_0(\bm p)\rangle$ is the intraband Berry connection, and $\hbar\bm k_c=\hbar\bm p_c+e\bm A(\bm r_c)$ is the gauge-independent physical momentum. These two constraints complete the construction of the wave packet. Although the wave packet is for Bloch states from a single band, it can be generalized to account for the general multiband case [@Culcer2005; @Shindou2005].
Semiclassical dynamics up to first order
----------------------------------------
With the knowledge of the wave packet, we now derive its evolution. The coefficient $C_0$ in the wave packet can be determined through the variational principle, i.e. the least action principle. Since $C_0$ is a complex function, with independent magnitude and phase, determined by $\bm k_c$ (or equivalently, $\bm p_c$) and $\bm r_c$ respectively, applying the variational principle to the Lagrangian will yield the dynamics of $\bm r_c$ and $\bm k_c$, i.e. the phase space equations of motion.
By evaluating the Lagrangian under the wave packet up to first order, we obtain [@Sundaram1999] $$\begin{aligned}
\label{eq_lag}
L&=\langle W|i\hbar\partial_t-\hat{H}_0-\hat{H}_1|W\rangle\notag\\
&=-(\bm r_c-\bm {\mathcal{A}}_0)\cdot \hbar\dot{\bm k}_c-\frac{1}{2}e\bm B\times \bm r_c\cdot \dot{\bm r}_c-\tilde{\varepsilon}_0\,,\end{aligned}$$ where $\hat{H}_1=\frac{e}{4} \bm B\cdot [(\bm r-\bm r_c)\times \hat{\bm v}-\hat{\bm v}\times (\bm r-\bm r_c)]+e\bm E\cdot (\bm r-\bm r_c)$ is the first order correction to the local Hamiltonian $\hat{H}_c$ and $\tilde{\varepsilon}=\varepsilon_0-\bm B\cdot \bm m+e\bm E\cdot \bm r_c$ with $\bm m=\bm m_{orb}-(g\mu_B/\hbar)\langle u_0|\hat{\bm s}|u_0\rangle$ is the modified Band energy after taking account of both the magnetic moment and electric dipole of the wave packet, which will be discussed later. Here we use the symmetric gauge for the vector potential.
The Berry connection $\bm {\mathcal{A}}_0$ is an essential ingredient in the Lagrangian. In the absence of $\bm {\mathcal{A}}_0$, the above Lagrangian reduces to the conventional canonical form for electrons under electromagnetic fields: $$L=\left(\hbar \bm k_c-\frac{1}{2}e\bm B\times \bm r_c\right)\cdot \dot{\bm r}_c-\tilde{\varepsilon}_0\,.$$ The appearance of $\bm {\mathcal{A}}_0$ indicates that $\bm r_c$ and $\bm k_c$ are not canonical variables. To make this statement clearer, we make the following transformation [@Chang2008] $$\begin{aligned}
\label{eq_rc}\bm r_c&=\bm q+\bm {\mathcal{A}}_0+\frac{1}{2}\frac{e}{\hbar}(\bm B\times \bm {\mathcal{A}}_0\cdot \bm \partial_{\bm p})\bm {\mathcal{A}}_0+\frac{1}{2}\frac{e}{\hbar}\bm \Omega_0\times (\bm B\times \bm {\mathcal{A}}_0)\,,\\
\label{eq_kc}\bm k_c&=\bm p+\frac{1}{2}\frac{e}{\hbar}\bm B\times \bm q+\frac{e}{\hbar}\bm B\times (\bm r_c-\bm q)\,,\end{aligned}$$ where $\bm \Omega_0=\bm \nabla_{\bm p}\times \bm {\mathcal{A}}_0$ is the Berry curvature, which can be viewed as the geometrical Berry phase per unit area [@Berry1984]. Here the argument of $\bm {\mathcal{A}}_0$ and $\bm \Omega_0$ is $\bm p+\frac{e}{2\hbar}\bm B\times \bm q$. Then the Lagrangian in Eq. recovers the canonical form for $\bm p$ and $\bm q$ $$L=\hbar\bm p\cdot \dot{\bm q}-\tilde{\varepsilon}_0\,.$$ Eq. and thus describe the connection between physical variables and canonical variables.
One consequence of the noncanonicality is that the phase space measure for the volume element $d\bm r_cd\bm k_c$ has to change based on the Jacobian $\partial(\bm r_c,\bm k_c)/\partial(\bm q,\bm p)$. The resulting phase space density of states reads [@Xiao2005] $$\mathcal{D}=1+\frac{e}{\hbar}\bm B\cdot \bm \Omega_0(\bm k_c)\,.$$ Since the physical position and momentum take the crystal volume and the Brillouin zone as their range, it is natural to evaluate the statistical average of any operator in the physical phase space spanned by $(\bm r_c,\bm k_c)$. Therefore, the modified density of states $\mathcal{D}$ should always be used if the magnetic field is involved.
The noncanonicality of $\bm r_c$ and $\bm k_c$ will also lead to a nontrivial symplectic form, and hence affects the structure of the equations of motion. From Eq. , the Euler-Lagrangian equations of motion yield the following phase space dynamics [@Sundaram1999] $$\begin{aligned}
\label{eq_dotrc}\dot{\bm r}_c&=\frac{\partial \tilde{\varepsilon}_0}{\hbar\partial \bm k_c}-\dot{\bm k}_c\times \bm \Omega_0(\bm k_c)\,,\\
\label{eq_dotkc}\hbar\dot{\bm k}_c&=-e\bm E-e\dot{\bm r}_c\times \bm B\,.\end{aligned}$$ The second term in the velocity equation is the anomalous velocity arising from the noncanonicality. The above equations of motion have wide applicability in the static spin, heat, and charge transport phenomena [@Xiao2010] and optical phenomena [@Moore2010; @Ma2015; @Zhong2015; @Morimoto2016; @Zhong2016; @Morimoto2016a; @Deyo2019].
Electromagnetic Dipoles
-----------------------
One direct application of the semiclassical theory is to derive the electromagnetic multipoles in crystals. This derivation confirms the validity of the semiclassical theory as the results are consistent with those derived using other methods. As the wave packet is sharply localized in the momentum space, it has a finite width in the real space. The charge and current distributions across such spread need not to be uniform. To represent such internal anisotropy of the wave packet, it is natural to use electromagnetic multipoles. For illustration purpose, we will consider the electromagnetic dipoles.
The electric dipole of the wave packet reads $\langle W|-e\bm r|W\rangle=-e\bm r_c$, with $\bm r_c$ given in Eq. . Under the periodic gauge [@Resta2000; @Xiao2010], the Bloch wave function is periodic in the momentum space, so is the coefficient $C_0$. Therefore, by integrating $\bm r_c$ in the Brillouin zone, the first term in Eq. from the phase of $C_0$ can at most yield an integer multiples of $2\pi$ and hence can be ignored. The remaining term yields the electric polarization in crystals $$\label{eq_dipole}
\bm P=-e\int \frac{d\bm k_c}{(2\pi)^3} \bm {\mathcal{A}}_0(\bm k_c)\,.$$ Originally, the electric polarization is derived by first considering the adiabatic charge pumping current and then identifying it as the displacement current from the temporal variation of the polarization [@Kingsmith1993; @Resta1994]. Here our method yields the same electric polarization under the periodic gauge for Bloch functions.
The magnetic dipole can be evaluated as follows [@Chang1996; @Sundaram1999] $$\begin{aligned}
\label{eq_m}
\bm m_\text{orb}&=-e\langle W|\frac{1}{2}[(\bm r-\bm r_c)\times\hat{\bm v}-\hat{\bm v}\times (\bm r-\bm r_c)]|W\rangle\notag\\
&=-\frac{e}{2} \sum_{n\neq 0}\bm {\mathcal{A}}_{0n}\times \bm v_{n0}\,,\end{aligned}$$ where $\bm {\mathcal{A}}_{0n}=\langle u_0|i\bm \partial_{\bm k_c}|u_n\rangle$ is the interband Berry connection and $\bm v_{n0}=\langle u_n|\hat{\bm v}|u_0\rangle$ is the interband velocity element. In this work, we use the convention that for intraband matrix elements, we only keep one band index. The magnetic moment $\bm m_\text{orb}$ represents how a wave packet couples to an external magnetic field, i.e. shifting the band energy through an effective Zeeman coupling $-\bm B\cdot \bm m_\text{orb}$. As such, $\bm m_\text{orb}$ contributes to the orbital magnetization as shown later. $\bm m_\text{orb}$ can usually be accessed through the optical activity of crystals. For example, in ferromagnetic materials, the cross-gap part of $\bm m_\text{orb}$ (with $0$ standing for occupied bands and $n$ for the unoccupied bands in Eq. ) can be measured through the $f$-sum rule for the circular dichroism [@Souza2008; @Yao2008]. Moreover, in noncentrosymmetric metals, the natural optical activity in the semiclassical regime is determined by the integration of the momentum space dipole of the magnetic moment (defined as $v_i (m_\text{orb})_j$ with $\bm v$ being the band velocity) on the Fermi surface [@Ma2015; @Zhong2015; @Zhong2016].
Interestingly, if one starts from the relativistic Dirac Hamiltonian and construct a coherent wave packet for the upper two bands, the orbital magnetic moment of this wave packet naturally recovers the spin magnetic moment [@Chang2008]. Mathematically, this is equivalent to the Foldy-Wouthuysen transformation that reduces the Dirac Hamiltonian to the non-relativistic Schr$\ddot{\rm o}$dinger Hamiltonian [@Foldy1950; @Blount1962]. Here the semiclassical theory offers a heuristic and clear picture for the emergence of spin.
To derive the orbital magnetization in crystals, one should evaluate the free energy under magnetic field. On one hand, the magnetic moment of the wave packet couples to the magnetic field and modifies the band energy. On the other hand, the phase space density of states is changed by the magnetic field. Taking both corrections into consideration, one obtains the following free energy $$F=\int \mathcal{D}\frac{d\bm k}{(2\pi)^3} (-k_BT)\ln \left[1+\exp\left(\frac{\tilde{\varepsilon}_0-\mu}{k_BT}\right)\right]\,.$$ Therefore, the orbital magnetization can be derived by taking the derivative of the free energy with respect to the magnetic field [@Xiao2006] $$\begin{aligned}
\bm M=-\frac{\partial F}{\partial B}=\int \frac{d\bm k}{(2\pi)^3} \left(f_0 \bm m-\frac{e}{\hbar}g_e\bm \Omega_0\right)\,,\end{aligned}$$ where $g_e=-(k_BT)\ln[1+\exp((\varepsilon_0-\mu)/k_BT)]$ is the grand potential density, and $f_0$ is the equilibrium Fermi distribution function. Here and hereafter we will ignore the subscript $c$ in the integration over $\bm k_c$. This is the contribution of the orbital magnetization from band $0$. The total contribution can be obtained by summing over the band index. The orbital magnetization thus contains two contributions. The first one is due to the relative motion inside the wave packet, or the self-rotation of the wave packet. The second one is due to the rotation of the wave packet as a whole, or the revolution of the wave packet. This same magnetization can also be obtained through the Wannier function approach [@Thonhauser2005; @Ceresoli2006], the exact Hofstadter spectrum [@Gat2003a; @Gat2003b], or the linear response theory [@Shi2007; @Qin2011].
Semiclassical dynamics up to second order
-----------------------------------------
Although the semiclassical dynamics in Eq. and is very powerful, it cannot fully account for the second order response functions such as the magnetoresistance, magnetoelectric coefficient, and so on. For this purpose, we need to extend the above semiclassical dynamics up to second order.
The construction of the wave packet has to be modified accordingly. The external electromagnetic fields generally affect the eigenfunction. As a result, the wave packet should be the superposition of the true eigenstates instead of the unperturbed ones. We can still expand the true eigenstate in the basis of the unperturbed ones, and obtain the following wave packet [@Gao2014] $$\label{eq_wp2}
|W\rangle=\int d\bm p e^{i\bm p\cdot \bm r}\left(C_0(\bm p) |u_0\rangle+\sum_{n\neq 0}C_n(\bm p)|u_n\rangle\right)\,.$$ Here for simplicity we drop the argument $\bm p+(e/\hbar)\bm A(\bm r_c)$ of $|u_0\rangle$ and $|u_n\rangle$. If $|W\rangle$ satisfies the Schr$\ddot{\rm o}$dinger equation, we have $$\begin{aligned}
\langle u_n|e^{-i\bm p\cdot \bm r}(i\hbar \partial_t -\hat{H}_\text{f})|W\rangle=0\,,\quad \forall\; n\neq 0\,.\end{aligned}$$ Therefore, we use this as an additional constraint on the wave packet in Eq. , which determines the connection between $C_n$ and $C_0$. Specifically, solving this constraint at the linear order of electromagnetic fields yields [@Gao2014] $$\label{eq_cn}
C_n=\frac{G_{n0}}{\varepsilon_0-\varepsilon_n}C_0-\frac{i}{2}\frac{e}{\hbar}[\bm B\times (i\bm \partial_{\bm p}+\bm {\mathcal{A}}_0 -\bm r_c)C_0]\cdot \bm {\mathcal{A}}_{n0}\,,$$ where $G_{n0}=-\bm B\cdot \bm M_{n0}+e\bm E\cdot \bm {\mathcal{A}}_{n0}$ with $\bm M_{n0}=\frac{e}{2}\sum_{m\neq 0} (\bm v_{nm}+\bm v_0\delta_{mn})\times \bm {\mathcal{A}}_{m0}-g\mu_B \bm s_{n0}/\hbar$ being the interband element of the total (spin plus orbital) magnetic moment. We comment that different from the discussion here, in Ref. [@Gao2014] and [@Gao2015], the spin magnetic moment has not been explicitly added.
![The correction to wave packet from magnetic field. $\hat{\bm q}$ has the same meaning of $\bm r$. From Ref. [@Gao2015].[]{data-label="fig_perturb"}](Perturb.pdf "fig:"){width="\columnwidth"}\
Equation indicates that the correction to the wave packet from the magnetic field has two contributions with distinct structures and hence different origins. It is useful to first analyze the nature of these two perturbations as they prelude the form of the second order correction to the phase space dynamics. The first term has the conventional form of the perturbation correction with the energy gap as the denominator. It is called the vertical mixing as it mixes Bloch states from different bands but at the same $\bm p$ point in the Brillouin zone. It also contains both adiabatic and nonadiabatic perturbation, as the wave packet $|W\rangle$ is time-dependent through the argument $\bm p+(e/\hbar)\bm A(\bm r_c)$ of the Bloch state.
In comparison, the second term in $C_n$ modifies the wave packet in the following way: $$\begin{aligned}
\label{eq_hoz}
&\int d\bm p e^{i\bm p\cdot \bm r} \frac{-i}{2} \frac{e}{\hbar}[\bm B\times (i\bm \partial_{\bm p}+\bm {\mathcal{A}}_0-\bm r_c)C_0]\cdot \sum_{n\neq 0}\bm {\mathcal{A}}_{n0}|u_n\rangle\notag\\
=&\int d\bm p e^{i\bm p\cdot \bm r} \frac{e}{2\hbar}\left(\bm B\times (\bm r-\bm r_c)\cdot \hat{\bm D}|u_0\rangle+\bm B\cdot \bm\Omega_0 |u_0\rangle\right)\,.\end{aligned}$$ where $\hat{D}=\bm \partial_{\bm p}+i\bm {\mathcal{A}}_0$ is the covariant derivative that ensures the gauge invariance. Notice that the momentum argument of $|u_0\rangle$ is $\bm p+(e/\hbar)\bm A(\bm r_c)$. The first term in Eq. has the meaning of shifting the momentum to $\bm p+(e/\hbar)\bm A(\bm r)$. This is a unique property of the perturbation from the magnetic field, and it suggests that the correction to the wave function also obeys the Peierls substitution. The same property is also derived using the Moyal product and phase space formulation of quantum mechanics [@Blount1962b]. The second term in Eq. makes the total correction normal to the original state $e^{i\bm p\cdot \bm r} |u_0\rangle$ to eliminate redundancy in the perturbative correction. Since the second term in $C_n$ effectively mixes Bloch states in the same band but at neighbouring $\bm p$ points, it is referred to as the horizontal mixing. Both corrections can be easily visualized in Fig. \[fig\_perturb\].
Using the modified wave packet, we obtain the following Lagrangian $$\label{eq_lag2}
L=-(\bm r_c-\bm {\mathcal{A}}_0^\text{t})\cdot \hbar\dot{\bm k}_c-\frac{1}{2}e\bm B\times \bm r_c\cdot \dot{\bm r}_c-\tilde{\varepsilon}_0\,,$$ where $\bm {\mathcal{A}}_0^\text{t}$ is the Berry connection evaluated in the true eigen state instead of the unperturbed one. It can be put in the form of the original Berry connection plus a positional shift correction: $\bm {\mathcal{A}}_0^\text{t}=\bm {\mathcal{A}}_0+\bm {\mathcal{A}}_0^\prime$. Before discussing the expression of the positional shift $\bm {\mathcal{A}}_0^\prime$ and modified energy $\tilde{\varepsilon}$, we comment that the Lagrangian up to second order has the exact same form with the previous one in Eq. . Consequently, the connection between physical and canonical variables also has the same form except that the Berry connection $\bm {\mathcal{A}}_0^\text{t}$ in the exact eigenstate should be used $$\begin{aligned}
\label{eq_rc2}\bm r_c&=\bm q+\bm {\mathcal{A}}_0^\text{t}+\frac{e}{2\hbar}(\bm B\times \bm {\mathcal{A}}_0^\text{t}\cdot \bm \partial_{\bm p})\bm {\mathcal{A}}_0^\text{t}+\frac{e}{2\hbar}\bm \Omega_0^\text{t}\times (\bm B\times \bm {\mathcal{A}}_0^\text{t})\,,\\
\label{eq_kc2}\bm k_c&=\bm p+\frac{e}{2\hbar}\bm B\times \bm q+\frac{e}{\hbar}\bm B\times (\bm r_c-\bm q)\,,\end{aligned}$$ where $\bm {\Omega}_0^\text{t}=\bm \nabla_{\bm p}\times \bm{\mathcal{A}}_0^\text{t}$ is the Berry curvature evaluated using the exact eigenstate. As a result, the phase space density of states reads $$\mathcal{D}=1+\frac{e}{\hbar}\bm B\cdot \bm {\Omega}_0^\text{t}\,.$$ More importantly, the equations of motion up to second order also keep the same form [@Gao2014] $$\begin{aligned}
\label{eq_dotrc2}\dot{\bm r}_c&=\frac{\partial \tilde{\varepsilon}_0}{\hbar\partial \bm k_c}-\dot{\bm k}_c\times \bm {\Omega}_0^\text{t}(\bm k_c)\,,\\
\label{eq_dotkc2}\hbar\dot{\bm k}_c&=-e\bm E-e\dot{\bm r}_c\times \bm B\,.\end{aligned}$$
The positional shift is gauge-independent. It represents the additional shift of the wave packet center after applying external electromagnetic fields $$\bm r_c=\langle W|\bm r|W\rangle=\left.\frac{\partial \gamma}{\partial \bm p}\right |_{\bm p=\bm p_c}+ \bm {\mathcal{A}}_0(\bm k_c)+\bm {\mathcal{A}}_0^\prime(\bm k_c)\,.$$ The positional shift at first order reads $$\bm {\mathcal{A}}_0^\prime=2\sum_{n\neq 0}{\rm Re}\frac{\bm {\mathcal{A}}_{0n} G_{n0}}{\varepsilon_0-\varepsilon_n}-\frac{1}{2}\frac{eB}{\hbar}\epsilon_{ijk} \gamma_{ji\ell}\hat{e}_\ell\,,$$ where $\epsilon_{ijk}$ is the Levi-Civita symbol and $\gamma_{ji\ell}=\frac{1}{2}(\partial_\ell g_{ij}+\partial_{i}g_{j\ell}-\partial_jg_{i\ell})$ is the Christoffel symbol. Here and hereafter, we use the Einstein summation convention for repeated indices and the partial derivative is with respect to the crystal momentum unless otherwise specified. $g_{ij}$ is a Fubini-Study metric tensor, usually called quantum metric [@Provost1980; @Neupert2013; @Anandan1990; @Resta2011]. It measures the distance between two neighbouring Bloch states in the momentum space. In fact, it is a part of a more general concept, the quantum geometrical tensor, which, for band $0$, is defined as $$\mathcal{G}_{ij}=\langle \partial_{i}u_0|\partial_{j}u_0\rangle-(\mathcal{A}_{i})_0(\mathcal{A}_{j})_0\,.$$ The real and imaginary part of $\mathcal{G}_{ij}$ yield the quantum metric and Berry curvature, respectively [@Provost1980; @Neupert2013].
In the positional shift, the first term is due to the vertical mixing and the second term is due to the horizontal mixing. It is interesting that the horizontal mixing yields a purely geometric correction to the Berry connection. This is not coincidental, as the horizontal mixing addresses neighbouring Bloch states in the momentum space, whose difference is proportional to the Berry connection and whose distance is determined by the quantum metric. We comment that this positional shift is also envisioned in Ref. [@Berry1987], using a similar technique.
The positional shift reflects the change in the electric dipole moment of the wave packet. When external fields are applied, the electric polarization has to pick up the following change. First, the density of states has to be changed to $\mathcal{D}$. Second, the Berry connection $\bm {\mathcal{A}}_0\rightarrow \bm {\mathcal{A}}_0^\text{t}+\frac{e}{2\hbar}(\bm B\times \bm {\mathcal{A}}_0^\text{t}\cdot \bm \partial_{\bm p})\bm {\mathcal{A}}_0^\text{t}+\frac{e}{2\hbar}\bm { \Omega}_0^\text{t}\times (\bm B\times \bm{\mathcal{A}}_0^\text{t})$ according to Eq. . As a result, the first order change in electric polarization reads $$\bm {\delta P}=-e\int \frac{d\bm k}{(2\pi)^3} \left[\frac{e}{2\hbar}(\bm \Omega_0\cdot \bm {\mathcal{A}}_0)\bm B+\bm {\mathcal{A}}_0^\prime\right]\,.$$ The first term is the Abelian Chern-Simons 3-form, which yields the topological part of the orbital magnetoelectric coefficient. The second term yields the electric polarizability and the cross-gap part of the orbital magnetoelectric polarizability, consistent with the calculation using the linear response theory [@Essin2010]. Such consistency confirms the validity of the first order correction to the Berry phase.
The modified band energy in Eq. should contain the correction up to second order, i.e. it can be put in the following form: $\tilde{\varepsilon}_0=\varepsilon_0-\bm B\cdot \bm m+e\bm E\cdot \bm r_c+\varepsilon_{(2)}$. The second order correction describes the change in the electric dipole and magnetic moment of the wave packet in response to external fields. It reads [@Gao2015] $$\begin{aligned}
\label{eq_eng2}
\varepsilon_{(2)}=&\frac{1}{ 4}\frac{e}{\hbar} (\bm B\cdot \bm{\Omega}_0) (\bm B\cdot \bm m)-{1\over 8}\frac{e^2}{\hbar^2} \epsilon_{s i k} \epsilon_{t j \ell}B_{s} B_{t}g_{ij} \alpha_{k\ell} \notag\\
&+\bm \nabla\cdot\bm P_\text{E}+\sum_{n\neq 0}{G_{0n} G_{n0}\over \varepsilon_{0}-\varepsilon_{n}}\notag\\
&-e\bm B\cdot (\bm {\mathcal{A}}_0^\prime \times \bm v_0)+{e^2\over 8m} (B^2 g_{ii}-B_i g_{ij} B_j),\end{aligned}$$ where $\alpha_{k\ell}=\partial_k \partial_\ell \varepsilon_0/\hbar^2$ is the inverse effective mass tensor, $m$ is the free electron mass, $G_{0n}=(G_{n0})^\star$, and $$\begin{aligned}
\bm P_\text{E}=\frac{1}{2}e \sum_{n\neq 0}{\rm Re}[(\bm B\times \bm {\mathcal{A}}_{0n})G_{n0}]\,.\end{aligned}$$ In the energy correction, the first two terms are purely due to the horizontal mixing and hence are geometrical corrections involving Berry curvature and quantum metric. The third term is due to the cross effect of the horizontal and vertical mixing. $\bm P_\text{E}$ is called energy polarization as it accounts for the energy dipole in the momentum space. The fourth term is purely due to the horizontal mixing and has the conventional form of the perturbation correction to the energy. The fifth term shifts the momentum argument of $\varepsilon_0$. The last term is the expectation of the second order correction to the local Hamiltonian.
We comment that if one starts from an effective tight-binding Hamiltonian, the last term of Eq. will change. Instead of the free electron mass, one has to use the Hessian operator $\hat{\Gamma}_{ij}=\partial^2 \hat{H}_0/\partial \hat{p}_i\partial \hat{p}_j$. Then the last term in Eq. should be replaced by the following two terms $$\begin{aligned}
\label{eq_hessian}
\varepsilon_\text{H}=&-{e^2\over 16} (\bm B\times \bm \partial)_i (\bm B\times \bm \partial)_j \langle u_0|\hat{\Gamma}_{ij}|u_0\rangle\notag\\
&+{e^2\over 8} \sum_{(m,n)\neq 0} (\bm B\times \bm {\mathcal{A}}_{0m})_i (\hat{\Gamma}_{ij})_{mn} (\bm B\times \bm {\mathcal{A}}_{n0})_j\,.\end{aligned}$$ Here $(\hat{\Gamma}_{ij})_{mn}=\langle u_m|\hat{\Gamma}_{ij}|u_n\rangle$.
Landau level quantization and magnetic susceptibility
-----------------------------------------------------
If one starts from a continuum tight-binding Hamiltonian, electronic states will be quantized into Landau levels under magnetic field. Therefore, as long as the semiclassical theory can successfully reproduce the Landau level quantization up to second order, it is enough to account for various response functions at second order in magnetic field.
There are various quantization rules for Landau levels, such as the Onsager’s rule [@Onsager1952], the Maslov canonical operator method [@Reijnders2013], the Gutzwiller trace formula [@Gutzwiller1971], and so on. For semiclassical theory, the most relevant one is the Onsager’s rule [@Onsager1952], which reads $$\begin{aligned}
S(\varepsilon_n)=2\pi\left(n+{1\over 2}\right) {eB\over \hbar}\,,\end{aligned}$$ where $S(\varepsilon_n)$ is the area in the momentum space enclosed by an equal-energy contour. Later, this quantization rule is generalized up to first order to take account of corrections from Berry phase and magnetic moment [@Wilkinson1984; @Rammal1985; @Mikitik1999; @Chang1996; @Carmier2008; @Xiao2010; @Fuchs2010].
![Density quantization rule for Landau levels. The true electron density in the blue line experiences a jump at each Landau level. From Ref. [@Gao2017a].[]{data-label="fig_ll"}](LL.pdf "fig:"){width="\columnwidth"}\
To further generalize the Onsager’s rule, one observes that the area $S(\varepsilon_n)$ is proportional to the total electron density below $\varepsilon_n$. Since the area $S$ as a function of energy is a smooth function, such electron density should be understood as the smooth semiclassical electron density $\rho_\text{semi}$. Then it has been proved that the Onsager’s rule can be replaced by the following density quantization rule [@Gao2017a] $$\rho_\text{semi}(\varepsilon_n)=\left(n+{1\over 2}\right) {eB\over h}\,.$$ This suggests that the smooth semiclassical electron density always intersects with the true stepwise electron density at the half-filling points, as shown in Fig. \[fig\_ll\].
In general $\rho_\text{semi}$ can be expanded near $B=0$ in power series of $B$. Up to second order we have $$\label{eq_qr}
\left(n+{1\over 2}\right) {eB\over h}=\frac{S(\varepsilon_n)}{4\pi^2}+B\left.\frac{\partial M}{\partial \mu}\right |_{\mu=\varepsilon_n}+\frac{1}{2}B^2 \left.\frac{\partial \chi}{\partial \mu}\right|_{\mu=\varepsilon_n}\,,$$ where $M$ and $\chi$ are the magnetization and magnetic susceptibility respectively, evaluated at zero magnetic field and zero temperature. It is interesting to note that Eq. indicates the nonlinearity in the Landau level fan diagram when plotted as $B$ against $1/n$ due to the appearance of the susceptibility. Such nonlinearity is consistent with experiments [@Taskin2010; @Analytis2010; @Ren2010; @Sacepe2011; @Bruene2011; @Xiu2011; @Xiang2015]. Based on Eq. , as long as the semiclassical theory can yield the correct susceptibility, it can give correct second order responses in magnetic field.
To evaluate the magnetic susceptibility, we need to calculate the free energy up to second order. In the semiclassical framework, the free energy can be expressed as follows $$\begin{aligned}
\label{eq_free}
F&=\int \frac{d\bm k_c}{8\pi^3} \mathcal{D}(g_e(\tilde{\varepsilon})+g_\text{L})\,.\end{aligned}$$ Here the second term $g_\text{L}$ is the Peierls-Landau magnetic free energy: $g_\text{L}=-(e^2f_0^\prime/48\hbar^2)B_s B_t \epsilon_{s i k}\epsilon_{t j \ell} \alpha_{i j} \alpha_{k\ell}\,,$ where $f_0^\prime$ is the energy derivative of the Fermi distribution function $f_0$. For isotropic bands, the effective mass tensor $\alpha$ is diagonal, and $g_\text{L}$ will reduce to its familiar form [@Mermin1976]. This term originates from higher order corrections to the replacement of the quantum mechanical commutator with the classical Poisson bracket [@Blount1962b].
After collecting terms at second order in $B$, one can find the following free energy $F=\int g^{\prime\prime} (d\bm k/8\pi^3)$ with $$\begin{aligned}
\label{eq_g2}
g^{\prime\prime}&=g_\text{L}+{f_0^\prime\over 2}(\bm B\cdot \bm m)^2-f_0^\prime \bm v_0\cdot \bm P_\text{E}\notag\\
&+f_0\sum_{n\neq 0}{G_{0n}G_{n0}\over \varepsilon_0-\varepsilon_n} +{e^2f_0\over 8m} (B^2 g_{ii}-B_i g_{ij} B_j)\notag\\
&-{3ef_0\over 4\hbar}(\bm B\cdot \bm{\Omega}_0) (\bm B\cdot \bm m)-{e^2f_0\over 8\hbar^2} \epsilon_{s i k} \epsilon_{t j \ell}B_{s} B_{t}g_{ij}\alpha_{k\ell}\,.\end{aligned}$$ The magnetic susceptibility can be readily obtained through its definition $$\chi_{ij}=-\left.\frac{\partial^2 F}{\partial B_i \partial B_j}\right |_{\mu, T, B=0}\,.$$
The first two terms in Eq. are Peierls-Landau diamagnetic and Pauli paramagnetic contributions. The physical meaning of the other terms in Eq. can be illustrated by re-expressing it in the Wannier function basis and then taking the atomic insulator limit. In particular, the two terms in the second line reduce to the familiar form of the Van-Vleck paramagnetic free energy and Langevin diamagnetic free energy in atomic physics, respectively. The last term is purely geometrical as it solely comes from the horizontal mixing. The energy polarization and geometrical magnetic free energy approach zero in the atomic limit, indicating that they have no analog in atomic physics and are novel terms in crystals.
The susceptibility obtained from Eq. in the semiclassical framework is consistent with the result derived using other methods. For example, the susceptibility has been calculated in the honeycomb lattice using Hofstadter spectrum or linear response theory [@Santos2011; @Raoux2014]. Eq. yields the exactly same result as shown in Fig. \[fig\_sus\]. In Ref. [@Ogata2015], it is also proved that Eq. is consistent with the susceptibility from linear response theory [^2]. This confirms the validity of the semiclassical dynamics under magnetic field.
Evaluating positional shift and energy correction in tight-binding models
-------------------------------------------------------------------------
The positional shift and energy correction can be implemented in first-principles codes. For this purpose, besides the energy spectrum, one needs three additional matrix elements: the velocity matrix element $\bm v_{mn}=\langle m|\hat{\bm v}|n\rangle$, the Hessian matrix element $(\Gamma_{ij})_{mn}=\langle u_m|\hat{\Gamma}_{ij}|u_n\rangle$, and the spin Pauli matrix element $\bm \sigma_{mn}=\langle m|\hat{\bm \sigma}|n\rangle$, all of which can be evaluated in first-principles codes in principle [@Marzari1997; @Souza2001; @Yates2007; @Marzari2012]. In the following, we sketch the process of re-expressing the positional shift and energy correction in terms of the energy spectrum and these three matrix elements.
We first examine the positional shift. The interband part of the Berry connection is related to the interband velocity element: $$\begin{aligned}
\label{eq_va}
\bm v_{mn}=\frac{i}{\hbar}(\varepsilon_m-\varepsilon_n)\bm {\mathcal{A}}_{mn}\,,\forall \; m\neq n\,.\end{aligned}$$ Using this identity, we manipulate the interband mixing element $G_{n0}$ $$\begin{aligned}
\label{eq_gn0}
G_{n0}=\frac{ie\bm B\hbar}{2}\cdot\sum_{m\neq 0} \frac{(\bm v_{nm}+\bm v_0\delta_{mn})\times \bm v_{m0}}{\varepsilon_m-\varepsilon_0}+\frac{g\mu_B}{2} \bm B\cdot\bm \sigma_{n0}\,.\end{aligned}$$ Then the first term in the positional shift can be put in the desired form $$\label{eq_ps1}
2\hbar{\rm Im}\sum_{n\neq 0}\frac{\bm v_{0n}G_{n0}}{(\varepsilon_0-\varepsilon_0)^2}\,.$$ For the remaining term we note that $$\begin{aligned}
\label{eq_ps2}
\epsilon_{ijk}\gamma_{ji\ell}&=\epsilon_{ijk}\partial_ig_{j\ell}\notag\\
&=-2\hbar^3\epsilon_{ijk} \sum_{n\neq 0}{\rm Re}\frac{[(v_i)_0-(v_i)_n](v_j)_{0n}(v_\ell)_{n0}}{(\varepsilon_0-\varepsilon_n)^3}\notag\\
&\quad+\hbar^2\epsilon_{ijk}\sum_{n\neq 0} {\rm Re}\frac{\partial_i(v_j)_{0n}(v_\ell)_{n0}+(j\leftrightarrow \ell)}{(\varepsilon_0-\varepsilon_n)^2}\,.\end{aligned}$$ The only unknown quantity has the form $\partial_i(v_j)_{mn}$, which can be manipulated as follows $$\begin{aligned}
\label{eq_dvmn}
\partial_i(v_j)_{mn}=&\sum_{m^\prime\neq m}\frac{\hbar(v_i)_{mm^\prime}(v_j)_{m^\prime n}}{\varepsilon_m-\varepsilon_{m^\prime}}-\sum_{m^\prime\neq n}\frac{\hbar(v_j)_{mm^\prime}(v_i)_{m^\prime n}}{\varepsilon_{m^\prime}-\varepsilon_n}\notag\\
&+\hbar(\Gamma_{ij})_{mn}+i[(\mathcal{A}_i)_m-(\mathcal{A}_i)_n] (v_j)_{mn}\,.\end{aligned}$$ The first three terms are in the desired form. The last term contains the intraband Berry connection which is gauge-dependent. Therefore, it cannot be effectively evaluated in the first-principles codes. However, it can be shown that it does not contribute to Eq. and hence need not to be evaluated. Equations to are enough to transform the positional shift in the desired form.
We now examine the energy correction. Using the following identities $$\begin{aligned}
\bm \Omega_0&=-\hbar^2\sum_{n\neq 0} {\rm Im}\frac{\bm v_{0n}\times \bm v_{n0}}{(\varepsilon_0-\varepsilon_n)^2}\,,\\
\bm m&=-\frac{e\hbar}{2}\sum_{n\neq 0} {\rm Im}\frac{\bm v_{0n}\times \bm v_{n0}}{\varepsilon_0-\varepsilon_n}-\frac{g\mu_B \bm \sigma_{0}}{2}\,,\\
g_{ij}&=\hbar^2\sum_{n\neq 0}{\rm Re}\frac{(v_i)_{0n}(v_j)_{n0}}{(\varepsilon_0-\varepsilon_n)^2}\,,\\
\alpha_{ij}&=(\Gamma_{ij})_{0}+2\sum_{n\neq 0}{\rm Re}\frac{(v_i)_{0n}(v_j)_{n0}}{\varepsilon_0-\varepsilon_n}\,,\end{aligned}$$ together with Eq. to , it is straightforward to put all terms except the energy polarization contribution in Eq. in the desired form. We comment that for the tight-binding Hamiltonian, one may have to substitute the last term in Eq. with Eq. which has realistic Hessian matrix. In this case, we have $$\begin{aligned}
\varepsilon_\text{H}=&\frac{e^2\hbar^2}{ 8} \sum_{(m,n)\neq 0} {\rm Re}\frac{(\bm B\times \bm v_{0m})_i (\hat{\Gamma}_{ij})_{mn}(\bm B\times \bm v_{n0})_j}{(\varepsilon_0-\varepsilon_m)(\varepsilon_n-\varepsilon_0)}\notag\\
&+\frac{e^2}{8}\epsilon_{sik}\epsilon_{tj\ell}B_sB_t\sum_{n\neq 0}{\rm Im}[(\Gamma_{ij})_{n0}\partial_\ell (\mathcal{A}_k)_{0n}]\notag\\
&+\frac{e^2}{8}\epsilon_{sik}\epsilon_{tj\ell}B_sB_t\sum_{n\neq 0}{\rm Im}[(\mathcal{A}_k)_{0n} \partial_\ell(\Gamma_{ij})_{n0}]\,.\end{aligned}$$ To evaluate the last two terms, one notes that $$\begin{aligned}
\label{eq_da}\partial_\ell (\mathcal{A}_k)_{0n}&=-\frac{i\hbar^2[(v_\ell)_0-(v_\ell)_n](v_k)_{0n}}{(\varepsilon_0-\varepsilon_n)^2}
+\frac{i\hbar\partial_\ell (v_k)_{0n}}{\varepsilon_0-\varepsilon_n}\,,\\
\label{eq_dgamma}\partial_\ell (\Gamma_{ij})_{n0}&=\sum_{m\neq n}\frac{\hbar(v_\ell)_{nm}(\Gamma_{ij})_{m0}}{\varepsilon_n-\varepsilon_m}-\sum_{m\neq 0}\frac{\hbar(\Gamma_{ij})_{nm}(v_\ell)_{m0}}{\varepsilon_m-\varepsilon_0}\notag\\
&+(\partial_\ell \Gamma_{ij})_{n0}+i[(\mathcal{A}_\ell)_n-(\mathcal{A}_\ell)_0](\Gamma_{ij})_{n0}\,.\end{aligned}$$ Using Eq. and Eq. , one can put the second term in $\varepsilon_\text{H}$ in the desired form plus extra terms from intraband Berry connection. After inserting Eq. into $\varepsilon_\text{H}$, the first term in the second line of Eq. does not contribute due to the anti-symmetrization of the spatial indices. The last term in Eq. will cancel the extra gauge-dependent terms in Eq. . This completes the manipulation of $\varepsilon_\text{H}$.
The manipulation of energy polarization contribution in Eq. can be done as follows: $$\begin{aligned}
\bm \nabla\cdot \bm P_\text{E}=\frac{1}{2}eB_i\epsilon_{\ell ij}{\rm Re}\sum_{n\neq 0}[\partial_\ell (\mathcal{A}_j)_{0n}G_{n0}+(\mathcal{A}_j)_{0n}\partial_\ell G_{n0}]\,,\end{aligned}$$ where $$\begin{aligned}
\partial_\ell G_{n0}&=-\frac{eB}{2}\sum_{m\neq 0}[\partial_\ell(\bm v_{nm}+\bm v_0\delta_{nm})]\times \bm {\mathcal{A}}_{m0}\notag\\
&-\frac{eB}{2}\sum_{m\neq 0}(\bm v_{nm}+\bm v_0\delta_{nm})\times (\partial_\ell\bm {\mathcal{A}}_{m0})\notag\\
&+\frac{g\mu_B \hbar\bm B}{2}\cdot \left[\sum_{m\neq n}\frac{(v_\ell)_{nm}\bm \sigma_{m0}}{\varepsilon_n-\varepsilon_m}
-\sum_{m\neq 0}\frac{(\bm \sigma)_{nm}(v_\ell)_{m0}}{\varepsilon_m-\varepsilon_0}\right]\notag\\
&+\frac{g\mu_B \hbar\bm B\cdot\bm \sigma_{n0}}{2} i[(\mathcal{A}_\ell)_n-(\mathcal{A}_\ell)_0]\,.\end{aligned}$$ Again, terms contain intraband Berry connection can be ignored as they will cancel each other in the end.
We comment that if one starts from the Schr$\ddot{\rm o}$dinger Hamiltonian, the Hessian matrix can be directly evaluated without using wave functions in first-principles codes. For Schrodinger Hamiltonian, $\hat{\Gamma}_{ij}=1/m\delta_{ij}$. Therefore, one has $(\Gamma_{ij})_{mn}=1/m \delta_{ij}\delta_{mn}$. Then for the semiclassical dynamics, one only needs to evaluate the velocity and spin matrix elements using output from first-principles codes.
![Band structure for (a) Dirac semimetal, (b) Weyl semimetal by breaking time reversal symmetry, (c) Weyl semimetal by breaking inversion symmetry, and (d) semiconductor. From Ref. [@Vazifeh2013].[]{data-label="fig_eng0"}](eng0.pdf "fig:"){width="\columnwidth"}\
As a concrete example, we consider the following tight-binding Hamiltonian describing a 3D TI in the Bi$_2$Se$_3$ family on a simple cubic lattice [@Fu2010; @Qi2011; @Vazifeh2013] $$\begin{aligned}
\label{eq_h0}
\hat{H}_\text{tb}=2\lambda \sigma_z(s_x\sin k_y-s_y \sin k_x)+2\lambda_z \sigma_y \sin k_z+\sigma_x M_k\,,\end{aligned}$$ where $\bm \sigma$ and $\bm s$ are Pauli matrices in orbital and spin space respectively, and $M_k=\epsilon-2t(\cos k_x+\cos k_y +\cos k_z)$. At $\epsilon=6t$, the above Hamiltonian is at the phase transition point between trivial phase and topological insulator phase, and supports a Dirac point at the $\Gamma$ point. The Dirac point can split into Weyl point if the following additional term is added to the Hamiltonian $$\label{eq_h1}
\hat{H}_\text{break}=b_0 \sigma_ys_z+b_zs_z\,.$$ These two terms break inversion and time reversal symmetry, respectively. The resulting Weyl semimetal phase is displayed in Fig. \[fig\_eng0\].
![First order (Panel a) and second order (Panel b) correction to the band energy. Parameters are chosen as follows: $\lambda=\lambda_z=1$, $t=0.5$, $\epsilon=2.8$, $b_z=0$, $b_0=0.8$, the flux of $B$ through the unit cell $\phi=eBa^2/\hbar=3.79\times 10^{-4}$, the spin Zeeman energy is taken to be $g\mu_BB=3.86\times 10^{-4}\lambda$ which is roughly at the same order of the orbital Zeeman energy. For Panel (a), energy is in units of $10^{-3}\lambda$ and for Panel (b) energy is in units of $10^{-4}\lambda$. The $x$-axis is along $Z$-$\Gamma$ direction, similar with Fig. \[fig\_eng0\]. Here we take the total length of $Z\Gamma$ to be unity and record the relative position along $Z\Gamma$ direction. In Panel (a), the energy correction is plotted along all the $Z\Gamma$ line while in Panel (b), only a portion of $Z\Gamma$ line near the band gap and band crossing points are plotted, to better illustrate the structure of the energy correction. The corrections for lowest to highest bands are represented in black, blue, red, pink colors, respectively. Near the $\Gamma$ point, black and red curves coincide, and so do the blue and pink curves.[]{data-label="fig_eng12"}](neng1.pdf "fig:"){width="\columnwidth"}\
Here we choose the semiconductor phase and calculate the first and second order correction to the band energy, as shown in Fig. \[fig\_eng12\]. It can be seen that both first and second order correction vary drastically near the band crossing points and the small band gap. In fact, both corrections diverge at the $\Gamma$ point due to the band crossing. Across the small global band gap as shown in Fig. \[fig\_eng0\], both first and second order corrections change sign. It is also interesting to note that near the $\Gamma$ point, the lowest two bands have similar first order corrections, but are opposite to those of the remaining two bands. This property is consistent with the calculation of magnetic moment for the low-energy Dirac model. In comparison, the second order correction have different properties. The lowest and the second highest band have similar second order corrections, opposite to those of the remaining two bands. This can be most easily seen from Eq. . The Berry curvature, effective mass tensor and energy gap reverse sign for the two bands forming a Dirac cone, leading to the sign change in their second order energy correction.
Nonlinear charge current
========================
With the semiclassical theory up to second order at hand, we can derive the nonlinear charge current up to third order, i.e. conductivity up to second order in fields. We focus on the semiclassical regime with weak electromagnetic fields. There is one important issue that we want to discuss before the detailed derivation of nonlinear currents. In the previous section, the semiclassical theory is generally derived for a single Bloch band that is well separated from all the other bands. In reality, this condition is hardly met as band-crossing points are generally present. In this case, the previous semiclassical theory can still be used to derive the conductivity as long as the Fermi surface is not close to those band-crossing points. The reason is as follows. As a perturbation theory, the semiclassical theory may fail when the band gap is small, such as near the band crossing points. However, the conductivity is a Fermi surface property. As a result, as long as the band-crossing points are deep inside the Fermi sea, the perturbation theory still works near the Fermi surface.
In this section, we will first establish the general theory of the charge current beyond the linear order. Then we will discuss several important examples, including the response of the anomalous Hall conductivity to electromagnetic fields, linear magnetoresistance in time-reversal-broken materials, chiral anomaly in Weyl semimetals and two different mechanisms for the negative longitudinal magnetoresistance: the intrinsic quadratic magnetoresistance and the current jetting.
General theory of nonlinear charge currents
-------------------------------------------
The current in the semiclassical theory reads $$\label{eq_cur}
\bm J=-e\int \frac{d\bm k}{8\pi^3} \mathcal{D}\dot{\bm r} f \,,$$ where $f$ is the electron distribution function, which is the Fermi function in equilibrium. Here and hereafter, we drop the subscript $c$ in the center of mass position $\bm r_c$ and momentum $\bm k_c$ for simplicity. From the semiclassical dynamics in Eq. and , one can show that $$\label{eq_dr}
\mathcal{D}\dot{\bm r}=\tilde{\bm v}+e\bm E\times {\bm \Omega}^\text{t}+\frac{e}{\hbar}\left(\tilde{\bm v}\cdot {\bm \Omega}^\text{t}\right)\bm B\,,$$ where $\tilde{\bm v}=\partial \tilde{\varepsilon}/\hbar\partial \bm k$ is the modified band velocity. Here and hereafter, we ignore the band index $0$ in relevant intraband quantities for simplicity and the subscript $0$ only has the meaning of zeroth order.
The remaining factor in the current is the distribution function $f$, which is typically solved from the Boltzmann equation. If the sample is homogeneous and reaches a steady state under external fields, under the relaxation time approximation, the Boltzmann equation reads $$\dot{\bm k}\cdot \frac{\partial f}{\partial \bm k}= \left. \frac{df}{dt}\right |_\text{collision}=-\frac{f-f_0}{\tau}\,.$$ Here the argument of equilibrium distribution $f_0$ is the modified band energy $\tilde{\varepsilon}$. Solving the Boltzmann equation perturbatively, one has [@Pal2010] $$\label{eq_dis}
f=\sum_{m=0}^\infty (-\tau \dot{\bm k}\cdot \bm \partial_{\bm k})^m f_0(\tilde{\varepsilon})\,.$$
Equations , and are enough to derive the current except the anomalous Hall current up to third order of the external fields, i.e. the conductivity up to second order. Other than the anomalous Hall current, the equilibrium distribution $f_0$ alone does not contribute to the current. Therefore, at least the first order correction to $f_0$ is required, which automatically contains an external field due to $\dot{\bm k}$. As a result, one only needs to evaluate the factor $\mathcal{D}\dot{\bm r}$ up to second order, which is given in the semiclassical dynamics. For the anomalous Hall conductivity, the semiclassical theory can only yields its first order correction, as only the first order correction to Berry curvature is obtained in the semiclassical theory.
To explicitly write out the solution in Eq. up to third order in fields, we note that $$\begin{aligned}
-\tau\dot{\bm k}&=\frac{\tau}{\hbar} \frac{e\bm E+e\tilde{\bm v}\times \bm B+\frac{e^2}{\hbar}(\bm E\cdot \bm B)\bm \Omega^\text{t}}{1+\frac{e}{\hbar}\bm B\cdot \bm \Omega^\text{t}}\,.\end{aligned}$$ Therefore, we write the distribution function as follows $$f=f_0(\tilde{\varepsilon})+f_1+f_2+f_3\,,$$ where the subscript $1$, $2$, and $3$ stand for different orders in $\tau$, and
$$\begin{aligned}
\label{eq_f1}f_1=&\frac{\tau}{\hbar} \frac{e\bm E+\frac{e^2}{\hbar}(\bm E\cdot \bm B)\bm \Omega^\text{t}}{1+\frac{e}{\hbar}\bm B\cdot \bm \Omega^\text{t}}\cdot \bm \partial f_0(\tilde{\varepsilon})\,,\end{aligned}$$
$$\begin{aligned}
\label{eq_f2}f_2=&\frac{e^2\tau^2 [E_i+(\bm v\times \bm B)_i]E_j}{\hbar^2}\partial_i\partial_j f_0(\varepsilon-\bm B\cdot \bm m)\notag\\
&+\frac{2e^2\tau^2 [E_j+(\bm v\times \bm B)_j]\partial_i\partial_j f_0(\varepsilon)}{\hbar^2}\left[\frac{E_i+(\bm v\times \bm B)_i}{1+\frac{e}{\hbar}\bm B\cdot \bm \Omega}+\frac{e}{\hbar}(\bm E\cdot \bm B)\Omega_i\right]\notag\\
&+\frac{e^2\tau^2 [E_j+(\bm v\times \bm B)_j]\partial_i f_0(\varepsilon)}{\hbar^2}\partial_j\left[\frac{E_i+(\bm v\times \bm B)_i}{1+\frac{e}{\hbar}\bm B\cdot \bm \Omega}+\frac{e}{\hbar}(\bm E\cdot \bm B)\Omega_i\right]\,,\\
\label{eq_f3}f_3=&\frac{e^3\tau^3E_i}{\hbar^3}[(\bm E+\bm v\times \bm B)\cdot \bm \partial]^2 \partial_{i}f_0(\varepsilon)\,.\end{aligned}$$
There is another contribution to the nonlinear conductivity. The system can reach a thermodynamic equilibrium together with a uniform magnetic field. During such process, if the sample remains charge neutral, the carrier density is fixed. As the magnetic field modifies the band energy, the chemical potential has to vary accordingly.
Without loss of generality, we assume $\bm B$ is along the $z$ direction, i.e. $B=B_z$. At first order in $B$, the change in chemical potential can be fixed using the following expansion of the electron density $n$ $$\label{eq_nexp1}
dn=\left.\frac{\partial n}{\partial B}\right |_{\mu,T,B=0} B+\left.\frac{\partial n}{\partial \mu}\right |_{T,B=0} \mu_{(1)}=0\,.$$ Note that $\left.\frac{\partial n}{\partial \mu}\right |_{T,B=0}=g(\mu)$, which is simply the density of states at the chemical potential for the unperturbed band. Moreover, $\left.\frac{\partial n}{\partial B}\right |_{\mu,T}$ can be connected to the magnetization through the Maxwell equation: $$\label{eq_dndb1}
\left.\frac{\partial n}{\partial B}\right |_{\mu,T}=\left. \frac{\partial M_z}{\partial \mu} \right |_{T,B}\,.$$ We comment that for insulators at zero temperature, $-\frac{\partial M_z}{\partial \mu}\rightarrow \sigma_{xy}$, and Eq. reduces to the familiar Streda formula. Combining Eq. and , we obtain $$\label{eq_mu1}
\mu_{(1)}=-\frac{(\partial M_z/\partial \mu)_{T,B=0}}{g(\mu)}\,.$$
At second order in $B$, the expansion of $n$ reads $$\begin{aligned}
dn&=\left.\frac{\partial n}{\partial \mu}\right |_{T,B=0}\mu_{(2)}+\frac{1}{2} \left.\frac{\partial^2n}{\partial B^2} \right |_{T,\mu,B=0} B^2\notag\\
&+\frac{1}{2}\left.\frac{\partial^2 n}{\partial \mu^2}\right |_{T,B=0}[\mu_{(1)}]^2+\left.\frac{\partial^2n}{\partial B\partial \mu}\right |_{T,B=0} B\mu_{(1)}\,.\end{aligned}$$ $\mu_{(2)}$ can be solved by setting the above equation equal to zero. The result can be simplified using Eq. and the definition of the isothermal magnetic susceptibility $\chi_{zz}=\partial M_z/\partial B$, yielding $$\label{eq_mu2}
\mu_{(2)}=-\frac{1}{2}\frac{B^2\frac{\partial \chi_{zz}}{\partial \mu}+\frac{\partial g}{\partial \mu}[\mu_{(1)}]^2+2\frac{\partial^2 M_z}{\partial \mu^2} B\mu_{(1)}}{g(\mu)}\,.$$
These change in chemical potential will also contribute to the nonlinear conductivity: $$\begin{aligned}
\label{eq_jm}
\bm J_\mu=&-e\mu_{(1)} \frac{\partial}{\partial\mu} \int \frac{d\bm k}{8\pi^3}\mathcal{D}\dot{\bm r}f\notag\\
&+\frac{e^2\tau}{\hbar} \int \frac{d\bm k}{8\pi^3} \bm v (\bm E\cdot \bm v)\left\{\mu_{(2)}f_0^{\prime\prime}-\frac{1}{2}[\mu_{(1)}]^2 f_0^{\prime\prime\prime}\right\}\,,\end{aligned}$$ where $f_0^{\prime\prime}$ and $f_0^{\prime\prime\prime}$ are second and third order energy derivatives of $f_0$.
Equation with Eq. and -, combined with Eq. , , and yields a complete description of the nonlinear conductivity up to second order in the framework of the semiclassical theory. This procedure gives a complete account for the drifting part in the Boltzmann equation. However, it ignores the dependence of the collision integral on electromagnetic fields. Nevertheless, the semiclassical theory can yield a qualitatively valid result and offers a fresh perspective of how the band properties beyond the spectrum affects the conductivity.
Finally, we comment that to get the corresponding change in the resistivity, one can invert the conductivity tensor. For example, if change in conductivity $\delta \sigma_{xx}$ is obtained, the corresponding change in resistivity is $$\frac{\delta \rho_{xx}}{\rho_{xx}}=-\frac{\delta \sigma_{xx}}{\sigma_{xx}}\,,$$ provided that the Hall conductivity $\sigma_{xy}$ is much smaller than the longitudinal conductivity $\sigma_{xx}$.
Nonlinear anomalous Hall conductivity
-------------------------------------
The anomalous Hall effect refers to a Hall-type current in ferromagnets solely driven by an electric field. It is a topic under extensive studies [@Nagaosa2010]. In the past, there are three mechanisms identified: the intrinsic contribution, the skew-scattering contribution, and the side-jump contribution. Here we focus on the intrinsic contribution, which is due to the anomalous velocity in Eq. [@Xiao2010]. From the perspective of the semiclassical theory, the Berry curvature acts as the magnetic field in the momentum space and bends the electron trajectory likewise, leading to a Hall-type current. The anomalous Hall effect has also been studied in noncolinear antiferromagnets, in which the net spin magnetization vanishes but the orbital magnetization is still present [@Ohgushi2000; @Shindou2001; @Taillefumier2006; @Kalitsov2009; @Takatsu2010; @Udagawa2013; @Chen2014; @Suzuki2017; @Guo2017].
From symmetry consideration, the anomalous Hall effect requires broken time-reversal symmetry. In other words, it requires either a net spin magnetization as in ferromagnets or orbital magnetization as in noncolinear antiferromagnets. It is forbidden if time reversal symmetry is present. However, the nonlinear Hall effect may still be present, due to the manipulation of the magnetization through electromagnetic fields.
In systems with both time reversal and inversion symmetry, a net magnetization can be induced through the magnetic susceptibility by a magnetic field. This magnetization can then lead to a Hall-type current. The analytical expression for this correction can be obtained by plugging the anomalous velocity (second term in Eq. ) into the current in Eq. . The result reads [@Gao2014] $$\label{eq_aheb}
\bm J=\frac{e^2}{\hbar}\bm E\times \int [\hbar\bm v\times \bm {\mathcal{A}}^\prime(\bm B)+\bm \Omega(\bm B\cdot \bm m)] f_0^\prime \frac{d\bm k}{8\pi^3}\,,$$ where $\bm {\mathcal{A}}^\prime(\bm B)$ stands for the part of $\bm {\mathcal{A}}^\prime$ solely dependent on $\bm B$ and $\bm m$ is the spin plus orbital magnetic moment. It can be explicitly checked that compared with the susceptibility in Eq. , the second term in Eq. is part of the geometric contribution to the magnetic susceptibility. Moreover, the first term is also part of the Van-Vleck and energy polarization contributions to the susceptibility. This current yields the conductivity $\sigma_{(ij,k),0}$ in Eq. .
This current has exactly the same dependence on electromagnetic fields with the ordinary Hall current. However, they have different origins. The ordinary Hall current is due to the Lorentz force and has the form of $\omega_c \tau$ with $\omega_c$ being the cyclotron frequency. In contrast, the current in Eq. requires a nontrivial structure in the momentum space and does not involve the relaxation time. In fact, the ratio of the resistivity $\rho_{xy}^\prime$ from Eq. and the ordinary Hall resistivity $\rho_{xy}^\text{ord}$ can be put in the following form $$\begin{aligned}
\label{eq_scale}
\frac{\rho^\prime_{xy}}{\rho_{xy}^\text{ord}}=\left(\rho_{xx} \frac{e^2}{4h}\right)^2 S(\mu)\,,\end{aligned}$$ where the first factor is universal and $S(\mu)$ is a model-dependent factor but independent of the scattering process. The universal scaling factor means that $\rho^\prime_{xy}$ will dominate the ordinary Hall effect when $\rho_{xx}$ is large, i.e. in dirty metals/semiconductors with relatively small relaxation time. In a typical Hall-bar measurement set-up, both $\rho_{xy}^\prime$ and $\rho_{xy}^\text{ord}$ will contribute to the Hall current. Therefore, the total Hall resistivity should be dependent on the relaxation time. To differentiate one contribution from the other, one should change the universal scaling factor through temperature, film thickness, or doping, and measure the scaling behaviour based on Eq. .
![Magnetoelectric effect (Panel a) and the related nonlinear anomalous Hall induced by the electric field (Panel b). Originally the magnetoelectric effect is shown as the polarization induced by a magnetic field, which is the same as a magnetization along $\hat{z}$ induced by an electric field along $\hat{y}$ through the magnetoelectric coefficient $G_{y}$. From Ref. [@Gao2014].[]{data-label="fig_nahe"}](NAHE.pdf "fig:"){width="\columnwidth"}\
In systems that simultaneously break the time reversal and inversion symmetry but preserve the combined symmetry, the anomalous Hall current also vanishes. This symmetry is exactly the magnetoelectric symmetry which allows a magnetization to be induced by an electric field through the magnetoelectric coefficient [@Landau1984]. This magnetization can then lead to a Hall-type current. The analytical expression can be obtained in a similar way with the magnetic-field-correction to anomalous Hall effect. The result reads [@Gao2014] $$\label{eq_nahe}
\bm J=e^2\bm E\times \int [\bm v\times \bm {\mathcal{A}}^\prime(\bm E)] f_0^\prime \frac{d\bm k}{8\pi^3}\,,$$ where $\bm {\mathcal{A}}^\prime(\bm E)$ stands for the part of $\bm {\mathcal{A}}^\prime$ solely dependent on $\bm E$. This current yields the conductivity $\sigma_{ijk,0}$ in Eq. . The connection between this Hall-type current and the magnetoelectric effect can be mostly seen for a two-band model, as shown in Fig. \[fig\_nahe\].
In noncentrosymmetric but time-reversal-invariant materials, a net magnetization can be induced in the nonequilibrium steady state through the Edelstein effect. Such magnetization can further lead to a Hall-type current. The analytical expression can be obtained by plugging Eq. and into Eq. and keeping terms at second order in $\bm E$, and reads [@Inti2015] $$\label{eq_eahe}
\bm J=\frac{e^3}{\hbar}\tau\bm E\times \int \bm \Omega (\bm E\cdot \bm v) f_0^\prime \frac{d\bm k}{8\pi^3}\,.$$ This current yields the conductivity $\sigma_{ijk,1}$ in Eq. . Interestingly, the requirement of the mirror symmetry in Fig. \[fig\_nahe\](b) also works for this current. But one should keep in mind that for the current in Eq. , as the time reversal symmetry is broken, the inherent spin texture in the sample is also subjected to the mirror operation. The observation of this nonlinear Hall current has been reported in several recent experiments [@Xu2018; @Kang2018; @Ma2018]. We comment that in Eq. , $\bm v\bm \Omega$ together constitutes a pseudotensor which represents the first order moment of the Berry curvature in the momentum space. Hence it is referred to as the Berry curvature dipole. This Berry curvature dipole, together with the magnetic moment dipole also play essential roles in the study of the natural optical activity [@Landau1984; @Malashevich2010; @Orenstein2013; @Ma2015; @Zhong2015; @Zhong2016].
Finally, using the quantum kinetic equations for the evolution of the density matrix under both electric field and disorders, it has been found that in noncentrosymmetric materials, besides the Berry curvature dipole contribution, there are also nonlinear side jump and skew scattering mechanisms responsible for the nonlinear anomalous Hall effect at second order of electric field [@Nandy2019]. It can compete with the nonlinear anomalous Hall effect due to Berry curvature dipole.
Linear Magnetoresistance
------------------------
Linear magnetoresistance, referring to the linear dependence of the resistance on magnetic field, was first observed in simple metals [@Reitz1967; @Penz1968; @Jones1969] and under continuous studies afterwards. There is a surge of interest in recent years, as it appears in various novel materials, such as multilayer graphene [@Friedman2010], topological insulators [@Qu2010; @Tang2011; @Wang2012; @Tian2014], and Weyl/Dirac semimetals [@He2014; @Liang2014; @Feng2015; @Novak2015; @Narayanan2015; @Huang2015]. It has been shown that the linear magnetoresistance can arise in the ultra-quantum regime for Dirac systems when only the lowest Landau level is partially filled [@Abrikosov1988; @Abrikosov2000; @Wang2012a], as well as in inhomogeneous samples with mobility fluctuation [@Herring1960; @Parish2003; @Porter2012; @Kozlova2012]. In the semiclassical regime, linear magnetoresistance is subject to a symmetry requirement. Generally speaking Onsager relation requires that $\sigma_{xx}(\bm B)=\sigma_{xx}(-\bm B)$, which forbids odd power dependence of the conductivity on magnetic field. However, if the time reversal symmetry is broken and a magnetization is present, Onsager relation becomes $\sigma_{xx}(\bm B,\bm M)=\sigma_{xx}(-\bm B,-\bm M)$, which can be satisfied in principle for terms with odd powers of $B$.
To derive the linear magnetoresistance, or equivalently the linear magnetoconductivity in the semiclassical regime, one collects terms linear in $B$ in the current in Eq. . To simplify the result, we first consider the transverse magnetoresistance and assume that the electric field is along $\hat{x}$ direction and $\bm B$ is along $\hat{z}$ direction. This configuration will yield the following conductivity (here we use the definition in Eq. ) [@Chen2015] $$\begin{aligned}
\sigma_{(xx,z),1}&=\frac{e^2}{\hbar}\int\frac{d\bm k}{8\pi^3}\left[2v_x\frac{\partial m_z}{\partial k_x}+\frac{e}{\hbar}(v_x)^2\Omega_z\right]f_0^\prime\notag\\
&+\frac{e^2}{\hbar}\int\frac{d\bm k}{8\pi^3}(v_x)^2 (m_z+\delta \mu_{(1)}/B_z) f_0^{\prime\prime}\,.\\
\sigma_{(xx,z),2}&=-e^3 \int\frac{d\bm k}{8\pi^3}(v_xv_y\alpha_{xx}-v_x^2\alpha_{xy}) f_0^\prime=0\,.\end{aligned}$$
For longitudinal magnetoresistance, we assume that the electric field is parallel to the magnetic field, i.e. it is along $\hat{z}$ direction. Then the magnetoconductivity reads $$\begin{aligned}
\sigma_{(zz,z),1}&=\frac{e^2}{\hbar}\int\frac{ d\bm k}{8\pi^3}\left[2v_z\frac{\partial m_z}{\partial k_z}+\frac{e}{\hbar}v_z(v_z\Omega_z-2\bm v\cdot \bm \Omega)\right]f_0^\prime\notag\\
&+\frac{e^2}{\hbar}\int\frac{d\bm k}{8\pi^3}(v_z)^2 (m_z+\delta \mu_{(1)}/B_z) f_0^{\prime\prime}\,.\\
\sigma_{(zz,z),2}&=-e^3 \int\frac{d\bm k}{8\pi^3}(v_zv_y\alpha_{xz}-v_zv_x\alpha_{yz}) f_0^\prime=0\,.\end{aligned}$$
Both $\sigma_{(xx,z),1}$ and $\sigma_{(zz,z),1}$ are clearly violations to the Kohler’s rule [@Pippard1989]. In other words, the magnetic field affects the conductivity not through the Lorentz force, but through the two extra corrections in the first order semiclassical equations of motion. On one hand, it couples to the $k$-dependent magnetic moment to modifies the band energy and hence band velocity. On the other hand, it couples to the Berry curvature and changes the phase of space density of states, or equivalently, the carrier density. In the longitudinal magnetoconductivity, there is an additional term with the factor $\bm v\cdot \bm \Omega$. This actually measures the flux strength of the momentum space magnetic field $\bm \Omega$. This unique contribution to the longitudinal magnetoresistance is due to the coupling between the real and momentum space lorentz force, i.e. the magnetic field along $\hat{z}$ can first bend the velocity along $\hat{x}$ (or $\hat{y}$) to the direction $\hat{y}$ (or $\hat{x}$), which is further bent to the $\hat{z}$ direction due to the momentum space magnetic field $\Omega_x$ (or $\Omega_y$). Finally, we comment that both $\sigma_{(xx,z),1}$ and $\sigma_{(zz,z),1}$ are linear in magnetic moment and Berry curvature, which are odd in time reversal operation. As a result, these two conductivities only appear in materials that break time reversal symmetry.
The fact that $\sigma_{(xx,z),2}$ and $\sigma_{(zz,z),2}$ vanish identically is consistent with the Onsager’s relation, as formally they do not need to break the time reversal symmetry according to Table. 1. We comment that if a detailed consideration of scattering process is performed, there is a contribution to $\sigma_{(xx,z)}$ that is of second order in $\tau$, due to the skew scattering process [@Chen2015]. Therefore, it requires a nontrivial antisymmetric part of the scattering probability $W_{\bm k\bm k^\prime}$ and hence the breaking of time reversal symmetry.
![Linear magnetoresistance from $\sigma_{(xx,z),1}$ as a function of chemical potential. It is calculated for the surface state of the ferromagnetic topological insulator with the Hamiltonian $\hat{H}=\hbar v_\text{F} (\sigma_xk_x+\sigma_yk_y)+\Delta \sigma_z$. $\hbar v_\text{F}=4.1{\rm eV}\cdot {\rm A}$. $\Delta=20{\rm meV}$. From Ref. [@Chen2015].[]{data-label="fig_lmr"}](lmr.pdf "fig:"){width="0.9\columnwidth"}\
When the chemical potential approaches the hot spot of the Berry curvature and the magnetic moment, the latter can have a large contribution to the linear magnetoresistance, as shown in Fig. \[fig\_lmr\]. As the chemical potential approaches the band edge, the Berry curvature and magnetic moment increases and so does the linear magnetoresistance. However, the semiclassical theory is only valid when $(\mu-\Delta)\tau/\hbar>1$. Otherwise, the scattering from different impurities will be correlated and a fully quantum mechanical treatment should be used.
Negative longitudinal magnetoresistance induced by chiral anomaly
-----------------------------------------------------------------
It is well known that the chemical potential difference in real space can lead to a current. Its counterpart in momentum space, i.e. the current due to the chemical potential difference in momentum space, has completely different origins and attracts great attention recently as it manifests in topological semimetals.
To understand this effect, we note that from the semiclassical theory, the Berry curvature is the analog of the magnetic field in the momentum space. The flux of the Berry curvature, defined as $(1/2\pi)\oint d\bm S\cdot \bm \Omega$, can be nonzero through a closed surface, indicating the appearance of a monopole charge in the momentum space. This is most easily realized by a pair of Weyl points $$\begin{aligned}
\label{eq_dirac}
\hat{H}=\lambda_i v\bm k\cdot \bm \sigma\,.\end{aligned}$$ where $\lambda_i=\pm 1$ stands for a positive or negative charge, or equivalently, the chirality of the Weyl node. For such pair of Weyl nodes, there is a current driven by the magnetic field. This is the chiral magnetic current, derived using the last term in Eq. $$\begin{aligned}
\label{eq_cme}
J_\text{CME}=-\frac{e^2}{\hbar}\int \frac{d\bm k}{8\pi^3} (\bm v\cdot \bm \Omega)\bm Bf_0=-\frac{e^2\bm B}{4\pi^2\hbar^2}\sum_i \lambda_i \mu_i\,,\end{aligned}$$ where $\mu_i$ is the chemical potential for each Weyl node. This current has to vanish in equilibrium [@Vazifeh2013], even when the two Weyl nodes are shifted in energy and the relative chemical potential is different.
However, away from equilibrium, the chemical potential for different valleys can respond to external fields in different ways, leading to an inhomogeneous chemical potential distribution in the momentum space and hence a net current through $J_\text{CME}$ [@Son2013]. To see this, we first write down the Boltzmann equation: $$\label{eq_ca}
\dot{\bm k}\cdot \bm \partial f_0=-\int d\bm k^\prime \mathcal{D}(\bm k^\prime) W_{\bm k\bm k^\prime}[f(\bm k)-f(\bm k^\prime)]\,,$$ where $W_{\bm k\bm k^\prime}$ is the scattering probability. Using the semiclassical equations of motion in Eq. and , one finds that $$\label{eq_dk}
\mathcal{D}\dot{\bm k}=-\frac{e}{\hbar}\bm E-\frac{e}{\hbar}\tilde{\bm v}\times \bm B-\frac{e^2}{\hbar^2}(\bm E\cdot \bm B)\bm \Omega\,.$$ Therefore, the Boltzmann equation becomes $$\begin{aligned}
\label{eq_bz}
&\frac{e}{\mathcal{D}(\bm k)\hbar}\left[ \bm E+\frac{e}{\hbar}(\bm E\cdot \bm B)\bm \Omega\right]\cdot \bm \partial f_0\notag\\
=&\int d\bm k^\prime \mathcal{D}(\bm k^\prime) W_{\bm k\bm k^\prime}[f(\bm k)-f(\bm k^\prime)]\,.\end{aligned}$$
In multivalley systems such as a pair of Weyl nodes, a special solution to the above equation may emerge. To see this, we note that usually the transport current is determined by the fastest scattering rate which is the intravalley scattering. For the intravalley scattering with $f_0(\bm k)$ and $f_0(\bm k^\prime)$ from the same type of Weyl node, one can multiply Eq. by $\mathcal{D}(\bm k)$ and integrate it around the Weyl node. The result is $$\begin{aligned}
\label{eq_cond}
&\frac{e}{\hbar}\int \frac{d\bm k}{8\pi^3}\bm E\cdot\left[ \bm vf_0^\prime- \bm \partial(\bm B\cdot \bm m f_0^\prime)+\frac{e}{\hbar}\bm B(\bm \Omega\cdot \bm v)f_0^\prime\right]=0\,.\end{aligned}$$ However, for a Weyl node, although the integration of the first two terms vanish, the integration of the third term is proportional to the charge of the Weyl node and hence does not vanish. It indicates that charge can be pumped from the positive Weyl node to the negative Weyl node. This contradiction shows that the intervalley scattering is indispensable in solving the above Boltzmann equation.
To explicitly derive this solution, one can take $f_0(\bm k)$ to be from the positive Weyl node and $f_0(\bm k^\prime)$ from the negative Weyl node. Then under the relaxation time approximation, Eq. for the positive Weyl node becomes $$\begin{aligned}
\label{eq_ca2}
\frac{e^2}{\hbar^2}\int \frac{d\bm k}{8\pi^3}(\bm E\cdot\bm B)(\bm \Omega\cdot \bm v)f_0^\prime=\frac{\Delta N_+-\Delta N_-}{\tau_\text{int}}\,,\end{aligned}$$ where $\Delta N_+$ and $\Delta N_-$ is the density of particles near the Fermi surface, and $\tau_\text{int}$ is the intervalley scattering time. Eq. means that the pumping of particles from one Weyl node to another through drifting has to be balanced by the diffusive collision through the intervalley scattering process. It leads to an inhomogeneous chemical potential distribution in the momentum space $$\begin{aligned}
\delta \mu_+-\delta\mu_-=-\frac{e^2}{4\pi^2\hbar^2}\frac{\tau_\text{int}}{g(\mu)}(\bm E\cdot \bm B)\,,\end{aligned}$$ where $g(\mu)=\mu^2/(2\pi^2\hbar^3v^3)$ is the density of states at the Fermi surface and $\tau_\text{int}$ is the intervalley scattering time. This chemical potential difference for Weyl nodes with opposite chiralities is the chiral anomaly effect [@Nielsen1983; @Aji2012; @Son2013; @Burkov2015a; @Armitage2018].
This imbalance in the chemical potential can lead to a current through Eq. $$\bm J_\text{CME}=\frac{e^4v^3\tau_\text{int}}{2\pi^3\hbar\mu^2}(\bm E\cdot \bm B)\bm B\,.$$ This current is only present when $\bm E$ is parallel to $\bm B$. The corresponding conductivity is quadratic in magnetic field and increases with it, indicating a negative longitudinal magnetoresistance. It corresponds to $\sigma_{(ij,kk),1}$ in Table. 1 and does not need to break time reversal or inversion symmetry.
Besides the semiclassical theory, the negative magnetoresistance has also been confirmed using chiral kinetic theory [@Stephanov2012; @Son2013a; @Spivak2016; @Hidaka2017; @Sekine2017] and exact quantized Landau levels [@Burkov2014]. In the semiclassical regime, the negative magnetoresistance for one pair of Weyl nodes is also extended in other situations. It has been shown that samples with Dirac points also have negative magnetoresistance, provided that the chiral charge is approximately conserved and the chiral relaxation time is large [@Burkov2015]. Moreover, when the Dirac point opens a gap due to the suppression of the lattice symmetry, the chiral relaxation time is still present, so is the negative magnetoresistance [@Andreev2018]. Experimentally, the negative longitudinal magnetoresistance in Weyl and Dirac semimetals has been reported and interpreted as the chiral magnetic effect [@Huang2015; @Shekhar2015; @Yang2015; @Wang2016; @Zhang2016; @Kim2013; @Xiong2015; @Feng2015a; @Li2015; @Liang2014; @Li2016; @Zhang2017], but similar signals could also arise from two different mechanisms discussed in the following two sections.
Finally, we comment that the Weyl nodes is gapless and near those band-crossing points, the semiclassical theory may fail. However, we can set the Fermi surface to be well above the Weyl point. The flux through the Fermi surface does not change with the Fermi energy. As a result, the discussion using the semiclassical theory still works.
Intrinsic quadratic longitudinal magnetoresistance
--------------------------------------------------
The magnetoconductivity due to chiral anomaly and chiral magnetic effect is one contribution to $\sigma_{(ij,kk)}$ in Eq. . It is not a complete description of $\sigma_{(ij,kk)}$. To fully account for the drifting contribution to $\sigma_{(ij,kk)}$, one has to use the semiclassical theory up to second order, as presented in Sect. III (A). Specially, one can plug Eq. and into Eq. and collect terms up to second order in magnetic field. The current responsible for $\sigma_{(ij,kk),1}$ is complicated but it can be put in several groups: $$\begin{aligned}
\bm J=\bm J_1+\bm J_2+\bm J_3+\bm J_4+\bm J_5\,.\end{aligned}$$
The first contribution has the form $$\begin{aligned}
\bm J_1&=\frac{e^4}{\hbar^2}\tau\int \frac{d\bm k}{8\pi^3}[(\bm v\times \bm B)\cdot (\bm E\times \bm \Omega)](\bm v\times \bm B)\times \bm \Omega f_0^\prime\,.\end{aligned}$$ This term is purely due to the anomalous velocity as both $\bm E\times \bm \Omega$ and $(\bm v\times \bm B)\times \bm \Omega$ are parts of the anomalous velocity. This current is particularly interesting. If the electromagnetic fields are along the same direction, which can be assume to be the $\hat{z}$ direction without loss of generality, this current becomes $$\begin{aligned}
J_{1,z}&=-\frac{e^4}{\hbar^2}EB^2\tau\int \frac{d\bm k}{8\pi^3}(v_x\Omega_x+v_y\Omega_y)^2 f_0^\prime\,.\end{aligned}$$ This will always lead to a negative longitudinal magnetoresistance.
The second contribution has the form $$\begin{aligned}
\bm J_2=&\frac{e^2\tau}{2}\int \frac{d\bm k}{8\pi^3} \hat{e}_i\alpha_{ij}E_j(\bm B\cdot \bm m)^2 f_0^{\prime\prime}\notag\\
&-\frac{e^2\tau}{\hbar^2}\int \frac{d\bm k}{8\pi^3} \frac{\partial(\bm B\cdot \bm m)}{\partial \bm k} (\bm E\cdot \bm \partial_{\bm k})[(\bm B\cdot \bm m)f_0^\prime]\,.\end{aligned}$$ This is solely due to the Zeeman coupling between magnetic field and the magnetic moment, containing both the spin and orbital part.
The third contribution has the form $$\begin{aligned}
\bm J_3&=\frac{e^3}{\hbar^2}\tau \int \frac{d\bm k}{8\pi^3}(\bm E\cdot \bm B)\bm v (\bm \Omega\cdot \bm \partial_{\bm k})[(\bm B\cdot \bm m)f_0^\prime]\notag\\
&+\frac{e^3}{\hbar^2}\tau \int \frac{d\bm k}{8\pi^3}(\bm E\cdot \bm B)(\bm \Omega\cdot \bm v)\frac{\partial (\bm B\cdot \bm m)}{\partial \bm k}f_0^\prime\notag\\
&+\frac{e^3}{\hbar^2}\tau \int \frac{d\bm k}{8\pi^3}[(\bm v\times \bm B)\times \bm \Omega] (\bm E\cdot \bm \partial_{\bm k})[(\bm B\cdot \bm m)f_0^\prime]\notag\\
&+\frac{e^3}{\hbar^2}\tau \int \frac{d\bm k}{8\pi^3}\left\{\left[\frac{\partial (\bm B\cdot \bm m)}{\partial \bm k}\times \bm B\right]\times \bm \Omega\right\} (\bm E\cdot \bm v)f_0^\prime]\,.\end{aligned}$$ This current is due to the coupling between the anomalous velocity and the Zeeman energy correction.
The fourth contribution has the form $$\begin{aligned}
\bm J_4=&e^2\tau \int \frac{d\bm k}{8\pi^3}(2\hat{e}_i \alpha_{ij}E_jf_0^\prime+\bm v(\bm E\cdot \bm v) f_0^{\prime\prime})\varepsilon_{(2)}\notag\\
&+\frac{e^3\tau}{\hbar}\int \frac{d\bm k}{8\pi^3}[(\bm v\times \bm B)\cdot (\bm E\times \bm \Omega^\prime)]\bm v_0 f_0^\prime\notag\\
&-\frac{e^3\tau}{\hbar}\int \frac{d\bm k}{8\pi^3} (\bm v\cdot \bm \Omega^\prime)\bm B (\bm E\cdot \bm v) f_0^\prime\,.\end{aligned}$$ The first line is due to the second order correction to the band energy and the remaining terms are due to the first order correction to the Berry curvature, i.e. $\bm \Omega^\prime$.
From the expressions of $\bm J_1$ to $\bm J_4$, we find that $\bm J_1$ is purely due to the orbital motion of Bloch electrons, while $\bm J_2$ to $\bm J_4$ can contain both spin and orbital contribution. The orbital part has been calculated in Ref. [@Gao2017b] and the spin part has been calculated in Ref. [@Dai2017]. Here $\bm J_1$ to $\bm J_4$ are complete expressions that contain the spin part, the orbital part, and the spin-orbital part contribution.
The last contribution is due to the correction to the chemical potential: $$\begin{aligned}
\bm J_5&=\frac{e^2\tau}{\hbar} \int \frac{d\bm k}{8\pi^3} \bm v (\bm E\cdot \bm v)\left\{\mu_{(2)}f_0^{\prime\prime}-\frac{1}{2}[\mu_{(1)}]^2 f_0^{\prime\prime\prime}\right\}\,.\end{aligned}$$
![(a) Intrinsic magnetoconductivity versus the chemical potential. It is calculated for the model $\hat{H}=v(k_x\sigma_x+k_y\sigma_y)+(\Delta+k_z^2/2m)\sigma_z$, which can be realized by stacked Honeycomb lattices. Here $\sigma_{\perp 0}$ and $\sigma_{\parallel 0}$ are zero-magnetic-field conductivities under transverse and longitudinal configurations respectively. (b) Ratio between magnetoconductivity $\delta\sigma$ and zero-magnetic-field conductivity $\sigma_0$ versus the angle $\theta$ between $E$ and $B$ fields, as illustrated in the inset. Here the model parameters are chosen as $B=2$T, $\Delta=50{\rm meV}$, $v_F=9.2\times 10^5 {\rm m/s}$, and $m^*=0.1m_e$ ($m_e$ is the free electron mass), and $\mu=60 {\rm meV}$ in (b). From Ref. [@Gao2017b].[]{data-label="fig_nlmr"}](nlmr.pdf "fig:"){width="\columnwidth"}\
In nonmagnetic metals, $\sigma_{(ij,kk),3}$ is also nonzero. The corresponding current reads $$\begin{aligned}
\bm J^\prime=-\frac{e^4}{\hbar^2}\tau^3\int \frac{d\bm k}{8\pi^3} \bm v[(\bm v\times \bm B\cdot \bm \partial_{\bm k})^2(\bm E\cdot \bm v)] f_0^\prime\,.\end{aligned}$$ This current requires a peculiar Fermi surface such that the integration does not vanish [@Pal2010].
The current $\bm J$ and $\bm J^\prime$ complete the description of the quadratic magnetoresistance. In $\bm J^\prime$ the magnetic field always couples to the relaxation time. Therefore, the corresponding magnetoresistance follows Kohler’s rule [@Pippard1989]. In comparison, $\bm J$ violates Kohler’s rule and its corresponding magnetoresistance does not depend on the relaxation time. Moreover, $\bm J$ does not need a highly anisotropic Fermi surface. As long as the Berry curvature, magnetic moment, and magnetic susceptibility does not vanish at the same time, $\bm J$ will be nonzero. This intrinsic contribution to magnetoresistance is also confirmed by a fully quantum mechanical treatment [@Wang2018].
The current $\bm J$ can yield a negative longitudinal magnetoresistance. In fact, for the Dirac Hamiltonian in Eq. , the ratio of the quadratic longitudinal magnetoresistance $\delta \rho$ from $\bm J$ to that from chiral anomaly reads $$\begin{aligned}
\frac{\delta \rho}{\delta \rho_\text{CA}}=+\frac{8}{45}\frac{\tau}{\tau_\text{int}}\,.\end{aligned}$$ The global positive sign means that $\delta \rho$ is negative, like $\delta\rho_\text{CA}$. Usually $\tau\ll \tau_\text{int}$. Therefore, in Weyl semimetals, the negative magnetoresistance should be dominated by the contribution from the chiral anomaly in the semiclassical regime. However, the negative magnetoresistance from $\bm J$ can persist in metals without any Dirac/Weyl point, as shown in Fig. \[fig\_nlmr\]. Panel (a) suggests that the longitudinal magnetoresistance is negative while the transverse one is positive. This causes the sign change of the effective magnetoresistance as the angle between $\bm E$ and $\bm B$ continuously changes from $0$ to $\pi/2$, as shown in Panel (b).
![Transverse (Red) and longitudinal (Black) magnetoconductivity as a function of chemical potential. $\mu$ is in units of $\lambda$. For the red curve, $\sigma_{(xx,zz),1}\tau/\sigma_{xx}$ is plotted and for the black curve, $\sigma_{(zz,zz),1}\tau/\sigma_{zz}$ is plotted. The flux in one unit cell is $\phi=eBa^2/\hbar=3.79\times 10^{-3}$.[]{data-label="fig_mr"}](mr.pdf "fig:"){width="\columnwidth"}\
It is interesting to further check the behaviour of the magnetoconductivity in a lattice model. We use the model Hamiltonian in Eq. and and choose the parameters such that it is in the semiconductor phase as shown in Fig. \[fig\_eng0\] (d). We then calculate the quadratic magnetoconductivity for the transverse and longitudinal configuration, using the expression $\bm J_1$ to $\bm J_5$. Both spin and orbital part of the magnetic moment is considered. The result is presented in Fig. \[fig\_mr\]. It can be found that both magnetoconductivity show strong dependence on the chemical potential. Consistent with the result in the low-energy model, the magnetoresistance will be strongly enhanced when chemical potential is near the band edge, as both the anomalous velocity and energy correction are enhanced. Moreover, the longitudinal magnetoconductivity is generally positive, indicating a negative longitudinal magnetoresistance, while the transverse magnetoconductivity is negative. Near the band edge, the longitudinal magnetoconductivity is larger than the transverse one, indicating a sizeable transition angle $\theta$ from negative to positive magnetoresistance, as the relative angle between $\bm E$ and $\bm B$ changes from $0$ to $\pi/2$. This is similar to the negative magnetoresistance induced by the chiral anomaly.
Current Jetting
---------------
Beyond the Boltzmann transport theory, the magnetoresistance is also greatly affected when the current distribution in the sample is inhomogeneous. For example, for a matchstick sample with two leads for current input and output, as shown in Fig. \[fig\_cj\], the current in the sample can be highly inhomogeneous provided that the transverse magnetoresistance is sufficiently large [@Yoshida1980; @Pippard1989]. This is the current jetting problem.
![The current jetting problem. $\Sigma_L$ and $\Sigma_R$ stand for the regions of current input and output. $L_x$, $L_y$, and $L_z$ are the length of the sample along $x$, $y$, and $z$ directions.[]{data-label="fig_cj"}](cj.pdf "fig:"){width="\columnwidth"}\
The current jetting problem can be solved through the charge continuity equation: $\partial \rho/\partial t+\bm \nabla\cdot \bm J=0$. In the steady state $\partial \rho/\partial t=0$. Therefore, in the sample, the following equation for the electrostatic potential is satisfied $$\begin{aligned}
\sigma_{ij}\partial_i\partial_j V=0\,.\end{aligned}$$ Since the Hall conductivity is antisymmetric, it does not affect the above equation. However, it can affect the boundary condition. If the Hall conductivity can be ignored, and the longitudinal conductivity is determined by the Drude formula: $\sigma_{xx}=\sigma_{yy}=\sigma_0/(1+\omega_c^2\tau^2)$ and $\sigma_{zz}=\sigma_0$, the above equation can be put in the form of the Poisson equation $$\label{eq_pos}
\frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}+R^2 \frac{\partial^2 V}{\partial z^2}=0\,,$$ where $R=\sqrt{\sigma_{zz}/\sigma_{xx}}$. The boundary condition should also be determined by the charge continuity equation.
A large $R$ will greatly stretch the equipotential contour along $\hat{z}$ direction. It can cause a great portion of current lies in a sharp cone centered around the $\hat{z}$ axis [@Pippard1989]. This is similar to the behaviour of the longitudinal magnetoresistance due to chiral anomaly. In fact, if the injection of the current only localized near a small region in the surface and does not spread to the whole surface, the current injection may cause a strong negative magnetoresistance [@Huang2015; @Reis2016; @Yuan2016].
Outlook
=======
We have shown how to derive various nonlinear currents in the framework of the semiclassical theory and Boltzmann transport theory. Band properties beyond the spectrum play essential roles in those nonlinear currents. However, the semiclassical theory ignores the electron interaction and the modification to electron-impurity scattering from electromagnetic fields. The semiclassical theory has been extended in Fermi liquid using Keldysh formalism up to first order [@Shindou2006; @Shindou2008]. Moreover, the Boltzmann transport theory is also extended to treat the drifting and collision in equal-footing up to first order using the Wigner distriubtion function and quantum kinetic theory [@Sekine2017; @Culcer2017]. It is tempting to extend such theories up to second order to give a complete treatment of various nonlinear currents.
For the three nonlinear anomalous Hall effects, the one due to Berry curvature dipole has attracted several experimental efforts, while no experiments are reported to measure the other two corrections. The correction due to magnetic field should be generally present in any system and competes with the ordinary Hall effect. Distinguishing the magneto nonlinear Hall effect from the ordinary Hall effect is similar to extracting the intrinsic anomalous Hall from total anomalous Hall signal [@Tian2009; @Chun2007; @Lee2004; @Mathieu2004; @Sales2006; @Zeng2006].
Moreover, in samples that have combined time reversal and inversion symmetry (but break each one separately), the linear anomalous Hall effect as well as the nonlinear Hall effect due to Berry curvature dipole vanishes identically and only the intrinsic nonlinear Hall current contributes. This type of materials is ideal to test the intrinsic nonlinear Hall effect. If the time reversal and inversion symmetry, as well as their combined symmetry are all broken, the anomalous Hall and its two electric nonlinear version should all be present. There are still methods to distinguish them. One can first perform a scaling of the relaxation time to distinguish the intrinsic contribution from the extrinsic contribution. This is similar to extracting intrinsic, skew-scattering and side-jump contribution from the total anomalous Hall signal [@Tian2009; @Chun2007; @Lee2004; @Mathieu2004; @Sales2006; @Zeng2006]. The nonlinear part can be further extracted by reversing the direction of the electric field and finding the difference in the Hall signal.
For the negative longitudinal magnetoresistance, the difficulty is to distinguish the contribution from chiral anomaly from the intrinsic and current jetting contributions. The current jetting effect may be reduced by attaching the lead across the whole end of the sample so that the current is uniformly injected. In comparison, the main difference between the chiral anomaly and the intrinsic contribution is a hierarchy of relaxation time. One needs to control the ratio of the intervalley or chiral relaxation time to the transport relaxation time to distinguish these two contributions.
There is another issue in evaluating nonlinear currents in first-principles codes. Depending on the type of the unperturbed Hamiltonian, the second order energy has different forms for Schrodinger Hamiltonian or effective tight-binding Hamiltonian, mainly due to different Hessian matrices. Then there are two methods to evaluate the second order energies and related effects: one can start from the Schr$\ddot{\rm o}$dinger Hamiltonian and use the wave function from first-principles codes as approximation to evaluate the analytical expression derived for the Schr$\ddot{\rm o}$dinger Hamiltonian; on the other hand, one can start from the effective tight-binding Hamiltonian and uses its response to electromagnetic fields to approach the real one. The second one requires to evaluate the effective Hessian matrices while that of the first one is simply $1/m\delta_{ij}$. It is tempting to study which method is more accurate.
We acknowledge useful discussions with Xiao Di. This work is supported by the Department of Energy, Basic Energy Sciences, Grant No. DE-SC0012509.
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[^1]: the highest order of $\tau$ is the same as the highest order of the field, which is the property of the asymptotic solution to the Boltzmann equation, as implied in Eq. .
[^2]: In Ref. [@Ogata2015], it is found that except the energy polarization contribution, all the other terms are consistent. For the energy polarization contribution, Ref. [@Gao2015] contains a typo. When inserting the second order energy in Eq. , the energy polarization in Ref. [@Gao2015] has an additional $1/4$ factor by mistake. After removing such factor as given in Eq. , the energy polarization has the same expression with Eq. (2.31) in Ref. [@Ogata2015]
|
---
abstract: 'We discuss the possibility of a class of gauge theories, in four Euclidean dimensions, to describe gravity at quantum level. The requirement is that, at low energies, these theories can be identified with gravity as a geometrodynamical theory. Specifically, we deal with de Sitter-type groups and show that a Riemann-Cartan first order gravity emerges. An analogy with quantum chromodynamics is also formulated. Under this analogy it is possible to associate a soft BRST breaking to a continuous deformation between both sectors of the theory, namely, ultraviolet and infrared. Moreover, instead of hadrons and glueballs, the physical observables are identified with the geometric properties of spacetime. Furthermore, Newton and cosmological constants can be determined from the dynamical content of the theory.'
address:
- 'UFF $-$ Universidade Federal Fluminense, Instituto de Física, Campus da Praia Vermelha, Avenida General Milton Tavares de Souza s/n, 24210-346, Niterói, RJ, Brasil.'
- 'IFSEMG $-$ Instituto Federal de Educação, Ciência e Tecnologia, Rua Bernardo Mascarenhas 1283, 36080-001, Juiz de Fora, MG, Brasil.'
author:
- 'R. F. Sobreiro and A. A. Tomaz'
- 'V. J. Vasquez Otoya'
title: Induced gravity from gauge theories
---
Introduction {#intro}
============
Since the advent of general relativity (GR) [@Einstein:1916vd], gravity has been showing itself to be very different from all other fundamental interactions. The three fundamental interactions contemplated by the standard model are quantum mechanically stable gauge theories. In fact, quantum field theory (QFT) [@Itzykson:1980rh] describes fluctuating fields which are parameterized by a set of rigid spacetime coordinates $X^\mu$ while spacetime itself is a Minkowski space (To perform actual computations, spacetime must be Wick-rotated to the Euclidean space). On the other hand, GR is a geometrodynamical theory [@Hojman:1976vp] that describes spacetime as a dynamical entity, *i.e.*, spacetime coordinates gain a dynamical character. If one declare that the metric tensor $g_{\mu\nu}(X)$ is the fundamental field of gravity, a standard quantization is automatically forbidden because its quantum nature would be transferred to spacetime. For instance, if $\hat{g}_{\mu\nu}(X)$ is the quantum version of the metric tensor then, $\hat{X}_\mu=\hat{g}_{\mu\nu}(X)X^\nu$, which implies on $\hat{X}^\mu=\hat{g}^{\mu\nu}(X)\hat{X}_\nu$. Thus, spacetime coordinates would acquire a quantum character. The inconsistency rises because a quantum field cannot be parameterized by “quantum coordinates”. Axiomatic QFT states that spacetime coordinates must be a rigid set of well-defined parameters [@Streater:1989vi]. Essentially, geometrodynamics appears to be an exclusively classical description of gravity.
Another property that ruins a quantum description of gravity is the success of Einstein equations in describing macroscopic phenomena. At macroscopic scale, the *rhs* of Einstein equations is the energy-momentum tensor of the matter distribution, which is an effective object. Thus, the *lhs* of Einstein equations should also be an effective object and not a fundamental tensor that could be easily quantized by standard techniques.
To avoid the incompatibility between the principles of QFT and GR, the main method adopted is to expand the metric tensor around a background Minkowski metric $g_{\mu\nu}=\eta_{\mu\nu}+kh_{\mu\nu}$, where $k$ is related to Newton’s constant and carry ultraviolet dimension $-1$. The perturbation $h_{\mu\nu}$ is interpreted as the graviton field. However, it is widely known that the linearized Einstein-Hilbert (EH) action, known as the Pauli-Fierz action [@Fierz:1939ix], does not define a stable action at quantum level [@'tHooft:1974bx; @Deser:1974cz; @Deser:1974cy]. It is necessary then to introduce higher derivative terms which, however, destroy the unitarity of the theory [@Stelle:1976gc].
To get gravity closer to the rest of the fundamental interactions, it is convenient to write it as a gauge theory [@Utiyama:1956sy; @Kibble:1961ba; @Sciama:1964wt]. This description of gravity is known as the first order formalism (The formalism based on the metric tensor is called second order formalism), see also [@Mardones:1990qc; @Zanelli:2005sa]. In this approach, gravity is described by two fundamental 1-form fields, the vierbein[^1] $e^\mathfrak{a}$ and the spin-connection ${\omega^\mathfrak{a}}_\mathfrak{b}$. The geometric properties of spacetime are obtained from specific gauge invariant composite fields [@Sobreiro:2010ji; @Sobreiro:2010qf]. In particular, the metric tensor is obtained from $$g_{\mu\nu}=\eta_\mathfrak{ab}e^\mathfrak{a}_\mu e^\mathfrak{b}_\nu\;,\label{geom0}$$ while the affine connection is determined by $$\Gamma^\alpha_{\mu\nu}=g^{\alpha\beta}\eta_\mathfrak{ab}\left(e^\mathfrak{b}_\beta\partial_\mu e^\mathfrak{a}_\nu+e^\mathfrak{b}_\beta\omega^\mathfrak{a}_{\mu\mathfrak{c}}e^\mathfrak{c}_\nu\right)\;.\label{geom1}$$ The vierbein is a map between coordinates $X^\mu$ at a point $X$, in spacetime manifold $\mathbb{M}^4$, and coordinates $X^a$, in the tangent space $T_X(\mathbb{M})$, at the very same point $X$, *i.e.*, $dX^\mathfrak{a}=e^\mathfrak{a}_\mu dX^\mu$. The spin-connection is related to parallel transport between near tangent spaces $T_X(\mathbb{M})$ and $T_{X+dX}(\mathbb{M})$ and is recognized as the gauge field of the theory. Typically, the gauge group is the Lorentz group $SO(1,3)$ and describes the local isometries of spacetime. Nevertheless, since the fundamental fields are still related to spacetime properties, any attempt of standard quantization of these fields would lead to the same inconsistency that occur in the second order formalism. Another attempt to solve the QFT-GR incompatibility is to declare that the fundamental fields are not immediately related to spacetime. Instead, the geometric properties of spacetime emerge as effective phenomena [@Zanelli:2005sa; @Sobreiro:2010ji; @Sobreiro:2010qf]. For instance, $$\begin{aligned}
g_{\mu\nu}&=&\eta_\mathfrak{ab}\left<e^\mathfrak{a}_\mu e^\mathfrak{b}_\nu\right>\;,\nonumber\\
\Gamma^\alpha_{\mu\nu}&=&g^{\alpha\beta}\eta_\mathfrak{ab} \left<e^\mathfrak{b}_\beta\partial_\mu e^\mathfrak{a}_\nu+e^\mathfrak{b}_\beta\omega^a_{\mu \mathfrak{c}}e^\mathfrak{c}_\nu\right>\;.\label{geom2}\end{aligned}$$ The idea is to work with geometric variables with no specific relation to the metric tensor nor the affine connection. Although promising, it is unclear to work with a ill defined coordinate system.
Sticking to the gauge theoretical approach, many other theories have been proposed by generalizing the gauge groups and their respective actions. In particular [@Stelle:1979va; @Stelle:1979aj; @Pagels:1983pq; @Tresguerres:2008jf; @Mielke:2010zz], it is worth mentioning the de Sitter groups $SO(m,n)$, with $(m+n)=5$, in four-dimensional spacetime, supplemented with a spontaneous symmetry breaking mechanism [@Higgs:1964pj; @Higgs:1964ia]. In the work [@MacDowell:1977jt], gravity and supergravity are considered under the framework of a dynamical breaking instead of a spontaneous breaking. In any of these cases, the Yang-Mills action [@Yang:1954ek] is not considered. When the Yang-Mills action is taken into account, the starting theory is already metric dependent, see for instance [@Tseytlin:1981nu; @Mahato:2004zi]. Only in [@Sobreiro:2007pn], the Yang-Mills was considered independently of the metric. The Lorentz group was taken as the gauge group of a theory in flat spacetime. A relation with a Palatini-type gravity emerges from a color symmetry breaking generated by a condensation mechanism.
More recently [@Sobreiro:2011hb], the Yang-Mills action for de Sitter groups was studied independently of the geometric properties of spacetime. Essentially, asymptotic freedom [@Gross:1973id; @Politzer:1973fx], mass parameters and an Inönü-Wigner contraction [@Inonu:1953sp] to Lorentz-type groups account for the emergence of a gravity theory. The absence of mass parameters in the starting action is important because it prevents the identification of the gauge field with the vierbein. Thus, it is important that the masses are dynamically generated. One possibility is the Gribov parameter [@Gribov:1977wm; @Zwanziger:1992qr; @Sobreiro:2005ec] which is required for quantum consistency at low energy scales. Moreover, the presence of the Gribov parameter induces a soft BRST breaking [@Baulieu:2008fy; @Baulieu:2009xr; @Dudal:2012sb], which continuously deforms the massless theory in flat spacetime into a gravity theory in the first order formalism. Extra dynamical masses can also contribute to the model [@Dudal:2005na; @Dudal:2011gd]. Remarkably, Newton and cosmological constants can be determined from these masses and the Yang-Mills coupling parameter. Thus, since pure Yang-Mills theory is renormalizable [@'tHooft:1972fi; @Piguet:1995er], this model can be regarded as a possible theory for quantum gravity.
In the present work, we exploit some physical properties of the theory developed in [@Sobreiro:2011hb]. In particular, we discuss the analogy between this theory and quantum chromodynamics (QCD), the soft BRST breaking, the dynamical masses contribution and the possibility of the theory in providing reliable physical results.
Section 2 is devoted to the foundations of Yang-Mills theories, BRST symmetry and confinement. Section 3 is dedicated to a QCD-Gravity analogy. In Section 4, the details of de Sitter gauge theories in four-dimensional Euclidean spacetime and their relation with gravity are discussed. In Section 5, some consistency checks of the model are provided. Our final considerations can be found in Section 6.
Yang-Mills theories, BRST symmetry and confinement
==================================================
Preliminary concepts {#gauge}
--------------------
The so called Yang-Mills theories [@Yang:1954ek] are gauge theories based on a gauge symmetry described by a semi-simple Lie group $G$. This class of theories consists of a generalization of electrodynamics, which is an Abelian gauge theory for the group $U(1)$. The fundamental field is the gauge connection 1-form $Y=Y^{\mathsf{a}}J_{\mathsf{a}}$, where $J_{\mathsf{a}}$ are the generators of the group and the group indices vary as $\{\mathsf{a},\mathsf{b},\mathsf{c},\ldots\}\in\{1,2,\ldots,\dim{G}\}$. The curvature 2-form is defined as $F=\nabla^2=\mathrm{d}Y+\kappa YY$, where $\nabla=\mathrm{d}+\kappa Y$ is the covariant derivative, $\mathrm{d}$ is the exterior derivative in spacetime and $\kappa$ is the coupling parameter.
The Yang-Mills action is constructed as $$S_{\mathrm{YM}}=\frac{1}{2}\int\;F^{\mathsf{a}}\ast F_{\mathsf{a}}\;,\label{ym0}$$ where $\ast$ is the Hodge operator. The action is invariant under gauge transformations, $$Y\longmapsto g^{-1}\left(\frac{1}{\kappa}\mathrm{d}+Y\right)g\;,\label{gt0}$$ where $g\in G$. To define a path integral for the Yang-Mills action, a gauge fixing must be imposed [@Itzykson:1980rh; @Faddeev:1967fc]. For a consistent introduction of a gauge fixing, it is convenient to adopt the BRST quantization method [@Piguet:1995er; @Tyutin:1975qk; @Becchi:1975nq]. For simplicity, we adopt the Landau gauge, $\mathrm{d}\ast Y=0$. In this framework, the action is replaced by $$S_0=S_{\mathrm{YM}}+\int\left(ib^\mathsf{a}\mathrm{d}\ast Y_\mathsf{a}+\overline{c}_\mathsf{a}\mathrm{d}\ast\nabla^\mathsf{ab}c_\mathsf{b}\right)\;,\label{ym1}$$ where the fields $c^\mathsf{a}$ and $\overline{c}^\mathsf{a}$ are the ghost and anti-ghost fields, respectively, and $b^\mathsf{a}$ is a Lagrange multiplier which enforces the Landau gauge. The components of the covariant derivative are $\nabla^\mathsf{ab}=\delta^\mathsf{ab}\mathrm{d}-\kappa f^\mathsf{abc}Y_\mathsf{c}$, where $f^\mathsf{abc}$ are the structure constants of the gauge group. The action , although not gauge invariant, is invariant under BRST transformations, namely, $$\begin{aligned}
sY^\mathsf{a}&=&\nabla^\mathsf{ab}c_\mathsf{b}\;,\nonumber\\
sc^\mathsf{a}&=&\frac{\kappa}{2}f^\mathsf{abc}c_\mathsf{b}c_\mathsf{c}\;,\nonumber\\
s\overline{c}^\mathsf{a}&=&b^\mathsf{a}\;,\nonumber\\
sb^\mathsf{a}&=&0\;,\label{brst0}\end{aligned}$$ where $s$ is the nilpotent BRST operator.
Let us make some important remarks about Yang-Mills theories.
- Although the fundamental fields and the action are not gauge invariant, physical observables must be gauge invariant. This statement is perhaps the most sacred principle in gauge theories, it is called *gauge principle*. This principle can be extended to BRST invariant operators in order to include more general operators that might be related to physical quantities. At classical level, the BRST invariance principle reduces to the usual gauge principle.
- Together with quantum electrodynamics [@Tomonaga:1946zz; @Schwinger:1948yk; @Feynman:1950ir], and the Higgs mechanism [@Higgs:1964pj; @Higgs:1964ia], Yang-Mills theories are the theoretical basis of the Standard model. Thus, it is a successful theory with a deep connection with reality.
- The action is a renormalizable theory, at least to all orders in perturbation theory [@'tHooft:1972fi; @Piguet:1995er]. Renormalizability states that ultraviolet divergences can be consistently eliminated from perturbative computations, *i.e.*, Yang-Mills theories are stable at quantum level. It is worth mention that BRST symmetry is very important with respect to the renormalizability of a gauge theory.
- Another feature of the action is unitarity [@Kugo:1979gm]. It states that the $S$-matrix conserves probability and, thus, asymptotic states can be defined. For pure gauge theories, unitarity requires a well defined BRST symmetry and that the gauge group is compact.
- The renormalization of the coupling parameter leads to the concept of asymptotic freedom [@Gross:1973id; @Politzer:1973fx]. This property predicts that, at high energies, the coupling is very small and perturbation theory can be safely employed. However, as the energy decreases the coupling increases. At this strong coupling regime, the theory is non-perturbative and yet to be fully understood.
- Confinement [@Baulieu:2009xr; @Dudal:2012sb; @Kugo:1979gm] states that quarks and gluons cannot be observable states. Actually, the physical observables must be, not only gauge invariant, but also colorless. At high energies, confinement manifests itself as the quark-gluon plasma, a state where quarks and gluons are almost free. However, they cannot be distinguished from the plasma. At low energies, the strong coupling enforces hadronization phenomena, *i.e.*, the physical spectrum is composed by hadrons and glueballs. Although experimentally established, it is a theoretical challenge to show that the Yang-Mills action leads to an effective theory composed by hadrons and glueballs.
- Another possible effect is that, at low energies, dynamical mass generation [@Dudal:2005na; @Dudal:2011gd] takes place. It originates from the condensation of dimension-2 operators. The mass parameters that emerge drastically change the behavior of the theory at infrared scale and are relevant for confinement.
- It was shown in [@Gribov:1977wm; @Singer:1978dk] that, at low energies, gauge fixing is not possible by the simple introduction of a constraint. This problem is known as Gribov ambiguities problem. The main point in the elimination of the Gribov ambiguities is that BRST quantization (and also the standard Faddeev-Popov quantization [@Faddeev:1967fc]) shows itself to be incomplete at low energy level. In fact, the implementation of a gauge fixing does not eliminate completely the gauge symmetry [@Gribov:1977wm], a residual symmetry survives. Moreover, it was shown that this problem occur for any gauge choice [@Singer:1978dk]. The spurious gauge configurations are called Gribov copies and they are the kernel of the Gribov ambiguities. Remarkably, these copies gain relevance only at low energies, keeping the high energy sector untouched. To eliminate the Gribov ambiguities is a hard step that is not yet fully understood. However, at the Landau gauge, the inifinitesimal copies can be eliminated through the introduction of a soft BRST breaking[^2] related to a mass parameter known as Gribov parameter. Outstandingly, the treatment of this technical problem leads to exceptional evidences of quark-gluon confinement [@Zwanziger:1992qr; @Baulieu:2008fy; @Baulieu:2009xr; @Dudal:2012sb; @Dudal:2005na].
The infrared puzzle, Gribov ambiguities and BRST soft breaking {#ir}
--------------------------------------------------------------
Basically, at the infrared regime, there are several techniques leading to confinement evidences. For instance, non-trivial solutions of the Schwinger-Dyson equations [@Binosi:2011zz], lattice simulations [@Greensite:2003bk; @Cucchieri:2011ig], AdS/QCD duality [@Maldacena:1997zz; @Witten:1998qj; @Miranda:2009qp], dual superconductivity and defects [@Nambu:1974zg; @'tHooft:1975; @Mandelstam:1974pi; @Faddeev:2001dda] and dimension-2 condensates and Gribov ambiguities [@Gribov:1977wm; @Zwanziger:1992qr; @Sobreiro:2005ec; @Baulieu:2008fy; @Baulieu:2009xr; @Dudal:2012sb; @Dudal:2005na; @Dudal:2011gd]. The puzzle consists in merging all evidences and formalisms in order to obtain a consistent theory in terms of hadrons and glueballs. As mentioned, we confine ourselves to explore the Gribov problem and its relation to confinement.
The treatment of Gribov ambiguities has the effect of changing the action to the so called Gribov-Zwanziger action, $$S=S_0+\int\left[\overline{\phi}_\mathsf{ac}\mathrm{d}\ast\nabla^\mathsf{ab}\phi_\mathsf{bc}-\overline{\omega}_\mathsf{ac}\mathrm{d}\ast\nabla^\mathsf{ab}\omega_\mathsf{bc}+\gamma^2f^\mathsf{abc}Y_\mathsf{a}\left(\overline{\phi}+\phi\right)_\mathsf{bc}+4M\gamma^4\right]\;,\label{ym2}$$ where $M=\dim{G}$, $\gamma$ is the Gribov parameter which has dimension of a mass, the fields $\overline{\phi}_\mathsf{ab}$ and $\phi_\mathsf{ab}$ are 1-form bosonic fields while $\overline{\omega}_\mathsf{ab}$ and $\omega_\mathsf{ab}$ are fermionic 1-form fields. These fields obey the following transformation rules, $$\begin{aligned}
s\overline{\omega}_\mathsf{ab}&=&\overline{\phi}_\mathsf{ab}\;,\nonumber\\
s\overline{\phi}_\mathsf{ab}&=&0\;,\nonumber\\
s\phi_\mathsf{ab}&=&\omega_\mathsf{ab}\;,\nonumber\\
s\omega_\mathsf{ab}&=&0\;,\label{brst1}\end{aligned}$$ and can be eliminated through field equations in favor of a nonlocal term, the horizon-function [@Zwanziger:1992qr; @Dudal:2005na; @Dudal:2011gd].
We now establish a few properties of the action .
- The action is renormalizable, at least to all orders in perturbation theory, in such a way that the ultraviolet sector of the theory is unchanged. In fact, no extra renormalizations are required [@Zwanziger:1992qr; @Dudal:2005na]; all extra fields and Gribov parameter renormalization factors depend on the gluon and coupling parameter renormalization factors.
- The action is not BRST invariant, and the breaking is proportional to the Gribov parameter, $sS\propto\gamma^2$. Since $\gamma$ has dimension of a mass, the breaking is soft, and thus, harmless to the UV sector [@Baulieu:2008fy; @Baulieu:2009xr].
- The Gribov parameter is determined from the minimization of the quantum action $\delta \Sigma/\delta \gamma^2=0$. A simple computation at the tree level predicts that $\gamma^2=\mu^2\exp\{-\frac{64\pi^2}{3N\kappa^2}\}$, where $\mu^2$ is a cutoff and $N$ is the Casimir of the gauge group, $f^\mathsf{acd}f_\mathsf{bcd}=-N\delta^\mathsf{a}_\mathsf{b}$. Thus, at the perturbative regime, $\kappa\rightarrow0$, we have that $\gamma\rightarrow0$. To take this limit is equivalent to take the high energy limit. Thus, the presence of the Gribov parameter allows the action to be continuously deformed into action . Then, the UV and IR sectors can be continuously deformed to each other. Moreover, the BRST is asymptotically restored at the UV limit.
- One of the evidences of confinement emerges from the gauge propagator which, at finite values of $\gamma$, acquires imaginary poles. The consequence is that it violates positivity of the spectral representation of this propagator and thus, no physical particles can be associated with this propagator [@Dudal:2005na].
- More recently [@Dudal:2011gd], the union of the Gribov-Zwanziger formalism and the condensation of dimension-2 operators has been developed. This is called the *refined Gribov-Zwanziger formalism* and improves the results obtained from the Gribov-Zwanziger action. For instance, the propagators of the RGZ action coincide with the results obtained from lattice simulations [@Cucchieri:2011ig].
- We also remark that, some advances on the determination of the physical spectrum of the infrared sector of Yang-Mills theories have been made [@Dudal:2010cd; @Capri:2012hh]. Although very difficult, the main requirement is gauge invariance, *i.e.*, observables are related with gauge invariant operators. In QCD, these operators must describe its low energy spectrum, *i.e.*, hadrons and glueballs.
QCD-Gravity analogy {#qcdgrav}
===================
This work is based on the idea that gravity can be described at quantum level by a gauge theory in a Euclidean four-dimensional spacetime. Classical gravity emerges if one could possibly show that a geometrodynamical theory rises at the infrared sector. This is performed by applying the soft BRST breaking technique. Let us elaborate this idea in more detail.
We assume that quantum gravity can be described by a Yang-Mills theory in four-dimensional Euclidean spacetime, $\mathbb{R}^4$. This choice, instead of a Minkowski spacetime, has three main reasons: (i) first, because QFT is only solvable in Euclidean spaces. Reliable computations require a Wick rotation from the Minkowkian to the Euclidean space [@Itzykson:1980rh]. However, a Wick rotation is known to be valid only at perturbative level. At non-perturbative scales, there are no indications that a Wick rotation can be performed. Thus, since we are interested in non-perturbative effects, it is necessary to start with a Euclidean spacetime. (ii) The second reason, and perhaps the most important, is that the Euclidean space is the simplest geometry. Since we are constructing a theory that will determine spacetime geometry, to start with the simplest geometry is the right choice. (iii) In four dimensions, the coupling parameter of a Yang-Mills theory is dimensionless. As a consequence, pure Yang-Mills action is massless[^3]. This fact is important because it prevents the gauge field, which has dimension 1, to be associated with the vierbein, which has vanishing dimension.
The gauge field is an algebra-valued 1-form and thus, carries $\dim{G}$ components with respect to the algebra of the gauge group. The gauge group $G$ has to fulfill a few requirements [@Sobreiro:2011hb; @Sobreiro:2012book]: (i) the first one is that $\dim{G}\ge\dim{ISO(4)}$, in such a way that the degrees of freedom of gravity are covered. (ii) The universal principal bundle that describes this theory must be non-trivial, so the Gribov problem is inherent to the theory and has to be properly treated [@Singer:1978dk; @Daniel:1979ez; @Nakahara:1990th]. (iii) The group must decompose at least[^4] as $G=H+Q$ where $Q=G/H$ must be a symmetric space and $H$, obviously, a stability group. The stable group must share a morphism with Lorentz-type groups and $Q$ must define a vector representation of $H$. Thus, $H$ can be identified with local Lorentz transformations and $Q$ with a sector that expand the vierbein. As a consequence, the gauge components can be identified with the spin-connection and the vierbein. (iv) It can be very convenient to have a symmetry breaking $G\rightarrow H$, in such a way that the field at the $Q$ sector acquires a matter-type transformation with respect to $H$. That is why $Q$ must be a symmetric space. It is important to mention that, depending on the gauge group, a symmetry breaking might not be needed. This occurs if the gauge transformations could be partially identified with local Lorentz transformations and partially with diffeomorphisms. A renowned example can be found in [@Witten:1988hc].
It is important that the theory develops at least one mass parameter, possibly the Gribov parameter, in order to rescale the gauge field at the sector $Q$. As mentioned, this field, being a connection, carries dimension 1 and thus, cannot be directly associated with the vierbein. Only after the emergence of these mass parameters is that this identification can be performed. Moreover, the Gribov parameter and the consequent BRST breaking are important concepts that must be interpreted as follows:
*The theory has two sectors, the UV, which is a massless gauge theory in Euclidean four-dimensional space, and the IR sector, which presents soft BRST symmetry breaking and dynamically generated mass parameters. The UV sector is a standard, non-Abelian, asymptotic free, gauge theory of spin-1 excitations. Although the degrees of freedom coincide in number with a first order gravity, this identification is forbidden unless a mass parameter arises, so that a vierbein can be defined. Once the energy starts to decrease, the soft BRST symmetry breaking takes place, the Gribov parameter and other possible masses appear. At this stage, propagators of the fundamental fields develop complex poles, a fact that is interpreted as an evidence that these excitations are ruled out from the physical spectrum of the theory (in QCD, this is recognized as confinement). Physical observables must be defined at this point. In QCD, the observables are hadrons and glueballs. In gravity, the low energy observables must be . Thus, one possibility is to identify the $H$ gauge field with the spin-connection and the $Q$ field with the vierbein. The consequence is a first order gravity where geometry is determined by the usual relations and .*
If all above requirements are fulfilled, we gain a deep analogy between gravity and quantum chromodynamics. At high energies, both theories are well defined quantum gauge theories in a four-dimensional flat space. Both theories present soft BRST breaking and have to be redefined in order to establish the physical content at the infrared regime. In the case of QCD, the physical content are confined states identified with hadrons and glueballs. In gravity, the physical content are identified with geometric properties of spacetime.
We now turn to a specific model when these ideas are realized.
Realization of the idea through de Sitter groups {#realiz}
================================================
de Sitter-Yang-Mills theory
---------------------------
One possible realization of the above ideas occurs in a gauge theory based on the group $SO(m,n)$ with $m+n=5$ and $m\in\{0,1,2\}$. For $m=0$, the gauge group is the orthogonal one. For $m=1$ and $m=2$, we have a de Sitter and anti de Sitter groups, respectively[^5]. Spacetime is a Euclidean four-dimensional differential manifold $\mathbb{R}^4$. The algebra of the group is given by $$\left[J^{AB},J^{CD}\right]=-\frac{1}{2}\left[\left(\eta^{AC}J^{BD}+\eta^{BD}J^{AC}\right)-\left(\eta^{AD}J^{BC}+\eta^{BC}J^{AD}\right)\right]\;,\label{alg1}$$ where $J^{AB}$ are the $10$ anti-hermitian generators of the gauge group, antisymmetric in their indices. Capital Latin indices are chosen to run as $\{5,0,1,2,3\}$. Comparing with the generic gauge theory described in Sect. \[gauge\], the indices identification is $\mathsf{a}\equiv AB=-BA$. The $SO(m,n)$ group defines a five-dimensional flat space, $\mathbb{R}^{m,n}_S$, with invariant Killing metric given by $\eta^{AB}\equiv\mathrm{diag}(\epsilon,\varepsilon,1,1,1)$ with $\epsilon=(-1)^{(2-m)!}$ and $\varepsilon=(-1)^{m!+1}$. The spaces $\mathbb{R}^4$ and $\mathbb{R}^{m,n}_S$ have no dynamical relation whatsoever.
The $SO(m,n)$ Yang-Mills action is renormalizable and is defined over a non-trivial universal bundle [@Sobreiro:2011hb; @Sobreiro:2012book; @Assimos:2012zzz]. Thus, BRST soft breaking takes place. Moreover, a few extra mass parameters may emerge. The presence of these masses will be forwardly used. However, we will fix our attention only at the Yang-Mills action and the gauge field $Y$.
The de Sitter group may be decomposed as a direct product, $SO(m,n)\equiv SO(m!-1,n)\otimes S(4)$ where $S(4)\equiv SO(m,n)/SO(m!-1,n)$ is a symmetric coset space with four degrees of freedom. This decomposition is carried out by projecting the group space in the fifth coordinate $A=5$. Defining then $J^{5a}=J^a$, where small Latin indices vary as $\{0,1,2,3\}$, the algebra decomposes as $$\begin{aligned}
\left[J^{ab},J^{cd}\right]&=&-\frac{1}{2}\left[\left(\eta^{ac}J^{bd}+\eta^{bd}J^{ac}\right)-\left(\eta^{ad}J^{bc}+\eta^{bc}J^{ad}\right)\right]\;,\nonumber\\
\left[J^a,J^b\right]&=&-\frac{\epsilon}{2}J^{ab}\;,\nonumber\\
\left[J^{ab},J^c\right]&=&\frac{1}{2}\left(\eta^{ac}J^b-\eta^{bc}J^a\right)\;,\label{alg2}\end{aligned}$$ where $\eta^{ab}\equiv\mathrm{diag}(\varepsilon,1,1,1)$.
The gauge connection, follows the same decomposition, $Y=Y^A_{\phantom{A}B}J_A^{\phantom{A}B}=A^a_{\phantom{a}b}J_a^{\phantom{a}b}+\theta^aJ_a$ and the gauge transformations in Eq. are decomposed, at infinitesimal level, as $$\begin{aligned}
{A}^a_{\phantom{a}b}&\longmapsto& {A}^a_{\phantom{a}b}+\mathrm{D}\alpha^a_{\phantom{a}b}-\frac{\epsilon\kappa}{4}\left(\theta^a\xi_b-\theta_b\xi^a\right)\;,\nonumber\\
\theta^a&\longmapsto&\theta^a+\mathrm{D}\xi^a+\kappa\alpha^a_{\phantom{a}b}\theta^b\;.\label{gt2}\end{aligned}$$ where the full gauge parameter splits as $\zeta=\alpha^a_{\phantom{a}b}J_a^{\phantom{a}b}+\xi^aJ_a$ and $\mathrm{D}=\mathrm{d}+\kappa A$ is the covariant derivative with respect to the sector $SO(m!-1,n)$. The field strength also decomposes, $F=\left(\Omega^a_{\phantom{a}b}-\frac{\epsilon\kappa}{4}\theta^a\theta_b\right)J_a^{\phantom{a}b}+K^aJ_a$, where $\Omega^a_{\phantom{a}b}=\mathrm{d}A^a_{\phantom{a}b}+\kappa A^a_{\phantom{a}c}A^c_{\phantom{c}b}$ and $K^a=\mathrm{D}\theta^a=\mathrm{d}\theta^a-\kappa A^a_{\phantom{b}b}\theta^b$. Under this decomposition, the Yang-Mills action is written as $$S=\frac{1}{2}\int\left[\Omega^a_{\phantom{a}b}{*}\Omega_a^{\phantom{a}b}+\frac{1}{2}K^a{*}K_a-\frac{\epsilon}{2}\Omega^a_{\phantom{a}b}{*}(\theta_a\theta^b)+\frac{1}{16}\theta^a\theta_b{*}(\theta_a\theta^b)\right]\;.\label{ym0a}$$
Once the energy starts to decrease the set of mass parameters, together with BRST soft breaking, dynamically arises. At this point, it is convenient to perform the following redefinitions[^6] $$\begin{aligned}
A&\longmapsto&\kappa^{-1}A\;,\nonumber\\
\theta&\longmapsto&\kappa^{-1}m\theta\;,\label{resc1}\end{aligned}$$ where $m$ is a mass scale depending on the mass parameters of the theory. The action is then rescaled to $$S=\frac{1}{2\kappa^2}\int\left[\overline{\Omega}^a_{\phantom{a}b}{*}\overline{\Omega}_a^{\phantom{a}b}+\frac{m^2}{2}\overline{K}^a{*}\overline{K}_a-\frac{\epsilon m^2}{2}\overline{\Omega}^a_{\phantom{a}b}{*}(\theta_a\theta^b)+\frac{m^4}{16}\theta^a\theta_b{*}(\theta_a\theta^b)\right]\;,\label{ym1a}$$ where $\overline{\Omega}^a_{\phantom{a}b}=\mathrm{d} {A}^a_{\phantom{a}b}+ {A}^a_{\phantom{a}c} {A}^c_{\phantom{c}b}$, $\overline{K}^a=\overline{\mathrm{D}}\theta^a$, and the covariant derivative is now $\overline{\mathrm{D}}=\mathrm{d}+A$. Moreover, a reparameterization of the $SO(m,n)$ generators is required due to the existence of a mass scale, *i.e.*, a stereographic projection is now allowed if one identifies the mass parameter with the radius of the gauge manifold $\mathbb{R}^{m,n}_S$, see \[ap1\]. The consequence for de Sitter algebra is $$\begin{aligned}
\left[J^{ab},J^{cd}\right]&=&-\frac{1}{2}\left[\left(\eta^{ac}J^{bd}+\eta^{bd}J^{ac}\right)-\left(\eta^{ac}J^{bc}+\eta^{bc}J^{ad}\right)\right]\;,\nonumber\\
\left[J^a,J^b\right]&=&-\frac{\epsilon m^2}{2\kappa^2}J^{ab}\;,\nonumber\\
\left[J^{ab},J^c\right]&=&\frac{1}{2}\left(\eta^{ac}J^b-\eta^{bc}J^a\right)\;.\label{alg3}\end{aligned}$$
Contraction and symmetry breaking {#cont}
---------------------------------
The main trick, in order to obtain gravity, is to consider that the rate $m^2/\kappa^2$ tends to vanish at low energy scales[^7]. Due to asymptotic freedom, this is a very consistent hypothesis. Thus, we can perform an Inönü-Wigner contraction [@Inonu:1953sp] through $m^2/\kappa^2\rightarrow 0$, at the algebra , enforcing the de Sitter group to be contracted down to the Poincaré group. The second of relations deforms to $\left[P^a,P^b\right]=0$, where $J^a\longmapsto -\kappa\gamma^{-1}P^a$ and $\theta\longmapsto-\theta^aP_a$ (See the limit of in \[ap1\]). The gauge symmetry is then dynamically deformed to the Poincaré group, $SO(m,n)\longrightarrow ISO(m!-1,n)$, for some values $\kappa$ in the strong coupling regime. The details of this deformation can be understood as a stereographic projection, see \[ap1\].
Typically, an Inönü-Wigner contraction is a deformation of the group to another group which is not a subgroup of the former. Thus, since the Poincaré group is not a symmetry of the action , a dynamical symmetry breaking takes place. In fact, the Lorentz-type group $SO(m!-1,n)$ is a common subgroup, $ISO(m!-1,n)\supset SO(m!-1,n)\subset SO(m,n)$, and also a stability subgroup for both groups. Thus, the theory suffers a symmetry breaking $SO(m,n)\longrightarrow SO(m!-1,n)$. Under the $SO(m!-1,n)$ gauge symmetry, the transformations reduce to $$\begin{aligned}
A^a_{\phantom{a}b}&\longmapsto& {A}^a_{\phantom{a}b}+\overline{\mathrm{D}}\alpha^a_{\phantom{a}b}\;,\nonumber\\
\theta^a&\longmapsto&\theta^a-\alpha^a_{\phantom{a}b}\theta^b\;,\label{gt3}\end{aligned}$$ where was assumed. Thus, at action , the field $A$ is a gauge field with respect to the Lorentz group while $\theta$ has migrated to the matter sector (it suffers only group rotations under infinitesimal Lorentz transformations, *i.e.*, it is a vector representation). Now, the theory is ready to be identified with gravity.
Effective gravity {#grav}
-----------------
The broken theory described in Sect. \[cont\] can generate an effective geometry if [@Sobreiro:2011hb; @Sobreiro:2012book; @Assimos:2013zzz]: (i) every configuration $(A,\theta)$ defines an effective geometry $(\omega,e)$; (ii) there exists a mapping from each point $x\in\mathbb{R}^4$ to a point $X\in\mathbb{M}^4$ of the deformed space. In order to preserve the algebraic structure already defined in $\mathbb{R}^4$, it is demanded that this mapping is an isomorphism; (iii) The local gauge group $SO(m!-1,n)$ defines, at each point of the mapping, the isometries of the tangent space $T_X(\mathbb{M})$. Thus, $\theta$ and $A$ can be identified with the vierbein $e$ and spin connection $\omega$, respectively, through $$\begin{aligned}
\omega^{\mathfrak{ab}}_\mu(X)dX^\mu&=&\delta^{\mathfrak{a}}_a\delta^{\mathfrak{b}}_bA^{ab}_\mu(x)dx^\mu\;,\nonumber\\
e^{\mathfrak{a}}_\mu(X)dX^\mu&=&\delta_a^{\mathfrak{a}}\theta^a_\mu(x)dx^\mu\;.\label{id2}\end{aligned}$$ In expressions , latin indices $\{\mathfrak{a},\mathfrak{b}\ldots\}$ belong to the tangent space $T_X(\mathbb{M})$. Moreover, it is always possible to impose that the space of all $p$-forms in $\mathbb{R}^4$ is mapped into the space of $p$-forms in $\mathbb{M}^4$, namely, $\Pi^p\longmapsto\widetilde{\Pi}^p$, and the same for the Hodge spaces, $\ast\Pi^p\longmapsto\star\widetilde{\Pi}^p$, where $\star$ is the Hodge dual in $\mathbb{M}^4$. This mapping, together with the identifications in Eq. , provides $$S=\frac{1}{8\pi G}\int\left[\frac{1}{2\Lambda^2}R^\mathfrak{a}_{\phantom{a}\mathfrak{b}}\star R_\mathfrak{a}^{\phantom{a}\mathfrak{b}}+T^\mathfrak{a}\star T_\mathfrak{a}-\frac{\epsilon}{2}\epsilon_\mathfrak{abcd}R^\mathfrak{ab}e^\mathfrak{c}e^\mathfrak{d}+\frac{\Lambda^2}{4}\epsilon_\mathfrak{abcd}e^\mathfrak{a}e^\mathfrak{b}e^\mathfrak{c}e^\mathfrak{d}\right]\;,\label{ym3}$$ where $R^\mathfrak{a}_{\phantom{a}\mathfrak{b}}=\mathrm{d}\omega^\mathfrak{a}_{\phantom{a}\mathfrak{b}}+\omega^\mathfrak{a}_{\phantom{a}\mathfrak{c}} \omega^\mathfrak{c}_{\phantom{c}\mathfrak{b}}$ and $T^\mathfrak{a}=\mathrm{d}e^\mathfrak{a}-\omega^\mathfrak{a}_{\phantom{a}\mathfrak{b}}e^\mathfrak{b}$ are the curvature and torsion, respectively. Newton and cosmological constants are determined by $m^2=\kappa^2/2\pi G$ and $\Lambda^2=m^2/4$.
Action is a gravity action in the first order formalism presenting (in order of appearance) a quadratic Yang-Mills-type curvature term, a quadratic torsion term, the Einstein-Hilbert term, and the cosmological constant term. Moreover, it is easy to check that the vacuum solution is a de Sitter-type spacetime with $T^\mathfrak{a}=0$ and $R^\mathfrak{ab}=2\epsilon\Lambda^2 e^\mathfrak{a}e^\mathfrak{b}$.
It is remarkable that, in the present theory, Newton and cosmological constants are related through $\Lambda^2=\kappa^2/8\pi G$. Obviously, from asymptotic freedom, $\kappa^2$ is a big quantity. And, by assumption, $G$ must be small. Thus, $\Lambda$ should be very big. In fact, if this is true, we can make two important remarks: (i) The first term in can be safely neglected. Moreover, due to the absence of matter fields, torsion can be taken as very small. The resulting theory is then, the usual Einstein-Hilbert theory with cosmological constant; (ii) although astrophysical predictions [@Perivolaropoulos:2008pg; @Padmanabhan:2012gv] determine that $\Lambda^2_{obs}$ is very small, quantum field theory predicts [@Nelson:1982kt; @Toms:1983qr; @Buchbinder:1986gj; @Parker:1985kc] a very large $\Lambda^2_{qft}$. Thus, the contribution of a pure gravitational cosmological constant, which is big in our case, can drive the cosmological puzzle to a final consistent answer. In fact, following [@Sobreiro:2011hb; @gr-qc/0611055; @Shapiro:2009dh], the renormalized cosmological constant of our model could be determined through $\Lambda^2_{ren}=\Lambda^2_{obs}-\Lambda^2_{qft}$.
Some consistency checks
=======================
The explicit mapping {#map}
--------------------
The formal aspects of the mapping between a gauge theory in Euclidean spacetime and a gravity theory can be discussed in terms of fiber bundle theory. The details can be found in [@Sobreiro:2011hb; @Sobreiro:2012book; @Assimos:2013zzz]. In Sect. \[qcdgrav\], it was demanded that the mapping between the original space $\mathbb{R}^4$ and the effective space $\mathbb{M}^4$ is an isomorphism. In fact, it can be also shown that this mapping is unique and has an inverse, which ensures the absence of ambiguities and thus, its isomorphic character. The details of the proof can also be found in \[ap2\].
Let us summarize this result. We suppose that, in the $d$-dimensional original manifold, the metric tensor is $g_{\mu\nu}(x)$, where $x$ are the set of coordinates at a point in this space. The effective metric tensor is defined as $\widetilde{g}_{\mu\nu}(X)$, where $X$ is the set of coordinates at a point in the target space. Thus, the matrix that defines the map $x\longmapsto X$ for all points in the effective manifold is given by $$L_{\phantom{\nu}\mu}^\nu=\left(\frac{\tilde{g}}{g}\right)^{1/2d}\tilde{g}^{\nu\alpha}g_{\alpha\mu}\;.\label{eq13}$$ Its inverse is given by $$\left(L^{-1}\right)_{\phantom{\nu}\mu}^\nu=\left(\frac{g}{\tilde{g}}\right)^{1/2d}g^{\nu\alpha}\tilde{g}_{\alpha\mu}\;.\label{eq15}$$ In and , $g=\det{g_{\mu\nu}}$ and $\widetilde{g}=\det{\widetilde{g}_{\mu\nu}}$ are taken as non-vanishing quantities. Thus, ambiguities are absent in this mapping.
Weinberg-Witten theorems and emergent gravity
---------------------------------------------
In [@Weinberg:1980kq], Weinberg and Witten established two very powerful theorems. Essentially, they forbid: (i) massless charged states with helicity $j>1/2$ which have a conserved Lorentz-covariant current and (ii) massless states with helicity $j>1$ which have conserved Lorentz-covariant energy-momentum tensor.
At first sight, the first theorem would forbid gauge theories to exist since, at high energies, vector bosons and gluons are charged massless states which carry helicity $j=1$. However, the conserved gauge current associated with these states are not Lorentz covariant [@Weinberg:1980kq]. Similarly, the second theorem forbids spin-2 states with conserved Lorentz-covariant energy-momentum tensor to be defined. In traditional GR the metric tensor $g_{\mu\nu}$ is not a Lorentz covariant tensor since it is constructed over a Riemannian manifold. The linearized version of GR, on the other hand, is also a gauge theory and the same principles of the first case apply [@Weinberg:1980kq]. Moreover, the energy-momentum tensor of the graviton field identically vanishes.
The Weinberg-Witten theorem (ii) applies, however, to emergent gravity theories for which massless spin-2 states are generated as effective or composite states that could be associated with graviton excitations. Nevertheless, one may argue that the Weinberg-Witten theorem (ii) applies to the present mechanism. However, this is not the case here. First, the theory has a few mass parameters and the theorem holds for massless states only. Second, and more important, there are no spin-2 states in this model. The fields are identified with geometry and not with spin-2 composite fields. Gravity emerges as geometrodynamics and not as a field theory for spin-2 particles in flat space. In fact, the mapping discussed in Sects. \[grav\] and \[map\] has been employed at classical level. Nevertheless, renormalizability establishes that the quantum action has the same form of its classical version. The difference between the classical and quantum actions lies on the fields and parameters, which are their respective renormalized versions. Hence, although each configuration $(A,\theta)$ can define a geometry $(\omega,e)$, the mapping should not be applied to each configuration in the path integral, but at the quantum action itself. In this way, the resulting geometrodynamical theory is obtained from the full dynamical content of the original gauge theory. Moreover, the identification of the renormalized fields are made with respect to geometric quantities and not with spin-2 states. The conclusion is that there is no violation of the Weinberg-Witten theorem.
Nevertheless, after the emergence of gravity as geometrodynamics, linearization is allowed for weak gravitation and the spin-2 states might also be considered. In that case, these states are classical states associated to propagating fields. Quantization of these states can be done since it is not the fundamental theory, but an effective theory. Clearly, this does not violate the theorems because it fits in the same category of GR, [@Weinberg:1980kq].
Unitarity and equivalence principle {#unit}
-----------------------------------
One of the most important features of $SU(N)$ gauge theories is unitarity [@Itzykson:1980rh], a property that, among other features, follows from the compact character of the gauge group. In the case of the present theory, unitarity is only ensured for $m=0$. In that case, the resulting gravity theory is an $SO(4)$ local isometric gravity. The local Euclidean character of spacetime provides a kind of incomplete prediction of the equivalence principle because it lacks the split between space and time. On the other hand, if we set[^8] $m=2$ from the beginning, the resulting theory has $SO(1,3)$ local isometries, *i.e.*, localy, spacetime is a Minkowski space. Thus, there is a split between space and time originated from the breaking that lead to the mapping. This is nothing else than the rising of the equivalence principle.
Thus, if on the one hand, we start with a unitary gauge theory, the resulting gravity is not exactly the desired one because space and time are still indistinguishable. On the other hand, by giving up unitarity, the split between space and time correctly emerges. To solve the paradox is just a mathematical problem that can be fixed by imposing a Wick rotation during the mapping between $\mathbb{R}^4$ and $\mathbb{M}^4$. See [@Sobreiro:2011hb].
Another argument can be provided by simply saying that unitarity is irrelevant because quantum gravity is far beyond Planck scale. Above Planck scale gravity is already a classical phenomenon. Moreover, due to confinement of Higgless non-Abelian gauge theories (such as QCD and the present gravity model), unitarity might be an overestimated property, mainly because the high energy state is a plasma with no observable singlet whatsoever (in the case of QCD, this state is the quark-gluon plasma).
Final considerations {#end}
====================
We have discussed that is possible to make a deep analogy between quantum chromodynamics and gravity. This analogy is realized if gravity could be described, at quantum level, by a gauge theory in a Euclidean four-dimensional spacetime while, at classical level, it is deformed to a geometrodynamical theory [@Sobreiro:2011hb; @Sobreiro:2012book]. We have established the conditions for this mechanism to be realized and an example based on de Sitter-type groups was provided. In this mechanism, gravity arises as an emergent geometric theory. The requirements are asymptotic freedom, dynamical mass generation, and BRST soft breaking due to Gribov ambiguities.
The starting action , for the $SO(m,n)$ gauge groups, lives in a four-dimensional Euclidean space, and thus, at high energies, it is a well defined quantum theory of spin-1 asymptotic free states. As the energy decreases, mass generation takes place, as well as the Gribov parameter and soft BRST breaking. The consequence is that the propagators acquire complex poles and are ruled out from the physical spectrum of the theory. At this point, an Inönü-Wigner contraction is assumed and a symmetry breaking $SO(m,n)\rightarrow SO(m!-1,n)$ drives the theory to a gravity theory described by the action . This action defines a first order gravity which has Einstein-Hilbert and cosmological constant terms. To obtain this action, a mapping has to be imposed. The consistency of this mapping was shown and an improvement has been made in order to harmonize the concepts of unitarity and the emergence of the equivalence principle.
A remarkable feature of this mechanism is that Newton and cosmological constant could be explicitly computed from the usual QFT techniques. Moreover, there is a constraint between these constants, namely, $\Lambda^2\propto\kappa^2/G$. Since $G$ is supposedly small and $\kappa$ is a big quantity at low energies, it is then expected that $\Lambda$ is very big. This property can be combined with the predictions of QFT for the cosmological constant in order to provide a value that agrees with the observed values. Also, from the fact that $\Lambda$ is big and the absence of matter in this work, it is possible to conclude that the action can be safely approximated to the Einstein-Hilbert action with cosmological constant.
Let us make a comparison between the present gravity theory and the standard model. Strong and electroweak interactions are described by gauge theories. At high energies, these theories are very similar (except for the gauge groups). At low energies, however, these theories tend to behave in very different ways. Electroweak interactions suffer spontaneous symmetry breaking through the Higgs mechanism, giving rise to perturbative electrodynamics and massive gauge bosons. On the other hand, quark-gluon confinement shows up in chromodynamics, and hadronization phenomena take place. Specifically, confinement and the gauge principle state that physical observables must be gauge invariant and colorless. These states are recognized as hadrons and glueballs. Now, if the present theory can describe gravity, then: (i) at high energies, gravity is a gauge theory which is very similar to the other fundamental interactions; (ii) at low energies, instead of hadrons and glueballs, the physical observables are identified with geometry, and spacetime itself is affected by this theory. Thus, geometry appears as the low energy manifestation of gravity, in the same way that hadronization and spontaneous symmetry breaking are the low energy manifestations of chromodynamics and electroweak interactions.
Finally, for the moment, we can only say that a standard four-dimensional renormalizable Yang-Mills theory can generate a gravity theory at low energy regime. Obviously, many computations and tests must be performed before we recognize this theory (or some variation) as *the* quantum gravity theory or only an academic exercise.
Acknowledgements {#acknowledgements .unnumbered}
================
R. F. S. is grateful to the organizing committee of *NEB 15 - Recent Developments in Gravity* for the opportunity to deliver this talk. Conselho Nacional de Desenvolvimento Científico e Tecnológico[^9] (CNPq-Brazil) and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) are acknowledged for financial support.
Stereographic projection {#ap1}
========================
The group $SO(m,n)$ defines a flat space, $\mathbb{R}_S^{m,n}$, with metric given by $\eta^{AB}\equiv\mathrm{diag}(\epsilon,\varepsilon,1,1,1)$. In the presence of a mass scale $m$, we can define the radius of this space by $$\mathcal{R} = \frac{2\epsilon}{\left(\epsilon\varepsilon\right)^{\frac{1}{2}}} \left(\frac{\kappa}{m}\right)\;,\label{eq:radius}$$ in such a way that $$\eta_{AB} \xi^{A}\xi^{B} = \varepsilon \mathcal{R}^{2} =\eta_{ab}\xi^{a}\xi^{b} + \epsilon(\xi^{5})^{2}\;,\label{aB:etaAB}$$ where $\xi^A$ are Cartesian coordinates in $\mathbb{R}_S^{m,n}$. Thus, the stereographic projection is defined by $$\begin{aligned}
\label{aB:proj}
\xi^{a} &=& n \overline{x}^{a}, \nonumber\\
\xi^5 &=& \left(\epsilon\varepsilon\right)^{\frac{1}{2}}\mathcal{R}\left(1-2n\right)\;,\end{aligned}$$ where $\eta_{ab} = diag\ (\varepsilon,1,1,1)$. In order to have a projection conformally flat, *i.e.*, $g_{ab} = n^2 \eta_{ab} $, it is demanded that $$n=\frac{1}{\left(1+\frac{\varepsilon\sigma^2}{4\mathcal{R}^2} \right)}\;,$$ where $\sigma^{2} = \eta_{ab}\overline{x}^a\overline{x}^b$.
The effect of the projection on the group generators can also be studied. The group generators can be written as $$\label{aB:ger}
J^A_{\phantom{A}B} = \frac{1}{2}\left(\eta^{AC}\xi_{C}\partial_{B} - \eta^{BC}\xi_{C}\partial_{A}\right)\;.$$ Thus, it is a straightforward computation to show that $$\begin{aligned}
J^a_{\phantom{a}b} &=& \frac{1}{2} \left( \overline{x}^a P^b - \overline{x}^b P^a \right)\;,\nonumber\\
J^a &=&- \frac{\kappa}{m} P^{a} + \frac{\epsilon}{16}\frac{m}{\kappa} \left(2 \overline{x}_a \overline{x}^b P_b +\sigma^2 P^a \right)\;,\label{aB:Jabproj}\end{aligned}$$ where and were used. Thus, from the second of relations , we have $$\theta\longmapsto\kappa^{-1}m\theta=-\theta^aP_a+\frac{\epsilon}{16}\frac{m^2}{\kappa^2}\theta^a\left(2\overline{x}_a\overline{x}_bP^b + \sigma^2 P_a\right). \label{eq:thetaJa}$$
Explicit derivation of the mapping {#ap2}
==================================
Let us only show how the explicit mapping between the spaces $\mathbb{R}^4$ and $\mathbb{M}^4$ can be obtained. We have considered that $$\begin{aligned}
\Pi^p&\longmapsto&\tilde{\Pi}^p\;,\nonumber\\
*\Pi^p&\longmapsto&\star\tilde{\Pi}^p\;.\label{mapfull}\end{aligned}$$ For generality purposes, we assume a generic original metric $g_{\mu\nu}$ which, eventually, we can set as a Euclidean metric. The effective metric is denoted by $\tilde{g}_{\mu\nu}$. Moreover, we can also consider manifolds with an arbitrary dimension $d$. Obviously, a necessary extra condition is that both $g=|\det{g_{\mu\nu}}|$ and $\tilde{g}=|\det{\tilde{g}_{\mu\nu}}|$ are non-vanishing quantities. To find the explicit mapping, we apply the first of to a generic $p$-form, $$f_{\mu_1\ldots\mu_p}(x)dx^{\mu_1}\ldots dx^{\mu_p}=\tilde{f}_{\mu_1\ldots\mu_p}(X)dX^{\mu_1}\ldots dX^{\mu_p}\;,\label{eq1}$$ where $x\in\mathbb{R}^d$ and $X\in\mathbb{M}^d$. From , one easily obtain $$f_{\mu_1\ldots\mu_p}(x)=L_{\phantom{\nu_1}\mu_1}^{\nu_1}\ldots L_{\phantom{\nu_p}\mu_p}^{\nu_p}\tilde{f}_{\nu_1\ldots\nu_p}(X)\;,\label{eq3}$$ where $L_{\phantom{\nu}\mu}^\nu=\frac{\partial X^\nu}{\partial x^\mu}$. For the corresponding Hodge dual we have, $$\begin{aligned}
& &\sqrt{g}\epsilon_{\mu_1\ldots\mu_p\nu_{p+1}\ldots\nu_d}f^{\mu_1\ldots\mu_p}(x)dx^{\nu_{p+1}}\ldots dx^{\nu_d}=\nonumber\\
&=&\sqrt{\tilde{g}}\epsilon_{\mu_1\ldots\mu_p\nu_{p+1}\ldots\nu_d}\tilde{f}^{\mu_1\ldots\mu_p}(X)dX^{\nu_{p+1}}\ldots dX^{\nu_d}\;,\label{eq4}\end{aligned}$$ from which one can find that $$f^{\mu_1\ldots\mu_p}=\left({\frac{\tilde{g}}{g}}\right)^{1/2}\left(\frac{L}{d}\right)^{d-p}\tilde{f}^{\mu_1\ldots\mu_p}\;,\label{eq8}$$ with $L=L^\mu_{\phantom{\mu}\mu}$. A comparison of and leads to $$f_{\mu_1\ldots\mu_p}=\left({\frac{\tilde{g}}{g}}\right)^{1/2}\left(\frac{L}{d}\right)^{d-p}\tilde{g}^{\nu_1\alpha_1}g_{\alpha_1\mu_1}\ldots\tilde{g}^{\nu_p\alpha_p}g_{\alpha_p\mu_p}\tilde{f}_{\nu_1\ldots\nu_p}\;.\label{eq10}$$ Combining and , we achieve $$L_{\phantom{\nu}\mu}^\nu=\left({\frac{\tilde{g}}{g}}\right)^{1/2p}\left(\frac{L}{d}\right)^{(d-p)/p}\tilde{g}^{\nu\alpha}g_{\alpha\mu}\;,\label{eq11}$$ which is not valid for $p=0$. In that case it is easy to find that is valid only if $$\left({\frac{\tilde{g}}{g}}\right)^{1/2}\left(\frac{L}{d}\right)^d=1\;.\label{0f}$$ The constraint implies that $$L_{\phantom{\nu}\mu}^\nu=\frac{d}{L}\;\tilde{g}^{\nu\alpha}g_{\alpha\mu}\;.\label{eq11a}$$ The trace $L$ can now be calculated from or providing $$\begin{aligned}
L&=&d\left({\frac{g}{\tilde{g}}}\right)^{1/2d}\;,\nonumber\\
L&=&d^{1/2}(\tilde{g}^{\mu\nu}g_{\mu\nu})^{1/2}\;,\label{trace}\end{aligned}$$ respectively. The relations enforce the extra constraint $$(\tilde{g}^{\mu\nu}g_{\mu\nu})^{1/2}=d^{1/2}\left({\frac{g}{\tilde{g}}}\right)^{1/2d}\;.\label{constx}$$ As a consequence, we obtain the final expression for the transformation matrix, which is given by Eq. . We recall that the effective metric is computed from the gravity field equations, while the original metric is a given quantity[^10]. It turns out that the mapping has an inverse, given by Eq. . The existence of the inverse ensures the non-degeneracy of the mapping.
References {#references .unnumbered}
==========
[90]{}
Einstein A 1916 The foundation of the general theory of relativity [*Annalen Phys.*]{} [**49**]{} 769 Itzykson C and Zuber J B 1980 [*Quantum Field Theory*]{} (New York: Mcgraw-Hill)
Hojman S A, Kuchar K and Teitelboim C 1976 Geometrodynamics Regained [*Annals Phys.*]{} [**96**]{} 88 Streater R F and Wightman A F 2000 [*PCT, spin and statistics, and all that*]{} (Princeton: Princeton University Press)
Fierz M and Pauli W 1939 On relativistic wave equations for particles of arbitrary spin in an electromagnetic field [*Proc. Roy. Soc. Lond.*]{} A [**173**]{} 211 ’t Hooft G and Veltman M J G 1974 One loop divergencies in the theory of gravitation [*Annales Poincaré Phys. Theor.*]{} [**A20**]{} 69-94
Deser S and van Nieuwenhuizen P 1974 One loop divergences of quantized Einstein-Maxwell fields [*Phys. Rev.*]{} D [**10**]{} 401
Deser S and van Nieuwenhuizen P 1974 Nonrenormalizability of the quantized Dirac-Einstein system [*Phys. Rev.*]{} D [**10**]{} 411
Stelle K S 1977 Renormalization of higher derivative quantum gravity [*Phys. Rev.*]{} D [**16**]{} 953 Utiyama R 1956 Invariant theoretical interpretation of interaction [*Phys. Rev.*]{} [**101**]{} 1597 Kibble T W B 1961 Lorentz invariance and the gravitational field [*J. Math. Phys.*]{} [**2**]{} 212 Sciama D W 1964 The physical structure of general relativity [*Rev. Mod. Phys.*]{} [**36**]{} 463-469
Mardones A and Zanelli J 1991 Lovelock-Cartan theory of gravity [*Class. Quant. Grav.*]{} [**8**]{} 1545 Zanelli J 2005 Lecture notes on Chern-Simons (super-)gravities [*Preprint*]{} hep-th/0502193
Sobreiro R F and Vasquez Otoya V J 2011 On the topological reduction from the affine to the orthogonal gauge theory of gravity [*J. Geom. Phys.*]{} [**61**]{} 137-150
Sobreiro R F and Vasquez Otoya V J 2011 Affine gauge theory of gravity and its reduction to the Riemann-Cartan geometry [*J. Phys.: Conf. Ser.*]{} [**283**]{} 012032
Stelle K S and West P C 1979 de Sitter gauge invariance and the geometry of the Einstein-Cartan theory [*J. Phys.*]{} A [**12**]{} L205-L210
Stelle K S and West P C 1980 Spontaneously broken de Sitter symmetry and the gravitational holonomy group [*Phys. Rev.*]{} D [**21**]{} 1466
Pagels H R 1984 Gravitational gauge fields and the cosmological constant [*Phys. Rev.*]{} D [**29**]{} 1690
Tresguerres R 2008 Dynamically broken Anti-de Sitter action for gravity [*Int. J. Geom. Meth. Mod. Phys.*]{} [**5**]{} 171-183
Mielke E W 2010 Einsteinian gravity from a spontaneously broken topological BF theory [*Phys. Lett.*]{} B [**688**]{} 273-277
Higgs P W 1964 Broken symmetries and the masses of gauge bosons [*Phys. Rev. Lett.*]{} [**13**]{} 508 Higgs P W 1964 Broken symmetries, massless particles and gauge fields [*Phys. Lett.*]{} [**12**]{} 132 MacDowell S W and Mansouri F 1977 Unified geometric theory of gravity and supergravity [*Phys. Rev. Lett.*]{} [**38**]{} 739
Yang C N and Mills R L 1954 Conservation of isotopic spin and isotopic gauge invariance [*Phys. Rev.*]{} [**96**]{} 191 Tseytlin A A 1982 On the Poincaré and de Sitter gauge theories of gravity with propagating torsion [*Phys. Rev.*]{} D [**26**]{} 3327
Mahato P 2004 de Sitter group and Einstein-Hilbert Lagrangian [*Phys. Rev.*]{} D [**70**]{} 124024 ([*Preprint*]{} gr-qc/0603100) Sobreiro R F and Vasquez Otoya V J 2007 Effective gravity from a quantum gauge theory in Euclidean space-time [*Class. Quant. Grav.*]{} [**24**]{} 4937
Sobreiro R F, Tomaz A A and Vasquez Otoya V J 2012 de Sitter gauge theories and induced gravities [*Eur. Phys. J.*]{} C [**72**]{} 1991 ([*Preprint*]{} hep-th/1109.0016) Gross D J and Wilczek F 1973 Ultraviolet behavior of non-Abelian gauge theories [*Phys. Rev. Lett.*]{} [**30**]{} 1343 Politzer H D 1973 Reliable perturbative results for strong interactions? [*Phys. Rev. Lett.*]{} [**30**]{} 1346 Inönü E and Wigner E P 1953 On the contraction of groups and their representations [*Proc. Nat. Acad. Sci.*]{} [**39**]{} 510-524
Gribov V N 1978 Quantization of nonabelian gauge theories [*Nucl. Phys.*]{} B [**139**]{} 1
Zwanziger D 1993 Renormalizability of the critical limit of lattice gauge theory by BRS invariance [*Nucl. Phys.*]{} B [**399**]{} 477-513
Sobreiro R F and Sorella S P 2005 Introduction to the Gribov ambiguities in Euclidean Yang-Mills theories [*Preprint*]{} hep-th/0504095 Baulieu L and Sorella S P 2009 Soft breaking of BRST invariance for introducing non-perturbative infrared effects in a local and renormalizable way [*Phys. Lett.*]{} B [**671**]{} 481 ([*Preprint*]{} hep-th/0808.1356) Baulieu L, Capri M A L, Gomez A J, Lemes V E R, Sobreiro R F and Sorella S P 2010 Renormalizability of a quark-gluon model with soft BRST breaking in the infrared region [*Eur. Phys. J.*]{} C [**66**]{} 451 ([*Preprint*]{} hep-th/0901.3158) Dudal D and Sorella S P 2012 The Gribov horizon and spontaneous BRST symmetry breaking [*Phys. Rev.*]{} D [**86**]{} 045005 ([*Preprint*]{} hep-th/1205.3934) Dudal D, Sobreiro R F, Sorella S P and Verschelde H 2005 The Gribov parameter and the dimension two gluon condensate in Euclidean Yang-Mills theories in the Landau gauge [*Phys. Rev.*]{} D [**72**]{} 014016
Dudal D, Sorella S P and Vandersickel N 2011 The dynamical origin of the refinement of the Gribov-Zwanziger theory [*Phys. Rev.*]{} D [**84**]{} 065039
’t Hooft G and Veltman M J G 1972 Regularization and renormalization of gauge fields [*Nucl. Phys.*]{} B [**44**]{} 189 Piguet O and Sorella S P 1995 Algebraic renormalization: Perturbative renormalization, symmetries and anomalies [*Springer Lect. Notes Phys.*]{} M [**28**]{} Faddeev L D and Popov V N 1967 Feynman diagrams for the Yang-Mills field [*Phys. Lett.*]{} B [**25**]{} 29 Tyutin I V 1975 Gauge invariance in field theory and statistical physics in operator formalism [*Lebedev Physics Institute: Preprint*]{} 39 ([*Preprint*]{} hep-th/0812.0580) Becchi C, Rouet A and Stora R 1976 Renormalization of gauge theories [*Annals Phys.*]{} [**98**]{} 287 Tomonaga S 1946 On a relativistically invariant formulation of the quantum theory of wave fields [*Prog. Theor. Phys.*]{} [**1**]{} 27 Schwinger J S 1948 Quantum electrodynamics I A covariant formulation [*Phys. Rev.*]{} [**74**]{} 1439 Feynman R P 1950 Mathematical formulation of the quantum theory of electromagnetic interaction [*Phys. Rev.*]{} [**80**]{} 440 Kugo T and Ojima I 1979 Local covariant operator formalism of nonabelian gauge theories and quark confinement problem [*Prog. Theor. Phys. Suppl.*]{} [**66**]{} 1 Singer I M 1978 Some Remarks on the Gribov Ambiguity [*Commun. Math. Phys.*]{} [**60**]{} 7 ed Binosi D, Aguilar A C, [*et al*]{} 2011 QCD Green’s functions, confinement and phenomenology [*Proc. Int. Workshop QCD-TNT-II*]{} (Italy :PoS QCD-TNT-II)
Greensite J 2003 The confinement problem in lattice gauge theory [*Prog. Part. Nucl. Phys.*]{} [**51**]{} 1 ([*Preprint*]{} hep-lat/0301023) Cucchieri A, Dudal D, Mendes T and Vandersickel N 2012 Modeling the gluon propagator in Landau gauge: Lattice estimates of pole masses and dimension-two condensates [*Phys. Rev.*]{} D [**85**]{} 094513 ([*Preprint*]{} hep-lat/1111.2327) Maldacena J M 1999 The Large N limit of superconformal field theories and supergravity [*AIP Conf. Proc.*]{} [**484**]{} 51 [*Int. J. Theor. Phys.*]{} [**38**]{} 1113 Witten E 1988 Anti-de Sitter space and holography [*Adv. Theor. Math. Phys.*]{} [**2**]{} 253 ([*Preprint*]{} hep-th/9802150) Miranda A S, Ballon Bayona C A, Boschi-Filho H and Braga N R F 2010 Glueballs at finite temperature from AdS/QCD [*Nucl. Phys. Proc. Suppl.*]{} [**199**]{} 107 ([*Preprint*]{} hep-th/0910.4319) Nambu Y 1974 Strings, monopoles and gauge fields [*Phys. Rev.*]{} D [**10**]{} 4262 ’t Hooft G 1975 [*High Energy Physics EPS Int. Conf.*]{} ed. A. Zichichi
Mandelstam S 1976 Vortices and quark confinement in nonabelian gauge theories [*Phys. Rept.*]{} [**23**]{} 245 Faddeev L D and Niemi A J 2002 Aspects of electric magnetic duality in SU(2) Yang-Mills theory [*Phys. Lett.*]{} B [**525**]{} 195 ([*Preprint*]{} hep-th/0101078) Dudal D, Guimaraes M S and Sorella S P 2011 Glueball masses from an infrared moment problem and nonperturbative Landau gauge [*Phys. Rev. Lett.*]{} [**106**]{} 062003 ([*Preprint*]{} hep-th/1010.3638) Capri M A L, Dudal D, Guimaraes M S, Palhares L F and Sorella S P 2012 Physical spectrum from confined excitations in a Yang-Mills-inspired toy model [*Preprint*]{} hep-th/1208.5676 Sobreiro R F 2012 Fiber bundles, gauge theories and gravity [*Quantum Gravity*]{} ed Sobreiro R F (Croatia: InTech)
Daniel M and Viallet C M 1980 The geometrical setting of gauge theories of the Yang-Mills type [*Rev. Mod. Phys.*]{} [**52**]{} 175 Nakahara M 1990 [*Geometry, topology and physics*]{} (Bristol, UK: Hilger)
Witten E 1988 (2+1)-dimensional gravity as an exactly soluble system [*Nucl. Phys.*]{} B [**311**]{} 46 Assimos T S, Pereira Jr A D, Santos T R S, Sobreiro R F and Tomaz A A To appear 2013
Assimos T S, Pereira Jr A D, Santos T R S, Sobreiro R F, Tomaz A A and Vasquez Otoya VJ To appear 2013
Perivolaropoulos L 2008 Vacuum energy, the cosmological constant, and compact extra dimensions: constraints from Casimir effect experiments [*Phys. Rev.*]{} D [**77**]{} 107301 ([*Preprint*]{} astro-ph/0802.1531) Padmanabhan T 2012 The physical principle that determines the value of the cosmological constant [*Preprint*]{} hep-th/1210.4174 Nelson B L and Panangaden P 1982 Scaling behavior of interacting quantum fields in curved space-time [*Phys. Rev.*]{} D [**25**]{} 1019 Toms D J 1983 The effective action and the renormalization group equation in curved space-time [*Phys. Lett.*]{} B [**126**]{} 37 Buchbinder I L 1984 Renormalization group equations in curved space-time [*Theor. Math. Phys.*]{} [**61**]{} 1215 ([*\[Teor. Mat. Fiz.*]{} [**61**]{} 393) Parker L and Toms D J 1985 Renormalization group and nonlocal terms in the curved space-time effective action: Weak field results [*Phys. Rev.*]{} D [**32**]{} 1409 Shapiro I L and Sola J 2007 Cosmological constant problems and renormalization group [*J. Phys.*]{} A [**40**]{} 6583 Shapiro I L and Sola J 2009 On the possible running of the cosmological “constant” [*Phys. Lett.*]{} B [**682**]{} 105-113
Weinberg S and Witten E 1980 Limits on massless particles [*Phys. Lett.*]{} B [**96**]{} 59
[^1]: The indices conventions are: $\{\mathfrak{a},\mathfrak{b},\mathfrak{c},\ldots\}\in\{0,1,2,3\}$ are frame indices; and $\{\mathrm{greek}\}\in\{0,1,2,3\}$ are world indices.
[^2]: A soft breaking is breaking whose field operators have dimension lower than the spacetime dimension.
[^3]: In Yang-Mills theories, the four-dimensional case is the only case where the coupling parameter is dimensionless.
[^4]: Larger groups are also allowed. The extra decompositions will generate a matter sector in the resulting gravity theory, [@Assimos:2012zzz].
[^5]: Except when necessary, we will indistinguishably call the generic $SO(m,n)$ group, with arbitrary $m$, by de Sitter group.
[^6]: The transformations are not accidental. Both sectors are rescaled with $\kappa^{-1}$ in order to factor out the coupling parameter outside the action, a standard procedure in Yang-Mills theories [@Itzykson:1980rh]. On the other hand, the mass parameter affects only the $\theta$-sector, transforming it in a field with dimensionless components. It turns out that this is the unique possibility if one wishes to identify $\theta$ with a vierbein field. If also $A$ is rescaled with a mass factor, then it would never be possible to identify it with the spin connection.
[^7]: We remark that, due to the hierarchy of fundamental interactions, for gravity, a low energy scale would be very high when compared to all other basic interactions. The rate $m^2/\kappa^2$ may, become small again at lower scales.
[^8]: The case $m=1$ provides a non-unitary theory and also a local Euclidean spacetime with no difference between space and time.
[^9]: RFS is a level PQ-2 researcher under the program *Produtividade em Pesquisa*, 304924/2009-1.
[^10]: For the present case we actually have ($d=4$ and $g_{\mu\nu}=\delta_{\mu\nu}$) $
L_{\phantom{\nu}\mu}^\nu=\left(\tilde{g}\right)^{1/8}\tilde{g}^{\nu\alpha}\delta_{\alpha\mu}$.
|
---
abstract: 'Theories with a curved momentum space, which became recently of interest in the quantum-gravity literature, can in general violate many apparently robust aspects of our current description of the laws of physics, including relativistic invariance, locality, causality and global momentum conservation. We here explore some aspects of the particularly severe pathologies arising in generic theories with curved momentum space for what concerns causality and momentum conservation. However, we also report results suggesting that when momentum space is maximally symmetric, and the theory is formulated (DSR-)relativistically, with the associated relativity of spacetime locality, momentum is globally conserved and there is no violation of causality.'
author:
- '[**Giovanni AMELINO-CAMELIA**]{}'
- '[**Stefano BIANCO**]{}'
- '[**Francesco BRIGHENTI**]{}'
- '[**Riccardo Junior BUONOCORE**]{}'
title: Causality and momentum conservation from relative locality
---
Introduction
============
Over the last decade several independent arguments suggested that the Planck scale might characterize a non-trivial geometry of momentum space (see, [*e.g.*]{}, Refs. [@majidCURVATURE; @dsr1; @dsr2; @jurekDSMOMENTUM; @girelliCURVATURE; @schullerCURVATURE; @changMINIC; @principle]). Among the reasons of interest in this possibility we should mention approaches to the study of the quantum-gravity problem based on spacetime noncommutativity, particularly when considering models with “Lie-algebra spacetime noncommutativity", $[x_\mu , x_\nu]= i \zeta^\sigma_{\mu \nu} x_\sigma$, where the momentum space on which spacetime coordinates generate translations is evidently curved (see, [*e.g.*]{}, Ref. [@gacmaj]). Also in the Loop Quantum Gravity approach [@rovelliLRR] one can adopt a perspective suggesting momentum-space curvature (see, [*e.g.*]{}, Ref. [@leeCURVEDMOMENTUM]). And one should take notice of the fact that the only quantum gravity we actually know how to solve, quantum gravity in the 2+1-dimensional case, definitely does predict a curved momentum space (see, [*e.g.*]{}, Refs. [@matschull; @BaisMullerSchroers; @dsr3FREIDLIVINE; @bernschr2012; @stefanoCQG2013]).
In light of these findings it is then important to understand what are the implications of curvature of momentum space. Of course the most promising avenue is the one of accommodating this new structure while preserving to the largest extent possible the structure of our current theories. And some progress along this direction has already been made in works adopting the “relative-locality curved-momentum-space framework", which was recently proposed in Ref.[@principle]. Working within this framework it was in particular shown [@GiuliaFlavio; @GACarXiv11105081; @cortes] that some theories on curved momentum spaces can be formulated as relativistic theories. These are not special-relativistic theories, but they are relativistic within the scopes of the proposal of “DSR-relativistic theories" [@dsr1; @dsr2] (also see Refs. [@jurekDSRfirst; @leejoaoPRDdsr; @leejoaoCQGrainbow; @jurekDSRreview; @gacSYMMETRYreview]), theories with two relativistic invariants, the speed-of-light scale $c$ and a length/inverse-momentum scale $\ell$: the scale that characterizes the geometry of momentum space must in fact be an invariant if the theories on such momentum spaces are to be relativistic.
For what concerns locality some works based on Ref.[@principle] have established that, while for generic theories on curved momentum spaces locality is simply lost, in some appropriate cases the curvature of momentum space is compatible with only a relatively mild weakening of locality. This is the notion of relative spacetime locality, such that [@whataboutbob] events observed as coincident by nearby observers may be described as non-coincident by some distant observers. In presence of relative spacetime locality one can still enforce as a postulate that physical processes are local, but needing the additional specification that they be local for nearby observers.
The emerging assumption is that research in this area should give priority to theories on curved momentum space which are (DSR-)relativistic and have relative locality. Of course, it is important to establish whether these two specifications are sufficient for obtaining acceptable theories. Here acceptable evidently means theories whose departures from current laws are either absent or small enough to be compatible with the experimental accuracy with which such laws have been so far confirmed experimentally. In this respect some noteworthy potential challenges have been exposed in the recent studies in Ref.[@linq] and in Ref.[@andrb]. Ref.[@linq] observed that, in general, theories on curved momentum space do not preserve causality, whereas Ref.[@andrb] observed that, in general, theories on curved momentum space, even when one enforces momentum conservation at interactions, may end up loosing global momentum conservation.
The study we here report intends to contribute to the understanding of theories formulated in the relative-locality curved-momentum-space framework proposed in Ref.[@principle]. Like Refs.[@linq; @andrb] we keep our analysis explicit by focusing on the case of the so-called $\kappa$-momentum space, which is known to be compatible with a (DSR-)relativistic formulation of theories. Our main focus then is on establishing whether enforcing relative locality is sufficient for addressing the concerns for causality reported in Ref.[@linq] and the concerns for momentum conservation reported in Ref.[@andrb]. This is indeed what we find: enforcing relative locality for theories on the $\kappa$-momentum space is sufficient for excluding the causality-violating processes of Ref.[@linq] and the processes violating global momentum conservation of Ref.[@andrb]. And we find further motivation for adopting a DSR-relativistic setup, with relative locality, by showing that instead for a generic curved momentum space (non-relativistic, without relative locality) the violations of causality are even more severe than previously established.
A key role in our analysis is played by translation transformations in relativistic theories with a curved momentum space. As established in previous works [@anatomy; @cortes] the relevant laws of translation transformations are in some sense less rigid than in the standard flat-momentum-space case, but still must ensure that all interactions are local as described by nearby observers. It is of course only through such translation transformations that one can enforce relative spacetime locality for chains of events such as those considered in Refs.[@linq; @andrb]. In presence of a chain of events any given observer is at most “near" one of the events (meaning that the event occurs in the origin of the observer’s reference frame) and, because of relative locality, that observer is then not in position to establish whether or not other events in the chain are local. Enforcing the principle of relative locality [@principle] then requires the use of translation transformations connecting at least as many observers as there are distant events in the chain: this is the only way for enforcing the spacetime locality of each event in the chain, in the sense of the principle of relative locality.
The main issues and structures we are here concerned with are already fully active and relevant in the case of $1+1$ spacetime dimensions and at [leading order]{} in the scale $\ell$ of curvature of momentum space. We shall therefore mainly focus on the 1+1-dimensional case and on [leading-in-$\ell$-order]{} results, so that our derivations can be streamlined a bit and the conceptual aspects are more easily discussed.
Preliminaries on classical particle theories on the $\kappa$-momentum space {#digitalsec}
===========================================================================
As announced our analysis adopts the relative-locality curved-momentum-space framework proposed in Ref.[@principle], and for definiteness focuses on the $\kappa$-momentum space. This $\kappa$-momentum space is based on a form of on-shellness and a form of the law of composition of momenta inspired by the $k$-Poincaré Hopf algebra [@majrue; @lukieANNALS], which had already been of interest from the quantum-gravity perspective for independent reasons [@gacmaj; @dsr3FREIDLIVINE; @leeCURVEDMOMENTUM]. The main characteristics of this momentum space are that, at leading order in the deformation scale $\ell$, the on-shellness of a particle of momentum $p$ and mass $m_p$ is $$\label{geomassJ}
\mathcal C_p \equiv p_0^2-p_1^2 -\ell p_0 p_1^2 = m^2_p \ ,$$ while the composition of two momenta $p$, $q$ is $$\begin{split}
(p \oplus q)_0 &= p_0 + q_0 \ ,\\
(p \oplus q)_1 &= p_1 + q_1 -\ell p_0 q_1 \ .
\end{split}
\label{jjj}$$ Useful for several steps of the sort of analyses we are here interested in is the introduction of the “antipode" of the composition law, denoted by $\ominus$, such that $(q \oplus (\ominus q))_\mu =0 = ((\ominus q) \oplus q)_\mu$. For the $\kappa$-momentum case one has that $$(\ominus q)_0 = - q_0 ~,~~~(\ominus q)_1 = - q_1 - \ell q_0 q_1$$
We shall not review here the line of analysis which describes these rules of kinematics as the result of adopting on momentum space the de Sitter metric and a specific torsionful affine connection. These points are discussed in detail in Refs. [@GiuliaFlavio; @anatomy].
In light of our objectives it is useful for us to briefly summarize here the description of events within the relative-locality curved-momentum-space framework. More detailed and general discussions of this aspect can be found in Refs. [@principle; @anatomy]. Here we shall be satisfied with briefly describing the illustrative case of the event in Fig.1, for which we might think for example of the event of absorption of a photon by an atom. The case of interest in the recent literature on the relative-locality framework is the one of events of this sort analyzed within classical mechanics (so, in particular, the diagram shown here in Fig.1 should not be interpreted in the sense of quantum theory’s Feynman diagrams, bur rather merely as a schematic description of a classical-physics event).
![[We here show schematically a 3-valent event marked by a $\mathcal{K}^{(0)}$ that symbolizes a boundary term conventionally located at value $s_0$ of the affine parameter $s$. The boundary term enforces (deformed) momentum conservation at the event.]{}[]{data-label="vertex"}](vertice.pdf)
The formalism introduced in Ref. [@principle] allows the description of such an event in terms of the law of on-shellness, which for the $\kappa$-momentum space is (\[geomassJ\]), and the law of composition of momenta, which for the $\kappa$-momentum space is (\[jjj\]). This is done by introducing the action [@principle]
$$\label{actionj}
\begin{array}{lll}
\mathcal{S}
&=&
\int_{-\infty}^{s_{0}}ds\left(z^{\mu}\dot{k}_{\mu}+\mathcal{N}_k [\mathcal{C}_{k} - m_k^2]\right)+
\int_{-\infty}^{s_{0}}ds\left({x}^{\mu}\dot{p}_{\mu}+\mathcal{N}_p[\mathcal{C}_{p} - m_p^2]\right)
+\int_{s_{0}}^{+\infty}ds\left(y^\mu\dot q_{\mu}+\mathcal{N}_{q}[\mathcal{C}_{q} - m_q^2]\right)
-\xi_{(0)}^{\mu}\mathcal{K}_{\ \mu}^{(0)}
\, .
\end{array}$$
Here the Lagrange multipliers $\mathcal{N}_k$,$\mathcal{N}_p$,$\mathcal{N}_q$ enforce in standard way the on-shellness of particles. The most innovative part of the formalization introduced in Ref. [@principle] is the presence of boundary terms at endpoints of worldlines, enforcing momentum conservation. In the case of (\[actionj\]), describing the single interaction in Fig.1, there is only one such boundary term, and the momentum-conservation-enforcing $\mathcal{K}_{\ \mu}^{(0)}$ takes the form[^1] $$\begin{split}
{\cal K}^{(0)}_{\ \mu}&= (k\oplus p)_\mu-q_\mu \ .
\label{constraints}
\end{split}$$
Relative spacetime locality is an inevitable feature of descriptions of events governed by curvature of momentum space of the type illustrated by our example (\[actionj\]). To see this we vary the action (\[actionj\]) keeping the momenta fixed at $s = \pm \infty$, as prescribed in Ref. [@principle], and we find the equations of motion $$\begin{gathered}
\dot k_\mu =0~,~~\dot p_\mu =0~,~~\dot q_\mu =0~,~~\\
\mathcal{C}_k=m_k^2~,~~\mathcal{C}_p=m_p^2~,~~\mathcal{C}_q=m_q^2~,~~\\
\mathcal{K}_\mu^{(0)}=0,\\
\dot z^\mu = \mathcal{N}_k \frac{\partial \mathcal{C}_k}{\partial k_\mu}~,~~~
\dot x^\mu = \mathcal{N}_p \frac{\partial \mathcal{C}_p}{\partial p_\mu}~,~~~
\dot y^\mu = \mathcal{N}_q \frac{\partial \mathcal{C}_q}{\partial q_\mu} \, ,
\label{jjjbbb}\end{gathered}$$ and the boundary conditions at the endpoints of the 3 semi-infinite worldlines $$z^\mu(s_{0}) = \xi^\nu_{(0)} \frac{\partial \mathcal{K}_\nu^{(0)}}{\partial k_\mu}~,~~~
x^\mu(s_{0}) = \xi^\nu_{(0)} \frac{\partial \mathcal{K}_\nu^{(0)}}{\partial p_\mu}~,~~~
y^\mu(s_{0}) =- \xi^\nu_{(0)} \frac{\partial \mathcal{K}_\nu^{(0)}}{\partial q_\mu} \ .
\label{jjjbbbRRR}$$
The relative locality is codified in the fact that for configurations such that $\xi_{(0)}^{\mu} \neq 0$ the boundary conditions (\[jjjbbbRRR\]) impose that the endpoints of the worldlines do not coincide, since in general $$\frac{\partial \mathcal{K}_\nu^{(0)}}{\partial k_\mu} \neq \frac{\partial \mathcal{K}_\nu^{(0)}}{\partial p_\mu} \neq -\frac{\partial \mathcal{K}_\nu^{(0)}}{\partial q_\mu} \ ,$$ so that in the coordinatization of the (in that case, distant) observer the interaction appears to be non-local. However, as shown in Fig.2, for observers such that the same configuration is described with $\xi_{(0)}^{\mu} =0$ the endpoints of the worldlines must coincide and be located in the origin of the observer ($x^\mu(s_{0}) = y^\mu(s_{0}) = z^\mu(s_{0})=0$).
And it is important to notice that taking as starting point of the analysis some observer Alice for whom $\xi_{(0)[A]}^\mu \neq 0$, [*i.e.*]{} an observer distant from the interaction who sees the interaction as non-local, one can obtain from Alice an observer Bob for whom $\xi_{(0)[B]}^\mu =0$ if the transformation from Alice to Bob for endpoints of coordinates has the form $$\begin{split}
z^\mu_{B}(s_{0}) &= z^\mu_{A}(s_{0})
- \xi_{A}^\nu \frac{\partial \mathcal{K}_\nu^{(0)}}{\partial k_\mu} \ ,
\\
x^\mu_{B}(s_{0}) &= x^\mu_{A}(s_{0})
- \xi_{A}^\nu \frac{\partial \mathcal{K}_\nu^{(0)}}{\partial p_\mu} \ ,\\
y^\mu_{B}(s_{0}) &= y^\mu_{A} (s_{0})
+ \xi_{A}^\nu \frac{\partial \mathcal{K}_\nu^{(0)}}{\partial q_\mu} \ .
\end{split}$$ Such a property for the endpoints is produced of course, for the choice $b^\nu = \xi_{A}^\nu$, by the corresponding prescription for the translation transformations: $$\begin{split}
x^\mu_{B}(s) &= x^\mu_{A}(s) -b^\nu \frac{\partial \mathcal{K}_\nu}{\partial p_\mu}= x^\mu_{A}(s)+b^\nu\{(k\oplus p)_\nu,x^\mu(s)\} \ ,\\
z^\mu_{B}(s) &= z^\mu_{A}(s) -b^\nu \frac{\partial \mathcal{K}_\nu}{\partial k_\mu}=z^\mu_{A}(s)+b^\nu\{(k\oplus p)_\nu,z^\mu(s)\}\ ,\\
y^\mu_{B}(s) &= y^\mu_{A}(s) +b^\nu \frac{\partial \mathcal{K}_\nu}{\partial q_\mu}=y^\mu_{A}(s)+b^\nu\{q_\nu,y^\mu(s)\} \ ,\\
\label{traslprl}
\xi_{B}^\mu &= \xi_{A}^\mu -b^\mu\ .
\end{split}$$ where it is understood that $\{x^\mu,p_\nu\}=\delta^\mu_\nu$, $\{z^\mu,k_\nu\}=\delta^\mu_\nu$, $\{y^\mu,q_\nu\}=\delta^\mu_\nu$. This also shows that in this framework one can enforce the “principle of relative locality" [@principle] that all interactions are local according to nearby observers (observers such that the interaction occurs in the origin of their reference frame).
Cause and effect, with relative locality {#digitalsec}
========================================
Technically our goal is to work within the framework briefly reviewed in the previous section (and described in more detail and generality in Refs.[@principle; @anatomy]), specifically assuming the laws (\[geomassJ\]) and (\[jjj\]) for the $\kappa$-momentum space, and show that the concerns for causality reported in Ref.[@linq] and the concerns for momentum conservation reported in Ref.[@andrb] do not apply once the principle of relative locality is enforced. We start with the causality issue and before considering specifically the concerns discussed in Ref.[@linq] we devote this section to an aside on the relationship between cause and effect in our framework. We just intend to show that relative locality, though weaker than ordinary absolute locality, is strong enough to ensure the objectivity of the causal link between a cause and its effect. An example of situation where this is not [*a priori*]{} obvious with relative locality is the one in Fig.\[process\], where we illustrate schematically two causal links: a pair of causally-connected events is shown in red and another pair of causally-connected events is shown in blue, but there is no causal connection (in spite of the coincidence of the events ${\cal K}^{(0)}$ and ${\cal K}^{(1)}$) between events where blue lines cross and events where red lines cross. An example of situation of the type shown in Fig.\[process\] is the one of two atoms getting both coincidently excited by photon absorption, then both propagating freely and ultimately both getting de-excited by emitting a photon each.
![[We here show schematically two causal links: a pair of causally-connected events is shown in red and another pair of causally-connected events is shown in blue, but there is no causal connection (in spite of the coincidence of the events ${\cal K}^{(0)}$ and ${\cal K}^{(1)}$) between events where blue lines cross and events where red lines cross. We analyze this situation with the simplifying assumption that some of the particles involved (those described by dashed lines) have energies small enough that the Planck-scale effects here of interest can be safely neglected.]{}[]{data-label="process"}](process-1.pdf)
A problem might arise when (as suggested in Fig.3) events on two different causal links happen to be rather close in spacetime: because of relative locality observers distant from such near-coincident (but uncorrelated) events might get a sufficiently distorted picture of the events that the causal links could get confused. We will arrange for a particularly insightful such situation by the end of this section. And ultimately we shall find that no confusion about causal links arises if information on the different events is gathered by nearby observers. Specifically for the situation in Fig.\[process\] it will be necessary to rely on at least two observers: an observer Alice near events $\mathcal{K}_{\ \mu}^{(0)}$ and $\mathcal{K}_{\ \mu}^{(1)}$ and an observer Bob near events $\mathcal{K}_{\ \mu}^{(2)}$ and $\mathcal{K}_{\ \mu}^{(3)}$.
We shall do this analysis in detail but making some simplifying assumptions about the energies of the particles involved. For the particles described by dashed lines in Fig.\[process\] we assume that they are “soft" [@anatomy], [*i.e.*]{} their energies $E$ are small enough that terms of order $\ell E^2$ are negligible in comparison to all other energy scales that we shall instead take into account. The particles described by solid lines in Fig.\[process\] are instead “hard", meaning that for them $\ell$ corrections must be taken into account. We also adopt the simplification that all particles are ultrarelativistic, [*i.e.*]{} for massive particles the mass can be neglected.
The action describing the situation in Fig.\[process\] within the relative-locality curved-momentum-space framework proposed in Ref. [@principle] is $$\label{action}
\begin{array}{lll}
\mathcal{S}
&=&
\displaystyle
\int_{-\infty}^{s_{1}}ds\left(z^{\mu}\dot{k}_{\mu}+\mathcal{N}_k\mathcal{C}_{k}^{(0)}\right)+
\int_{-\infty}^{s_{1}}ds\left({x}^{\mu}\dot{p}_{\mu}+\mathcal{N}_p(\mathcal{C}_{p}-m_p^2)\right)+
\int_{-\infty}^{s_{0}}ds\left(y^{\mu}\dot{q}_{\mu}+\mathcal{N}_q(\mathcal{C}_{q}^{(0)}-m^2_q)\right)+\\
&&
\displaystyle
\int_{-\infty}^{s_{0}}ds\left({u}^{\mu}\dot{r}_{\mu}+\mathcal{N}_r\mathcal{C}_{r}^{(0)}\right)+
\int_{s_{1}}^{s_{2}}ds\left({x'}^{\mu}\dot p'_{\mu}+\mathcal{N}_{p'}(\mathcal{C}_{p'}-m_{p'}^2)\right)
+\int_{s_{0}}^{s_{3}}ds\left(y'^{\mu}\dot q'_{\mu}+\mathcal{N}_{q'}(\mathcal{C}_{q'}^{(0)}-m_{q'}^2)\right)+\\
&&
\displaystyle
\int_{s_{3}}^{+\infty}ds\left(y''^\mu\dot q''_{\mu}+\mathcal{N}_{q''}(\mathcal{C}_{q''}^{(0)}-m^2_{q''})\right)+
\int_{s_{3}}^{+\infty}ds\left({u'}^{\mu}\dot r'_{\mu}+\mathcal{N}_{r'}\mathcal{C}_{r'}^{(0)}\right)+\int_{s_{2}}^{+\infty}ds\left(x''^\mu\dot p''_\mu+\mathcal{N}_{p''}(\mathcal{C}_{p''}-m_{p''}^2)\right)+\\
&&
\displaystyle
\int_{s_{2}}^{+\infty}ds\left(z'^{\mu}\dot{k'}_{\mu}+\mathcal{N}_{k'}\mathcal{C}_{k'}^{(0)}\right)
-\xi_{(0)}^{\mu}\mathcal{K}_{\ \mu}^{(0)}
-\xi_{(1)}^{\mu}\mathcal{K}_{\ \mu}^{(1)}
-\xi_{(2)}^{\mu}\mathcal{K}_{\ \mu}^{(2)}
-\xi_{(3)}^{\mu}\mathcal{K}_{\ \mu}^{(3)}\ ,
\end{array}$$ where the ${\cal K}^{(i)}_{\ \mu}$ appearing in the boundary terms enforce the relevant conservation laws $$\begin{split}
{\cal K}^{(0)}_{\ \mu}&= (r \oplus q)_\mu-q'_\mu \ ,\\
{\cal K}^{(1)}_{\ \mu}&= (k\oplus p)_\mu-p'_\mu \ ,\\
{\cal K}^{(2)}_{\ \mu}&= p'_\mu - (k'\oplus p'')_\mu \ ,\\
{\cal K}^{(3)}_{\ \mu}&= q'_\mu-(r' \oplus q'')_\mu \ .
\label{constraints}
\end{split}$$ Several aspects of (\[action\]) are worth emphasizing. First we notice that the action in (\[action\]) is just the sum of two independent pieces, one for each (two-event-)chain of causally-connected events. For soft particles we codified the on-shellness in terms of $\mathcal{C}^{(0)}_{p}=p_0^2-p_1^2$, while for hard particles we have $\mathcal C_p \equiv p_0^2-p_1^2 -\ell p_0 p_1^2$, appropriate for the $\kappa$-momentum case. For conceptually clarity massive particles in (\[action\]) are identifiable indeed because we write a mass term for them, even though, as announced, we shall assume throughout this section that all particles are ultrarelativistic. Also note that the action (\[action\]) is not specialized to the case which will be here of interest from the causality perspective, which is the case of coincidence of the two events ${\cal K}^{(0)}$ and ${\cal K}^{(1)}$: we shall enforce that feature later by essentially focusing on cases such that $\xi^\mu_{(0)}=\xi^\mu_{(1)}$.
By varying the action (\[action\]), one obtains the following equations of motion $$\begin{gathered}
\dot p_\mu =0~,~~\dot q_\mu =0~,~~\dot k_\mu =0~,~~\dot r_\mu =0~,~~\dot p'_\mu =0~,~~\dot q'_\mu =0~,~~\dot p''_\mu =0~,~~\dot q''_\mu =0~,~~\dot k'_\mu =0~,~~\dot r'_\mu =0\ ,\\
\mathcal{C}_p=m_p^2~,~~\mathcal{C}_q^{(0)}=m^2_q~,~~\mathcal{C}_k^{(0)}=0~,~~\mathcal{C}_r^{(0)}=0~,~~\mathcal{C}_{p'}=m_{p'}^2~,~~\mathcal{C}_{q'}^{(0)}=m_{q'}^2~,~~\mathcal{C}_{p''}=m_{p''}^2~,~~\mathcal{C}_{q''}^{(0)}=m^2_{q''}~,~~\mathcal{C}_{k'}^{(0)}=0~,~~\mathcal{C}_{r'}^{(0)}=0\ ,\\
{\cal K}^{(0)}_\mu=0 ~,~~~{\cal K}^{(1)}_\mu=0 ~,~~~{\cal K}^{(2)}_\mu=0 ~,~~~{\cal K}^{(3)}_\mu=0\ ,\\
\dot x^\mu = \mathcal{N}_p {\frac{\partial \mathcal{C}_p}{\partial p_\mu}}~,~~~
\dot y^\mu = \mathcal{N}_q {\frac{\partial \mathcal{C}_q^{(0)}}{\partial q_\mu}}~,~~~
\dot z^\mu = \mathcal{N}_k {\frac{\partial \mathcal{C}_k^{(0)}}{\partial k_\mu}}~,~~~
\dot u^\mu = \mathcal{N}_{r} {\frac{\partial \mathcal{C}_r^{(0)}}{\partial r_\mu}}~,~~~\\
\dot x'^\mu = \mathcal{N}_{p'} {\frac{\partial \mathcal{C}_{p'}}{\partial p'_\mu}}~,~~~
{\dot y}'^\mu = \mathcal{N}_{q'} {\frac{\partial \mathcal{C}_{q'}^{(0)}}{\partial q_\mu'}}~,~~~
\dot x''^\mu = \mathcal{N}_{p''} {\frac{\partial \mathcal{C}_{p''}}{\partial p''_\mu}}~,~~~
\dot y''^\mu = \mathcal{N}_{q''} {\frac{\partial \mathcal{C}_{q''}^{(0)}}{\partial q''_\mu}}~,~~~
\dot z'^\mu = \mathcal{N}_{k'} {\frac{\partial \mathcal{C}_{k'}^{(0)}}{\partial k'_\mu}}~,~~~
\dot u'^\mu = \mathcal{N}_{r'} {\frac{\partial \mathcal{C}_{r'}^{(0)}}{\partial r'_\mu}}~,~~~\\\end{gathered}$$ and the boundary conditions for the endpoints of the worldlines $$\begin{gathered}
x^\mu(s_{1}) = \xi^\nu_{(1)} {\frac{\partial \mathcal{K}^{(1)}_\nu}{\partial p_\mu}}~,~~~
z^\mu(s_{1}) = \xi^\nu_{(1)} {\frac{\partial \mathcal{K}^{(1)}_\nu}{\partial k_\mu}}~,~~~
y^\mu(s_{0})= \xi^{\nu}_{(0)} {\frac{\partial \mathcal{K}^{(0)}_\nu}{\partial q_\mu}}~,~~~
u^\mu(s_{0})= \xi^{\nu}_{(0)} {\frac{\partial \mathcal{K}^{(0)}_\nu}{\partial r_\mu}}~,~~~
x'^\mu(s_{1}) = -\xi^\nu_{(1)} {\frac{\partial \mathcal{K}^{(1)}_\nu}{\partial p'_\mu}}~,~~~ \\
\qquad x'^\mu(s_{2}) = \xi^\nu_{(2)} {\frac{\partial \mathcal{K}^{(2)}_\nu}{\partial p'_\mu}}~,~~~
y'^\mu(s_{0}) = -\xi^{\nu}_{(0)} {\frac{\partial \mathcal{K}^{(0)}_\nu}{\partial q'_\mu}}~,~~~
y'^\mu(s_{3}) = \xi^\nu_{(3)} {\frac{\partial \mathcal{K}^{(3)}_\nu}{\partial q'_\mu}}~,~~~
x''^\mu(s_{2}) = -\xi^\nu_{(2)} {\frac{\partial \mathcal{K}^{(2)}_\nu}{\partial p''_\mu}}~,~~~
z'^\mu(s_{2}) = -\xi^\nu_{(2)} {\frac{\partial \mathcal{K}^{(2)}_\nu}{\partial k'_\mu}}~,~~~\\
y''^\mu(s_{3}) = -\xi^\nu_{(3)} {\frac{\partial \mathcal{K}^{(3)}_\nu}{\partial q''_\mu}}~,~~~
u'^\mu(s_{3}) = -\xi^\nu_{(3)} {\frac{\partial \mathcal{K}^{(3)}_\nu}{\partial r'_\mu}}~ \ .\end{gathered}$$ It is easy to check that the above equations of motion and boundary conditions are invariant under the following translation transformations: $$\begin{split}
x^\mu_B&= x^\mu_A+ b^\nu\{(k\oplus p)_\nu,x^\mu\}\ ,\\
z^\mu_B&= z^\mu_A+ b^\nu\{(k\oplus p)_\nu ,z^\mu\} \ ,\\
y^\mu_B&= y^\mu_A+ b^\nu\{(r \oplus q)_\nu ,y^\mu\} \ ,\\
u^\mu_B&= u^\mu+ b^\nu\{(r \oplus q)_\nu ,u^\mu\} \ ,\\
{x'}^\mu_B&={x'}_{A}^{\mu}+ b^\nu\{p'_\nu ,{x'}^\mu\}\ ,\\
{y'}_{B}^{\mu}&={y'}_{A}^{\mu}+ b^\nu\{q'_\nu,{y'}^{\mu}\}\ ,\\
{x''}^\mu_B&= x''^\mu_A+ b^\nu\{(k'\oplus p'')_\nu ,x''^\mu\} \ ,\\
{z'}^\mu_B&= z'^\mu_A+ b^\nu\{k'\oplus p'')_\nu ,z'^\mu\} \ ,\\
{y''}^\mu_B&=y''^\mu_A+ b^\nu\{(r' \oplus q'')_\nu ,y''^\mu\} \ ,\\
{u'}^\mu_B&= u'^\mu_A+ b^\nu\{(r' \oplus q'')_\nu ,u'^\mu\} \ ,
\label{translations}
\end{split}$$ where $b^\mu$ are the translation parameters and it is understood that $\{z^{\mu} , k_{\nu} \} = \delta^{\mu}_{\nu}$, $\{x^{\mu} , p_{\nu} \} = \delta^{\mu}_{\nu}$, $\{y^{\mu} , q_{\nu} \} = \delta^{\mu}_{\nu}$, $\{u^{\mu} , r_{\nu} \} = \delta^{\mu}_{\nu}$, $\{z'^{\mu} , k'_{\nu} \} = \delta^{\mu}_{\nu}$, $\{x'^{\mu} , p'_{\nu} \} = \delta^{\mu}_{\nu}$, $\{y'^{\mu} , q'_{\nu} \} = \delta^{\mu}_{\nu}$, $\{u'^{\mu} , r'_{\nu} \} = \delta^{\mu}_{\nu}$, $\{x''^{\mu} , p''_{\nu} \} = \delta^{\mu}_{\nu}$, $\{y''^{\mu} , q''_{\nu} \} = \delta^{\mu}_{\nu}$.
Because of relative locality we evidently need here two observers Alice and Bob chosen so that the questions here of interest can be investigated in terms of the locality of interactions near them. As announced we focus on the case in which the interactions ${\cal K}^{(0)}$ and ${\cal K}^{(1)}$ are coincident, and we take as Alice an observer for whom these two interactions occur in the origin of her reference frame. This in particular allows us to restrict our attention to cases with $x'^\mu_A(s_1)=y'^\mu_A(s_0)=0$. We take the other observer, Bob, at rest with respect to Alice and such that the event ${\cal K}^{(3)}$ occurs in the origin of Bob’s reference frame, so that $y'^\mu_B(s_3)=0$. Since in the $\kappa$-momentum case the physical speed of ultrarelativistic particles depends on their energy [@anatomy] the interaction ${\cal K}^{(2)}$ cannot be coincident with the interaction ${\cal K}^{(3)}$ (since ${\cal K}^{(0)}$ and ${\cal K}^{(3)}$ exchange a soft particle whereas ${\cal K}^{(1)}$ and ${\cal K}^{(2)}$ exchange a hard particle one must take into account the difference in physical speed between the hard and the soft exchanged particle). But this dependence on energy of the physical speed of ultrarelativistic particles is anyway a small $\ell$-suppressed effect, so we can focus on a situation where ${\cal K}^{(2)}$ and ${\cal K}^{(3)}$ are nearly coincident, and we study that situation assuming ${\cal K}^{(2)}$ occurs in spatial origin of Bob’s reference frame (but at time different from ${\cal K}^{(3)}$). This allows us to specify $x'^1_B(s_2)=0$. Also note that as long as the distance of ${\cal K}^{(2)}$ from the spacetime origin of Bob’s reference frame is an $\ell$-suppressed feature Bob’s description of the locality (or lack thereof) of the interaction ${\cal K}^{(2)}$ is automatically immune from relative-locality effects at leading order in $\ell$, which is the order at which we are working.
Equipped with this choice of observers and these simplifying assumptions about the relevant events, we can quickly advance with our analysis of causal links from the relative-locality perspective. We start by noticing that from the equations of motion it follows that both for Alice and Bob[^2] $$\frac{\dot{x}'^1}{\dot{x}'^0}=1-\ell p'_1 \, , \qquad \frac{\dot{y}'^1}{\dot{y}'^0}=1\, .$$ This implies that according to Alice (for whom the events ${\cal K}^{(0)}$ and ${\cal K}^{(1)}$ occur in the origin of the reference frame) the worldlines of the two exchanged particles are $$\begin{split}
{x'}_{A}^{1}&=(1-\ell p'_1){x'}_{A}^{0}\ ,\\
{y'}_{A}^{1}&={y'}_{A}^{0}\ .
\label{eq-motion-alice}
\end{split}$$ A key aspect of the analysis we are reporting in this section is establishing how these two worldlines are described by the distant observer Bob. On the basis of (\[translations\]) one concludes that the relevant translation transformation is undeformed: $$\label{translations_atoms}
\begin{split}
{x'}_{B}^{\mu}(s)&={x'}_{A}^{\mu}(s)+b^{\nu} \{ p'_\nu , x'^\mu\} = {x'}_{A}^{\mu}(s)-b^{\mu} \ , \\
{y'}_{B}^{\mu}(s)&={y'}_{A}^{\mu}(s)+b^{\nu} \{ q'_\nu , y'^\mu\}= {y'}_{A}^{\mu}(s)-b^{\mu}\ .
\end{split}$$ So the worldlines in Bob’s coordinatization must have the form $$\begin{split}
x'^1_B&=(1-\ell p'_1)x'^0_B -b^1+b^0-b^0\ell p'_1\, , \\
y'^1_B&=y'^0_B -b^1+b^0 \, .
\label{eq-motion-bob}
\end{split}$$ Since we have specified for Bob that ${\cal K}^{(3)}$ occurs in the origin of his reference frame, $y'^\mu_B(s_3)=0$, we must have that $b^0=b^1$. And then finally we establish that the event ${\cal K}^{(2)}$, occurring in the spatial origin of Bob’s reference frame, $x^1_B(s_2)=0$, is timed by Bob at $$x'^0_B(s_2)=b^1\ell p'_1 \ .
\label{arrival-time}$$ In particular for positive $\ell$ one has that according to Bob ${\cal K}^{(2)}$ occurs before ${\cal K}^{(3)}$ in his spatial origin, with a time difference between them given by $\Delta t=b^1\ell p'_1$.
This was just preparatory material for the point we most care about in this section, which concerns possible paradoxes for causality and their clarification. For that we need to look at how Alice describes the two events distant from her, ${\cal K}^{(2)}$ and ${\cal K}^{(3)}$. ${\cal K}^{(3)}$ is an interaction involving only soft particles so nothing noteworthy can arise from looking at ${\cal K}^{(3)}$, but ${\cal K}^{(2)}$ involves hard particles and therefore the inferences about ${\cal K}^{(2)}$ by observer Alice, who is distant from ${\cal K}^{(2)}$, will give a description of ${\cal K}^{(2)}$ as an apparently non-local interaction. This is the main implication of relative locality, and we can see that it does give rise to a combined description of ${\cal K}^{(2)}$ and ${\cal K}^{(3)}$ that at first may appear puzzling from the causality perspective. We show this by noting down the values of coordinates of particles involved in ${\cal K}^{(2)}$ and ${\cal K}^{(3)}$ according to Alice. For the particles with coordinates $y''^{\mu}$ and $u'^\mu$ on the basis of (\[translations\]) one finds that the translation is completely undeformed, and since $y''^{\mu}_B (s_3)=u'^\mu_B (s_3)=0$ one has that $$y''^{\mu}_A (s_3)=u'^\mu_A (s_3)=\xi_{(3)A}^{\mu}=b^1 \ .$$ For the particles involved in the hard vertex ${\cal K}^{(2)}$, with coordinates $x''^\mu$ and $z'^\mu$, on the basis of (\[translations\]) one finds that the translation is deformed, and starting from the fact that $x''^0_B=b^1\ell p'_1$, $x''^1_B=0$, $z'^0_B=b^1\ell p'_1$, $z'^1_B=0$ one arrives at finding that $$\begin{split}
x''^0_A (s_2)&=b^1+b^1\ell p'_1 \ , \\
x''^1_A (s_2)&=b^1 - b^1 \ell k'_1\ \approx b^1, \\
z'^0_A (s_2)&= b^1+b^1\ell p'_1 -b^1\ell p''_1\ , \\
z'^1_A (s_2)&= b^1 \ .
\end{split}$$ As shown in Fig.4 the most striking situation from the viewpoint of causality arises when $p'_1 \simeq p''_1$, in which case according to Alice $z'^0_A (s_2)= z'^1_A (s_2) =b^1$, which means that the particle with coordinates $z'^\mu$, who actually interacts at ${\cal K}^{(2)}$, in the coordinatization by distant observer Alice appears to come out of the interaction ${\cal K}^{(3)}$. This is an example of the sort of apparent paradoxes for causality that can be encountered with relative locality: they all concern the description of events by distant observers. Of course, there is no true paradox since a known consequence of relative locality is that inferences about distant events are misleading. Indeed, as also shown in Fig.4, Bob’s description of the interactions ${\cal K}^{(2)}$ and ${\cal K}^{(3)}$ (which are near Bob) is completely unproblematic. However, in turn, Bob’s inferences about the events ${\cal K}^{(0)}$ and ${\cal K}^{(1)}$ (which are distant from Bob) are affected by peculiar relative-locality features, as also shown in Fig.4. In looking at Fig.4 readers should also keep in mind that for that figure we magnified effects in order to render them visible: actually all noteworthy features in Fig.4 are Planck-scale suppressed, and would amount to time intervals no greater than $10^{-19}s$ for Earth experiments (over distances of, say, $10^6m$) involving particles with currently accessible energies (no greater than, say, $1TeV$).
Causal Loops
============
The observations on relative locality reported in the previous two sections illustrate how misleading the characterization of events and chains of events can be, if not based on how each event is seen by a nearby observer. For chains of events this imposes that the analysis be based on more than one observer: at least one observer for each interaction in the chain.
Equipped with this understanding we now progress to the next level in testing causality: we consider the possibility of a “causal loop", [*i.e.*]{} a chain of events that form a loop in such a way that causality would be violated.
The starting point for being concerned about these causal loops is the analysis reported in Ref.[@linq], which considered a loop diagram of the type here shown in Fig.5. Ref.[@linq] works on a curved momentum space, but without enforcing relative locality, and finds that a causal loop of the type here shown in Fig.5 could be possible. Our objective is to show that such causal loops are excluded if one enforces relative locality. In light of the observations reported in the previous two sections we shall of course need to study the loop diagram in Fig.5 on the basis of the findings of two observers, one near the first interaction and one near the second interaction (whereas the analysis of Ref.[@linq] only considered the perspective of one observer, in which case the principle of relativity of spacetime locality cannot be enforced or investigated).
![We here show schematically a pair of events causally connected by the exchange of two particles arranged in such a way that one would have a causal loop. Such causal loops are allowed, if one assumes curvature of momentum space without enforcing (DSR-)relativistic covariance and the associated relativity of spacetime locality.[]{data-label="looplinq"}](causal_loop.pdf)
We stress that here, just as in Ref.[@linq], we are working at the level of classical mechanics, so the loop diagram in Fig.\[looplinq\] involves all particles on-shell and merely keeps track of the causal links among different events, assigning worldlines exiting/entering each event (one should not confuse such loop diagrams with the different notion arising in Feynman’s perturbative approach to quantum field theory).
We start by writing down an action of the type already considered in the previous two sections, which gives the description of the loop diagram in Fig.\[looplinq\] within the relative-locality curved-momentum-space formalism proposed in Ref.[@principle]. We shall see that our action does reproduce the equations of motion and the boundary conditions which were at the basis of the analysis reported in Ref.[@linq]. This action giving the diagram in Fig. \[looplinq\] is $$\begin{array}{lll}
\mathcal S &=& \displaystyle
\int_{-\infty}^{s_0}ds\left(y^\mu\dot q_\mu+\mathcal N_q(\mathcal C_q-m^2_q)\right)
+\int_{s_0}^{+\infty}ds\left(y'^\mu\dot q_\mu'+\mathcal N_{q'}(\mathcal C_{q'}-m^2_{q'})\right)
+\int_{-\infty}^{s_1}ds\left(z^\mu\dot k_\mu+\mathcal N_k(\mathcal C_k-m^2_k)\right)+\\
&&\displaystyle
+\int_{s_1}^{+\infty}ds\left(z'^\mu\dot k'_\mu+\mathcal N_{k'}(\mathcal C_{k'}-m^2_{k'})\right)
+\int_{s_0}^{s_1}ds\left(x'^\mu\dot p'_\mu+\mathcal N_{p'}(\mathcal C_{p'}-m^2_{p'})\right)
+\int_{s_1}^{s_0}ds\left(x^\mu\dot p_\mu+\mathcal N_p(\mathcal C_p-m^2_p)\right)+\\
&&\displaystyle
-\xi_{(0)}^{\mu}\mathcal K_{\mu}^{(0)}-\xi_{(1)}^{\mu}\mathcal K_{\mu}^{(1)} \ ,
\end{array}$$ where $\mathcal K_\mu^{(0)}=\left[(q\oplus p)\oplus(\ominus(p'\oplus q'))\right]_\mu$ and $\mathcal K_\mu^{(1)}=\left[(p'\oplus k)\oplus(\ominus(k'\oplus p))\right]_\mu$. It is important for us to stress, since this is the key ingredient for seeking a violation of causality, that the last integral, which stands for the free propagation of the particle which is travelling back in time, has inverted integration extrema.
By varying this action we obtain equations of motion
$$\begin{gathered}
\dot p_\mu=0 \ , \qquad \dot p'_\mu=0 \ , \qquad \dot q_\mu=0 \ , \qquad \dot q'_\mu=0 \ ,\qquad \dot k_\mu=0 \ , \qquad \dot k'_\mu=0 \ ,\\
\mathcal C_p=m^2_p \ , \qquad \mathcal C_{p'}=m^2_{p'} \ , \qquad \mathcal C_q=m^2_q \ , \qquad \mathcal C_{q'}=m^2_{q'} \ , \qquad \mathcal C_{k'}=m^2_{k'} \ , \qquad \mathcal C_k=m^2_k \ ,\\
\dot x^\mu(s)=\mathcal N_p{\frac{\partial \mathcal C_p}{\partial p_\mu}} \ ,\label{dotcoordinates1}\qquad
\dot x'^\mu(s)=\mathcal N_{p'}{\frac{\partial \mathcal C_{p'}}{\partial p'_{\mu}}} \ ,\qquad
\dot y^\mu(s)=\mathcal N_q{\frac{\partial \mathcal C_q}{\partial q_\mu}} \ ,\\
\dot y'^\mu(s)=\mathcal N_{q'}{\frac{\partial \mathcal C_{q'}}{\partial q'_\mu}} \ ,\label{dotcoordinates2}\qquad
\dot z^\mu(s)=\mathcal N_k{\frac{\partial \mathcal C_k}{\partial k_\mu}} \ ,\qquad
\dot z'^\mu(s)=\mathcal N_{k'}{\frac{\partial \mathcal C_{k'}}{\partial k'_\mu}} \ , \\
\mathcal K_\mu^{(0)}=0 \ , \qquad \mathcal K_\mu^{(1)}=0 \ ,\label{boundaries1}\end{gathered}$$
and boundary terms
$$\begin{gathered}
y^\mu(s_0)=\xi_{(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial q_\mu}} \ ,\qquad
y'^\mu(s_0)=-\xi_{(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial q'_{\mu}}} \ ,\qquad
z'^\mu(s_1)=-\xi_{(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial k'_\mu}} \ ,\qquad
z^\mu(s_1)=\xi_{(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial k_\mu}} \ ,\label{boundaries2}\\
x^\mu(s_0)=\xi_{(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\mu}} \ ,\qquad
x^\mu(s_1)=-\xi_{(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\mu}} \ ,\qquad
x'^\mu(s_0)=-\xi_{(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}} \ ,\qquad
x'^\mu(s_1)=\xi_{(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}} \ ,\label{boundaries3}\end{gathered}$$
which indeed reproduce the ones used in the analysis reported in Ref.[@linq].
Aside on the absence of causal loops in Special Relativity
----------------------------------------------------------
We find it useful to start by first considering the $\ell \rightarrow 0$ limit of the problem of interest in this section: the causal loop in Special Relativity ([*i.e.*]{} with Minkowskian geometry of momentum space). This allows us to assume temporarily that the on-shellness is governed by $\mathcal C^{(0)}=p_0^2-p_1^2$ and that therefore the following relationship holds $$\label{minkowskianeom}
\dot x^\mu(s)=\left(\dot x^\nu\dot x_\nu\right)^\frac{1}{2}\frac{p^{\mu}}{m_p} \ .$$ We take advantage of some simplification of analysis, without loosing any of the conceptual ingredients here of interest, by focusing on $\dot x_\mu\dot x^\mu>0,\ \dot x^0>0;\ p^2=m_p^2>0,\ p^0\geq m_p>0$, [*i.e.*]{} our particles travel along timelike worldlines. We have that the proper time of a particle is given by $$d\tau=\left(\dot x^\mu\dot x_\mu\right)^{\frac{1}{2}}ds=
\dot x^0\sqrt{1-\left(\frac{\dot x^1}{\dot x^0}\right)^{2}}ds=
\dot x^0\sqrt{1-\left(\frac{p^1}{p^0}\right)^2}ds=
\dot x^0\gamma_p^{-1}ds \ ,$$ where $\gamma_p$ is the usual Lorentz factor and in the third equality we used (\[minkowskianeom\]).
Going back to the diagram in Fig.\[looplinq\] we have that for the particle with phase-space coordinates $(p',x')$, whose worldline is exchanged between the interaction $\mathcal K^{(0)}$ and the interaction $\mathcal K^{(1)}$ (and therefore travels from $x'^\mu(s_0)$ to $x'^\mu(s_1)$) the following chain of equalities holds $$\label{fin-iniz(x')}
\begin{array}{lll}
x'^\mu(s_1)-x'^\mu(s_0)
&=&\displaystyle
\int_{s_0}^{s_1}ds\,\frac{dx'^\mu}{ds}=\int_{s_0}^{s_1}ds\left(\dot x'^\nu\dot x'_\nu\right)^\frac{1}{2}\frac{p'^\mu}{m_{p'}}=\\
&=&\displaystyle
\int_{\tau'(s_0)}^{\tau'(s_1)}d\tau'\frac{p'^\mu}{m_{p'}}=\Delta\tau'u'^\mu \ ,
\end{array}$$ with $\displaystyle u'^\mu=\frac{p'^\mu}{m_{p'}}$ .
Similarly, for the other particle exchanged between $\mathcal K^{(0)}$ and $\mathcal K^{(1)}$, the one with phase-space coordinates $(p,x)$, one has that $$\label{fin-iniz(x)}
\begin{array}{lll}
x^\mu(s_0)-x^\mu(s_1)
&=&\displaystyle
\int_{s_1}^{s_0}ds\,\frac{dx^\mu}{ds}=\int_{s_1}^{s_0}ds\left(\dot x^\nu\dot x_\nu\right)^\frac{1}{2}\frac{p^\mu}{m_p}=\\
&=&\displaystyle
\int_{\tau(s_1)}^{\tau(s_0)}d\tau\frac{p^\mu}{m_p}=\Delta\tau\ u^\mu \ .
\end{array}$$ Since in this subsection we are working in the $\ell \rightarrow 0$ limit we have that $\mathcal K_\mu^{(0)}=q_\mu+p_\mu-p'_\mu-q'_\mu$ and $\mathcal K_\mu^{(1)}=p'_\mu+k_{\mu}-k'_\mu-p_\mu$ , in which case it is easy to see that our boundary conditions simply enforce $$\xi_{(0)}^\mu=x'^\mu(s_0)~,~~~
\xi_{(0)}^\mu=x^\mu(s_0)~,~~~
\xi_{(1)}^\mu=x'^\mu(s_1)~,~~~
\xi_{(1)}^\mu=x^\mu(s_1)~.$$ So evidently $$\begin{gathered}
\xi_{(1)}^\mu-\xi_{(0)}^\mu=x'^\mu(s_1)-x'^\mu(s_0)=\Delta\tau'u'^\mu \ ,\\
\xi_{(0)}^\mu-\xi_{(1)}^\mu=x^\mu(s_0)-x^\mu(s_1)=\Delta\tau\ u^\mu \ ,\end{gathered}$$ and $$\label{speesrelationSpRel}
\Delta\tau\ u^\mu+\Delta\tau'u'^\mu=0 \ .$$ Since the relevant proper-time intervals are positive and the zero components of the four-velocities are positive this requirement can never be satisfied: as well known causal loops are forbidden in Special Relativity.
Another way to see that causal loops are forbidden in Special Relativity can be based on deriving the relationship between the relevant proper-time intervals and the interaction coordinates $\xi_{(0)}^\mu$, $\xi_{(1)}^\mu$. One easily finds that $$\begin{gathered}
\Delta\tau=\int_{s_1}^{s_0}ds\ \dot x^0\gamma_p^{-1}=
\gamma_p^{-1}\left(x^0(s_0)-x^0(s_1)\right)=
\gamma_p^{-1}\left(\xi_{(0)}^0-\xi_{(1)}^0\right) \ ,\label{deltatauSR}\\
\Delta\tau'=\int_{s_0}^{s_1}ds\ \dot x'^0\gamma_{p'}^{-1}=
\gamma_{p'}^{-1}\left(x'^0(s_1)-x'^0(s_0)\right)=
\gamma_{p'}^{-1}\left(\xi_{(1)}^0-\xi_{(0)}^0\right) \ .\label{deltatau'SR}\end{gathered}$$ So again the fact that $\Delta\tau \geq 0$ and $\Delta\tau'\geq 0$ excludes the causal loop, since on the basis of (\[deltatauSR\])-(\[deltatau’SR\]) this would require $\xi_{(0)}=\xi_{(1)}$: by construction $\left(\xi_{(1)}-\xi_{(0)}\right)_\mu \left(\xi_{(1)}-\xi_{(0)}\right)^\mu \geq 0$ (the interval between the two interactions is timelike or null) and then $\xi_{(0)}^0=\xi_{(1)}^0$ implies $\xi_{(0)}^\mu=\xi_{(1)}^\mu$, *i.e.* the loop can only collapse into a single event (no causality issue, not a causal loop).
Causal loop with curved momentum space
--------------------------------------
Our next step is to introduce leading-order-in-$\ell$ corrections, but without enforcing the principle of relative locality. Such setups in general do allow causal loops, as we shall now show (in agreement with what was already claimed in Ref.[@linq]). What changes with respect to the special-relativistic case of the previous subsection is that (for the $\kappa$-momentum case, which we chose as illustrative example) the on-shellness is governed by $\mathcal C_p=p_0^2-p_1^2-\ell p_0p_1^2$ while conservation laws at first order take the form
$$\begin{aligned}
&\mathcal K_ {\ 0}^{(0)}=q_0+p_0-q'_0-p'_0 \ ,\\
&\mathcal K_{\ 1}^{(0)}=q_1+p_1-q'_1-p'_1-\ell\left[q_0p_1-(q_0+p_0-q'_0-p'_0)p'_1-(q_0+p_0-q'_0)q'_1\right] \ ,\\
&\mathcal K_ {\ 0}^{(1)}=p'_0+k_0-p_0-k'_0 \ ,\\
&\mathcal K_{\ 1}^{(1)}=p'_1+k_1-p_1-k'_1-\ell\left[p'_0k_1-(p'_0+k_0-p_0-k'_0)k'_1-(p'_0+k_0-p_0)p_1\right] \ .\end{aligned}$$
Also the equations of motion are $\ell$-deformed, as shown in (\[dotcoordinates1\])-(\[dotcoordinates2\]), and for example one has that $$\label{kminkowskianeom}
\dot x^\mu(s)=\mathcal N_p\left[2p^\mu -\ell\left(\delta^\mu_0p_1^2+\delta^\mu_12p_0p_1\right)\right] \ .$$ This still allows one to write a relationship analogous to (\[minkowskianeom\]) from the previous subsection, $$\dot x^\mu(s)=\left(\dot x^\nu\dot x_\nu\right)^\frac{1}{2}u^\mu \ ,$$ but with $$u^\mu=\frac{p^\mu}{m_p}-\frac{\ell}{2m_p}\left(-2p^\mu\frac{p_0p_1^2}{m_p^2}+\delta_0^\mu p_1^2+\delta^\mu_1 2p_0p_1\right)\ .$$ Analogously, for $x'^\mu$ one has that $$\dot x'^\mu(s)=\left(\dot x'^\nu\dot x'_\nu\right)^\frac{1}{2}u'^\mu \ ,$$ with $$u'^\mu=\frac{p'^\mu}{m_{p'}}-\frac{\ell}{2m_{p'}}\left(-2p'^\mu\frac{p'_0p_1'^2}{m_{p'}^2}+\delta_0^\mu p_1'^2+\delta^\mu_1 2p'_0p'_1\right)\ .$$
In close analogy with (\[fin-iniz(x’)\]) and (\[fin-iniz(x)\]) one easily finds that $$\begin{aligned}
&x'^\mu(s_1)-x'^\mu(s_0)=\Delta\tau'u'^\mu, \label{kfin-iniz(x')}\\
&x^\mu(s_0)-x^\mu(s_1)=\Delta\tau\ u^\mu ~, \label{kfin-iniz(x)}\end{aligned}$$ and from (\[boundaries3\]) it follows that $$\begin{aligned}
&\xi_{(0)}^\nu=-x'^\mu(s_0)\left({\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}}\right)^{-1}=x^\mu(s_0)\left({\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\mu}}\right)^{-1} \ ,\label{boundaries3new}\\
&\xi_{(1)}^\nu=x'^\mu(s_1)\left({\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}\right)^{-1}=-x^\mu(s_1)\left({\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\mu}}\right)^{-1} \ .\label{boundaries4new}\end{aligned}$$ Combining (\[boundaries3new\]) with (\[kfin-iniz(x)\]) one finds that $$\label{eq:-3}
-x'^\mu(s_0)\left({\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}}\right)^{-1}{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\rho}}=x^\rho(s_0)=x^\rho(s_1)+\Delta\tau u_p^\rho \ ,$$ while combining (\[boundaries4new\]) with (\[kfin-iniz(x’)\]) one finds that $$\label{eq:-4}
x^\rho(s_1)=
-x'^\mu(s_1)\left({\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}\right)^{-1}{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\rho}}=
-(x'^\mu(s_0)+\Delta\tau'u'^\mu)\left({\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}\right)^{-1}{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\rho}} \ .$$ Finally, combining (\[eq:-4\]) with (\[eq:-3\]), we obtain the same condition given in [@linq], $$\label{eq:-5}
\left[{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\rho}}\left({\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}\right)^{-1}-{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\rho}}\left({\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}}\right)^{-1}\right]x'^\mu(s_0)=
-{\frac{\partial \mathcal K _\nu^{(1)}}{\partial p_\rho}}\left({\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}\right)^{-1}\Delta\tau'u'^\mu+\Delta\tau u^\rho \ ,$$ which takes the following form upon expanding $\mathcal K_\nu^{(0)}$ and $\mathcal K_\nu^{(1)}$ to leading order in $\ell$: $$\label{eq:-5'}
\ell\left[\delta_1^\rho\left(k'_0-q_0\right)+\delta_0^\rho\left(q'_1-k_1\right)\right]x'^1(s_0)=
\Delta\tau u^\rho+\Delta\tau'\left[u'^\rho+u'^1\ell\left(\delta_0^\rho k_1-\delta_1^\rho k'_0\right)\right] \ .$$ This (\[eq:-5’\]) is what replaces (\[speesrelationSpRel\]) when the causal loop is analyzed on a curved momentum space without enforcing relative locality.
Notice that this (\[eq:-5’\]), when its left-hand side does not vanish, can have solutions with positive $\Delta\tau$ and $\Delta\tau'$ and positive zero components of the four-velocities, which was not possible with (\[speesrelationSpRel\]). This means that contrary to the special-relativistic case (Minkowski momentum space) causal loops are possible on a curved momentum space, at least if one does not enforce relative locality.
We also note down some equalities that follow from (\[eq:-5’\]) and therefore must hold for the causal loop to be allowed $$\label{sistemalinqcompletoA}
\Delta\tau=-\Delta\tau'\frac{u'^0}{u^0}+\ell x'^1(s_0)\left(\frac{q'_1-k_1}{u^0}\right)-\ell\Delta\tau'\left(\frac{u'^1k_1}{u^0}\right) \ ,$$ $$\label{sistemalinqcompletoB}
\ell x'^1(s_0)=\Delta\tau'\frac{u^1u'^0-u^0u'^1+\ell u'^1(k_1u^1+k'_0u^0)}{u^0(q_0-k'_0)+u^1(q'_1-k_1)}$$ and we note that in order for (\[sistemalinqcompletoA\]) to have acceptable solutions one must have that $$x'^1(s_0)>\frac{\Delta\tau' (u'^0+\ell u'^1k_1)}{\ell|q'_1-k_1|} \ .
\label{reqextra}$$ This is in good agreement with the results of Ref. [@linq], but we find useful to add some observations to those reported in Ref. [@linq]. A first point to notice is that Eq. (\[reqextra\]) appears to suggest that $x'^1$ should take peculiarly large values, as in some of the estimates given in Ref. [@linq], since $x'^1$ has magnitude set by a formula with the small scale $\ell$ in the denominator. If one could conclude that only cases with ultralarge $x'^1$ allowed such a causal loop, then the violations of causality would be to some extent less concerning (if confined to a range of values of $x'^1$ large enough to fall outside our observational window). However, it is easy to see that (\[reqextra\]) does not really impose any restriction on the size of $x'^1$: one will have that typically $x'^1$ is much larger than $\Delta\tau'$ but there are causal loops for any value of $x'^1$ (under the condition of taking suitable values of $\Delta\tau'$ and $\Delta\tau$). So when momentum space is curved and one does not enforce the relativity of spacetime locality the violations of causality are rather pervasive.
There is also a technical point that deserves some comments and is related to this pervasiveness of the violations of causality: it might appear to be surprising that within a perturbative expansion, assuming small $\ell$, one arrives at a formula like (\[reqextra\]), with $\ell$ in the denominator. This is however not so surprising considering the role of $x'^1$ in this sort of analysis. The main clarification comes from observing that in the unperturbed theory (the $\ell=0$ theory, [*i.e.*]{} special relativity) $x'^1$ is completely undetermined: as shown in the previous subsection the only causal loops allowed in special relativity are those that collapse (no violation of causality) and such collapsed causal loops are allowed for any however large or however small value of $x'^1$. As stressed above this fact that $x'^1$ can take any value is preserved by the $\ell$ corrections. The apparently surprising factor of $1/\ell$ only appears in a relationship between $x'^1$ and $\Delta\tau'$. If $x'^1$ and $\Delta\tau'$ both had some fixed finite value in the $\ell=0$ theory than at finite small $\ell$ their values should change by very little. But since in the $\ell=0$ theory $x'^1$ is unconstrained (in particular it could take any however large value) and its value is not linked in any way to the value $\Delta\tau'$, then it is not surprising that the $\ell$ corrections take the form shown for example in (\[reqextra\]).
Causal loop analysis in 3+1 dimensions
--------------------------------------
So far we examined the 1+1-dimensional case, but it is rather evident that the features we discussed in the previous subsection are not an artifact of that dimensional reduction. Nonetheless it is worth pausing briefly in this subsection for verifying that indeed those features are still present in $3+1$ dimensions. In this case the on-shellness is governed by $\mathcal C_p=p_0^2-\vec{p}^2-\ell p_0\vec{p}^2$ while conservation laws at first order take the form
$$\begin{aligned}
&\mathcal K_ {\ 0}^{(0)}=q_0+p_0-q'_0-p'_0 \ ,\\
&\mathcal K_{\ i}^{(0)}=q_i+p_i-q'_i-p'_i-\ell\delta_i^j\left[q_0p_j-(q_0+p_0-q'_0-p'_0)p'_j-(q_0+p_0-q'_0)q'_j\right] \ ,\\
&\mathcal K_ {\ 0}^{(1)}=p'_0+k_0-p_0-k'_0 \ ,\\
&\mathcal K_{\ i}^{(1)}=p'_i+k_i-p_i-k'_i-\ell\delta_i^j\left[p'_0k_j-(p'_0+k_0-p_0-k'_0)k'_j-(p'_0+k_0-p_0)p_j\right] \ ,\end{aligned}$$
where $i,j=1,2,3$.
Adopting these expressions, eq.(\[eq:-5\]), keeping only terms up to first order in $\ell$ in the matrices like ${\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\rho}}$ and their products, takes the form $$\label{eq:-5''}
\ell\left[\delta_i^\rho\left(k'_0-q_0\right)+\delta_0^\rho\left(q'_i-k_i\right)\right]x'^i(s_0)=\left[u'^\rho+u'^i\ell\left(\delta_0^\rho k_i-\delta_i^\rho k'_0\right)\right]\Delta\tau'+
u^\rho\Delta\tau \ ,$$ or, more clearly, using the energy conservation laws, $$\label{4Dloop}
\begin{split}
\ell (q'_1-k_1)x'^1(s_0)+\ell (q'_2-k_2)x'^2(s_0)+\ell (q'_3-k_3)x'^3(s_0)&=(u'^0+\ell k_1 u'^1+\ell k_2 u'^2+\ell k_3 u'^3)\Delta \tau'+u^0\Delta \tau,\\
\ell(k_0-q'_0)x'^1(s_0)&=(1-\ell k'_0)u'^1\Delta\tau'+u^1\Delta\tau,\\
\ell(k_0-q'_0)x'^2(s_0)&=(1-\ell k'_0)u'^2\Delta\tau'+u^2\Delta\tau,\\
\ell(k_0-q'_0)x'^3(s_0)&=(1-\ell k'_0)u'^3\Delta\tau'+u^3\Delta\tau.
\end{split}$$
Without really loosing any generality we can analyze the implications of this for an observer orienting her axis of the reference frame so that $p_i=0$ and $p'_i=0$ for $i=2,3$. As a result we also have that $u^i=0$ and $u'^i=0$ for $i=2,3$. For what concerns the other momenta involved in the analysis, $q,\,q',\,k,\,k'$. this choice of orientation of axis only affects rather mildly the conservation laws: $$\begin{gathered}
q_2=q'_2-\ell p'_0q'_2,\qquad q_3=q'_3-\ell p'_0q'_3,\qquad q'_2=q_2+\ell p'_0q_2,\qquad q'_3=q_3+\ell p'_0q_3,\\
k_2=k'_2+\ell p'_0k'_2,\qquad k_3=k'_3+\ell p'_0k'_3,\qquad k'_2=k_2-\ell p'_0k_2,\qquad k'_3=k_3-\ell p'_0k_3.\end{gathered}$$ Since $u^i=0$ and $u'^i=0$ for $i=2,3$ the last two equations of eq.(\[4Dloop\]) imply $x'^2=0$ and $x'^3=0$, which in turn (looking then at the first two equations of eq.(\[4Dloop\])) take us back to (\[sistemalinqcompletoA\])-(\[sistemalinqcompletoB\]) $$\label{sistemalinqcompletoA2}
\Delta\tau=-\Delta\tau'\frac{u'^0}{u^0}+\ell x'^1(s_0)\left(\frac{q'_1-k_1}{u^0}\right)-\ell\Delta\tau'\left(\frac{u'^1k_1}{u^0}\right),$$ $$\label{sistemalinqcompletoB2}
\ell x'^1(s_0)=\Delta\tau'\frac{u^1u'^0-u^0u'^1+\ell u'^1(k_1u^1+k'_0u^0)}{u^0(q_0-k'_0)+u^1(q'_1-k_1)} \ .$$ Evidently then all the features discussed for the 1+1-dimensional in the previous subsection are also present in the 3+1-dimensional case.
Enforcing Relative Locality
---------------------------
We shall now show that our causal loop is not allowed in theories with curved momentum space if one makes sure that these theories are (DSR-)relativistic, with associated relative locality. This suggests that relative locality is evidently a weaker notion than absolute locality but it is still strong enough to enforce causality.
By definition [@principle] relative locality is such that the locality of events may not be manifest in coordinatizations by distant observers, but for the coordinatizations by observers near an event (ideally at the event) it enforces locality in a way that is than ordinary locality.
Also notice that the definition of relative locality that translation transformations be formalized in the theory: since one must verify that events are local according to nearby observers (while they may be described as non-local by distant observers) one must use translation transformations in order to ensure that the principle of relative locality [@principle] is enforced. Since our interest is in (DSR-)relativistic theories, of course such translation transformations must be symmetries.
In Ref. [@anatomy] some of us introduced a prescription for having a very powerful implementation of translational invariance in relative-locality theories. One can easily see that the causal loop described in the previous subsections is not compatible with that strong implementation of translational invariance. Evidently then we have that causality is preserved in theories with curved momentum spaces if the strong notion of translational invariance of Ref. [@anatomy] is enforced by postulate.
What we here want to show is that the causal loop of Fig.5 is still forbidden even without enforcing such a strong notion of translational invariance. Causal loops are forbidden even by a minimal notion of translational invariance, the bare minimum needed in order to contemplate relative locality with a DSR-relativistic picture.
Consistently with this objective we ask only for the availability of some translation generator (with possibly complicated momentum dependence) that can enforce the covariance of the equations of motion and the boundary conditions. Let us call our first observer Alice and the second one Bob, purely translated by a parameter $b^\mu$ with respect to Alice. For the particles involved inside the loop we have $$\begin{gathered}
x_B^\mu(s)=x_A^\mu(s)-b^\nu\mathcal T_\nu^\mu \ ,\label{weaker_translation_x} \\
x_B'^\mu(s)=x_A'^\mu(s)-b^\nu\mathcal T'^\mu_\nu \ ,\label{weaker_translation_x'}\end{gathered}$$ where $\mathcal T_\nu^\mu$ and $\mathcal T_\nu'^\mu$ are to be determined through the request of translational invariance.
Combining the first two boundary conditions of (\[boundaries3\]) with (\[weaker\_translation\_x\]) we obtain $$\begin{gathered}
-\xi_{B(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\mu}}=
x_B^\mu(s_1)=
x_A^\mu(s_1)-b^\nu\mathcal T_\nu^\mu=
-\xi_{A(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\mu}}-b^\nu\mathcal T_\nu^\mu \label{trans_vs_boundaries_x1} \ ,\\
\xi_{B(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\mu}}=
x_B^\mu(s_0)=
x_A^\mu(s_0)-b^\nu\mathcal T_\nu^\mu=
\xi_{A(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\mu}}-b^\nu\mathcal T_\nu^\mu ~. \label{trans_vs_boundaries_x0}\end{gathered}$$ We find convenient to introduce $\delta\xi_{(i)}^\nu \equiv \xi_{B(i)}^\nu-\xi_{A(i)}^\nu$ and to rewrite equations (\[trans\_vs\_boundaries\_x1\]) and (\[trans\_vs\_boundaries\_x0\]) as follows $$\begin{gathered}
b^\nu\mathcal T_\nu^\mu=\delta\xi_{(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\mu}} \ ,\\
b^\nu\mathcal T_\nu^\mu=-\delta\xi_{(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\mu}} \ .\end{gathered}$$ This shows that any form one might speculate about for what concerns translational invariance will still inevitably require enforcing $$\label{eq:}
\delta\xi_{(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\mu}}=-\delta\xi_{(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\mu}} \ .$$ Similarly, combining the last two boundary conditions of (\[boundaries3\]) with the transformation (\[weaker\_translation\_x’\]) we obtain $$\begin{gathered}
-\xi_{B(0)}^{\nu}{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}}=
x_{B}'^\mu(s_{0})=
x_A'^\mu(s_0)-b^\nu\mathcal T_\nu'^\mu=
-\xi_{A(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}}-b^\nu\mathcal T_\nu'^\mu \ ,\\
\xi_{B(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}=
x_B'^\mu(s_1)=
x_A'^\mu(s_1)-b^\nu\mathcal T_\nu'^\mu=
\xi_{A(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}-b^\nu\mathcal T_\nu'^\mu \ ,\end{gathered}$$ from which it follows that $$\label{eq:-1}
-\delta\xi_{(1)}^\nu{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}=\delta\xi_{(0)}^\nu{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}} \ .$$ The fact that we are insisting only on a minimal requirement of translational invariance is reflected also in the fact that our requirements are more general (weaker) than the ones so far used for translational invariance in previous works on the relative-locality framework. Our requirements (\[eq:\]) and (\[eq:-1\]) reproduce the ones enforced in Ref. [@cortes] upon opting for boundary terms written in the form $\displaystyle\bigoplus_{i=1}^{i=n}P^i_{in}-\bigoplus_{i=1}^{i=m}P^i_{out}$, where $P^i_{in}$ are the ingoing momenta in a vertex and $P^i_{out}$ are the outgoing momenta. And our requirements (\[eq:\]) and (\[eq:-1\]) reproduce the strong translation transformations enforced in Ref. [@anatomy], by adopting $\delta\xi_{(1)}^\nu=\delta\xi_{(0)}^\nu=-b^\nu$, *i.e.*, momentum independence of the $\xi^\mu$.
Let us next observe that from equation (\[eq:-1\]) one has that $$\delta\xi_{(0)}^\nu=
-\delta\xi_{(1)}^\sigma{\frac{\partial \mathcal K_\sigma^{(1)}}{\partial p'_\mu}}\left({\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}}\right)^{-1} \ ,$$ and using this in equation (\[eq:\]) leads us to $$\delta\xi_{(1)}^\sigma
\left[{\frac{\partial \mathcal K_\sigma^{(1)}}{\partial p_\rho}}-{\frac{\partial \mathcal K_\sigma^{(1)}}{\partial p'_\mu}}\left({\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}}\right)^{-1}{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\rho}}\right]=0 \ .$$ Since $\delta\xi_{(1)}^\sigma\neq 0$ (in order for this to be a non-collapsed loop the two observers must be distant) we conclude that $$\label{eq:-2}
{\frac{\partial \mathcal K_\nu^{(1)}}{\partial p_\rho}}\left({\frac{\partial \mathcal K_\nu^{(1)}}{\partial p'_\mu}}\right)^{-1}-
{\frac{\partial \mathcal K_\nu^{(0)}}{\partial p_\rho}}\left({\frac{\partial \mathcal K_\nu^{(0)}}{\partial p'_\mu}}\right)^{-1}=0 \ .$$ This equation (\[eq:-2\]) plays a pivotal role in our analysis since it shows that an however weak requirement of translational invariance (required for relative locality in a relativistic setup) imposes a restriction on the possible choices of boundary terms. We shall now easily show that once the condition (\[eq:-2\]) on boundary terms is enforced the causal loop is forbidden. We start by showing that for the boundary terms used in Ref. [@linq] the condition (\[eq:-2\]) takes the shape of a condition on the momenta involved in the process, specifically, at leading order in $\ell$, $$\label{CLcovariance_condition}
\ell\delta_\mu^1\left[\delta_1^\rho\left(k'_0-q_0\right)+\delta_0^\rho\left(q'_1-k_1\right)\right]
=0 \ ,$$ which implies that $k'_0=q_0+\mathcal O(\ell)$ and $q'_1=k_1+\mathcal O(\ell)$. The fact that the causal loop is forbidden can then be seen easily for example by looking back at equation (\[eq:-5’\]), now enforcing (\[CLcovariance\_condition\]): one obtains $$\Delta\tau u^\rho+\Delta\tau'\left[u'^\rho+u'^1\ell\left(\delta_0^\rho k_1-\delta_1^\rho k'_0\right)\right]=0 \ .
\label{jocextra}$$ This excludes the causal loop for just the same reasons that, as observed earlier in this section, the causal loop is excluded in ordinary special relativity: for $\rho=0$ equation (\[jocextra\]), $$\Delta\tau=-\Delta\tau'\frac{u'^0}{u^0}-\ell\Delta\tau'\left(\frac{u'^1k_1}{u^0}\right),$$ does not admit solutions with positive $\Delta\tau$ and $\Delta\tau'$ and positive zeroth component of the two 4-velocities. This causal loop is indeed forbidden once a DSR-relativistic description, with associated relative locality, is enforced.
Möbius diagram and translational invariance
===========================================
Having shown that the causal loop of Ref.[@linq] is indeed allowed in generic theories on curved momentum spaces, but is forbidden when a DSR-relativistic description, with associated relative spacetime locality, is enforced, we now proceed to the next announced task, which concerns the diagram studied in Ref.[@andrb] as a possible source of violations of global momentum conservation. Ref.[@andrb] considered theories on a curved momentum space, without enforcing relative spacetime locality, and found that in general the diagram shown in our Fig.\[loopandrb\] can produce violations of global momentum conservation. These violations take the shape [@andrb] of $k' \neq k$, [*i.e.*]{} the momentum incoming into the diagram is not equal to the momentum outgoing from the diagram. Similarly to what we showed in the previous section for a causal loop, we shall find that these violations of global momentum conservation from the diagram in Fig.\[loopandrb\] do not occur if one enforces a DSR-relativistic description, with associated relative spacetime locality.
![We here show schematically two causally-connected events that form a “Möbius diagram". The laws of conservation at the two vertices are setup in such a way that the particle outgoing from the first vertex has its momentum appearing on the right-hand side of the composition law and its momentum also appears on the left-hand side of the composition of momenta at the second vertex.[]{data-label="loopandrb"}](twisted_loop.pdf)
The relative-locality-framework description of the diagram in Fig.\[loopandrb\] is obtained through the action $$\begin{array}{lll}
\mathcal S
&=&
\displaystyle \int_{-\infty}^{s_0}ds\left(z^\mu\dot k_\mu+\mathcal N_k(\mathcal C_k-m_k^2)\right)
+\int_{s_1}^{+\infty}ds\left(z'^\mu\dot k'_\mu+\mathcal N_{k'}(\mathcal C_{k'}-m_{k'}^2)\right)+\\
&+&
\displaystyle
\int_{s_0}^{s_1}ds\left(x'^\mu\dot p'_\mu+\mathcal N_{p'}(\mathcal C_{p'}-m_{p'}^2)\right)+
\int_{s_0}^{s_1}ds\left(x^\mu\dot p_\mu+\mathcal N_p(\mathcal C_p-m_p^2)\right)+\\
&-&
\xi_{(0)}^\mu\mathcal K_\mu^{(0)}-\xi_{(1)}^\mu\mathcal K_\mu^{(1)},
\end{array}$$ with
$$\begin{aligned}
&\mathcal K_{\ \mu}^{(0)}=
\left(k \oplus\left(\ominus\left(p\oplus p'\right)\right)\right)_\mu \simeq
k_\mu-p_\mu-p'_\mu+\delta_\mu^1 \ell \left[p_1\left(k_0-p_0-p'_0\right)+p'_1\left(k_0-p'_0\right)\right],\label{MDLconservation1}\\
&\mathcal K_{\ \mu}^{(1)}=
\left(\left( p'\oplus p\right) \oplus \left( \ominus k'\right)\right)_\mu \simeq
p'_\mu+p_\mu-k'_\mu+\delta_\mu^1 \ell \left[k'_1 \left(p'_0+p_0-k_0'\right)-p'_0p_1\right].\label{MDLconservation2}\end{aligned}$$
From the structure of (\[MDLconservation1\])-(\[MDLconservation2\]) it is clear why we choose to label the diagram in Fig.\[loopandrb\] as a “Möbius diagram": the laws of conservation at the two vertices use the noncommutativity of the composition law in such a way that the particle outgoing from the first vertex with momentum appearing on the right-hand side of the composition law enters the second vertex with momentum appearing on the left-hand side of the composition law. \[Of course, the opposite applies to the other particle exchanged between the vertices\]. If one then draws the diagram with the convention that the orientation of pairs of legs entering/exiting a vertex consistently reflects the order in which the momenta are composed, then the only way to draw the diagram makes it resemble a Möbius strip.
Evidently there is no room for such a structure when the momentum space has composition law which is commutative. In particular there is no way to contemplate such a Möbius diagram in special relativity. But on our $\kappa$-momentum space this structure is possible and its implications surely need to be studied.
Consistently with what we reported in the previous sections, our interest is into understanding how the properties of the Möbius diagram are affected if one enforces relative spacetime locality in DSR-relativistic theories on the $\kappa$-momentum space. In particular, we want to show that $k' = k$ (no violation of global momentum conservation).
As also already stressed above, relative spacetime locality in a relativistic theory on curved momentum space necessarily requires at least a weak form of translational invariance. This insistence on at least the weakest possible notion of translational invariance led us to find equations (\[eq:\]) and (\[eq:-1\]) for the causal loop, and, as the interested reader can easily verify, for the case of the Möbius diagram it leads us to the equations
$$\begin{gathered}
\delta \xi^\nu_{(0)}{\frac{\partial \mathcal K^{(0)}_\nu}{\partial p_\mu}}=
-\delta \xi^\nu_{(1)}{\frac{\partial \mathcal K^{(1)}_\nu}{\partial p_\mu}},\\
\delta \xi^\nu_{(0)}{\frac{\partial \mathcal K^{(0)}_\nu}{\partial p'_\mu}}=
-\delta \xi^\nu_{(1)}{\frac{\partial \mathcal K^{(1)}_\nu}{\partial p'_\mu}}.\end{gathered}$$
These allow us to deduce that $$\label{transl_mobius}
\left[{\frac{\partial \mathcal K^{(1)}_\sigma}{\partial p_\mu}}-{\frac{\partial \mathcal K^{(1)}_\sigma}{\partial p'_\rho}}\left({\frac{\partial \mathcal K^{(0)}_\nu}{\partial p'_\rho}}\right)^{-1}{\frac{\partial \mathcal K^{(0)}_\nu}{\partial p_\mu}}\right]=0.$$ The implications of this equation are best appreciated by exposing explicitly the momentum dependence of the terms appearing in (\[transl\_mobius\]):
\[derKandrb\] $$\begin{gathered}
{\frac{\partial \mathcal K^{(1)}_\sigma}{\partial p_\mu}}=
\delta_\sigma^\mu +\ell \delta_\sigma^1 \left(\delta_0^\mu k'_1-\delta_1^\mu p'_0\right),\\
{\frac{\partial \mathcal K^{(1)}_\sigma}{\partial p'_\rho}}=
\delta_\sigma^\rho+\ell \delta_\sigma^1\delta_0^\rho\left(k'_1-p_1\right),\\
\left({\frac{\partial \mathcal K^{(0)}_\nu}{\partial p'_\rho}}\right)^{-1}=
-\delta_\rho^\nu-\ell \delta_\rho^1\left[\delta^\nu_1\left(k_0-p'_0\right)-\delta_0^\nu\left( p_1+p'_1\right)\right],\\
{\frac{\partial \mathcal K^{(0)}_\nu}{\partial p_\mu}}=
-\delta^\mu_\nu-\ell \delta^\mu_0\delta^1_\nu p_1.\end{gathered}$$
These allow us to conclude that from (\[transl\_mobius\]) it follows that $$\label{MDLcovariance_condition}
\ell \left[\delta_1^\mu k_0-\delta_0^\mu \left(p_1+p'_1\right)\right]=0.$$ Using this result in combination with the conservation laws $\mathcal K^{(0)}_{\mu}=0$ and $\mathcal K^{(1)}_{\mu}=0$ one can easily establish that $$\label{MDLp+p'}
p_\mu+p'_\mu=0+\mathcal O(\ell)~,$$ and one can also rewrite those conservation laws as follows $$\begin{gathered}
0=k_\mu-p_\mu-p'_\mu-\delta_\mu^1\ell p'_1p'_0,\label{consJOCa}\\
0=p'_\mu+p_\mu-k'_\mu- \delta_\mu^1\ell p'_0p_1 \, \, .
\label{consJOCb}\end{gathered}$$ Summing these (\[consJOCa\]) and (\[consJOCb\]), also using (\[MDLp+p’\]), we get to the sought result $$k_\mu=k'_\mu+\mathcal O(\ell^2) \,\, ,$$ showing that indeed by insisting on a having a DSR-relativistic picture, with associated relative spacetime locality, one finds no global violation of momentum conservation (at least at leading order in $\ell$, which is the level of accuracy we are here pursuing).
Combinations of Möbius diagrams and implications for building a quantum theory
==============================================================================
In the previous section we reported results suggesting that when theories are (DSR-)relativistic, with the associated relativity of spacetime locality, momentum is globally conserved and there is no violation of causality. It should be noticed that the objective of enforcing relative spacetime locality led us to introduce some restrictions on the choice of boundary terms, particularly for causally-connected interactions. The relevant class of theories has been studied so far only within the confines of classical mechanics, and therefore such prescriptions concerning boundary terms are meaningful and unproblematic (they can indeed be enforced by principle, as a postulate). The quantum version of the theories we here considered is still not known, but if one tries to imagine which shape it might take it seems that enforcing the principle of relative locality in a quantum theory might be very challenging: think in particular of quantum field theories formulated in terms of a generating functional, where all such prescriptions are usually introduced by a single specification of the generating functional. While we do not have anything to report on this point which would address directly the challenges for the construction of such quantum theories, we find it worthy to provide evidence of the fact that combinations of diagrams on curved momentum space might have less anomalous properties, even without enforcing relative locality, than single diagrams.
In an appropriate sense we are attempting to provide first elements in support of a picture which we conjecture ultimately might be somewhat analogous to what happens, for example, in the analysis of the gauge invariance of the first contribution to the matrix element of the Compton scattering $e^-+\gamma\rightarrow e^-+\gamma$ in standard QED. In fact, in that case there are only two Feynman diagrams that need to be taken into account and the matrix element is given by $$\mathcal{M}_{fi}=(-ie)^2\left(\bar{u}_{p'}\slashed{\epsilon}(q')\frac{i}{\slashed{p}+\slashed{q}-m}\slashed{\epsilon} (q)u_p+\bar{u}_{p'}\slashed{\epsilon}(q)\frac{i}{\slashed{p}-\slashed{q}'-m}\slashed{\epsilon}(q')u_p \right),
\label{jocmm}$$ where $p$ and $q$ are the momenta of the electron and the photon respectively in the initial state, $p'$ and $q'$ are the momenta of the electron and the photon respectively in the final state, $u_p$ and $\bar{u}_p$ are Dirac spinors and $\epsilon_\mu$ is the photon polarization 4-vector. For a free photon described in the Lorentz gauge by a plane wave $A_\mu(x) \propto \epsilon_\mu(k)e^{\pm ik_\nu x^\nu}$ the gauge transformation $A_\mu^\Lambda(x)=A_\mu(x) +\partial_\mu \Lambda(x)$ with $\Lambda(x)=\tilde{\Lambda}(k)e^{\pm ik_\nu x^\nu}$ corresponds to a transformation of the polarization 4-vector $\epsilon_\mu ^\Lambda(k)=\epsilon_\mu(k)\pm ik_\mu \tilde{\Lambda}(k)$. Equipped with these observations one can easily see that the two terms in (\[jocmm\]) are not individually gauge invariant, but their combination is gauge invariant.
We are not going to provide conclusive evidence that a similar mechanism is at work for causality and global momentum conservation in theories on curved momentum space (it would be impossible without knowing how to formulate such a quantum theory), but it may be nonetheless interesting that we can find some points of intuitive connection with stories such as that of gauge invariance for Compton scattering.
For definiteness and simplicity we focus on the case of Möbius diagrams. In the previous section we analyzed a Möbius diagram using the choice of boundary terms adopted in Ref. [@andrb] since the appreciation of the presence of a challenge due to Möbius diagrams originated from the study reported in Ref. [@andrb]. In this section we look beyond the realm of considerations offered in Ref. [@andrb], so we go back to our preferred criterion for the choice of boundary conditions, the one first advocated in Ref.[@anatomy], which allows us to streamline the derivations. So we consider the Möbius diagram by adopting the following prescription for the boundary terms: $$\label{trevMob}
\begin{split}
\mathcal K^{(0)}_\mu=k_\mu-(p\oplus p')_\mu\simeq k_\mu-p_\mu-p'_\mu+\ell\delta_\mu^1p_0p'_1,\\
\mathcal K^{(1)}_\mu=(p'\oplus p)_\mu-k'_\mu\simeq p'_\mu+p_\mu-k_\mu-\ell\delta_\mu^1p'_0p_1.
\end{split}$$ From the conservation of four-momentum at each vertex $\mathcal K^{(0)}_\mu=0$, $\mathcal K^{(1)}_\mu=0$ we get $$\label{deltaMomentaMobius1}
k_\mu-k'_\mu=\ell\delta_\mu^1(p'_0p_1-p_0p'_1)=\ell\delta_\mu^1(\frac{m_{p}^2p'_1}{2p_1}-\frac{m_{p'}^2p_1}{2p'_1})\equiv \ell \delta_\mu^1\Delta ,$$ where, since we are considering particles of energy-momentum $\ell^{-1}\gg p_\mu\gg m$, from the on-shell relation (\[geomassJ\]) we expressed the energy of the particles as $p_0=\sqrt{p_1^2+m^2}+\frac{\ell p_1^2}{2}\approx |p_1|+\frac{m^2}{2|p_1|}+\frac{\ell p_1^2}{2}$.
At this point we must stress that evidently this is not the only way to have a Möbius diagram, since we can interchange the prescription for which particle enters the composition law for the first event on the right side of the composition law (then entering the second event on the left side of the composition law). This alternative possibility (which is the only other possibility allowed within the prescriptions of Ref. [@anatomy]) is characterized by boundary terms of the form $$\label{trevMob2}
\begin{split}
\tilde{\mathcal K}^{(0)}_\mu=k_\mu-(p'\oplus p)_\mu\simeq k_\mu-p'_\mu-p_\mu+\ell\delta_\mu^1p'_0p_1,\\
\tilde{\mathcal K}^{(1)}_\mu=(p\oplus p')_\mu-k'_\mu\simeq p'_\mu+p_\mu-k'_\mu-\ell\delta_\mu^1p_0p'_1.
\end{split}$$ Then the condition one obtains in place of (\[deltaMomentaMobius1\]) is $$\label{deltaMomentaMobius2}
k_\mu-k'_\mu=-\ell\delta_\mu^1\Delta.$$
Of course, in light of what we established in the previous section, both of these Möbius diagrams must be excluded if one enforces the principle of relative spacetime locality. But it is interesting to notice that if we were to allow these Möbius diagrams the violation of global momentum conservation produced by one of them, (\[deltaMomentaMobius1\]), is exactly opposite to the one produced by the other one, (\[deltaMomentaMobius2\]). In a quantum-field theory version of the classical theories we here analyzed one might have to include together these opposite contributions, in which case we conjecture that the net result would not be some systematic prediction of violation of global momentum conservation, but rather something of the sort rendering global momentum still conserved but fuzzy.
Going back to the classical-mechanics version of these theories it is amusing to notice that a chain composed of two Möbius diagrams, one of type (\[deltaMomentaMobius1\]) and one of type (\[deltaMomentaMobius2\]), would have as net result no violation of global momentum conservation.
Summary and outlook
===================
The study of Planck-scale-curved momentum spaces is presently at a point of balance between growing supporting evidence and concerns about its consistency with established experimental facts. On one side, as stressed in our opening remarks, the list of quantum-gravity approaches where these momentum-space-curvature effects are encountered keeps growing, and interest in this possibility is also rooted in some opportunities for a dedicated phenomenological programme with Planck-scale sensitivity[@principle; @gacLRR]. On the other hand it is increasingly clear that in general theories on curved momentum space may violate several apparently robust aspects of our current description of the laws of physics, including relativistic invariance, locality, causality and global momentum conservation. We here contributed to the characterization of how severe these challenges can be for generic theories on curved momentum spaces, but we also reported results suggesting that when the theory is formulated (DSR-)relativistically, with the associated relativity of spacetime locality, momentum is globally conserved and there is no violation of causality. It seems then that (at least in this first stages of exploration) it might be appropriate to restrict the focus of research on curved momentum space on this subclass with more conventional properties, which one should expect when the momentum space is maximally symmetric.
It should be noticed that here (just like in Refs. [@linq; @andrb]) we only considered the simplest chain of events that could have led to violations of causality and global momentum conservation. That already involved some significant technical challenges, but does not suffice to show that in general causality and glabal momentum conservation are ensured when these theories are formulated (DSR-)relativistically, with the associated relativity of spacetime locality. The fact that the violations are in general present for the simple chains of events we analyzed but disappear when relative locality is enforced is surely of strong encouragement but does not represent a general result.
Of course, the main challenge on the way toward greater maturity for this novel research programme is the development of a quantum-field-theory version. As we were in the final stages of the writeup of this manuscript a general framework for introducing such quantum field theories was proposed in Ref. [@freidelFT]. While presently this proposal appears to be still at too early and too formal a stage of development for addressing the challenges that were here of interest, it is legitimate to hope that, as its understanding deepens, a consistent quantum picture of causality and momentum conservation with curved momentum spaces will arise.
[50]{}
S. Majid, arXiv:hep-th/0006166, Lect. Notes Phys. 541 (2000) 227.
G. Amelino-Camelia, arXiv:gr-qc/0012051, Int. J. Mod. Phys. D11 (2002) 35
G. Amelino-Camelia, arXiv:hep-th/0012238, Phys. Lett. B510 (2001) 255.
J. Kowalski-Glikman, arXiv:hep-th/0207279, Phys. Lett. B547 (2002) 291.
F. Girelli, E.R. Livine, arXiv:gr-qc/0412079, Braz. J. Phys. 35 (2005) 432.
D. Ratzel, S. Rivera and F.P. Schuller, Phys. Rev. D83 (2011) 044047.
L.N. Chang, Z. Lewis, D. Minic and T. Takeuchi, arXiv:1106.0068, Adv. High Energy Phys. 2011 (2011) 493514.
G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, L. Smolin, arXiv:1101.0931, Phys. Rev. D84 (2011) 084010.
G. Amelino-Camelia, S. Majid, arXiv:hep-th/9907110, Int. J. Mod. Phys. A15 (2000) 4301.
C. Rovelli, Living Rev. Relativity 11, (2008) 5.
L. Smolin, Lect. Notes Phys. 669 (2005) 363.
L. Freidel, E. R. Livine, arXiv:hep-th/0512113, Phys. Rev. Lett. 96 (2006) 221301.
H. -J. Matschull, M. Welling, arXiv:gr-qc/9708054, Class. Quant. Grav. 15 (1998) 2981.
F. A. Bais, N. M. Muller, B. J. Schroers, arXiv:hep-th/0205021, Nucl. Phys. B640 (2002) 3.
P.K. Osei, B.J. Schroers J. Math. Phys. 53 (2012) 073510.
G. Amelino-Camelia, M. Arzano, S. Bianco, R. J. Buonocore, arXiv:1210.7834, Class.Quant.Grav. 30 (2013) 065012.
G. Gubitosi, F. Mercati, arXiv:1106.5710, Class. Quant. Grav. 30 (2013) 145002.
G. Amelino-Camelia, arXiv:1110.5081, Phys. Rev. D85 (2012) 084034.
J. M. Carmona, J. L. Cortes, D. Mazon, F. Mercati, arXiv:1107.0939, Phys. Rev. D84 (2011) 085010.
J. Kowalski-Glikman, arXiv:hep-th/0102098, Phys. Lett. A286 (2001) 391-394.
L. Smolin and J. Magueijo, arXiv:gr-qc/0207085, Phys. Rev. D67 (2003) 044017.
L. Smolin and J. Magueijo, arXiv:gr-qc/0305055, Class. Quant. Grav. 21 (2004) 1725-1736.
J. Kowalski-Glikman, arXiv:hep-th/0405273, Lect. Notes Phys. 669 (2005) 131-159.
G. Amelino-Camelia, arXiv:1003.3942, Symmetry 2 (2010) 230-271.
G. Amelino-Camelia, M. Matassa, F. Mercati, G. Rosati, arXiv:1006.2126, Phys. Rev. Lett. 106 (2011) 071301.
L. -Q. Chen, arXiv:1212.5233, Phys. Rev. D88 (2013) 024052.
A. Banburski, arXiv:1305.7289, Phys. Rev. D88 (2013) 076012.
G. Amelino-Camelia, M. Arzano, J. Kowalski-Glikman, G. Rosati and G. Trevisan, arXiv:1107.1724, Class. Quant. Grav. [29]{} (2012) 075007.
S. Majid, H. Ruegg, arXiv:hep-th/9405107, Phys. Lett. B334 (1994) 348-354.
J. Lukierski, H. Ruegg, W.J. Zakrzewski, arXiv:hep-th/9312153, Annals Phys. 243 (1995) 90.
G. Amelino-Camelia, arXiv:0806.0339, Living Rev. Rel. [16]{} (2013) 5.
L. Freidel and T. Rempel, arXiv:1312.3674.
[^1]: Note that for associative composition laws, as is the case of the $\kappa$-momentum-space composition law (\[jjj\]), on can rewrite $(k\oplus p)_\mu-q_\mu =0$ equivalently as $((k\oplus p) \oplus (\ominus q))_\mu =0$. This is due to the logical chain $((k\oplus p) \oplus (\ominus q))_\mu =0 ~ \Rightarrow ((k\oplus p) \oplus (\ominus q) \oplus q)_\mu=q_\mu ~ \Rightarrow
(k\oplus p)_\mu =q_\mu $.
[^2]: Note that within our conventions the direction of propagation and the sign of the spatial momentum with lower index, $p_1$, are opposite. So negative $p_1$ is actually for propagation along the positive direction of the $x^1$-axis.
|
---
abstract: |
We study weighted norm inequalities of $(p,r)$-type, $$\Vert \mathbf{G} (f \, d \sigma) \Vert_{L^r(\Omega, d\sigma)} \le C \Vert f \Vert_{L^p(\Omega, \sigma)}, \quad \text{ for all } f \in L^p(\sigma),$$ for $0 < r < p$ and $p>1$, where $\mathbf{G}(f d \sigma)(x)=\int_\Omega G(x, y) f(y) d \sigma(y)$ is an integral operator associated with a nonnegative kernel $G(x,y)$ on $\Omega\times \Omega$, and $\sigma$ is a locally finite positive measure in $\Omega$.
We show that this embedding holds if and only if $$\int_\Omega (\mathbf{G} \sigma)^{\frac{pr}{p-r}} d \sigma<+\infty,$$ provided $G$ is a quasi-symmetric kernel which satisfies the weak maximum principle.
In the case $p=\frac{r}{q}$, where $0<q<1$, we prove that this condition characterizes the existence of a non-trivial solution (or supersolution) $u \in L^r(\Omega, \sigma)$, for $r>q$, to the the sublinear integral equation $$u - \mathbf{G}(u^q \, d \sigma) = 0, \quad u \ge 0.$$
We also give some counterexamples in the end-point case $p=1$, which corresponds to solutions $u \in L^q (\Omega, \sigma)$ of this integral equation, studied recently in [@QV1], [@QV2]. These problems appear in the investigation of weak solutions to the sublinear equation involving the (fractional) Laplacian, $$(-\Delta)^{\alpha} u - \sigma \, u^q = 0, \quad u \ge 0,$$ for $0<q<1$ and $0 < \alpha < \frac{n}{2}$ in domains $\Omega \subseteq {{\mathbb R}}^n$ with a positive Green function.
address: 'Department of Mathematics, University of Missouri, Columbia, MO 65211, USA'
author:
- 'Igor E. Verbitsky'
title: |
Sublinear equations and Schur’s test\
for integral operators
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Remark]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Definition]{}
Introduction
============
Let $\Omega$ be a locally compact, Hausdorff space. For a positive, lower semicontinuous kernel $G\colon \Omega\times \Omega\to (0, +\infty]$, we denote by $${{\mathbf G}}(f \, d \sigma)(x) =\int_{\Omega} G(x, y) \, f(y) \, d \sigma(y), \quad x \in \Omega,$$ the corresponding integral operator, where $\sigma \in \mathcal{M}^+(\Omega)$, the class of positive locally finite Radon measures in $\Omega$.
We study the $(p, r)$-weighted norm inequalities $$\label{strong-type}
\Vert \mathbf{G} (f d\sigma) \Vert_{L^r(\Omega, \sigma)} \le C \, \Vert f \Vert_{L^p(\Omega, \sigma)}, \quad \forall f \in L^p(\Omega, \sigma),$$ in the case $0<r<p$ and $p \ge 1$, where $C$ is a positive constant which does not depend on $f$.
The main goal of this paper is to find explicit characterizations of in terms of $\mathbf{G} \sigma $ under certain assumptions on $G$. We also study connection of inequality with $p=\frac{r}{q}$, where $0<q<1$, to the existence of a positive function $u \in L^r(\Omega, \sigma)$ such that $$\label{super_sol}
u \ge {{\mathbf G}}(u^q \sigma) \quad d \sigma-\text{a.e.} \, \, \text{ in $\Omega$},$$ in the case $r>q$. In other words, $u$ is a supersolution for the sublinear integral equation $$\label{int_eqn}
u - {{\mathbf G}}(u^q \sigma) =0, \quad 0<u<+\infty \quad d \sigma-\text{a.e.} \, \, \text{ in $\Omega$},$$ where $0<q<1$.
In this paper, we assume that the kernel $G$ of the integral operator is quasi-symmetric, and satisfies a weak maximum principle (WMP); see Sec. \[background\]. Such restrictions are satisfied by the Green kernel associated with many elliptic operators, including the fractional Laplacian $(- \Delta)^{\alpha}$, as well as quasi-metric kernels, and radially symmetric, decreasing convolution kernels $G(x,y) = k(|x-y|)$ on $\mathbb{R}^n$ (see, e.g., [@AH], [@An], [@Maz], [@QV1], [@QV2] and the literature cited there).
If $G$ is Green’s kernel associated with the Laplacian in an open domain $\Omega \subseteq \mathbb{R}^n$, is equivalent to the sublinear elliptic boundary value problem $$\label{lap_eqn}
\begin{cases}
- \Delta u - \sigma u^q=0, &u>0 \, \text{ in } \Omega, \\
u = 0 & \text{ on } \partial \Omega,
\end{cases}$$ where $0 < q < 1$.
We observe that solutions $u \in L^r(\Omega, \sigma)$ to in the case $r=1+q$ correspond to finite energy solutions $u \in L^{1, 2}_{0} (\Omega)$ in the Dirichlet space, i.e., $$\int_\Omega |\nabla u|^2 dx <+\infty,$$ where $u$ has zero boundary values (see [@CV1]).
The more difficult end-point case $p=1$ of , along with solutions $u \in L^q(\Omega, \sigma)$ in the case $r=q$, was studied recently in [@QV1], [@QV2]. After a certain modification, it leads to solutions $u \in L^q_{{\rm loc}}(\Omega, \sigma)$, i.e., all solutions to , or understood in a weak sense (see [@MV]). For Riesz kernels on $\Omega={{\mathbb R}}^n$ such $(1, q)$-weighted norm inequalities, along with weak solutions to the sublinear problem $$\label{frac_lap_eqn}
\begin{cases}
(- \Delta)^{\alpha} u - \sigma u^q=0, \quad u>0 \, \text{ in } {{\mathbb R}}^n, \\
\displaystyle{\liminf_{x \to \infty}} \,\, u = 0, \quad u \in L^q_{{\rm loc}}(\sigma), &
\end{cases}$$ for $0<\alpha<\frac{n}{2}$, were treated earlier in [@CV1], [@CV2], [@CV3].
Our main result is the following theorem.
\[strong-thm\] Let $\sigma \in \mathcal{M}^{+}(\Omega)$. Suppose $G$ is a positive, quasi-symmetric, lower semicontinuous kernel on $\Omega\times\Omega$ which satisfies the weak maximum principle.
\(i) If $\, 1<p<+\infty$ and $0<r<p$, then the $(p, r)$-weighted norm inequality holds if and only if $$\label{r-p}
\int_\Omega (\mathbf{G} \sigma)^{\frac{pr}{p-r}} d \sigma<+\infty.$$
\(ii) If $\, 0<q<1$ and $q<r < \infty$, then there exists a positive (super)solution $u\in L^r(\Omega, d \sigma)$ to if and only if holds with $p=\frac{r}{q}$, or equivalently, $$\label{r-q}
\int_\Omega (\mathbf{G} \sigma)^{\frac{r}{1-q}} d \sigma<+\infty.$$
\[rm1\] [We observe that the “if” parts of statements (i) and (ii) of Theorem \[strong-thm\] fail if $p=1$, and $r=q$, respectively. The “only if” parts hold for all $0<r<p$ in statement (i), and $r>0$ in statement (ii).]{}
\[rm2\] [It is known that inequality with $p=\frac{r}{q}\ge 1$ in the case $0<q<1$ yields the existence of a positive supersolution $u\in L^r(\Omega, \sigma)$ for . This statement follows from a lemma due to Gagliardo [@G], and does not require $G$ to be quasi-symmetric or to satisfy the WMP (see Sec. \[sec3\] below). However, the converse statement does not hold without the WMP (see [@QV2] in the case $r=q$).]{}
\[rm3\] [Without the assumption that $G$ satisfies the WMP, the “only if” parts of statement (i) (with $p=\frac{r}{q}\ge 1$) and statement (ii) (with $r \ge q$) hold only for $0<r \le 1-q^2$ (see Lemma \[lemma2\] below).]{}
In particular, if there exists a positive (super)solution $u \in L^q(\Omega, \sigma)$, then holds with $r=q$ for $0<q \le q_0$, where $q_0 = \frac{\sqrt{5}-1}{2} = 0.61\ldots$ is the conjugate golden ratio. However, with $r=q$ generally fails (even for symmetric kernels) in the case $q_0<q<1$; the cut-off $q=q_0$ here is sharp [@QV2].
In Sec. \[sec2\] below, we discuss related results, and provide some counterexamples in the case $p=1$.
Kernels and potential theory {#background}
============================
Let $G\colon \Omega \times \Omega \rightarrow (0, +\infty]$ be a positive kernel. We will assume that $\Omega$ is a locally compact space Hausdorff space, and $G$ is lower semicontinuous, so that we can apply elements of the classical potential theory developed for such kernels (see [@Brelot], [@F]). Most of our results hold for *non-negative* kernels $G(x, y) \ge 0$. In that case, some statements concerning the existence of positive solutions (rather than supersolutions) require the additional assumption that $G$ is non-degenerate; see [@QV2].
By $\mathcal{M}^+(\Omega)$ we denote the class of all nonnegative, locally finite, Borel measures on $\Omega$. We use the notation $\operatorname{supp}(\nu)$ for the support of $\nu \in \mathcal{M}^+(\Omega)$ and $\Vert \nu \Vert = \nu (\Omega)$ if $\nu$ is a finite measure.
For $\nu \in \mathcal{M}^+(\Omega)$, the potential of $\nu$ is defined by $$\mathbf{G}\nu (x) {\mathrel{\mathop:}=}\int_{\Omega} G(x,y) d\nu(y), \quad \forall x \in \Omega,$$ and the potential with the adjoint kernel $$\mathbf{G}^*\nu (y) {\mathrel{\mathop:}=}\int_{\Omega} G(x, y) \, d\nu(x), \quad \forall y \in \Omega.$$
A positive kernel $G$ on $\Omega\times \Omega$ is said to satisfy the *weak maximum principle (WMP)* with constant $h\ge 1$ if, for any $\nu \in \mathcal{M}^+(\Omega)$, $$\label{wmp}
\sup \Big\{\mathbf{G}\nu (x) \colon x \in \operatorname{supp}(\nu) \Big\} \le M \, \Longrightarrow
\sup \Big\{ \mathbf{G}\nu (x) \colon x \in \Omega\Big\} \le h \, M,$$ for any constant $M>0$. When $h=1$, $G$ is said to satisfy the *strong maximum principle*. It holds for Green’s kernels associated with the classical Laplacian, or fractional Laplacian $(-\Delta)^{\alpha}$ in the case $0 < \alpha\le 1$, for all domains $\Omega$ with positive Green’s function. The WMP holds for Riesz kernels on ${{\mathbb R}}^n$ associated with $(-\Delta)^{\alpha}$ in the full range $0<\alpha<\frac{n}{2}$, and more generally for all radially non-increasing kernels on ${{\mathbb R}}^n$ (see [@AH]).
The WMP also holds for the so-called quasi-metric kernels (see [@FNV], [@FV], [@HN], [@QV2]). We say that $d(x,y)\colon \, \Omega \times \Omega \rightarrow [0, + \infty)$ satisfies the quasimetric triangle inequality with quasimetric constant $\kappa$ if $$\label{quasitr}
d(x,y) \le \kappa [d(x,z) + d(z, y)],$$ for any $x, y, z \in \Omega$. We say that $G$ is a *quasimetric* kernel (with quasimetric constant $\kappa>0$) if $G$ is symmetric and $d(x,y) = \frac{1}{G(x,y)}$ satisfies .
A kernel $G\colon \Omega \times \Omega \to (0, +\infty]$ is said to be *quasi-symmetric* if there exists a constant $a$ such that $$a^{-1} G(y,x) \le G(x,y) \le a \, G(y,x), \quad \forall x, y \in \Omega.$$ Many kernels associated with elliptic operators are quasi-symmetric and satisfy the WMP (see [@An]).
For $0 < q < 1$, and $\sigma \in \mathcal{M}^+(\Omega)$, we are interested in *positive solutions* $u \in L^r(\sigma)$ ($r>0$) to the integral equation $$\begin{aligned}
\label{int-eq}
u = \mathbf{G}(u^q \sigma), \quad u>0 \quad d\sigma-a.e.
\end{aligned}$$ and *positive supersolutions* $u \in L^r(\sigma)$ to the integral inequality $$\begin{aligned}
\label{int-sup}
u \ge \mathbf{G}(u^q \sigma), \quad u>0 \quad d\sigma-a.e.
\end{aligned}$$
In [@QV2], we characterized the existence of positive solutions $u \in L^q(\Omega, \sigma)$ and $u \in L^q_{\rm loc}(\sigma)$. The latter correspond to the so-called “very weak” solutions to the sublinear boundary value problem (see [@FV], [@MV]). It is easy to see that the condition $u \in L^q_{\rm loc}(\sigma)$ is necessary for the existence of any positive (super)solution, since otherwise $u\equiv+\infty$ $d \sigma$-a.e. (see [@QV2]).
For a measure $\lambda \in \mathcal{M}^+(\Omega)$, the *energy of $\lambda$* is given by $$\mathcal{E}(\lambda) {\mathrel{\mathop:}=}\int_\Omega \mathbf{G}\lambda \, d\lambda.$$ The notion of energy is closely related to another major tool of potential theory, the capacity of a set, and the associated equilibrium measure.
For a kernel $G \colon \Omega\times \Omega \to (0, +\infty]$, we consider the *Wiener capacity* $$\label{wiener}
\operatorname{cap}(K) {\mathrel{\mathop:}=}\sup \Big\{ \mu(K)\colon \, \, \mathbf{G}^*\mu(y) \le 1 \, \, \textrm{on} \, \, \operatorname{supp}(\mu), \, \, \mu \in \mathcal{M}^+(K)\Big \},$$ defined for compact sets $K \subset \Omega$.
The extremal measure $\mu$ for which the supremum in is attained is called the *equilibrium measure*. Alternatively, capacity can be defined as a solution to the following extremal problem involving energy: $$\label{cap-energy}
\operatorname{cap}(K) {\mathrel{\mathop:}=}\left[ \inf \Big \{ \mathcal{E}(\mu)\colon \, \, \, \mu \in \mathcal{M}^+(K), \quad \mu(K) = 1 \Big\}\right]^{-1}.$$
We say that a property holds *nearly everywhere* (or n.e.) on $K$ when the exceptional set $Z \subset K$ where this property fails has zero capacity, $\operatorname{cap}(Z) = 0$.
We will use the following fundamental theorem [@Brelot], [@F].
\[fuglede\_thm\] Let $G$ be a positive symmetric kernel on $\Omega\times \Omega$, and let $K\subset \Omega$ a compact set. The two extremal problems $$\begin{aligned}
& \max \Big\{\lambda( K ) \colon \, \, \mathbf{G}\lambda \le 1 \, \, \text{\textnormal{on}} \, \, \operatorname{supp}(\lambda), \, \, \lambda \in \mathcal{M}^+(K) \Big\}, \\
& \max \Big\{ 2 \lambda (K) - \mathcal{E}(\lambda) \colon \, \, \lambda \in \mathcal{M}^+(K)\Big\},
\end{aligned}$$ always have solutions, which are precisely the same, and each maximum coincides with the Wiener capacity $\operatorname{cap}K$. The class of all solutions consists of measures $\lambda \in \mathcal{M}^+(K)$ for which $$\mathcal{E}(\lambda) = \lambda(\Omega) = \operatorname{cap}(K).$$ The potential of any solution has the following properties:
1. $\mathbf{G}\lambda(x) \ge 1 $ $K$,
2. $\mathbf{G}\lambda(x) \le 1$ $\operatorname{supp}(\lambda)$,
3. $\mathbf{G}\lambda(x) = 1$ $d \lambda\text{\textnormal{-a.e. in}}$ $\Omega$.
The extremal measure $\lambda$ in Theorem \[fuglede\_thm\] is the equilibrium measure for the set $K$. We observe that since $G$ is a positive kernel, the capacity of all compact sets $K$ is finite. (This is true even for non-negative kernels if $G(x, x)>0$ for all $x \in \Omega$; see [@F]).
Weighted norm inequalities, supersolutions, and energy estimates {#sec3}
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We begin this section with a proof of Theorem \[strong-thm\]. We remark that the “only if” part of statement (i) of Theorem \[strong-thm\] is proved without using the assumption that $G$ is quasi-symmetric. Furthermore, the proof of this part works in the case $p=1$ as well.
We first prove statement (i). If the $(p, r)$-inequality holds for $0<r<p$, where $p \ge 1$, then assuming that $f=({{\mathbf G}}\sigma)^{\frac{r}{p-r}} \in L^{p}(\Omega, \sigma)$ and using it as a test function, we deduce $$\int_\Omega \left[ {{\mathbf G}}\Big( ({{\mathbf G}}\sigma)^{\frac{r}{p-r}} d \sigma\Big)\right]^r d \sigma \le C^r \
\left[ \int_\Omega ({{\mathbf G}}\sigma)^{\frac{pr}{p-r}} d \sigma \right]^{\frac{r}{p}},$$ where $C$ is the embedding constant in . We now use the pointwise inequality $$\label{iter-s}
\Big [{{\mathbf G}}\sigma(x) \Big]^{s} \le s \, h^{s-1} \, {{\mathbf G}}\Big( ({{\mathbf G}}\sigma)^{s-1} d \sigma\Big)(x), \quad x \in \Omega,$$ for all $s\ge 1$, established in [@GV2 Lemma 2.5 and Remark 2.6] for non-negative kernels satisfying the WMP with constant $h\ge 1$. Applying with $s=\frac{p}{p-r}$, we obtain $$\int_\Omega ({{\mathbf G}}\sigma)^{\frac{pr}{p-r}} d \sigma \le \Big(\frac{p}{p-r}\Big)^r h^\frac{r^2}{p-r} \, C^r
\left[ \int_\Omega ({{\mathbf G}}\sigma)^{\frac{pr}{p-r}} d \sigma \right]^{\frac{r}{p}}.$$ Since $0<r<p$, this estimate yields $$\int_\Omega ({{\mathbf G}}\sigma)^{\frac{pr}{p-r}} d \sigma\le \Big(\frac{p}{p-r}\Big)^{\frac{pr}{p-r}} h^\frac{pr^2}{(p-r)^2} \, C^{\frac{pr}{p-r}}.$$ The extra assumption that $f=({{\mathbf G}}\sigma)^{\frac{r}{p-r}} \in L^{p}(\Omega, \sigma)$ is easy to remove by using $\chi_K \, f$ in place of $f$, where $K$ is a compact subset of $\Omega$ on which ${{\mathbf G}}\sigma(x) \le n$, and then letting $n \to +\infty$ (see details in [@QV2]).
In the opposite direction, suppose that holds for $0<r<p$ and $p>1$. Without loss of generality we may assume that $f \ge 0$. By Hölder’s inequality, $$\begin{aligned}
\int_\Omega [{{\mathbf G}}( f d \sigma)]^r d \sigma & = \int_{\Omega}\left [\frac{{{\mathbf G}}( f d \sigma)}{{{\mathbf G}}\sigma}\right]^r
({{\mathbf G}}\sigma)^r d \sigma \\ & \le \left[\int_{\Omega}\left (\frac{{{\mathbf G}}( f d \sigma)}{{{\mathbf G}}\sigma}\right)^p d \sigma\right]^{\frac{r}{p}}
\left[\int_\Omega({{\mathbf G}}\sigma)^{\frac{pr}{p-r}} d \sigma \right]^{1-\frac{r}{p}}.
\end{aligned}$$
We next sketch a proof of a $(1,1)$-weak type estimate obtained in a more general context in [@QV2 Lemma 5.10]: $$\label{weak-1-1}
\left \Vert \frac{{{\mathbf G}}( f d \sigma)}{{{\mathbf G}}\sigma} \right \Vert_{L^{1, \infty} (\Omega, d \sigma)} \le
c \, ||f||_{L^1(\Omega, d \sigma)},$$ where $c=c(h, a)$ depends only on the constants $h\ge 1$ in the weak maximum principle, and $a>0$ in the quasi-symmetry condition.
Since $G$ is quasi-symmetric, we can assume without loss of generality that it is symmetric by replacing $G$ with $\frac{1}{2}(G+ G^*)$. Let $E_t = \{ x \in \Omega\colon \frac{{{\mathbf G}}( f d \sigma)}{\mathbf{G}\sigma}
(x) > t \}$, where $t>0$. For an arbitrary compact set $K \subset E_t$, we denote by $\mu \in \mathcal{M}^+(K)$ an equilibrium measure on $K$ (see Sec. \[background\] above) such that $\mathbf{G}\mu \ge 1$ n.e. on $K$ and $\mathbf{G}\mu \le 1$ on $\operatorname{supp} (\mu)$.
It is easy to see that in fact $$\label{n.e.}
\mathbf{G}\mu \ge 1 \quad d \sigma-\textrm{a.e. on} \,\, K.$$ Indeed, from it follows that $\mathbf{G} \sigma<+\infty$ $d \sigma$-a.e. Since $\mathbf{G}\mu \ge 1$ n.e. on $K$, the set $Z = \{ x \in K\colon \, \mathbf{G}\mu(x) < 1 \}$ has zero capacity, and consequently, $$\begin{aligned}
\sigma (Z) &=\sigma (\{x\in Z \colon \,\, \mathbf{G}\sigma(x)<+\infty\})
\\& \le \sum_{n=1}^{+\infty} \sigma(\{x\in Z \colon \, \mathbf{G}\sigma(x) \le n\}) \\
& \le \sum_{n=1}^{+\infty} n \, \operatorname{cap} (\{x \in Z \colon \, \mathbf{G}\sigma(x) \le n\}) =0.
\end{aligned}$$ Thus, $\sigma(Z)=0$, which proves .
Since $\mathbf{G}\mu \le 1$ on $\operatorname{supp}(\mu)$, it follows that $\mathbf{G}\mu \le h$ on $\Omega$ by the WMP. From this and , using Fubini’s theorem, we deduce $$\begin{aligned}
\sigma(K) & \le \int_K {{\mathbf G}}\mu \, d\sigma = \int_K \mathbf{G}\sigma_K \, d\mu \\
&\le \int_K \frac{{{\mathbf G}}( f d \sigma)}{t} \, d\mu = \frac{1}{t} \int_K \mathbf{G} \mu \, f \, d\sigma \\ &\le \frac{1}{t} \int_\Omega h \, f \, d\sigma = \frac{h}{t} \, ||f||_{L^1(\Omega, \sigma)}.
\end{aligned}$$ Taking the supremum over all $K\subset E_t$, we obtain $$\sigma(E_t) \le \frac{h}{t} \, ||f||_{L^1(\Omega, \sigma)},$$ which proves .
The corresponding $L^\infty$ estimate is obvious: $$\left \Vert \frac{{{\mathbf G}}( f d \sigma)}{{{\mathbf G}}\sigma} \right \Vert_{L^{\infty} (\Omega, d \sigma)} \le
||f||_{L^\infty(\Omega, d \sigma)}.$$ Thus, for $1<p<+\infty$, by the Marcinkiewicz interpolation theorem we obtain $$\left \Vert \frac{{{\mathbf G}}( f d \sigma)}{{{\mathbf G}}\sigma} \right \Vert_{L^{p} (\Omega, d \sigma)} \le
C \, ||f||_{L^p(\Omega, d \sigma)},$$ for all $f \in L^p(\Omega, d \sigma)$. Hence, combining the preceding estimates, we deduce $$\int_\Omega [{{\mathbf G}}( f d \sigma)]^r d \sigma \le C \, ||f||^{r}_{L^p(\Omega, \sigma)}
\left[\int_\Omega({{\mathbf G}}\sigma)^{\frac{pr}{p-r}} d \sigma \right]^{1-\frac{r}{p}}.$$ This proves statement (i).
We now prove statement (ii). Let $0<q<1$. Suppose there exists a positive supersolution $u \in L^r(\Omega, \sigma)$ with $r>q$. As shown in [@GV2 Corollary 3.6], if $G$ satisfies the WMP, then any nontrivial supersolution $u$ satisfies the global pointwise bound $$\label{lower-est}
u(x) \ge (1-q)^{\frac{1}{1-q}} h^{-\frac{q}{1-q}} \, [ {{\mathbf G}}\sigma (x)]^{\frac{1}{1-q}} \quad d\sigma-\textrm{a.e.}$$ Thus, holds.
Conversely, by statement (i), with $r>q$ implies the $(p, r)$-inequality with $p=\frac{r}{q}$. Letting $u_0 = c \, [ {{\mathbf G}}\sigma (x)]^{\frac{1}{1-q}}$ where $c>0$ is a positive constant, we get a sequence of iterations $$u_{j+1} = {{\mathbf G}}(u_j^q \, d \sigma), \quad j=0, 1, \ldots,$$ where by induction we see that $u_{j+1} \ge u_j$, provided the constant $c$ is small enough. Here the initial step $u_1\ge u_0$ follows from with $s=\frac{1}{1-q}$, since $$u_1 = {{\mathbf G}}(u_0^q d \sigma)=c^q \, {{\mathbf G}}\Big[({{\mathbf G}}\sigma)^{\frac{q}{1-q}}d \sigma\Big] \ge c \, [ {{\mathbf G}}\sigma (x)]^{\frac{1}{1-q}}=u_0,$$ for an appropriate choice of $c=c(q, h, a)$. By with $p=\frac{r}{q}$ and $f=u_j$, we have by induction, $$\begin{aligned}
||u_{j+1}||_{L^r(\Omega, \sigma)} =
\Big \Vert{{\mathbf G}}(u_j^q \, d \sigma)\Big \Vert_{L^r(\Omega, \sigma)}
\le C \, \Vert u_j\Vert^{q}_{L^r(\Omega, \sigma)}<+\infty.
\end{aligned}$$ Since $0<q<1$ and $u_j \le u_{j+1}$, it follows that $$||u_{j+1}||_{L^r(\Omega, \sigma)} \le C(r, q, h, a), \quad j=0, 1, \ldots.$$ Using the monotone convergence theorem, we obtain a positive solution $$u=\lim_{j\to \infty} u_j, \quad u\in L^r(\Omega, \sigma).$$
Theorem \[strong-thm\] makes use of energy conditions of the type $$\label{energy}
\int_{\Omega} ({{\mathbf G}}\sigma)^s d \sigma< \infty,$$ for some $s>0$. Note that when $s = 1$, this gives the energy $\mathcal{E}(\sigma)$ introduced above.
In the next lemma, we deduce for $s=\frac{r}{1-q}$ provided there exists a positive supersolution $u \in L^r(\Omega, \sigma)$ to , for non-negative, quasi-symmetric kenels $G$, without assuming that holds, or that $G$ satisfies the WMP. In the special case $r=q$ it was proved in [@QV2 Lemma 5.1].
\[lemma2\] Let $\sigma \in \mathcal{M}^+(\Omega)$, and let $0<q<1$. Suppose $G$ is a non-negative quasi-symmetric kernel on $\Omega\times \Omega$. Suppose there is a positive supersolution $u \in L^r(\Omega, \sigma)$ $(r>0)$, i.e., $\mathbf{G} (u^q d \sigma)\le u$ $d \sigma$-a.e. Let $0 < q \le 1-r^2$. Then $$\label{est1}
\int_{\Omega} ({{\mathbf G}}\sigma)^{\frac{r}{1-q}} d \sigma \le a^{\frac{rq}{(1-q)(1-r+q)}} \int_{\Omega}
u^{r}
d \sigma <+\infty,$$ where $a$ is the quasi-symmetry constant of $G$.
Suppose $u\in L^r(\Omega, \sigma)$, where $0<r<1$, is a positive supersolution. Let $\gamma\ge 1$. By Hölder’s inequality with exponents $\gamma$ and $\gamma'=\frac{\gamma}{\gamma-1}$, we estimate $$\begin{aligned}
{{\mathbf G}}\sigma (x) & = \int_{\Omega} u^{\frac{q}{\gamma}} u^{-\frac{q}{\gamma}} G(x, y) \, d \sigma(y)\\& \le \left[ {{\mathbf G}}(u^q
d \sigma(x)\right]^{\frac{1}{\gamma}} \left[{{\mathbf G}}(u^{-\frac{q}{\gamma-1}} d \sigma)(x)\right]^{\frac{1}{\gamma'}} \\&
\le [u(x)]^{\frac{1}{\gamma}} \left[{{\mathbf G}}(u^{-\frac{q}{\gamma-1}} d \sigma)(x)\right]^{\frac{1}{\gamma'}} .
\end{aligned}$$ Let $\gamma = 1+\frac{q}{1-r}$, where $0<r \le 1-q^2$. Then $\frac{(1-q)\gamma'}{r}\ge 1$. Using the preceding inequality, along with Hölder’s inequality with the conjugate exponents $$\frac{(1-q)(1-r+q)}{1-r-q^2} >1 \quad \textrm{and} \quad
\frac{(1-q)(1-r+q)}{rq} \ge 1,$$ and Fubini’s theorem, we estimate
$$\begin{aligned}
\int_\Omega ({{\mathbf G}}\sigma)^{\frac{r}{1-q}} d \sigma & \le \int_{\Omega} u^{\frac{r}{(1-q)\gamma}}
\left[{{\mathbf G}}(u^{r-1} d \sigma)\right]^{\frac{r}{(1-q)\gamma'}}
d \sigma \\&= \int_{\Omega} u^{\frac{r(1-r-q^2)}{(1-q)(1-r +q)}}
\left[ u^q \, {{\mathbf G}}(u^{r-1} d \sigma)\right]^{\frac{rq}{(1-q)(1-r+q)}}
d \sigma \\&
\le \left[ \int_{\Omega} u^{r} d \sigma\right]^{\frac{1-r-q^2}{(1-q)(1-r +q)}}
\left[\int_{\Omega} {{\mathbf G}}(u^{r-1} d \sigma) \, u^q \,d \sigma\right]^{\frac{rq}{(1-q)(1-r+q)}}
\\
& =
\left[ \int_{\Omega} u^{r} d \sigma\right]^{\frac{1-r-q^2}{(1-q)(1-r +q)}}
\left[ \int_{\Omega} {{\mathbf G}}^{*}(u^q d \sigma) u^{r-1} d \sigma\right]^{\frac{rq}{(1-q)(1-r+q)}}
\\&
\le a^{\frac{rq}{(1-q)(1-r+q)}} \left[ \int_{\Omega} u^{r} d \sigma\right]^{\frac{1-r-q^2+rq}{(1-q)(1-r +q)}}.
\end{aligned}$$
In the last estimate we used the inequality $ {{\mathbf G}}^{*}(u^q d \sigma) \le a \, u$. Since $1-r-q^2+rq=(1-q)(1-r +q)$, this completes the proof of .
We next show that, for general non-negative kernels $G$, the $(p, r)$-weighted norm inequality with $p=\frac{r}{q}\ge 1$ yields the existence of a supersolution $u \in L^r(\Omega, \sigma)$ to . This is deduced from Gagliardo’s lemma [@G] (see also [@Szeptycki]), as in the special case $r=q$ in [@QV2].
It will be convenient for us to construct a measurable function $\phi$ such that $$\label{phi}
0< [\mathbf{G}(\phi \, d\sigma)]^q \le \phi <+\infty \quad d \sigma-\textnormal{a.e.},$$ for $0<q<1$. Clearly, if $\phi$ satisfies the above estimate, then $u=\phi^{\frac{1}{q}}$ satisfies . Moreover, $u \in L^r(\Omega, \sigma)$ if $\phi \in L^p(\Omega, \sigma)$, where $p=\frac{r}{q}\ge 1$.
We recall that a convex cone $P\subset B$ is *strictly convex at the origin* if, for any $\phi, \psi \in P$, $\alpha \phi + \beta \psi = 0$ implies $\phi = \psi = 0$, for any $\alpha, \beta > 0$ such that $\alpha + \beta = 1$.
\[gag\_lemma\] Let $B$ be a Banach space, and let $P \subset B$ be a convex cone which is strictly convex at the origin and such that if $ (\phi_n) \subset P$, $\phi_{n+1} - \phi_n \in P$, and $\Vert \phi_n \Vert \le M$ for all $n = 1, 2, \dots$, then there exists $\phi \in P$ so that $\Vert \phi_n - \phi \Vert \rightarrow 0$.
Let $S\colon \, P \rightarrow P$ be a continuous mapping with the following properties:
1. For $\phi, \psi \in P$, such that $\phi - \psi \in P$, we have $S\phi - S\psi \in P$.
2. If $\Vert \phi \Vert \le 1$ and $\phi \in P$, then $\Vert Su \Vert \le 1$.
Then for every $\lambda > 0$ there exists $\phi \in P$ so that $(1 + \lambda)\phi - S\phi \in P$ and $0 < \Vert \phi \Vert \le 1$. Moreover, for every $\psi \in P$ such that $0<\Vert \psi \Vert_B \le \frac{\lambda}{1+\lambda}$, $\phi$ can be chosen so that $\phi =\psi + \frac{1}{1+\lambda} S \phi$.
We will apply this lemma to $B= L^p(\sigma)$, $p\ge 1$, and the cone of non-negative functions $P$ in $B$. In this case obviously one can ensure that $\phi>0$ $d \sigma$-a.e.
\[phi\_soln\] Let $(\Omega, \sigma)$ be a sigma-finite measure space, and let $G$ be a non-negative kernel on $\Omega\times \Omega$. Let $0<r<+\infty$ and $0<q<1$. Suppose holds for $p=\frac{r}{q}\ge 1$ with an embedding constant $C=\varkappa>0$. Then, for every $\lambda > 0$, there is a positive $\phi \in L^p(\sigma)$ satisfying so that $$\Vert \phi \Vert_{L^p(\sigma)} \le (1+\lambda)^{\frac{1}{1-q}} \varkappa^{\frac{q}{1-q}}.$$
The supersolution $\phi$ is constructed using Lemma \[gag\_lemma\]. Define $S\colon \, L^p(\sigma) \rightarrow L^p(\sigma)$ by $$S \phi {\mathrel{\mathop:}=}\Big[\frac{1}{\varkappa^q} \, \mathbf{G}(\phi \, d\sigma)\Big]^q,$$ for all $\phi \in L^p(\sigma)$, $\phi \ge 0$. Inequality gives that $S$ is a bounded continuous operator. In fact, by we see that if $\Vert \phi \Vert_{ L^p(\sigma)} \le 1$, then $$\begin{aligned}
\Vert S(\phi) \Vert^p_{L^p(\sigma)}
&= \frac{1}{\varkappa^r} \int_\Omega [\mathbf{G}(\phi \sigma)]^r \, d\sigma \\
&= \frac{1}{\varkappa^r} \varkappa^r \left(\int_\Omega \phi^p \, d\sigma\right)^q\le 1.
\end{aligned}$$ Therefore, by Lemma \[gag\_lemma\], there exists $\phi \in L^p(\sigma)$ such that $$(1+\lambda) \phi \ge \frac{1}{\varkappa^q} [\mathbf{G}(\phi \sigma)]^q,$$ $\Vert \phi \Vert_{L^p(\sigma)} \le 1$, and $\phi > 0$ $d \sigma$-a.e. Setting $\phi_0 = c \, \phi$, where $$c= \left[ \frac{1}{(1 + \lambda) \varkappa^q} \right]^{\frac{1}{1-q}},$$ we deduce that $\phi > 0$ $d \sigma$-a.e., and $$\phi_0 \ge \mathbf{G}(\phi_0 \sigma)^q, \quad \Vert \phi_0 \Vert_{L^p(\sigma)} \le (1 + \lambda)^{\frac{1}{1-q}} \varkappa^{\frac{q}{1-q}}.$$
Remark \[rm2\] follows immediately from Lemma \[phi\_soln\].
\[rm4\] [For $p=\frac{r}{q}$, a counterexample in [@QV2] demonstrates that, without the WMP, the existence of a supersolution $u\in L^r(\Omega, \sigma)$ to in the case $r=q$ does not imply the $(p, r)$-weighted norm inequality , even for positive symmetric kernels $G$. A slight modification of that counterexample shows that the same is true in the case $r>q$ as well. ]{}
A counterexample in the end-point case $p=1$ {#sec2}
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In the case $p=1$, $0<q<1$, the $(1, q)$-weighted norm inequality with $r=q$ follows from a similar inequality for the space of measures $\mathcal{M}^{+}(\Omega)$ in place of $L^1(\Omega, \sigma)$, $$\label{strong-meas}
\Vert \mathbf{G} \nu \Vert_{L^q(\Omega, \sigma)} \le C \, ||\nu||, \quad \forall \nu \in \mathcal{M}^{+}(\Omega),$$ where $||\nu||=\nu(\Omega)$. This inequality was shown in [@QV2] to be equivalent to the existence of a positive supersolution $u \in L^q(\Omega, \sigma)$ to for quasi-symmetric kernels $G$ satisfying the WMP. In this case, is equivalent to with $r=q$ and $p=1$ in view of Lemma \[phi\_soln\].
However, a characterization of , or with $r=q$ and $p=1$, in terms of the energy estimate with $r=q$ is not available, contrary to the case $r>q$: the condition $$\label{q-energy}
\int_\Omega (\mathbf{G} \sigma)^{\frac{q}{1-q}} d \sigma<+\infty$$ is not sufficient for .
On the other hand, it is not difficult to see that holds for all $\nu \in \mathcal{M}^{+}(\Omega)$ if and only if it holds for all finite linear combinations of point masses, $\nu =\sum_{j=1}^n a_j \, \delta_{x_j}$, $a_j>0$. It had been conjectured that, for $0<q<1$, condition combined with for single point masses $\nu=\delta_{x}$, i.e., $$\label{point-mass}
\int_\Omega G (x,y)^q \, d \sigma(y) \le C <+\infty, \quad \forall x \in \Omega,$$ was not only necessary, but also sufficient for . (Notice that in the case $q\ge 1$ is obviously necessary and sufficient for ; see [@QV2].)
In this section, we give a counterexample to this conjecture for Riesz potentials on ${{\mathbb R}}^n$, $$\mathbf{I}_{2 \alpha} \nu(x) = \int_{{{\mathbb R}}^n} \frac{d \nu(y)}{|x-y|^{n-2 \alpha}}, \quad x \in {{\mathbb R}}^n,$$ where $\nu \in \mathcal{M}^{+}({{\mathbb R}}^n)$, and $0<2\alpha<n$. Clearly, Riesz kernels $|x-y|^{2 \alpha-n}$ are symmetric, and satisfy the WMP.
Suppose $0<q<1$, $n \ge 1$, and $0< 2 \alpha < n$. We construct $\sigma \in {\mathcal{M}^{+}({{\mathbb R}}^n)}$ such that $$\label{A}
{{\mathcal E}}(\sigma)={{\mathcal E}}_{\alpha, q} (\sigma){\mathrel{\mathop:}=}\int_{{{\mathbb R}}^n} \Big (\mathbf{I}_{2 \alpha} \sigma\Big)^{\frac{q}{1-q}} d \sigma < +\infty,$$ and $$\label{B}
\mathcal{K} (\sigma)=\mathcal{K}_{\alpha, q} (\sigma){\mathrel{\mathop:}=}\sup_{x \in {{\mathbb R}}^n} \int_{{{\mathbb R}}^n} \frac{d \sigma(y)}{|x-y|^{(n-2\alpha)q}}< +\infty,$$ but $$\label{C}
\kappa(\sigma) = \kappa(\sigma)_{\alpha, q}{\mathrel{\mathop:}=}\sup \left\{ \frac{|| {{\mathbf I}}_{2 \alpha} \nu||_{L^q(\sigma)}}{||\nu||_{{\mathcal{M}^{+}({{\mathbb R}}^n)}}} \colon \, \, \nu \in {\mathcal{M}^{+}({{\mathbb R}}^n)}, \, \, \nu \not=0\right\} = +\infty.$$
In other words, we need to construct a measure $\sigma$ such that $ \mathcal{E}(\sigma)<+\infty$ (in the special case $q=\frac{1}{2}$ this means that $\sigma$ has finite energy), and holds for all $\delta$-functions $\nu = \delta_x$ ($x \in {{\mathbb R}}^n$), but fails for a linear combination of $\delta$-functions $$\label{delta}
\nu =\sum_{j=1}^{\infty} a_j \, \delta_{x_j}, \quad \textrm{where} \quad \sum_{j=1}^{\infty} a_j < +\infty, \quad a_j>0.$$ We will use a modification of the example considered in [@CV2] for other purposes.
We will need the following lemma and its corollary in the radially symmetric case (see [@CV2]).
\[lemvar\] Let $0<q<1$ and $0<2 \alpha < n$. If $d \sigma = \sigma(|x|) \, dx$ is radially symmetric, then $\kappa(\sigma)<+\infty$ if and only if $\mathcal{K}(\sigma)< +\infty$. Moreover, there exists a constant $c=c(q, \alpha, n)>0$ such that $\kappa(\sigma)$ satisfies $$\label{const-a}
\mathcal{K}(\sigma) \le \kappa(\sigma)^q \le c \, \mathcal{K}(\sigma),$$ where in the this case $$\label{rad-const}
\mathcal{K}(\sigma) = \int_{{{\mathbb R}}^n} \frac{d \sigma(y)}{|y|^{(n-2\alpha)q}}.$$
[For radially symmetric $\sigma$, condition $\mathcal{K}(\sigma)< +\infty$ is equivalent to $\sigma\in L^{\frac{1}{1-q},1}({{\mathbb R}}^n, \sigma)$, which is necessary and sufficient for in this case; see [@QV1], [@QV2]. Here $L^{s,1}({{\mathbb R}}^n, \sigma)$ denotes the corresponding Lorentz space with respect to the measure $\sigma$.]{}
\[cor-c\] Let $\sigma_{R, \gamma} = \chi_{B(0, R)} |x|^{-\gamma}$, where $0\le \gamma<n-q(n-2 \alpha)$ and $R>0$. Then $$\label{rad-K}
\mathcal{K}(\sigma) = \frac{ \omega_n \, R^{n-\gamma - q(n-2 \alpha)}} {n-\gamma - q(n-2 \alpha)},$$ and $$\label{rad-kappa}
\frac{\omega_n}{n-\gamma - q(n-2 \alpha)} \le \frac{\kappa(\sigma_{R, \gamma})^q} {R^{{n-\gamma-q(n-2 \alpha)}}}
\le \frac{c}{n-\gamma - q(n-2 \alpha)},$$ where $c=c(q, \alpha, n)$, and $\omega_n= |S^{n-1}|$ is the surface area of the unit sphere.
Let $$\label{sigma}
\sigma = \sum_{k=1}^\infty c_k \sigma_k,$$ where $$\label{r-k}
\sigma_k = \sigma_{R_k, \gamma_k} (x+x_k), \quad R_k=|x_k|=k, \quad \gamma_k = n-q(n-2 \alpha) - \epsilon_k,$$ and the positive scalars $c_k$, $\epsilon_k$ are picked so that $\sum_{k=1}^\infty c_k < \infty$, $\epsilon_k \to 0$, and $0<\gamma_k <n$. Notice that $\gamma_k \to n-q(n-2 \alpha)$ as $k \to \infty$, which is a critical exponent for the inequality (with $\sigma_k$ in place of $\sigma$) discussed below.
More precisely, for $0<q<1$ and $0<\delta<+\infty$, we set $$\label{choice}
a_k = \frac{1}{k (\log (k+1))^{\frac{1}{q}}}, \quad c_k=\frac{1}{k^{2-q+\delta}}, \quad \epsilon_k = \frac{1}{k^{1+\delta}}, \quad k=1, 2, \ldots ,$$ so that $$\label{sums}
\sum_{k=1}^{+\infty} a_k<+\infty, \quad \sup_{k\ge 1} \frac{c_k}{\epsilon_k} < +\infty, \quad \sum_{k=1}^{+\infty} \frac{c_k}{\epsilon_k^{1-q}}<+\infty, \quad \textrm{but} \quad \sum_{k=1}^{+\infty}
\frac{c_k \, a_k^q}{\epsilon_k}=+\infty.$$
We first verify condition . Notice that $$\label{const}
c_1 \, A \le [{{\mathcal E}}_{\alpha, q}(\sigma)]^{1-q} \le c_2 \, A,$$ where $A$ is the least constant in the inequality (see [@COV06]; [@CV1], Lemma 3.3) $$\label{sobolev}
\int_{{{\mathbb R}}^n} | {{\mathbf I}}_{\alpha} f |^{1+q} \, d \sigma \le A \, || f ||^{1+q}_{L^2(dx)}, \quad \text{for all} \, \, f \in L^2({{\mathbb R}}^n, dx),$$ or, equivalently, $$\label{sobolev2}
\int_{{{\mathbb R}}^n} | {{\mathbf I}}_{2 \alpha} (g d \sigma) |^{1+q} \, d \sigma \le A^{2} \, || g||^{1+q}_{L^{\frac{1+q}{q}}(d \sigma)}, \quad \text{for all} \, \, g \in L^2({{\mathbb R}}^n, \sigma),$$ where the constants of equivalence $c_1$, $c_2$ in depend only on $\alpha$, $q$, and $n$.
Consequently, $[{{\mathcal E}}_{\alpha, q}(\sigma)]^{1-q}$ is equivalent to a *norm* on a subset of ${\mathcal{M}^{+}({{\mathbb R}}^n)}$, so that $$\label{norm}
\Big[{{\mathcal E}}_{\alpha, q}\Big(\sum_k \sigma_k\Big)\Big]^{1-q}\le c \, \sum_k \Big[{{\mathcal E}}_{\alpha, q}(\sigma_k)\Big]^{1-q},$$ where $c=c(\alpha, q, n)$ is a positive constant which depends only on $\alpha$, $q$, and $n$.
We claim that, $$\label{norm-est}
{{\mathcal E}}_{\alpha, q}(\sigma_k) \le \frac{C \, {R_k}^{\frac{\epsilon_k}{1-q}}}{\epsilon_k},
\quad k=1, 2, \ldots ,$$ where $C=C(\alpha, q, n)$.
Indeed, by the semigroup property of Riesz kernels, $$\begin{aligned}
{{\mathbf I}}_{2 \alpha} \sigma_k (x) & = c(\alpha, n) \int_{B(0, R_k)} \frac{d t}{|x-t|^{n- 2 \alpha}
|t+x_k|^{\gamma_k} }\\ & \le c(\alpha, n) \int_{{{\mathbb R}}^n} \frac{d t}{|x-t|^{n- 2 \alpha}
|t+x_k|^{\gamma_k} }
= c \, |x+x_k|^{2 \alpha-\gamma_k},
\end{aligned}$$ where $c = c(n, 2 \alpha+n -\gamma_k)$ remains bounded by a constant $C(\alpha, q, n)$ as $k \to +\infty$, since $\lim_{k\to +\infty} (2 \alpha+n -\gamma_k) = 2 \alpha +q(n-2 \alpha)<n$.
Notice that $(\gamma_k - 2 \alpha)\frac{q}{1-q} + \gamma_k = n -\frac{\epsilon_k}{1-q}$. Hence, by the preceding estimate, $$\begin{aligned}
{{\mathcal E}}_{\alpha, q}(\sigma_k) & = \int_{{{\mathbb R}}^n} \Big({{\mathbf I}}_{2 \alpha} \sigma_k\Big)^{\frac{q}{1-q}} d \sigma_k
\\ & \le c^{\frac{q}{1-q}} \int_{|x+x_k|<R_k} \frac{dx}{|x+x_k|^{n-\frac{\epsilon_k}{1-q}}} \\& = c^{\frac{q}{1-q}} \omega_n \int_{0}^{R_k} r^{\frac{\epsilon_k}{1-q}-1} dr
\\& \le \frac{C(\alpha, q, n) \, {R_k}^{\frac{\epsilon_k}{1-q}}}{\epsilon_k},
\end{aligned}$$ which proves .
It follows from and the preceding estimate that, for $\sigma$ defined by , $$\label{sum}
\begin{aligned}
\Big[{{\mathcal E}}_{\alpha, q}(\sigma)\Big]^{1-q}
& \le c(\alpha, q, n) \, \sum_k c_k \Big[{{\mathcal E}}_{\alpha, q}(\sigma_k)\Big]^{1-q}\\
& \le c(\alpha, q, n) \, C(\alpha, q, n)^{1-q} \, \sum_k \frac{c_k \, R_k^{\epsilon_k}}{\epsilon_k^{1-q}}<+\infty,
\end{aligned}$$ by , since obviously $\sup_{k \ge 1} R_{k}^{\epsilon_k} < +\infty$ by . This proves .
To prove , we will need the following lemma.
\[lemma\] Let $R>0$, $0< \beta<n$, and $0<\epsilon<n-\beta$. For $\gamma = n-\beta-\epsilon>0$, we have $$\label{mathcal-Kb}
\phi_{R, \gamma} (x) {\mathrel{\mathop:}=}\int_{|t|<R} \frac{dt }{|x-t|^{\beta} |t|^{\gamma}} \approx
\left\{ \begin{array}{ll}
\frac{R^{\epsilon}-|x|^{\epsilon}}{\epsilon} & \quad \textrm{ if ~~} |x| \le \frac{R}{2},\\ R^{\epsilon} \Big(\frac{R}{|x|}\Big)^{\beta} & \quad \textrm{ if ~~} |x| >
\frac{R}{2},
\end{array} \right.$$ where the constants of equivalence depend only on $\beta$ and $n$.
Suppose first that $|x| > \frac{R}{2}$. Then $$\begin{aligned}
\phi_{R, \gamma} (x) & = \int_{|t|<\frac{R}{4}} \frac{dt }{|x-t|^{\beta} |t|^{\gamma}} +
\int_{\frac{R}{4}<|t|<R} \frac{dt }{|x-t|^{\beta} |t|^{\gamma}}
\\ & {\mathrel{\mathop:}=}I + II.
\end{aligned}$$ Clearly, in the first integral $ \frac{|x|}{2}\le |x-t|\le \frac{3 |x|}{2} $, and so $I$ is bounded above and below by $$\frac{\omega_n \, c(\beta)}{|x|^{\beta}} \int_{0}^{R} r^{n-1-\gamma} dr = \frac{c(\beta, n) R^{n-\gamma}}{|x|^{\beta}}.$$ To estimate the second term, notice that, for $|x|>2 R$ and $|t|<R$, we have $|x-t|>\frac{|x|}{2}$, so that $$II\le \frac{c(\beta, n)}{ R^{\gamma} \, |x|^{\beta}} \int_{\frac{R}{4}<|t|<R} dt =
\frac{c(\beta, n)R^{n-\gamma}} {|x|^{\beta}}.$$ For $\frac{R}{2}< |x|< 2 R$ and $|t|<R$, we have $|x-t|< 3R$, and consequently $$\begin{aligned}
II & \le
\frac{c(\beta, n)}{R^{\gamma}} \int_{|x-t|< 3 R} \frac{dt }{|x-t|^{\beta}} \\ &
= \frac{\omega_n c(\beta, n)}{R^{\gamma}} \int_{0}^{3R} r^{n-1-\beta} dr
\\ & = C(\beta, n) R^{n -\beta-\gamma} \\ & \le \frac{C(\beta, n) R^{n-\gamma}}{|x|^{\beta}}.
\end{aligned}$$ Thus, $I I \le c(n, \beta) \, I$, which proves in the case $|x|\ge \frac{R}{2}$.
Suppose now that $|x| \le \frac{R}{2}$. Then $$\begin{aligned}
\phi_{R, \gamma} (x) & = \int_{|t|<\frac{|x|}{2}} \frac{dt }{|x-t|^{\beta} |t|^{\gamma}} +
\int_{\frac{|x|}{2}<|t|<2 |x|} \frac{dt }{|x-t|^{\beta} |t|^{\gamma}} + \int_{2 |x|<|t|<R} \frac{dt }{|x-t|^{\beta} |t|^{\gamma}}
\\ & {\mathrel{\mathop:}=}I I I+ IV + V.
\end{aligned}$$
Clearly, in the first integral $\frac{|x|}{2} < |x-t|< \frac{3|x|}{2}$, and so $III$ is bounded above and below by $$\frac{c(\beta)}{|x|^\beta} \int_{|t|<\frac{|x|}{2}} \frac{dt }{|t|^{\gamma}}
= \frac{\omega_n c(\beta)}{|x|^\beta} \int_{0}^{\frac{|x|}{2}} r^{n-1-\gamma} dr
= \frac{c(\beta, n)}{(n-\gamma) 2^{n-\gamma}} |x|^{\epsilon}.$$
The second integral $IV$ is bounded above and below by $$\frac{c(\gamma)}{|x|^\gamma} \int_{\frac{|x|}{2}<|t|<2 |x|} \frac{dt }{|x-t|^{\beta}}.$$ Clearly, $$\begin{aligned}
I V & \le
\frac{c(\gamma)}{|x|^{\gamma}} \int_{|x-t|<3 |x|} \frac{dt }{|x-t|^{\beta}} \\ & = \frac{\omega_n c(\gamma)}{|x|^{\gamma}} \int_{0}^{3|x|} r^{n-1-\beta} dr \\ & =
\frac{\omega_n c(\gamma)}{|x|^{\beta + \gamma-n}} = c(\beta, \gamma, n)
|x|^{\epsilon},
\end{aligned}$$ so that $I V\le c(\beta, n) \, III$.
Finally, the integral $V$ is bounded above and below by $$\begin{aligned}
& c(\beta) \int_{2 |x|<|t|<R} \frac{dt }{|t|^{\gamma + \beta}}
\\ & = c(\beta) \int_{2 |x|}^{R} r^{n-1-\gamma-\beta} dr\\ & = c(\beta) \frac{R^{\epsilon} - (2 |x|)^{\epsilon}}{\epsilon}.
\end{aligned}$$ Combining these estimates we complete the proof of .
By Lemma \[lemma\] with $\beta=(n-2 \alpha)q$, $R=R_k$, $\epsilon=\epsilon_k$, and $\gamma=\gamma_k = n-\beta - \epsilon_k$, we obtain, for $k=2, 3, \ldots$, $$\label{mathcal-Ka}
\begin{aligned}
\phi_{R_k, \gamma_k} (x-x_k) & = \int_{|t+x_k|<R_k} \frac{dt }{|x-t|^{(n-2 \alpha) q}|t+x_k|^{\gamma}} \\& \le C(\alpha, q, n) \, \left\{ \begin{array}{ll}
\frac{R_k^{\epsilon_k}}{\epsilon_k} & \quad \textrm{ if ~~} \, \, |x-x_k|< 1,\\ & \\
\frac{R_k^{\epsilon_k}-1}{\epsilon_k} & \quad \textrm{ if ~~} \, \, 1\le |x-x_k|\le \frac{R_k}{2},\\ & \\ R_k^{\epsilon_k} & \quad \textrm{ if ~~} \, \, |x-x_k| >
\frac{R_k}{2}.
\end{array} \right.
\end{aligned}$$ In the case $k=1$, we use the estimate $\phi_{R_1, \gamma_1} (x-x_1)\le C(\alpha, q, n)
\frac{R_1^{\epsilon_1}}{\epsilon_1}$ for all $x \in {{\mathbb R}}^n$.
We next estimate $$\label{mathcal-K}
\begin{aligned}
{{\mathcal K}}(\sigma) & = \sup_{x\in {{\mathbb R}}^n} \sum_{k=1}^{+\infty} c_k \, \phi_{R_k, \gamma_k} (x-x_k)
\\
& \le
\sup_{x\in {{\mathbb R}}^n} \sum_{|x-x_k| \le 1} c_k \, \phi_{R_k, \gamma_k} (x-x_k) \\& + \sup_{x\in {{\mathbb R}}^n} \sum_{1< |x-x_k| <\frac{R_k}{2}} c_k \, \phi_{R_k, \gamma_k} (x-x_k) \\ & + \sup_{x\in {{\mathbb R}}^n} \sum_{|x-x_k| \ge \frac{R_k}{2}} c_k \, \phi_{R_k, \gamma_k} (x-x_k) \\ & {\mathrel{\mathop:}=}I + II + III.
\end{aligned}$$
Suppose that $j \le |x| \le j+1$ for some $j=0, 1, \ldots$. We first estimate $I$. Since $|x-x_k|\le 1$, and $|x_k|=k$, it follows that $$k=|x_k| \le 1 +|x| \le 1+ |x-x_k| + |x_k|= k+2.$$ Consequently, $j-1\le k \le j+2$ if $j \ge 2$, and $1\le k \le 3$ if $j=0, 1, 2$. Hence, the corresponding sum contains no more than four terms, and therefore $$\begin{aligned}
I & {\mathrel{\mathop:}=}\sup_{x\in {{\mathbb R}}^n} \sum_{|x-x_k| \le 1} c_k \, \phi_{R_k, \gamma_k} (x-x_k) \\ & \le C(\alpha, q, n) \, \sup_{j\ge 0} \, \, \sum_{\max(j-1, 1) \le k \le \max (j+2, 3)} \frac{c_k \, R_k^{\epsilon_k}}{\epsilon_k}\\ & \le C(\alpha, q, n),
\end{aligned}$$ since by and , $$\sup_{k\ge 1} \, R_k^{\epsilon_k}< +\infty, \quad {\rm and} \quad \sup_{k\ge 1} \, \frac{c_k}{\epsilon_k} < +\infty.$$
To estimate $II$, notice that $0< \epsilon_k \log R_k \le C$, and consequently $$\frac{R_k^{\epsilon_k}-1}{\epsilon_k}\le C \log R_k.$$ Hence, by and , $$\begin{aligned}
II & {\mathrel{\mathop:}=}\sup_{x\in {{\mathbb R}}^n} \sum_{1< |x-x_k| <\frac{R_k}{2}} c_k \, \phi_{R_k, \gamma_k} (x-x_k) \\ & \le
C(\alpha, q, n) \sup_{x\in {{\mathbb R}}^n} \sum_{1< |x-x_k| <\frac{R_k}{2}} \frac{c_k \, (R_k^{\epsilon_k}-1)}{\epsilon_k} \\ & \le C(\alpha, q, n) \sum_{k=1}^{+\infty} \, c_k \, \log R_k < +\infty.
\end{aligned}$$
Finally, we estimate $III$ using and . Since $\sup_k R_k^{\epsilon_k} < +\infty$, we deduce $$\begin{aligned}
III & {\mathrel{\mathop:}=}\sup_{x\in {{\mathbb R}}^n} \sum_{|x-x_k| \ge \frac{R_k}{2}} c_k \, \phi_{R_k, \gamma_k} (x-x_k)\\
& \le C(\alpha, q, n) \sup_{x\in {{\mathbb R}}^n} \sum_{|x-x_k| \ge \frac{R_k}{2}} c_k \, R_k^{\epsilon_k}
\\
& \le C(\alpha, q, n)
\sum_{k=1}^{+\infty} c_k \le C(\alpha, q, n).
\end{aligned}$$ This proves .
It remains to verify for $\sigma=\sum_{k=1}^{+\infty} c_k \sigma_k$ and $\nu = \sum_{j=1}^{+\infty} a_j \delta_{x_j}$ defined above. We estimate $$\begin{aligned}
\Vert {{\mathbf I}}_{2 \alpha} \nu\Vert^q_{L^q(\sigma)} & =
\sum_{k=1}^{+\infty} c_k \int_{{{\mathbb R}}^n} \Big ( \sum_{j=1}^{+\infty} \frac{a_j}{|x+x_j|^{n-2 \alpha} }
\Big)^q d \sigma_k \\ & \ge \sum_{k=1}^{+\infty} c_k \int_{{{\mathbb R}}^n} \frac{a_k^q}{|x+x_k|^{q(n-2 \alpha)} }
d \sigma_k
\\ & = \sum_{k=1}^{+\infty} c_k \, a_k^q \int_{|x+x_k|<R_k}
\frac{dx}{|x+x_k|^{(n-2 \alpha)q+\gamma_k}}.
\end{aligned}$$ Since $$\begin{aligned}
\int_{|x+x_k|<R_k} \frac{dx}{|x+x_k|^{(n-2 \alpha)q+\gamma_k}}& = \int_{|x|<R_k} \frac{dx}{|x|^{(n-2 \alpha)q+\gamma_k}} \\ &= \omega_n \int_{0}^{R_k} r^{-1+ \epsilon_k} dr=\omega_n \frac{R_k^{\epsilon_k}}{\epsilon_k},
\end{aligned}$$ and $R_k^{\epsilon_k}\ge 1$, it follows by that $$\begin{aligned}
\Vert {{\mathbf I}}_{2 \alpha} \nu\Vert^q_{L^q(\sigma)} & \ge \omega_n \sum_{k=1}^{+\infty} \frac{c_k \, a_k^q}{\epsilon_k}\\ & = \omega_n \sum_{k=1}^{+\infty}\frac{1}{k \log(k+1)}=+\infty.
\end{aligned}$$
[25]{}
, Grundlehren der math. Wissenschaften [**314**]{}, Berlin–Heidelberg–New York, Springer, 1996.
Nagoya Math. J. [**165**]{} (2002), 123–158.
, [*Lectures on Potential Theory*]{}, Lectures on Math. [**19**]{}, Tata Institute, Bombay, 1960.
, [*On [$L\sp p$]{}-[$L\sp
q$]{} trace inequalities*]{}, J. London Math. Soc. [**74**]{} (2006), 497–511.
, [*Finite energy solutions of quasilinear elliptic equations with sub-natural growth terms*]{}, Calc. Var. PDE [**52**]{} (2015), 529–546.
, [*Nonlinear elliptic equations and intrinsic potentials of Wolff type,*]{} J. Funct. Analysis [**272**]{} (2017), 112–165.
, [*Pointwise estimates of Brezis–Kamin type for solutions of sublinear elliptic equations,*]{} Nonlin. Analysis, Ser. A: Theory, Methods & Appl. [**146**]{} (2016), 1–19.
, [*Global estimates for kernels of Neumann series and Green’s functions*]{}, J. London Math. Soc. [**90**]{} (2014), 903–918.
, [*Positive solutions to Schrödinger’s equation and the exponential integrability of the balayage*]{}, Ann. Inst. Fourier (Grenoble) (published online), arXiv:1509.09005.
, [*Potentiel de masses à somme algébrique nulle*]{}, Kungl. Fysiogr. Sällskapets i Lund Förhandlingar \[Proc. Roy. Physiog. Soc. Lund\] [**20**]{} (1950), 1–21.
, [*On the theory of potentials in locally compact spaces*]{}, Acta Math. [**103**]{} (1960), 139–215.
, [*On integral trasformations with positive kernel*]{}, Proc. Amer. Math. Soc. [**16**]{} (1965), 429–434.
, J. d’Analyse Math. (to appear) arXiv:1511.03188.
, arXiv:1707.09596.
, [*On the Picard principle for ${\Delta} +
\mu$*]{}, Math. Z. [**270**]{} (2012) 783–807.
, Walter de Gruyter, Berlin–Boston, 2014.
, [*Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces [$L\sp{p}$]{}*]{}, Ast[é]{}risque [**11**]{} (1974) Soc. Math. France, Paris.
, [*Sobolev Spaces, with Applications to Elliptic Partial Differential Equations*]{}, 2nd, Augmented Edition. Grundlehren der math. Wissenschaften **342**, Springer, Berlin, 2011.
, [*Weighted norm inequalities of $(1,q)$-type for integral and fractional maximal operators*]{}, Harmonic Analysis, Partial Differential Equations and Applications, in Honor of Richard L. Wheeden, eds. S. Chanillo et al., Ser. Appl. Numer. Harmonic Analysis, Birkhäuser, 2017, 217–238.
, [*A sublinear version of Schur’s lemma and elliptic PDE*]{}, Analysis & PDE (to appear), arXiv:1702.02682.
, [*Introduction to Fourier Analysis on Euclidean Spaces*]{}, Princeton Univ. Press, Princeton, N.J., 1971.
, [*Notes on Integral Transformations*]{}, Dissert. Math. (Rozprawy Mat.), 231 (1984), p. 48.
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---
abstract: 'We perform a thorough analysis on the choice of estimators for random series path integral methods. In particular, we show that both the thermodynamic (T-method) and the direct (H-method) energy estimators have finite variances and are straightforward to implement. It is demonstrated that the agreement between the T-method and the H-method estimators provides an important consistency check on the quality of the path integral simulations. We illustrate the behavior of the various estimators by computing the total, kinetic, and potential energies of a molecular hydrogen cluster using three different path integral techniques. Statistical tests are employed to validate the sampling strategy adopted as well as to measure the performance of the parallel random number generator utilized in the Monte Carlo simulation. Some issues raised by previous simulations of the hydrogen cluster are clarified.'
author:
- Cristian Predescu
- Dubravko Sabo
- 'J. D. Doll'
- 'David L. Freeman'
title: 'Energy estimators for random series path-integral methods'
---
Introduction {#sec:intro}
============
Numerical path integral methods have proved to be highly useful tools in the analysis of finite temperature, many-body quantum systems.[@Nig99] A central theme in such studies is the conscious use of dimensionality, both in the reformulation of the original problem and in the subsequent numerical simulations.
As the scale of the problems under study continues to grow, it becomes increasingly important that the formal properties of the numerical methods that are utilized be properly characterized. Recently, Predescu and co-workers[@Pre02; @Pre03a; @Pre03] have presented a number of results concerning the convergence properties of random series-based path integral techniques. Important in their own right, these formal properties have also led to the development of a new class of path integral methods, the so-called reweighted techniques.[@Pre03] Reweighted approaches accelerate the convergence of “primitive” series methods by including the effects of “higher-order” path variables in a simple, approximate fashion. Reweighted methods achieve the convergence rate of related partial averaging approaches[@Dol85] without requiring the construction of the Gaussian transform of the underlying potential energy function.
Previous work on the reweighted method has focused principally on the construction of the quantum-mechanical density matrix.[@Pre03; @Pre03b] In the present work, we wish to examine estimators for various coordinate-diagonal and off-diagonal properties. While the present discussion is focused principally on reweighted methods, the results obtained are broadly applicable to more general random series approaches.
In Section II of the present article, we examine the thermodynamic (T-method) and direct (H-method) estimators for the total energy. In order to avoid any confusion with earlier estimators, we mention that in the present article by T-method and H-method estimators we understand the respective energy estimators introduced by Predescu and Doll in Ref. . Thus, the T-method estimator we employ does not have the variance difficulties associated with the Barker estimator for large numbers of path variables.[@Her82] As the low-temperature simulation presented in the second part of the article demonstrates, the present T-method estimator does not exhibit any of the difficulties sometimes associated with the virial estimator for low-temperature systems or for strongly correlated Monte Carlo sampling techniques.[@Gia88; @Fer95; @Cao89; @Kol96; @Jan97] The T-method estimator is closely related and similar in form to the centroid virial estimator.[@Cep95; @Gla02] We expect the two estimators to have similar behavior with the nature of the quantum system, the temperature, and the Monte Carlo sampling method. However, an important difference between the two estimators is the fact that the T-method estimator is a veritable thermodynamic estimator, in the sense that it is obtained by temperature differentiation of the quantum partition function. This observation is important because the temperature differentiation can be implemented numerically by a finite-difference scheme and, in principle, may lead to numerically stable algorithms that do not require derivatives of the potential. For large dimensional systems or systems described by complicated potentials, we expect such algorithms to be significantly faster than those based on explicit analytical formulas. The relative merits of such algorithms will be examined in future work.
In Section III, we examine the application of the reweighted methods to a model problem, that of simulating the thermodynamic properties of the (H$_2)_{22} $ molecular cluster. In Section IV, we summarize our present findings and clarify a number of issues raised in previous studies of this molecular hydrogen system.[@Cha98; @Dol99]
Energy estimators {#sec:estimators}
=================
In this section, we consider a one-dimensional quantum canonical system characterized by inverse temperature $\beta = 1/ (k_{B} T)$ and set forward the task of computing its average energy by Monte Carlo integration methods developed around several reweighted techniques.[@Pre03; @Pre03b] The physical system is made up of a particle of mass $m_0$ moving in the potential $V(x)$. We discuss the numerical implementation and the computational merits of both the T-method and H-method estimators. Any time the multidimensional extension is not obvious, we present the explicit formulas of the respective estimators.
We begin by presenting the general form of the path integral methods we employ in this paper. We remind the reader that in terms of a standard Brownian motion $\left\{B_u, u \geq 0\right\}$, the Feynman-Kaç formula has the expression[@Sim79] $$\begin{aligned}
\label{eq:1a}&& \nonumber
\rho(x,x';\beta) = P\left[\sigma B_1 = x' | \sigma B_0 = x \right] \\
&&\times \mathbb{E}\left[e^{-\beta \int_0^1 V(\sigma B_u){d}u } | \sigma
B_1 = x' , \sigma B_0 = x \right],\end{aligned}$$ where $\sigma = (\hbar^2\beta/m_0)^{1/2}$. In this paper, we shall use the symbol $\mathbb{E}$ to denote the expected value (average value) of a certain random variable against the underlying probability measure of the Brownian motion $B_u$. It is straightforward to see that the first factor of the product in Eq. (\[eq:1a\]) (which represents the conditional probability density that the rescaled Brownian motion $\sigma B_u$ reaches the point $x'$ provided that it starts at the point $x$) is the density matrix of a free particle of mass $m_0$ $$P\left[\sigma B_1 = x' | \sigma B_0 = x \right] =
\rho_{fp}(x,x';\beta).$$ Moreover, rather than using the conditional expectation appearing in the second factor of Eq. (\[eq:1a\]), one usually employs a stochastic process $\{ B_u^0; 0 \leq u \leq 1\}$, called a standard Brownian bridge,[@Sim79; @Dur96] which is defined as a standard Brownian motion conditioned on the end points such that $B_0^0 = 0$ and $B_1^0 = 0$. In terms of the newly defined process, the Feynman-Kaç formula reads $$\frac{\rho(x,x';\beta)}{\rho_{fp}(x,x';\beta)}= \mathbb{E}
\exp\left\{-\beta \int_0^1 V[x_r(u)+\sigma B_u^0]{d}u\right\},$$ where $x_r(u) = x + (x'-x)u$ is a straight line connecting the points $x$ and $x'$ and is called the reference path.
As discussed in Ref. , one of the most general constructions of the standard Brownian bridge is given by the Ito-Nisio theorem.[@Kwa92] Let $\{\lambda_k(\tau)\}_{k \geq 1}$ be a system of functions on the interval $[0,1]$, which together with the constant function $\lambda_0(\tau)=1$, make up an orthonormal basis in $L^2[0,1]$. Let $$\Lambda_k(t)=\int_0^t \lambda_k(u){d}u.$$ If $\Omega$ is the space of infinite sequences $\bar{a}\equiv(a_1,a_2,\ldots)$ and $$\label{eq:1}
{d}P[\bar{a}]=\prod_{k=1}^{\infty}{d}\mu(a_k)$$ is the probability measure on $\Omega$ such that the coordinate maps $\bar{a}\rightarrow a_k$ are independent identically distributed (i.i.d.) variables with distribution probability $$\label{eq:2}
{d}\mu(a_i)= \frac{1}{\sqrt{2\pi}} e^{-a_i^2/2}\,{d}a_i,$$ then $$\label{eq:2a} B_u^0(\bar{a})\stackrel{d}{=}
\sum_{k=1}^{\infty}a_k\Lambda_{k}(u),\; 0\leq u\leq1 ;$$ i.e., the right-hand side random series is equal in distribution to a standard Brownian bridge. The notation $B_u^0(\bar{a})$ in (\[eq:2a\]) is then appropriate and allows us to interpret the Brownian bridge as a collection of random functions of argument $\bar{a}$, indexed by $u$.
Using the Ito-Nisio representation of the Brownian bridge, the Feynman-Kaç formula takes the form $$\begin{aligned}
\label{eq:3}
\frac{\rho(x, x' ;\beta)}{\rho_{fp}(x, x' ;\beta)}&=&\int_{\Omega}{d}P[\bar{a}]\nonumber \exp\bigg\{-\beta
\int_{0}^{1}\! \! V\Big[x_r(u) \\& +& \sigma \sum_{k=1}^{\infty}a_k
\Lambda_k(u) \Big]{d}u\bigg\}.\end{aligned}$$ For a multidimensional system, the Feynman-Kaç formula is obtained by employing an independent random series for each additional degree of freedom.
A reweighted method constructed from the random series $\sum_{k=1}^\infty
a_k \Lambda_k(u)$ is any sequence of approximations to the density matrix of the form[@Pre03] $$\begin{aligned}
\label{eq:5}&&
\frac{\rho^{\text{RW}}_n(x, x' ;\beta)}{\rho_{fp}(x, x'
;\beta)}=\int_{\mathbb{R}}{d}\mu(a_1)\ldots \int_{\mathbb{R}}{d}\mu(a_{qn+p})\nonumber \\&& \times \exp\bigg\{-\beta \; \int_{0}^{1}\! \!
V\Big[x_r(u)+ \sigma \sum_{k=1}^{qn+p}a_k
\tilde{\Lambda}_{n,k}(u)\Big]{d}u\bigg\},\qquad\end{aligned}$$ where $q$ and $p$ are some fixed integers, where $$\label{eq:6}
\tilde{\Lambda}_{n,k}(u)= \Lambda_k(u) \quad \text{if} \ 1\leq k \leq n,$$ and where $$\label{eq:7}
\sum_{k=n+1}^{qn+p}\tilde{\Lambda}_{n,k}(u)^2=\sum_{k=n+1}^{\infty}
\Lambda_{k}(u)^2.$$ In Eq. (\[eq:5\]), $n$ indexes the sequence of reweighted approximations $\rho^{\text{RW}}_n(x, x' ;\beta)$, sequence that converges to the density matrix $\rho(x, x' ;\beta)$ in the limit $n\to \infty$. Remark that the approximation of index $n$ actually utilizes $qn+p$ variables for path parameterization. In the construction of a certain path, the first $n$ functions $\tilde{\Lambda}_{n,k}(u)$ coincide with the ones for the corresponding series representation, as shown by Eq. (\[eq:6\]). A number of $(q-1)n+p$ additional functions are constructed so that to maximize the order of convergence of the reweighted approximation. Notice that if the resulting approximation has a convergence of order $\alpha$ as measured against $n$, then it has the same order of convergence when measured against the total number of variables $qn+p$, though the convergence constant is $q^\alpha$ times larger. This explains why the number of additional functions is chosen to scale linearly with $n$. For additional information, the reader is advised to consult Ref. .
It is convenient to introduce the additional quantities $X_n(x,x',\bar{a};\beta)$ and $X_\infty(x,x',\bar{a};\beta)$, which are defined by the expressions $$\begin{aligned}
\label{8}&& \nonumber X_n(x,x',\bar{a};\beta)=\rho_{fp}(x,x';\beta)\\&&
\times \exp\bigg\{-\beta \; \int_{0}^{1}\! \! V\Big[x_r(u)+ \sigma
\sum_{k=1}^{qn+p}a_k
\tilde{\Lambda}_{n,k}(u)\Big]{d}u\bigg\}\qquad\end{aligned}$$ and $$\begin{aligned}
\label{9}&& \nonumber
X_\infty(x,x',\bar{a};\beta)=\rho_{fp}(x,x';\beta)\\&&
\times \exp\bigg\{-\beta \; \int_{0}^{1}\! \! V\Big[x_r(u)+ \sigma
\sum_{k=1}^{\infty}a_k
\Lambda_{k}(u)\Big]{d}u\bigg\},\qquad\end{aligned}$$ respectively. With the new notation, Eq. (\[eq:5\]) becomes $$\begin{aligned}
\label{eq:10}&&
\rho^{\text{RW}}_n(x, x' ;\beta)=\int_{\Omega}{d}P[\bar{a}]X_n(x,x',\bar{a};\beta),\end{aligned}$$ while the Feynman-Kaç formula reads $$\begin{aligned}
\label{eq:11}&&
\rho(x, x' ;\beta)=\int_{\Omega}{d}P[\bar{a}]X_\infty(x,x',\bar{a};\beta).\end{aligned}$$
The analytical expressions of the functions $\tilde{\Lambda}_{n,k}(u)$ depend on the nature of the reweighted techniques and are generally chosen to maximize the asymptotic convergence of the respective reweighted techniques.[@Pre03] To a large extent, the specific form of these functions is not important for the present development, but the reader is advised to consult Refs. and for quadrature techniques and additional clarifications.
The remainder of the present section is split into two parts. First, we discuss the problem of computing the ensemble averages of operators diagonal in coordinate representation. In particular, this resolves the problem of computing the average potential energy. Second, we consider the problem of evaluating the total energies (hence, also the kinetic energies) by means of the T-method and H-method estimators.
Operators diagonal in the coordinate representation
---------------------------------------------------
By definition, the ensemble average of an operator $\hat{O}$ diagonal in the coordinate representation is $$\label{eq:12}
\left\langle O \right\rangle_{\beta}=\frac{\int_{\mathbb{R}}
\rho(x;\beta)O(x){d}x}{\int_{\mathbb{R}} \rho(x;\beta){d}x}.$$ The quantity $\rho(x;\beta) = \rho(x,x;\beta)$ is the diagonal density matrix. By convention, we drop the second variable of the pair $(x,x')$ any time $x = x'$. For instance, we use $X_n(x,
\bar{a};\beta)$ instead of $X_n(x, x,\bar{a};\beta)$. By means of Eq. (\[eq:11\]), the average above can be recast as $$\label{eq:13}
\left\langle O \right\rangle_{\beta}=\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)O(x)}{\int_{\mathbb{R}} {d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)}.$$ This average can be recovered as the limit $n \to \infty$ of the sequence $$\label{eq:14}
\left\langle O
\right\rangle_{\beta,n}^{\text{pt}}=\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_n(x,\bar{a};\beta)O(x)}{\int_{\mathbb{R}}
{d}x \int_{\Omega}{d}P[\bar{a}]X_n(x,\bar{a};\beta)},$$ the terms of which are to be evaluated by Monte Carlo integration. The estimating function $O(x)$ appearing in the above formula is called the point estimating function of the operator $\hat{O}$.
An alternative to the point estimating function is the so-called path estimating function, the derivation of which is presented shortly. As demonstrated in Appendix A, the function $O(x)$ appearing in Eq. (\[eq:13\]) can be replaced by $O[x+ \sigma B_u^0(\bar{a})]$, without changing the value of the average $\left\langle O
\right\rangle_{\beta}$. That is, the equality $$\left\langle O \right\rangle_{\beta}=\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)O[x+ \sigma
B_u^0(\bar{a})]}{\int_{\mathbb{R}} {d}x \int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)}$$ is valid for all $0 \leq u \leq 1$. Averaging over the variable $u$, one obtains $$\label{eq:15}
\left\langle O \right\rangle_{\beta}=\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)\int_0^1 O[x+ \sigma
B_u^0(\bar{a})] {d}u}{\int_{\mathbb{R}} {d}x \int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)}.$$ Eq. (\[eq:15\]) shows that the ensemble average of the operator $\hat{O}$ can also be recovered as the limit $n \to \infty$ of the sequence $$\label{eq:16}
\left\langle O
\right\rangle_{\beta,n}^{\text{pth}}=\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_n(x,\bar{a};\beta)\int_0^1 O[x+ \sigma
\tilde{B}_{u,n}^0(\bar{a})] {d}u}{\int_{\mathbb{R}} {d}x
\int_{\Omega}{d}P[\bar{a}]X_n(x,\bar{a};\beta)},$$ where we have set $$\tilde{B}_{u,n}^0(\bar{a})= \sum_{k=1}^{qn+p}a_k
\tilde{\Lambda}_{n,k}(u)$$ for convenience of notation.
In the remainder of the present subsection, we discuss the relative merits of the point and path estimators. We first consider which of $\left\langle O \right\rangle_{\beta,n}^{\text{pt}}$ and $\left\langle O
\right\rangle_{\beta,n}^{\text{pth}}$ is closer to $\left\langle O
\right\rangle_\beta$ for a given $n$ assuming the averages given in Eqs. (\[eq:14\]) and (\[eq:16\]) are computed exactly. Let us notice that Eq. (\[eq:14\]) can be put in the form $$\left\langle O
\right\rangle_{\beta,n}^{\text{pt}}=\frac{\int_{\mathbb{R}}{d}x
\rho_n^{\text{RW}}(x;\beta)O(x)}{\int_{\mathbb{R}} {d}x
\rho_n^{\text{RW}}(x;\beta)}.$$ The probability distribution $$\label{eq:17}
\frac{\rho_n^{\text{RW}}(x;\beta){d}x}{\int_{\mathbb{R}}
\rho_n^{\text{RW}}(x;\beta) {d}x }$$ represents the marginal distribution of the variable $x$ regarded as a random variable on the space $\mathbb{R} \times \Omega$, which is endowed with the probability measure $$\label{eq:18}
\frac{X_n(x,\bar{a};\beta) {d}x \ {d}P[\bar{a}]}{\int_{\mathbb{R}} {d}x \int_{\Omega}{d}P[\bar{a}]X_n(x,\bar{a};\beta)}.$$ The reweighted techniques are designed so that the distribution given by Eq. (\[eq:17\]) is as close as possible to the quantum statistical one, which is given by the expression $$\frac{\rho(x;\beta){d}x}{\int_{\mathbb{R}}\rho(x;\beta){d}x }.$$ In designing the reweighted techniques, one seeks to optimize the rate of convergence of the sequence $\rho_n^{\text{RW}}(x,x';\beta) \to
\rho(x,x';\beta)$ for all $x$ and $x'$.[@Pre03]
For arbitrary $u$, the marginal distribution of $x+ \sigma
B_{u,n}^0(\bar{a})$ is usually different from the one given by Eq. (\[eq:17\]) and is not optimized. With few notable exceptions to be analyzed below, the points $x+ \sigma B_{u,n}^0(\bar{a})$ for different $u$ are not equivalent, and their probability distribution may differ significantly from the quantum statistical one. (However, as shown in Appendix A, they become equivalent in the limit $n \to \infty$.) Therefore, especially for those reweighted techniques having fast asymptotic convergence, we expect the point estimator to be more rapidly convergent with $n$ than the path estimator.
An additional issue appearing in Monte Carlo computations is the variance of the two estimating functions $O(x)$ and $\int_0^1 O[x+ \sigma
\tilde{B}_{u,n}^0(\bar{a})]{d}u$. In the limit $n \to \infty$, the variance of the point estimating function converges to $$\begin{aligned}
&&
\frac{\int_{\mathbb{R}}{d}x \int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)O(x)^2}{\int_{\mathbb{R}} {d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)}-\left\langle O
\right\rangle_{\beta}^2 \\ && =
\frac{\int_{\mathbb{R}}{d}x \int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)\int_0^1 O[x+ \sigma
B_{u}^0(\bar{a})]^2 {d}u}{\int_{\mathbb{R}} {d}x \int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)} \\ && \ \ \ \ \ \ \ \ \ \
-\left\langle O \right\rangle_{\beta}^2,\end{aligned}$$ while the variance of the path estimating function converges to $$\frac{\int_{\mathbb{R}}{d}x \int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)\left\{\int_0^1 O[x+ \sigma
{B}_{u}^0(\bar{a})] {d}u\right\}^2}{\int_{\mathbb{R}} {d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)} -\left\langle O
\right\rangle_{\beta}^2.$$ The Cauchy-Schwartz inequality implies $$\left\{\int_0^1 O[x+ \sigma {B}_{u}^0(\bar{a})] {d}u\right\}^2 \leq
\int_0^1 O[x+ \sigma {B}_{u}^0(\bar{a})]^2 {d}u$$ and shows that the variance of the path estimating function is always smaller than that of the point estimating function. The actual decrease in the variance is not always significant because the points $x+ \sigma
{B}_{u}^0(\bar{a})$ for different $u$ are strongly correlated. Depending on the nature of the function $O(x)$, the variance decrease may not compensate the effort required to compute the average $\int_0^1 O[x+
\sigma \tilde{B}_{u,n}^0(\bar{a})] {d}u$. However, if the function $O(x)$ is the potential $V(x)$, then the smaller variance of the path estimator is a desirable feature because the path average $\int_0^1 V[x+
\sigma \tilde{B}_{u,n}^0(\bar{a})] {d}u$, which also enters the expression of $X_n(x,\bar{a};\beta)$, is computed anyway.
To summarize the findings of the present subsection, the point estimator provides a more accurate value but has a larger variance than the path estimator. We next ask if there are any methods for which one may construct an estimator providing the same values as the point estimator but having the variance of the path estimator. More precisely, we seek methods for which there is a division $0= u_0 \leq u_1 \leq \ldots \leq u_{q_n} \leq u_{q_n+1} = 1$ such that the mesh $\max_{0 \leq i \leq q_n} |u_{i+1} - u_i|$ converges to zero as $n \to \infty$ and such that the points $\left\{x+ \sigma \tilde{B}_{u_i,n}^0(\bar{a}); 0 \leq i \leq
q_n+1\right\}$ have the same marginal distribution as $x$. For such methods, the expected value of the estimating function $$\label{eq:19}
\sum_{i = 0}^{q_n} O[x+ \sigma
\tilde{B}_{u_i,n}^0(\bar{a})](u_{i+1}-u_{i})$$ under the probability distribution given by Eq. (\[eq:18\]) is an estimator satisfying the criteria outlined in this paragraph.
There are two methods we employ in the present paper for which such an estimator exists. The first one, is the trapezoidal Trotter discrete path integral method (TT-DPI) obtained by the Trotter composition $$\begin{aligned}
\label{eq:20}
\rho_n^{\text{TT}}(x,x';\beta)=\int_{\mathbb{R}}{d}x_1 \ldots
\int_{\mathbb{R}}{d}x_n\;
\rho_0\left(x,x_1;\frac{\beta}{n+1}\right)\nonumber \\ \ldots
\rho_0\left(x_n,x';\frac{\beta}{n+1}\right)\qquad\end{aligned}$$ of the short-time approximation $$\rho_0^{\text{TT}}(x,x';\beta) = \rho_{fp}(x, x';\beta) \exp\left[-\beta
\frac{V(x)+V(x')}{2} \right].$$ It has been shown[@Pre02b] that for $n=2^k-1$, the TT-DPI method admits the following implementation
$$\begin{aligned}
\label{eq:21}\nonumber
\frac{\rho_n^{\text{TT}}(x, x' ;\beta)}{\rho_{fp}(x, x'
;\beta)}=\int_{\mathbb{R}}{d}a_{1,1}\ldots \int_{\mathbb{R}}{d}a_{k,2^{k-1}} \left( 2\pi \right)^{-n/2}
\exp\left({-\frac{1}{2}\sum_{l=1}^k\sum_{i=1}^{2^{l-1}} a_{l,i}^2}\right)
\\ \times \exp\left\{-{\beta}\sum_{i=0}^{2^k}\omega_i
V\left[x_r(u_i)+\sigma \sum_{l=1}^{k}F_{l,[2^{l-1}
u_i]+1}(u_i)a_{l,[2^{l-1} u_i]+1}\right]\right\},\end{aligned}$$
where $u_i= 2^{-k} i $ for $0\leq i \leq 2^k$ and $$\omega_i =\left\{ \begin{array}{l l}2^{-(k+1)},& \text{if}\ i\in \{0,
2^k\},\\ 2^{-k}, & \text{if}\ 1\leq i \leq 2^k-1.
\end{array} \right.$$ The functions $F_{l,k}(u)$ are the so-called Schauder functions,[@McK69] the definitions of which are presented in the cited references. We leave it for the reader to use Eq. (\[eq:20\]) and show that if $x = x'$, then all the points $ x+ \sigma
\tilde{B}_{u_i,n}^0(\bar{a})$ have identical marginal distribution given by the formula $$\frac{\rho_n^{\text{TT}}(x;\beta){d}x}{\int_{\mathbb{R}}\rho_n^{\text{TT}}(x;\beta){d}x }.$$ In this case, the point and the path estimators produce identical results for the ensemble average of a diagonal operator $\hat{O}$ $$\frac{\int_\mathbb{R}\rho_n^{\text{TT}}(x;\beta) O(x) {d}x}{\int_{\mathbb{R}}\rho_n^{\text{TT}}(x;\beta){d}x }.$$ At least for the ensemble average of the potential energy, one should always use the path estimator, which has smaller variance.
A second method for which there is an estimator giving the same values as the point estimator but having (asymptotically, as $n \to \infty$) the variance of the path estimator is the so-called Lévy-Ciesielski reweighted technique (RW-LCPI) defined by the formula[@Pre03]
$$\begin{aligned}
\label{eq:22} \nonumber
\frac{\rho_n^{\text{LC}}(x, x' ;\beta)}{\rho_{fp}(x, x'
;\beta)}=&&\int_{\mathbb{R}}{d}a_{1,1}\ldots \int_{\mathbb{R}}{d}a_{k+2,2^{k+1}} \left( 2\pi \right)^{-(4n+3)/2}
\exp\left({-\frac{1}{2}\sum_{l=1}^{k+2}\sum_{j=1}^{2^{l-1}}
a_{l,j}^2}\right)
\\&& \times \exp\left\{-\beta \int_0^1 V\left[x_r(u)+\sigma
\sum_{l=1}^{k+2} a_{l,[2^{l-1} u]+1} \;\tilde{F}^{(n)}_{l,[2^{l-1}
u]+1}(u)\right]{d}u\right\},\end{aligned}$$
where $[2^{l-1} u]$ is the integer part of $2^{l-1} u$. It has been shown that for $n=2^k-1$, the RW-LCPI method can be put in the Trotter product form[@Pre03] $$\begin{aligned}
\label{eq:23}
\rho_n^{\text{LC}}(x,x';\beta)=\int_{\mathbb{R}}{d}x_1 \ldots
\int_{\mathbb{R}}{d}x_n\;
\rho_0^{\text{LC}}\left(x,x_1;\frac{\beta}{n+1}\right)\nonumber \\ \ldots
\rho_0^{\text{LC}}\left(x_n,x';\frac{\beta}{n+1}\right),\qquad\end{aligned}$$ where $$\begin{aligned}
\nonumber
\frac{\rho_0^{\text{LC}}(x,x';\beta)}{\rho_{fp}(x,x';\beta)}=\frac{1}{\left(2\pi\right)^{3/2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}
e^{-\left(a_1^2+a_2^2+a_3^2\right)/2}\\\times \exp\bigg\{-\beta \int_0^1
V[x+(x'-x)u+a_1\sigma C_0(u)\\ +a_2 \sigma L_0(u)+ a_3 \sigma R_0(u)] {d}u\bigg\} {d}a_1 {d}a_2 {d}a_3. \nonumber\end{aligned}$$ The analytical expressions of the functions $\tilde{F}^{(n)}_{k,l}(u)$, $L_0(u)$, $R_0(u)$, and $C_0(u)$ can be found in Refs. and .
Again, we leave it for the reader to use Eq. (\[eq:23\]) and prove that if $x'=x$, then all the points $$x+\sigma \sum_{l=1}^{k+2} a_{l,[2^{l-1} u_i]+1}
\;\tilde{F}^{(n)}_{l,[2^{l-1} u_i]+1}(u_i)$$ with $u_i = 2^{-k}i$ for $0 \leq i \leq 2^k$ have identical marginal distributions equal to that of $x$. The estimator $$\label{eq:24} 2^{-k}\sum_{i = 0}^{2^{k}-1} O\left[ x+\sigma
\sum_{l=1}^{k+2} a_{l,[2^{l-1} u_i]+1} \;\tilde{F}^{(n)}_{l,[2^{l-1}
u_i]+1}(u_i)\right]$$ produces the same results as the point estimator, but it has the variance of the path estimator. As far as the evaluation of the average potential energy is concerned, in order to avoid unnecessary calls to the potential routine, it is desirable that the points $\{
2^{-k} i; 0 \leq i \leq 2^k\}$ be among the quadrature points utilized for the computation of the path averages appearing in Eq. (\[eq:22\]). The quadrature technique designed in Ref. shares this property. As opposed to the TT-DPI method, the point and the path estimators for the RW-LCPI method produce different results.
Estimators for the total energy
-------------------------------
In this subsection, we discuss the implementation of the thermodynamic (T) and the direct (H) estimators for the total energy. The T-method estimator is defined as the following functional of the diagonal density matrix: $$\label{eq:25}
\left\langle E \right\rangle^{T}_{\beta} =-\frac{\partial}{\partial \beta}
\ln{\left[ \int_{\mathbb{R}} \rho(x;\beta){d}x \right]}.$$ The above formula can be expressed as the statistical average $$\label{eq:26}
\left\langle E \right\rangle^{T}_{\beta} =\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)E_\infty^T(x,\bar{a};\beta) }
{\int_{\mathbb{R}}{d}x\int_{\Omega}{d}P[\bar{a}]X_\infty(x,
\bar{a};\beta)},$$ where the T-method estimating function $E_\infty^T(x,\bar{a};\beta)$ can be shown to be[@Pre02] $$\begin{aligned}
\label{eq:27} \nonumber
E_\infty^T(x, \bar{a};\beta)=\frac{1}{2\beta}+ \int_{0}^{1}\! \!
V\left[x+ \sigma {B}_{u}^0(\bar{a}) \right]\,{d}u \\ +
\frac{\sigma}{2}\int_{0}^{1}\! \! V'\left[x+ \sigma {B}_{u}^0(\bar{a})
\right]{B}_{u}^0(\bar{a})\,{d}u\end{aligned}$$ provided that $e^{-\beta V(x)}$ has (Sobolev) first order derivatives as a function of $x$. For a $d$-dimensional system, the expression of the T-method estimating function reads
$$\begin{aligned}
\label{eq:28} \nonumber
E_\infty^T(x_1,\ldots,x_d, \bar{a}_1, \ldots,
\bar{a}_d;\beta)=\frac{d}{2\beta}+ \int_{0}^{1}\! \! V\left[x_1+ \sigma_1
{B}_{u}^{0,1}(\bar{a}_1),\ldots, x_d+ \sigma_d {B}_{u}^{0,d}(\bar{a}_d)
\right]{d}u \\ + \sum_{i=1}^d\frac{\sigma_i}{2}\int_{0}^{1}\! \!
\left\{\frac{\partial}{\partial x_i}V\left[x_1+ \sigma_1
{B}_{u}^{0,1}(\bar{a}_1),\ldots, x_d+ \sigma_d {B}_{u}^{0,d}(\bar{a}_d)
\right]\right\}{B}_{u}^{0,i}(\bar{a}_i)\, {d}u.\end{aligned}$$
The ensemble average energy can be obtained as the limit $n \to \infty$ of the sequence $$\label{eq:29}
\left\langle E \right\rangle^{T}_{\beta,n} =\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_n(x,\bar{a};\beta)E_n^T(x,\bar{a};\beta) }
{\int_{\mathbb{R}}{d}x\int_{\Omega}{d}P[\bar{a}]X_n(x, \bar{a};\beta)},$$ where $$\begin{aligned}
\label{eq:30} \nonumber
E_n^T(x, \bar{a};\beta)=\frac{1}{2\beta}+ \int_{0}^{1}\! \! V\left[x+
\sigma \tilde{B}_{u,n}^0(\bar{a}) \right]\,{d}u \\ +
\frac{\sigma}{2}\int_{0}^{1}\! \! V'\left[x+ \sigma
\tilde{B}_{u,n}^0(\bar{a}) \right] \tilde{B}_{u,n}^0(\bar{a})\,{d}u.\end{aligned}$$ The finite-dimensional integral appearing in Eq. (\[eq:29\]) can be evaluated by Monte Carlo integration. In the limit $n \to
\infty$, the variance of the estimator is finite because the square of $E_\infty^T(x,\bar{a};\beta)$ given by Eq. (\[eq:20\]) is a well defined function, the average value of which is finite for smooth enough potentials.
A second energy estimator we employ in the present paper is the H-method estimator. This direct estimator is defined by the equation $$\label{eq:31}
\left\langle E \right\rangle^{H}_{\beta} =\frac{ \int_{\mathbb{R}}
\hat{H}_{x'} \rho(x,x';\beta)\big|_{x'=x} {d}x} { \int_{\mathbb{R}}
\rho(x;\beta) {d}x},$$ where the Hamiltonian of the system $\hat{H}_{x'}$ is assumed to act on the density matrix through the variable $x'$. By explicit computation and some integration by parts, the H-method estimator can be expressed as the statistical average $$\label{eq:32}
\left\langle E \right\rangle^{H}_{\beta} =\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)E_\infty^H(x,\bar{a};\beta) }
{\int_{\mathbb{R}}{d}x\int_{\Omega}{d}P[\bar{a}]X_\infty(x,
\bar{a};\beta)}$$ of the estimating function[@Pre02] $$\begin{aligned}
\label{eq:33} \nonumber
E_\infty^H(x, \bar{a};\beta)=\frac{1}{2\beta}+ V(x)+\frac{\hbar^2
\beta^2}{4 m_0}
\int_{0}^{1}\! \! \int_{0}^{1}\! \!(u-\tau)^2 \\
\times V'[x+\sigma B_u^0(\bar{a})]\, V'[x+\sigma B_\tau^0(\bar{a})]\,{d}u \,{d}\tau.\end{aligned}$$ The H-estimator is properly defined even for potentials that do not have second-order derivatives. For a $d$-dimensional system, the H-method estimating function reads
$$\begin{aligned}
\label{eq:34}
\nonumber E_\infty^H(x_1,\ldots,x_d, \bar{a}_1, \ldots,
\bar{a}_d;\beta)&=&\frac{d}{2\beta}+ V\left(x_1,\ldots, x_d \right) +
\sum_{i = 1}^d \frac{\hbar^2 \beta^2}{4 m_{0,i}}
\int_{0}^{1}\! \! \int_{0}^{1}\! \!(u-\tau)^2 \\ &\times& \left\{
\frac{\partial}{\partial x_i}V\left[x_1+ \sigma_1
{B}_{u}^{0,1}(\bar{a}_1),\ldots, x_d+ \sigma_d {B}_{u}^{0,d}(\bar{a}_d)
\right]\right\} \\&\times& \left\{
\frac{\partial}{\partial x_i}V\left[x_1+ \sigma_1
{B}_{\tau}^{0,1}(\bar{a}_1),\ldots, x_d+ \sigma_d
{B}_{\tau}^{0,d}(\bar{a}_d) \right]\right\}{d}u \, {d}\tau . \nonumber\end{aligned}$$
The reader should notice that the double integral appearing in Eq. (\[eq:33\]) is really a sum of products of one dimensional integrals. Indeed, one easily computes $$\begin{aligned}
\label{eq:35} \nonumber && E_\infty^H(x, \bar{a};\beta)=\frac{1}{2\beta}+
V(x)+\frac{\hbar^2 \beta^2}{2 m_0}\\&& \times
\left\{\int_{0}^{1} u^2 V'[x+\sigma B_u^0(\bar{a})]{d}u\right\} \nonumber
\left\{\int_{0}^{1} V'[x+\sigma B_u^0(\bar{a})]{d}u\right\}\\ &&
-\frac{\hbar^2 \beta^2}{2 m_0}\left\{\int_{0}^{1} u V'[x+\sigma
B_u^0(\bar{a})]{d}u\right\}^2.\end{aligned}$$ The H-method estimator is the sum of the “classical” energy and a “quantum” correction term. Equation (\[eq:32\]) shows that the total energy can also be recovered as the limit $n \to \infty$ from the sequence $$\label{eq:36}
\left\langle E \right\rangle^{H}_{\beta,n} =\frac{\int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_n(x,\bar{a};\beta)E_n^H(x,\bar{a};\beta) }
{\int_{\mathbb{R}}{d}x\int_{\Omega}{d}P[\bar{a}]X_n(x, \bar{a};\beta)},$$ where $$\begin{aligned}
\label{eq:37} \nonumber
E_n^H(x, \bar{a};\beta)=\frac{1}{2\beta}+ V(x)+\frac{\hbar^2 \beta^2}{4
m_0}
\int_{0}^{1}\! \! \int_{0}^{1}\! \!(u-\tau)^2 \\
\times V'[x+\sigma \tilde{B}_{u,n}^0(\bar{a})]\, V'[x+\sigma
\tilde{B}_{\tau,n}^0(\bar{a})]\,{d}u \,{d}\tau.\end{aligned}$$ The forms of the T- and the H-method estimators derived here with the reweighted techniques in mind extend naturally to the TT-DPI method by means of Eq. (\[eq:21\]). One just replaces the one dimensional integrals appearing in Eqs. (\[eq:30\]) and (\[eq:37\]) by appropriate trapezoidal quadrature sums.
For the reweigthed techniques, we anticipate that the kinetic energy estimator entering the H-method estimator provides more accurate results than the kinetic energy estimator entering the T-method estimator. As for the point and the path estimators of diagonal operators, the derivatives of the density matrix against the spatial coordinates, which measure fluctuations around the preferential points $x$ and $x'$ for which the reweighted density matrices are optimized, are expected to be reproduced in a better way than the temperature derivatives, which involve unoptimized path-averaged fluctuations. However, for sufficiently low temperatures, the variance of the H-method kinetic energy estimator is expected to be larger than the variance of its thermodynamic counterpart. This larger variance is due to the factor $\beta^2$ appearing in Eqs. (\[eq:33\]) and (\[eq:37\]).
There is one special property of the T- and H-method estimators that proves to be important in simulations. Let us notice that by virtue of the Bloch equation $$\hat{H}_{x'} \rho(x,x';\beta) = -\frac{\partial}{\partial
\beta}\rho(x,x';\beta),$$ we have the equality $$\left\langle E \right\rangle_{\beta}:= \left\langle E
\right\rangle^{H}_{\beta}=\left\langle E \right\rangle^{T}_{\beta}.$$ Here, the symbol $:=$ signifies that the average energy $\left\langle E \right\rangle_{\beta}$ is *defined* to be the common value of the T-method and the H-method energy estimators, provided that these are equal. However, since $\rho_n^{\text{RW}}(x,x';\beta)$ does not satisfy the Bloch equation (except for the free particle), in general $$\begin{aligned}
\left\langle E \right\rangle^{H}_{\beta,n}= \frac{ \int_{\mathbb{R}}
\hat{H}_{x'} \rho_n^\text{RW}(x,x';\beta)\big|_{x'=x} {d}x} {
\int_{\mathbb{R}} \rho_n^\text{RW}(x;\beta) {d}x} \\ \neq \left\langle
E \right\rangle^{T}_{\beta,n}= -\frac{\partial}{\partial \beta}
\ln{\left[ \int_{\mathbb{R}} \rho_n^\text{RW}(x;\beta){d}x \right]}\end{aligned}$$ and the T- and H-method estimators produce the same result only in the limit $n \to \infty$. Given that the two energy estimators discussed in the present section can be computed simultaneously without incurring any computational penalty, we recommend that the agreement between the T- and the H-method estimators be used as a verification tool in actual simulations in order to check the convergence of various path integral methods. However, we emphasize that the agreement between the T- and the H-method estimators is not a sufficient convergence criterion and in practice, the convergence of different ensemble averages with the number of path variables should also be monitored.
As Eqs. (\[eq:30\]) and (\[eq:37\]) show, the path and the point estimating functions for the potential energy enter naturally the expressions of the T- and H-method estimating functions, respectively. For the purpose of using the agreement between the two energy estimators as a verification tool for convergence, one should not replace the path estimating function for the potential energy in the expression of the T-method estimator with the point estimating function, even if this may improve the estimated energy. For special cases, as for instance the TT-DPI and RW-LCPI methods discussed in the previous subsection, one may replace the point estimating function for the potential energy appearing in the expression of the H-method estimator with other estimating functions that produce the same value but have smaller variance. In this paper, we replace the point estimating function with the path estimating function for the TT-DPI method and with the estimating function given by Eq. (\[eq:24\]) for the RW-LCPI method, respectively.
A numerical example
===================
We have tested the relative merits of the T- and H-method energy estimators on a cluster of 22 hydrogen molecules at a temperature of 6 K, using three different path integral methods. Two of these methods, the trapezoidal Trotter discrete path integral method and a Lévy-Ciesielski reweighted technique, have been already presented in the preceding section. The third method is a Wiener-Fourier reweighted (RW-WFPI) technique introduced in Ref. . The numerical implementation of the methods has been extensively discussed in Ref. by some of us and are not reviewed here.
The physical system we study has been recently examined by Chakravarty, Gordillo, and Ceperley[@Cha98] as well as by Doll and Freeman[@Dol99] in their comparison of Fourier and discrete path integral Monte Carlo methods. The total potential energy of the $(\text{H}_2)_{22}$ cluster is given by $$\label{2.1} V_{tot} = \sum_{i<j}^{N} V_{LJ}(r_{ij})+\sum_{i=1}^{N}
V_c(\mathbf{r_i}),$$ where $V_{LJ}(r_{ij})$ is the pair interaction of Lennard-Jones potential $$\label{2.2} V_{LJ}(r_{ij}) = 4\epsilon_{LJ}\left
[\left( \frac{\sigma_{LJ}}{r_{ij}}\right)^{12}
-\left( \frac{\sigma_{LJ}}{r_{ij}}\right)^{6}\right]$$ and V$_{c}(\mathbf{r_i})$ is the constraining potential $$\label{2.3}
V_c(\mathbf{r_i})=\epsilon_{LJ}\left(\frac{|\mathbf{r_i}-\mathbf{R_{cm}}|}{R_c}\right)^{20}.$$ The values of the Lennard-Jones parameters $\sigma_{LJ}$ and $\epsilon_{LJ}$ used are 2.96 [Å]{} and 34.2 K, respectively. [@Cha98] $\mathbf{R_{cm}}$ is the coordinate of the center of mass of the cluster and is given by $$\label{2.4}
\mathbf{R_{cm}}=\frac{1}{N}\sum_{i=1}^N \mathbf{r_i}.$$ Finally, $R_c=4\sigma_{LJ}$ is the constraining radius. The role of the constraining potential $V_c(\mathbf{r_i})$ is to prevent molecules from permanently leaving the cluster since the cluster in vacuum at any finite temperature is metastable with respect to evaporation.
At the temperature of $6$ K and at the small densities employed in our computation, the molecules of hydrogen can be described by spherical rotational wave functions, because the majority of the molecules are in the $J=0$ state. To a good approximation, the molecules can be regarded as spherical bosons interacting through isotropic pair potentials. However, a thorough study of parahydrogen clusters has showed that quantum exchange of molecules is small at temperatures greater than $2$ K and that the hydrogen molecules can be safely treated as distinguishable particles.[@Sin91]
The optimal choice of the parameter $R_c$ for the constraining potential has been discussed in recent work.[@Nei00] If $R_c$ is taken to be too small, the properties of the system become sensitive to its choice, whereas large values of $R_c$ can result in problems attaining an ergodic simulation. To facilitate comparisons, in the current work, $R_c$ has been chosen to be identical to that used in Ref. . While this choice of constraining potential can induce ergodicity problems in calculations of fluctuation quantities like the heat capacity, we provide evidence below that the simulations in the current work are ergodic.
The three path integral methods we have employed utilize different numbers of path variables for a given index $n$. For instance, the TT-DPI $n$-th order approximation to the density matrix $\rho_n^{\text{TT}}(x,x';\beta)$ utilizes $n$ path variables for each physical dimension, whereas $\rho_n^{\text{LC}}(x,x';\beta)$ and $\rho_n^{\text{WF}}(x,x';\beta)$ utilize $4n+3$ and $4n$ path variables, respectively. To ensure fair comparison with respect to the number of path variables employed, we have tabled the total number of variables $n_v$ used for each physical dimension and not the index $n$.
Sampling strategy
-----------------
We have discussed in Section II that the evaluation of the ensemble average of any observable eventually reduces to the evaluation of the average of a certain estimating function against the probability distribution $$\label{eq:3.1}
\frac{X_n(x,\bar{a};\beta) {d}x \, {d}P[\bar{a}]}{\int_{\mathbb{R}} {d}x \int_{\Omega}{d}P[\bar{a}]X_n(x,\bar{a};\beta)}$$ or its multidimensional counterpart. This probability distribution can be sampled with the help of the Metropolis algorithm, which comprises the following steps.[@Met53; @Kal86] One initializes the imaginary-time paths in some fashion. Then, one attempts a trial move of the paths, which may involve changing several coordinates at a time. The displacement of the new paths is usually chosen to be relative to the old paths. To ensure ergodicity, one makes sure that all variables of the system are eventually moved in a cyclic or a random fashion. The proposed path is then accepted or rejected with a certain probability. The average value of the quantity of interest is computed by averaging the values of the corresponding estimating function evaluated at the current paths.
To establish some notation necessary for our discussion, for each vector $\mathbf{r}_i= ( x_i, y_i, z_i )$ denoting the physical coordinates of the particle $i$, we let $\mathbf{\bar{a}}_i = \{\mathbf{a}_{i,1},
\ldots, \mathbf{a}_{i,n_v}\}$ be the collection of path variables associated with the respective particle. Each $$\mathbf{a}_{i,k}=\left(a^{x}_{i,k}, a^{y}_{i,k}, a^{z}_{i,k}\right)$$ is itself a three-dimensional vector whose components denote the $k$-th parameter of particle $i$ for the $x$, $y$, and $z$ coordinates, respectively. Going back to the description of the Metropolis algorithm, the full imaginary-time path has been initialized by choosing the physical coordinates $\bf{r}_i$ randomly in a sphere of radius $R_c$ centered about origin. The path variables $\bf{\bar{a}}_i$ have been initialized with zero.
Except for the Wiener-Fourier method with $n_v = 512$ ($n = 128$), we update the individual particles one at a time in a cyclic fashion. Each update of a particle consists of an attempt to move the physical coordinate $\mathbf{r}_i$ together with the first one quarter of the path variables $\mathbf{\bar{a}}_i$ (that is, together with the variables $\left\{\mathbf{a}_{i,k}; 1 \leq k \leq [n_v/4]\right\}$) followed by a separate attempt to move the rest of the path variables associated with the particle $i$. Both the physical coordinates and the path variables are moved in a cube centered about the old coordinates: $$\mathbf{r}'_i = \mathbf{r}_i + \Delta_r (2 \mathbf{u} - 1)$$ and $$\mathbf{a}'_{i,k} =\mathbf{a}_{i,k} + \Delta_a(2 \mathbf{u} -1),$$ where the three components of $\mathbf{u}$ are independent uniformly distributed random numbers on the interval $[0,1]$. Throughout our simulations, we have used the following maximum displacement values: $\Delta_r = 0.26$ Å and $\Delta_a = 0.15$. The sampling technique employed guaranties an acceptance ratio between 30% and 70% for all methods studied and for $n_v \leq 256$.
Because the acceptance ratio drops below 20% for the Wiener-Fourier reweighted technique with $n_v = 512$, each most basic step of the previously described algorithm has been decomposed into two successive steps. The first step is decomposed into an attempt to move the physical coordinate $\mathbf{r}_i$ together with the first $1/8$ of the path variables $\mathbf{\bar{a}}_i$, followed by an attempt to move the physical coordinate $\mathbf{r}_i$ together with the next $1/8$ path variables $\mathbf{\bar{a}}_i$. The second step is decomposed in a similar fashion; half of the remaining variables have been moved in a first attempt and then the other half in a second attempt. This restores the overall acceptance ratio to about 33%. In fact, we have monitored separately the acceptance ratio for the four different steps necessary to update all the coordinates associated with a given particle and have made sure that the sampling is well balanced in the sense that the acceptance for each individual step is about 30% or larger.
As a counting device, we define a *pass* as the minimal set of Monte Carlo attempts over all variables in the system. A pass consists of $2
\cdot 22 = 44$ basic steps for all simulations with $n_v \leq 256$. For the Wiener-Fourier reweighted technique with $n_v = 512$, a pass consists of $4 \cdot 22 = 88$ basic attempted moves. One also defines a *block* as a computational unit made up of ten thousand passes.
Embarrassingly parallel computation
-----------------------------------
In order to achieve a statistical error of about $0.1$ K/molecule for all computed energies, we have employed a large number of Monte Carlo passes (10.4 million) and we have divided the computation in $16$ independent tasks to be run in parallel. For the Wiener-Fourier reweighted method with $n_v = 512$, we have utilized a number of 40 million passes divided in $80$ independent tasks. The Monte Carlo simulations are embarrassingly parallel provided that one can generate independent streams of uniformly distributed random numbers. In this situation, there is no need for communication among the different code replica running on different nodes, and the program is an ideal candidate for use on a distributed computing cluster. However, to be mathematically rigorous, it is necessary to ensure that all the communication needed is already buried in the independence of the streams of random numbers. This underlies the need for “good” parallel random number generators.
The Mersenne Twister (MT) is a fast serial pseudorandom number generating algorithm with a long period and good $k$-distribution properties.[@Mat98] Quite interestingly, the algorithm allows for the development of random number generators meeting certain user specifications. For instance, one may specify the period (which must be a Mersenne prime number i.e., a prime number of the form $2^p-1$), the word size, or the memory size. Given a specified period, one may still produce various algorithms which differ by their characteristic polynomials. The dynamic creation of distributed random number generators is based on the hypothesis that MT random number generators having relatively prime characteristic polynomials produce highly independent streams of random numbers.[@Mat98a] Because the laws by which the numbers are generated are significantly different, it is very probable that the streams produced by the different generators are highly uncorrelated. In this paper, we have used the Dynamic Creator C-language library[@Mat98b] with the Mersenne number $2^{3217}-1$. The library outputs streams of 32-bit integers, which are easy to convert into real numbers on the interval $[0,1]$. Different streams are identified by different identification numbers. The streams have been initialized once at the beginning of the simulation with different seeds.
Given the $16$ streams of independent random numbers, the Monte Carlo simulation proceeds as follows. For each stream, one performs an independent simulation consisting of $65$ blocks. These blocks are preceded by $13$ equilibration blocks, which are needed to bring the system into probable configurations but do not contribute to the averages of the estimating functions. For the Wiener-Fourier reweighted method with $n_v = 512$, we use $80$ independent streams of $50$ blocks each, for a total of $40$ million passes. The equilibration phase consists of $10$ blocks for each stream. Ideally, the length of the individual streams should be chosen to be sufficiently large, that the averages of the computed property for different streams are independent and normally distributed, as dictated by the central limit theorem. This requirement is satisfied by all simulations we have performed.
We have collected individual averages for all blocks and streams and performed several statistical tests verifying the applicability of the central limit theorem as well as the independence between the block averages of same or different streams. Let $\{Z_{i,j}: 1\leq i \leq 16;
1\leq j \leq 65\}$ denote the block-averages of the property $Z$ for stream $i$ and block $j$ (the RW-WFPI simulation for $n_v = 512$ has been analyzed in a similar fashion). Under the assumption that the size of the blocks is large enough so that the correlation between different block-averages is negligible and under the assumption that the block-averages for different streams are highly uncorrelated, the values $Z_{i,j}$ should have a Gaussian distribution centered around the average value $$\label{eq:3.2}
\overline{Z} =\frac{1}{16 \cdot 65} \sum_{i=1}^{16} \sum_{j=1}^{65}
Z_{i,j}$$ with variance $$\label{eq:3.3}
\sigma^2(Z) = \frac{1}{16 \cdot 65} \left(\sum_{i=1}^{16} \sum_{j=1}^{65}
Z_{i,j}^2\right) - \overline{Z}^2.$$ The validity of this assumption can be verified with the help of the Shapiro-Wilks normality test.[@Sha65] If the collection of samples $Z_{i,j}$ does not pass the test, it does not necessarily follow that the samples $Z_{i,j}$ are not independent, as their distribution is normal only if the size of the blocks is sufficiently large. At a significance level of 5%, we do not reject the Gaussian distribution hypothesis for all computed average properties. To within the statistical significance of our calculations, the samples $Z_{i,j}$ can be assumed to be independent and have a Gaussian distribution.
A second set of tests consists in verifying that the row and column averages of $Z_{i,j}$ have Gaussian distributions centered around $\overline{Z}$ with variances $\sigma^2(Z)/65$ and $\sigma^2(Z)/16$, respectively. The validity of this distribution follows from the central limit theorem and the assumption that the samples $Z_{i,j}$ are independent and have a Gaussian distribution characterized by the average $\overline{Z}$ and the variance $\sigma^2(Z)$. It is important to emphasize that the row averages must pass this test. As previously discussed, the number of blocks in a stream should be sufficiently large so that the row averages have the required distribution even if the independent samples $Z_{i,j}$ do not have a Gaussian distribution. Again, under the assumption of independence only, the row averages should have a Gaussian distribution centered around $\overline{Z}$ and have variance $\sigma^2(Z)/N_\text{blocks}$ for a sufficiently large number of blocks $N_\text{blocks}$. We have employed the Kolmogorov-Smirnov test[@Pre92] to compare the distributions of the row and column averages with the theoretical Gaussian distributions. For all computed average properties, we find that the respective distributions are identical at a statistical significance level of 5%. The agreement for the distribution of the row averages is evidence that the streams generated by the Dynamic Creator package are sufficiently independent, whereas the agreement for the distribution of the column averages is evidence that the block averages of the same streams are independent.
For the third set of tests, we have considered two time-series $\{Z'_i, 1
\leq i \leq 16 \cdot 65\}$ and $\{Z''_i, 1 \leq i \leq 16 \cdot 65\}$ obtained by concatenating the rows of the matrix $Z_{i,j}$ and the columns, respectively. We then have studied the autocorrelation of the two time series for a maximum lag of $32$. The correlation coefficients for a lag $k \leq 32$ are computed with the formula $$r'_k = \frac{1}{\sigma^2(Z)} \frac{1}{16\cdot 65} \sum_{i = 1}^{16
\cdot 65} \left(Z'_i - \overline{Z}\right)
\left(Z'_{i+k}-\overline{Z}\right),$$ where $Z'_{i + k} = Z'_{i + k - 16 \cdot 65}$ if $i + k > 16 \cdot
65$. Under the independence hypothesis of the samples $Z'_i$, the statistics of the correlation coefficients for $1 \leq k \leq 32$ is normal with average zero and standard deviation $\sigma' = 1/\sqrt{16
\cdot 65}$. Moreover, the correlation coefficients can be regarded as independent samples of this normal distribution. By the binomial distribution, the probability that at most $m$ correlation coefficients lie outside the interval $[-2\sigma', 2\sigma']$ is given by the formula $$P(m) = \sum_{k = 0}^{m} \frac{32!}{k!(32 - k)!} q^k (1-q)^{32 -k },$$ where $q \approx 0.046$ is the probability that a normal distributed variable of mean zero and standard deviation $\sigma'$ lies outside the interval $[-2\sigma', 2\sigma']$. One computes $P(3) = 0.942$ and $P(4) =
0.985$ so at a level of significance of 5%, the hypothesis that the $r'_k$ are independent samples of a normally distributed variable of mean zero and standard deviation $\sigma' = 1/\sqrt{16 \cdot 65}$ should be rejected if $4$ or more correlation coefficients lying outside the interval $[-2\sigma', 2\sigma']$ are observed.
![\[Fig:1\] Correlograms for the time-series $Z'_i$ and $Z''_i$. The property $Z$ is the average ensemble energy computed by means of the H-method estimator using the RW-WFPI method with $n_v = 32$. One notices that both the correlation between the block averages ($r'_k$) and the correlation between the streams ($r''_k$) are negligible. ](502344JCP1.eps){width="8.5cm"}
The autocorrelation of the series $Z'_i$ is representative of the correlation between the block averages of same streams, whereas the autocorrelation of the time series $Z''_i$ is representative of the correlation between the blocks of similar rank corresponding to different streams. Fig. \[Fig:1\] shows the correlograms of the two series for a RW-WFPI Monte Carlo simulation with $n_v = 32$. The computed property is the H-method energy estimator. Both series $Z'_i$ and $Z''_i$ have only one point lying outside the interval $[-2\sigma', 2\sigma']$. These points are $r'_5$ and $r''_{21}$, respectively (of course, the points $r'_0 = r''_0 =1$ are not counted). Consequently, the simulation passes our third statistical test. In fact, all the simulations performed have passed this statistical test for all computed properties. We conclude that the correlation between the block averages of same or different streams is negligible. By the central limit theorem, the statistical error in the determination of the average of the property $Z$ is $$\label{eq:3.4}
\pm 2\sigma(Z) / \sqrt{16 \cdot 65},$$ where $\sigma^2(Z)$ is defined by Eq. (\[eq:3.3\]). (For the statistical error, we employ the $2 \sigma$ value, corresponding to an interval of 95% confidence. The 5% probability that the results lie outside the confidence interval is chosen to agree with the level of significance of the statistical tests).
The analysis performed in the present subsection demonstrates that the streams generated by the Dynamic Creator algorithm have negligible correlation at least for our purposes.
A separate advantage in the use of independent streams is to overcome the phenomenon of quasiergodicity,[@Val77] which might appear in Monte Carlo simulations whenever the distribution that is sampled has several well defined minima that are separated by walls of high energy. In this case, the random walker may be trapped in one of the wells and never sample the others, or sample them with the wrong frequency. The Monte Carlo simulation may pass all the aforementioned statistical tests but still produce the wrong results. For our system, the probability that such a situation may occur is quite low because the system is highly quantum mechanical with strong barrier tunneling. Moreover, the $16$ independent streams have been initialized randomly in configuration space. This makes it unlikely that all the streams are trapped precisely into the same local minimum or group of local minima. Evidence for quasiergodicity may be captured in the form of a few outlying averages among the stream averages. Such outlying averages have not been observed.
Summary and discussion of the computed averages
-----------------------------------------------
[ccccccc]{} $n_v$ & $\langle E \rangle^T_\beta$ & $\langle E
\rangle^H_\beta$ & $\langle V \rangle^T_\beta$ & $\langle V \rangle^H_\beta$ & $\langle K \rangle^T_\beta$ & $\langle K \rangle^H_\beta$\
\
4 & -57.66 $\pm$ 0.05 & -16.63 $\pm$ 0.18 & -82.14 $\pm$ 0.07 & -61.72 $\pm$ 0.12 & 24.48 $\pm$ 0.02 & 45.09 $\pm$ 0.15\
\
8 & -37.61 $\pm$ 0.05 & -17.77 $\pm$ 0.16 & -64.74 $\pm$ 0.06 & -53.07 $\pm$ 0.11 & 27.13 $\pm$ 0.02 & 35.29 $\pm$ 0.13\
\
16 & -25.68 $\pm$ 0.04 & -18.28 $\pm$ 0.13 & -54.27 $\pm$ 0.06 & -49.33 $\pm$ 0.10 & 28.60 $\pm$ 0.03 & 31.06 $\pm$ 0.11\
\
32 & -20.23 $\pm$ 0.04 & -18.00 $\pm$ 0.12 & -49.66 $\pm$ 0.06 & -48.05 $\pm$ 0.10 & 29.42 $\pm$ 0.03 & 30.05 $\pm$ 0.11\
\
64 & -18.29 $\pm$ 0.04 & -17.85 $\pm$ 0.11 & -48.19 $\pm$ 0.06 & -47.86 $\pm$ 0.09 & 29.90 $\pm$ 0.03 & 30.01 $\pm$ 0.11\
\
128& -17.75 $\pm$ 0.04 & -17.64 $\pm$ 0.12 & -47.83 $\pm$ 0.06 & -47.81 $\pm$ 0.09 & 30.08 $\pm$ 0.03 & 30.17 $\pm$ 0.11\
\
256& -17.71 $\pm$ 0.04 & -17.70 $\pm$ 0.12 & -47.85 $\pm$ 0.07 & -47.87 $\pm$ 0.10 & 30.14 $\pm$ 0.03 & 30.17 $\pm$ 0.12\
\
[ccccccc]{} $n_v$ & $\langle E \rangle^T_\beta$ & $\langle E
\rangle^H_\beta$ & $\langle V \rangle^T_\beta$ & $\langle V \rangle^H_\beta$ & $\langle K \rangle^T_\beta$ & $\langle K \rangle^H_\beta$\
\
3 & -70.46 $\pm$ 0.06 & 18.24 $\pm$ 0.20 & -93.47 $\pm$ 0.07 & -69.03 $\pm$ 0.09 & 23.01 $\pm$ 0.02 & 87.27 $\pm$ 0.19\
\
7 & -44.08 $\pm$ 0.05 & -10.81 $\pm$ 0.15 & -71.03 $\pm$ 0.06 & -55.08 $\pm$ 0.08 & 26.94 $\pm$ 0.02 & 44.28 $\pm$ 0.14\
\
15 & -29.84 $\pm$ 0.04 & -15.84 $\pm$ 0.12 & -58.33 $\pm$ 0.06 & -49.10 $\pm$ 0.07 & 28.50 $\pm$ 0.02 & 33.26 $\pm$ 0.12\
\
31 & -22.76 $\pm$ 0.04 & -17.40 $\pm$ 0.10 & -51.95 $\pm$ 0.06 & -47.83 $\pm$ 0.06 & 29.19 $\pm$ 0.03 & 30.43 $\pm$ 0.11\
\
63 & -19.50 $\pm$ 0.04 & -17.68 $\pm$ 0.10 & -49.15 $\pm$ 0.06 & -47.69 $\pm$ 0.06 & 29.65 $\pm$ 0.03 & 30.01 $\pm$ 0.11\
\
127& -18.25 $\pm$ 0.04 & -17.68 $\pm$ 0.10 & -48.20 $\pm$ 0.06 & -47.80 $\pm$ 0.06 & 29.95 $\pm$ 0.03 & 30.11 $\pm$ 0.11\
\
255& -17.84 $\pm$ 0.04 & -17.65 $\pm$ 0.11 & -47.93 $\pm$ 0.07 & -47.85 $\pm$ 0.07 & 30.09 $\pm$ 0.03 & 30.20 $\pm$ 0.12\
\
[ccccccc]{} $n_v$ & $\langle E \rangle^T_\beta$ & $\langle E
\rangle^H_\beta$ & $\langle V \rangle^T_\beta$ & $\langle V \rangle^H_\beta$ & $\langle K \rangle^T_\beta$ & $\langle K \rangle^H_\beta$\
\
3 & -68.54 $\pm$ 0.05 & 78.08 $\pm$ 0.30 & -89.88 $\pm$ 0.07 & -89.88 $\pm$ 0.07 & 21.34 $\pm$ 0.02 & 167.97 $\pm$ 0.32\
\
7 & -45.29 $\pm$ 0.05 & 7.22 $\pm$ 0.19 & -70.88 $\pm$ 0.06 & -70.88 $\pm$ 0.06 & 25.58 $\pm$ 0.02 & 78.10 $\pm$ 0.21\
\
15 & -30.61 $\pm$ 0.04 & -12.52 $\pm$ 0.13 & -58.53 $\pm$ 0.06 & -58.53 $\pm$ 0.06 & 27.92 $\pm$ 0.02 & 46.01 $\pm$ 0.15\
\
31 & -22.95 $\pm$ 0.04 & -16.86 $\pm$ 0.11 & -51.99 $\pm$ 0.06 & -51.99 $\pm$ 0.06 & 29.04 $\pm$ 0.03 & 35.14 $\pm$ 0.12\
\
63 & -19.55 $\pm$ 0.04 & -17.66 $\pm$ 0.10 & -49.19 $\pm$ 0.06 & -49.19 $\pm$ 0.06 & 29.65 $\pm$ 0.03 & 31.53 $\pm$ 0.11\
\
127& -18.29 $\pm$ 0.04 & -17.70 $\pm$ 0.10 & -48.27 $\pm$ 0.06 & -48.27 $\pm$ 0.06 & 29.97 $\pm$ 0.03 & 30.57 $\pm$ 0.11\
\
255& -17.86 $\pm$ 0.04 & -17.71 $\pm$ 0.11 & -47.94 $\pm$ 0.07 & -47.94 $\pm$ 0.07 & 30.07 $\pm$ 0.03 & 30.23 $\pm$ 0.12\
\
The computed averages for all methods and estimators utilized are presented in Tables \[table:rw-wfpi\], \[table:rw-lcpi\], and \[table:lc-ttpi\]. For a given number of path variables $n_v$, the RW-WFPI, RW-LCPI, and TT-DPI methods utilize $2n_v$, $2.25 n_v$, and $n_v$ quadrature points, respectively. \[For a discussion of the minimal number of quadrature points and of the nature of the quadrature schemes that must be employed for the first two methods, the reader should consult Ref. . For the RW-WFPI method, we have utilized $2n_v$ Gauss-Legendre quadrature points, though a number of $1.75n_v$ points would have sufficed.\] The observed overall computational time for the three methods have followed the ratios $2 : 2.25 : 1$, even though the time necessary to compute the paths is proportional to $n_v^2$ for the first method and to $n_v \log_2(n_v)$ for the other methods. The computation of the paths takes full advantage of the vector floating point units of the modern processors and is dominated by the calls to the potential, except for very large $n_v$.
As discussed in Ref. , the asymptotic convergence for the reweighted techniques is expected to be cubic, even for the Lennard-Jones potential that is not included in the class of potentials for which cubic convergence has been demonstrated formally. We find that the asymptotic convergence is attained only for very large $n_v$, as one may see by comparing for example the total, potential, and kinetic energies computed with the help of the T-method estimator for the RW-LCPI and the TT-DPI methods. Even if the latter method has only $1/n_v^2$ asymptotic convergence, the two methods produce almost equal results. In fact, a numerical analysis of the relationship $$\left\langle E \right\rangle_{\beta,n_v}^T \approx \left\langle
E\right\rangle_\beta + \frac{const}{(n_v)^\alpha},$$ in which the left-hand side quantity is plotted against $1/(n_v)^\alpha$ for different values of $\alpha$, suggests that, while the methods have converged within the statistical error, none of the three methods includes sufficiently large values of $n_v$ to attain the ultimate asymptotic rate of convergence.
When comparing the values of the H-method energy estimator and of the related potential and kinetic estimators for the three path integral techniques, one notices that the RW-LCPI technique provides better values than the TT-DPI method. The H-method estimator has a better behavior if used together with a reweighted technique. This behavior is consistent with the analysis we have performed in Section II on the values of the potential point-estimators and the excellent values found with the RW-WFPI method. For the reweighted techniques, the H-method estimator provides better energy values than the T-method estimator. This is also true of the potential and kinetic parts of the estimators. However, the variance of the H-method estimator is significantly larger than the variance of the T-method estimator and the difference is even more pronounced if one compares the corresponding kinetic estimators.
As discussed in Section II.A, the path estimator for the potential energy has a smaller variance than the point estimator. Indeed, the results from Table \[table:rw-wfpi\] show that the variance of the path estimator is approximately $(0.9/0.6)^2=2.25$ times smaller than the variance of the point estimator. In the case of the RW-LCPI and TT-DPI methods, we have employed the estimator given by Eq. (\[eq:24\]) and the path estimator, respectively. These were shown to produce values identical to the point estimator but have the variance of the path estimator. For the RW-WFPI and RW-LCPI methods, the point and the path estimators produce different results. Due to the very design of the reweighted techniques, we have argued that the point estimator results should be the more accurate ones. This theoretical prediction is well supported by the values presented in Tables \[table:rw-wfpi\] and \[table:rw-lcpi\].
While we have argued that the H-method estimator is a better estimator as value (but not necessarily as variance) than the T-method estimator for the reweighted methods, it is apparent from Table \[table:lc-ttpi\] that the same difference persists for the trapezoidal Trotter scheme. As discussed before, for the TT-DPI method, the point and path estimators provide the same value for the average potential. As opposed to the reweigthed techniques, the H-method kinetic estimator is less accurate than the T-method kinetic energy estimator. Quite interestingly, even if individually the potential and the kinetic parts are more accurate for the T-method estimator, it is the H-method energy estimator that provides a more accurate total energy. Clearly, a strong compensation of errors appears in the case of the H-method estimator. Such a compensation of errors is generally characteristic of variational methods. In this respect, notice that the TT-DPI density matrices are positive definite because they are obtained by Lie-Trotter composing a certain symmetrical short-time approximation. By the Ritz variational principle, the H-method energy estimator cannot have a value smaller than the ground-state energy. Thus, the Ritz variational principle provides some control on the values of the H-method estimator, but not on the individual components, nor on the T-method estimator. The RW-LCPI density matrices are also positive definite for $n \geq 2$ and indeed, the energy provided by the H-method estimator is still better than what the values of the potential and kinetic parts suggest. While a final resolution awaits further study, it is apparent that this finding is not related to the asymptotic rate of convergence of the path integral technique.
Among the three methods presented, the RW-WFPI has the fastest convergence for all properties studied. Moreover, for $n_v = 128$ and $n_v = 256$, there is a good agreement (within statistical noise) between the T- and the H-method energy estimators, as well as between their potential and kinetic energy components. For $n_v = 256$, one concludes that the systematic error is smaller than the statistical error for all properties computed. An additional RW-WFPI simulation with $n_v = 512$ in $40$ million Monte Carlo passes has produced results consistent with the findings above. The results are summarized in Table \[tab:1\] and represent the energy values we report.
-------------------------------- ---------- ------- ------------ ------------------------------- ----------- ------- ----------
$\langle E \rangle^T_\beta \ $ $-17.69$ $\pm$ $ 0.02 \ $ $ \ $-17.71 $ $\pm$ $ 0.06$
\langle E \rangle^H_\beta \ $
$\langle V \rangle^T_\beta \ $ $-47.82$ $\pm$ $ 0.03$ $ \ $-47.81 $ $\pm$ $ 0.05$
\langle V \rangle^H_\beta \ $
$\langle K \rangle^T_\beta \ $ $30.13$ $\pm$ $0.02$ $ \ $30.10 $ $\pm$ $ 0.06$
\langle K \rangle^H_\beta \ $
-------------------------------- ---------- ------- ------------ ------------------------------- ----------- ------- ----------
: Estimated energies in K/molecule for the $(\text{H}_2)_{22}$ cluster computed with the help of the Wiener-Fourier reweighted technique using $512$ path variables and $40$ million Monte Carlo passes. Listed are the average potential $\langle V \rangle_\beta$, kinetic $\langle K \rangle_\beta$, and total energies $\langle E
\rangle_\beta$ calculated with the help of the T-method (left column) and H-method (right column) estimators. The reported errors are two standard deviations.[]{data-label="tab:1"}
Conclusions {#sec:conclude}
===========
In the present work we have considered a number of issues related to the choice of estimators for random series path integral methods. We have illustrated our results by applying them to the problem of computing various thermodynamic properties of a model of the (H$_2)_{22}$ cluster using reweighted path integral techniques. The molecular hydrogen cluster is a strongly quantum mechanical system and is representative of the type of problems one is likely to encounter in many applications. Hence, it constitutes a useful benchmark for present and future path integral techniques and for this reason it is important that its physical properties be determined within advertized statistical error bars. Path integral methods capable of dealing with such highly quantum-mechanical systems in an efficient manner are needed, both for reliable determinations of the physical properties of the respective systems as well as for accurate parameterizations of the intermolecular potentials.
We wish to make a number of points concerning the present results and the methods we have utilized to obtain them. At a more general level, we would like to emphasize that the reweighted path integral methods discussed here provide a broadly applicable, simple, and formally well characterized set of techniques. As demonstrated by the present results, they are capable of producing high-quality numerical results for problems of appreciable physical complexity. Moreover, they do so without the assumption of a particular form for the underlying microscopic forces. Furthermore, the estimators described in the present paper are convenient, accurate, and easily implemented for any random series approach. As discussed in Section III, when used together, the T and H-method estimators provide an important consistency check on the quality of the path integral simulations. Such consistency checks are a valuable element in judging the reliability of particular simulations.
As previously mentioned, the cluster application discussed here provides a convenient test bed for the development of numerical methods. For this reason, we have exercised due diligence with respect to the quality of our final results summarized in Table \[tab:1\]. As discussed in Section III, we have subjected both the parallel random number generator employed and the numerical results obtained to a series of quality-control tests. Beyond these statistical checks, it is important to note there is an internal consistency check on the quality of the present results. Specifically, as is evident in Tables \[table:rw-wfpi\], \[table:rw-lcpi\], and \[table:lc-ttpi\], the kinetic, potential, and total energies from the three different path integral approaches (trapezoidal Trotter, reweighted Lévy-Ciesielski, and reweighted Wiener-Fourier) all agree. It is also important to note in this context that, while the presently computed total energies agree with those reported by Chakravarty *et al.*,[@Cha98] the individual kinetic and potential energies do not. The kinetic energy reported by Chakravarty *et al.*[@Cha98] is approximately 0.8 K/particle higher than found in the present simulations (with the potential energy being correspondingly lower). The magnitude of this difference is well outside the statistical error bars involved and appears to signal a systematic error. Based on the observed consistency between the results produced by three different path integral methods and on the agreement between the T and H-method estimators for each of these path integral formulations, we feel confident of the results we have reported in Table \[tab:1\].
*Note:* After the present simulations had been completed, we have learned from D. M. Ceperley that the off-diagonal pair density used as the starting point in the simulations reported in Ref. was truncated at first order in the expansion of off-diagonal displacements instead of second order and that the inclusion of this second order term resolves the kinetic and potential energy difference noted above.
The authors acknowledge support from the National Science Foundation through awards No. CHE-0095053 and CHE-0131114. They also wish to thank the Center for Advanced Scientific Computing and Visualization (TCASCV) at Brown University, especially Dr. James O’Dell, for valuable assistance with respect to the numerical simulations described in the present paper. They would also like to thank Mr. Cristian Diaconu for helpful discussions concerning the present work. Finally, the authors would like to express a special thanks to Professor David Ceperley for continuing discussions concerning the present simulations and for his efforts in tracking down the origin of the pair density issues noted in Section IV.
The main purpose of this section is to give a compact form for the integral $$\begin{aligned}
\label{eq:A1}&&
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,x',\bar{a};\beta)O[x_r(\theta)+
\sigma B_\theta^0(\bar{a})],\end{aligned}$$ where $\theta$ is an arbitrary point in the interval $[0,1]$. In terms of a standard Brownian motion \[see Eq. (\[eq:1a\])\], the above integral can be put into the form
$$\begin{aligned}
\label{eq:A2}&& \nonumber P \left[\sigma B_1 = x' \big| \sigma B_0 = x
\right]\mathbb{E} \left[e^{-\beta \int_0^1 V(\sigma B_u){d}u }O(\sigma
B_\theta) \Big| \sigma B_1 = x' , \sigma B_0 = x \right] \\ && =
\int_{\mathbb{R}} {d}y O(y) P \left[\sigma B_1 = x', \sigma B_\theta = y
\big| \sigma B_0 = x \right]\mathbb{E} \left[e^{-\beta \int_0^1 V(\sigma
B_u){d}u } \Big| \sigma B_1 = x' , \sigma B_\theta = y, \sigma B_0 = x
\right].\end{aligned}$$
Using the Markov property of the Brownian motion, one readily justifies the equalities $$\begin{aligned}
\label{eq:A3}&& \nonumber P\left[\sigma B_1 = x', \sigma B_\theta = y
\big| \sigma B_0 = x \right]\\ && = P\left[\sigma B_1 = x'\big| \sigma
B_\theta = y \right] P\left[ \sigma B_\theta = y \big| \sigma B_0 = x
\right] \qquad \\ && \nonumber =
\rho_{fp}(x,y, \theta \beta) \rho_{fp}[y,x'; (1-\theta)\beta]\end{aligned}$$ and $$\begin{aligned}
\label{eq:A4}&& \nonumber
\mathbb{E} \left[e^{-\beta \int_0^1 V(\sigma B_u){d}u } \Big| \sigma B_1
= x' , \sigma B_\theta = y, \sigma B_0 = x \right] \\ && =\mathbb{E}
\left[e^{-\beta \int_0^\theta V(\sigma B_u){d}u } \Big| \sigma B_\theta
= y , \sigma B_0 = x\right] \\ && \times \mathbb{E} \left[e^{-\beta
\int_{\theta}^1 V(\sigma B_u){d}u } \Big| \sigma B_1 = x' , \sigma
B_\theta = y\right].\nonumber\end{aligned}$$
Performing the transformation of coordinates $u'= u -\theta$ in the second factor of the right-hand side of Eq. (\[eq:A4\]) and employing the invariance of the Brownian motion under time translation $$\begin{aligned}
&&
\left\{\sigma B_{u+\theta}\big| \sigma B_\theta = y , \sigma B_1 = x', u
\geq 0\right\} \\ && \stackrel{d}{=} \left\{\sigma B_{u}\big| \sigma B_0
= y , \sigma B_{1-\theta} = x', u\geq 0\right\},\end{aligned}$$ one obtains $$\begin{aligned}
\label{eq:A5}\nonumber
\mathbb{E} \left[e^{-\beta \int_0^1 V(\sigma B_u){d}u } \Big| \sigma B_1
= x' , \sigma B_\theta = y, \sigma B_0 = x \right] \\ =\mathbb{E}
\left[e^{-\beta \int_0^\theta V(\sigma B_u){d}u } \Big| \sigma B_\theta
= y , \sigma B_0 = x\right] \\ \times \mathbb{E} \left[e^{-\beta
\int_0^{1-\theta} V(\sigma B_u){d}u } \Big| \sigma B_{1-\theta} = x' ,
\sigma B_0 = y\right].\nonumber\end{aligned}$$ Let us focus on the term $$\mathbb{E} \left[e^{-\beta \int_0^\theta V(\sigma B_u){d}u } \Big|
\sigma B_\theta = y , \sigma B_0 = x\right].$$ Performing the substitution of variables $u' = u / \theta$ and employing the scaling property of the Brownian motion $$\begin{aligned}
&&
\left\{\sigma B_{u\theta}\big| \sigma B_0 = x , \sigma B_\theta = y, u
\geq 0\right\} \\ && \stackrel{d}{=} \left\{\sigma
\theta^{1/2}B_{u}\Big| \sigma \theta^{1/2}B_0 = x , \sigma \theta^{1/2}
B_{1} = y, u\geq 0\right\},\end{aligned}$$ one proves $$\begin{aligned}
\label{eq:A6}\nonumber &&
\mathbb{E} \left[e^{-\beta \int_0^\theta V(\sigma B_u){d}u } \Big|
\sigma B_\theta = y , \sigma B_0 = x\right]\\&& \nonumber = \mathbb{E}
\left[e^{-\beta \theta \int_0^1 V(\sigma \theta^{1/2} B_u){d}u } \Big|
\sigma \theta^{1/2} B_1 = y , \sigma \theta^{1/2} B_0 = x\right] \\&& =
\rho(x,y; \theta \beta)/\rho_{fp}(x,y; \theta \beta).\end{aligned}$$ In a similar fashion, one demonstrates that $$\begin{aligned}
\label{eq:A7}\nonumber
\mathbb{E} \left[e^{-\beta \int_0^{1-\theta} V(\sigma B_u){d}u } \Big|
\sigma B_{1-\theta} = x' , \sigma B_0 = y\right]\\ = \rho\left[y,x';
(1-\theta) \beta\right]\big/\rho_{fp}\left[y,x'; (1-\theta) \beta\right].\end{aligned}$$
We now combine Eqs. (\[eq:A1\]), (\[eq:A2\]), (\[eq:A3\]), (\[eq:A5\]), (\[eq:A6\]), and (\[eq:A7\]) to obtain $$\begin{aligned}
\label{eq:A8} \nonumber
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,x',\bar{a};\beta)O[x_r(\theta)+
\sigma B_\theta^0(\bar{a})] \\ =
\int_{\mathbb{R}} \rho(x,y;\theta \beta) \rho[y,x'; (1-\theta)\beta] O(y)
{d}y.\end{aligned}$$ With the help of Eq. (\[eq:A8\]) and by cyclic invariance, $$\begin{aligned}
\label{eq:A9} &&\nonumber
\int_{\mathbb{R}}{d}x \int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)O[x+ \sigma B_\theta^0(\bar{a})] \\
&&\nonumber =
\int_{\mathbb{R}}{d}x\int_{\mathbb{R}} {d}y \rho(x,y;\theta \beta)
\rho[y,x; (1-\theta)\beta] O(y) \\&& = \int_{\mathbb{R}} {d}y
\rho(y,y;\beta) O(y) \\ &&\nonumber = \int_{\mathbb{R}}{d}x
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)O(x).\end{aligned}$$ Moreover, since the function $O(x)$ is arbitrary, the last identity also implies that the random variables $x$ and $x+ \sigma
B_\theta^0(\bar{a})$ have identical distribution functions under the probability measure $$\frac{X_\infty(x,\bar{a};\beta){d}x \, {d}P[\bar{a}]}{\int_{\mathbb{R}}{d}x \int_{\Omega}{d}P[\bar{a}]X_\infty(x,\bar{a};\beta)}.$$ By setting $O(x)=1$ in Eq. (\[eq:A8\]), one obtains the well-known product formula $$\begin{aligned}
\label{eq:A10}\rho(x,x';\beta) = \nonumber
\int_{\Omega}{d}P[\bar{a}]X_\infty(x,x',\bar{a};\beta)\\ =
\int_{\mathbb{R}} \rho(x,y;\theta \beta) \rho[y,x'; (1-\theta)\beta] {d}y,\end{aligned}$$ which is seen to be a consequence of some basic properties of the Brownian motion.
[99]{} *Quantum Monte Carlo Methods in Physics and Chemistry*, edited by M. P. Nightingale and C. J. Umrigar, (Kluwer, Drodrecht, 1999). C. Predescu and J. D. Doll, J. Chem. Phys. **117**, 7448 (2002). C. Predescu, J. D. Doll, and David L. Freeman, *Asymptotic convergence of the partial averaging technique,* e-print: http://arXiv.org/abs/cond-mat/0301525. C. Predescu, *Reweighted Methods: Definition and Asymptotic Convergence,* e-print: http://arXiv.org/abs/cond-mat/0302171. J. D. Doll, R. D. Coalson, and D. L. Freeman, Phys. Rev. Letts. **55**, 1 (1985); R. D. Coalson, D. L. Freeman, and J. D. Doll, J. Chem. Phys. **85**, 4567 (1986). C. Predescu, D. Sabo, and J. D. Doll, *Numerical implementation of some reweighted path integral techniques,* e-print: http://arXiv.org/abs/cond-mat/0305436. M. F. Herman, E. J. Bruskin, and B. J. Berne, J. Chem. Phys. **76**, 5051 (1982). A. Giansanti and G. Jacucci, J. Chem. Phys. **89**, 7454 (1988). P. A. Fernandes, A. P. Carvalho, and J. P. P. Ramalho, J. Chem. Phys. **103**, 5720 (1995). J. Cao and B. J. Berne, J. Chem. Phys. **91**, 6359 (1989). M. Kolar and S. F. O’Shea, J. Phys. A **29**, 3471 (1996). W. Janke and T. Sauer, J. Chem. Phys. **107**, 5821 (1997). D. M. Ceperley, Rev. Mod. Phys., **67**, 279 (1995). K. R. Glaesemann and L. E. Fried, J. Chem. Phys. **116**, 5951 (2002). C. Chakravarty, M. C. Gordillo, and D. M. Ceperley, J. Chem. Phys. **109**, 2123 (1998). J. D. Doll and David L. Freeman, J. Chem. Phys. **111**, 7685 (1999). B. Simon, *Functional Integration and Quantum Physics* (Academic, London, 1979). R. Durrett, *Probability: Theory and Examples,* 2nd ed. (Duxbury, New York, 1996), pp. 430-431. S. Kwapien and W.A. Woyczynski, *Random Series and Stochastic Integrals: Single and Multiple* (Birkhäuser, Boston, 1992), Theorem 2.5.1. C. Predescu and J. D. Doll, Phys. Rev. E **67**, 026124 (2003). H. P. McKean Jr., *Stochastic Integrals* (Academic, New York, 1969). P. Sindzingre, D. M. Ceperley, and M. L. Klein, Phys. Ref. Lett. **67**, 1871 (1991). J. P. Neirotti, D. L. Freeman, and J. D. Doll, Phys. Rev. E **62**, 7445 (2000). N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. M. Teller, and E. Teller, J. Chem. Phys. **21**, 1087 (1953). M. Kalos and P. Whitlock, *Monte Carlo Methods* (Wiley-Interscience, New York, 1986). M. Matsumoto and T. Nishimura, ACM Transactions on Modeling and Computer Simulation **8**, 3 (1998). M. Matsumoto and T. Nishimura, *Dynamic Creation of Pseudorandom Number Generators*, in *Monte Carlo and Quasi-Monte Carlo Methods 1998*, (Springer-Verlag, New York, 2000), pp 56–69. `http://www.math.keio.ac.jp/~matumoto/emt.html`. S. S. Shapiro and M. B. Wilk, Biometrika, **52**, 591 (1965). W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, *Numerical recipes* (Cambridge University, Cambridge, 1992) Ch. 14.3. J. P. Valleau and S. G. Whittington, in *Statistical Mechanics*, edited by B. J. Berne (Plenum, New York, 1977) Ch. 4, p. 145.
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---
abstract: 'Precision measurements of the vortex phase diagram in single crystals of the layered superconductor Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ in oblique magnetic fields confirm the existence of a second phase transition, in addition to the usual first order vortex lattice melting line $H_{m}(T)$. The transition has a strong first order character, is accompanied by strong hysteresis, and intersects the melting line in a tricritical point ($H_{m}^{\perp}$, $H^{\parallel}_{cr}$). Its field dependence and the changing character of the melting line at the tricritical point strongly suggest that the ground state for magnetic fields closely aligned with the superconducting layers is a lattice of uniformly tilted vortex lines.'
author:
- 'M. Konczykowski$^{a}$, C.J. van der Beek$^{a}$, A.E. Koshelev$^{b}$, V. Mosser$^{c}$, M. Dodgson$^{d}$, and P.H. Kes$^{e}$'
title: ' Composite to tilted vortex lattice transition in Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ in oblique fields'
---
The first order “vortex melting” transition from a solid (phase-ordered) state to a liquid state with only short range correlations is the main feature of the phase diagram of vortex lines in clean, layered high-temperature superconductors [@melting]. The application of a small field component $H^{\parallel}$, parallel to the superconducting layers, leads to a lattice of tilted vortex lines that melts in a similar fashion [@CrossLatPRL99]. However, in the more anisotropic (layered) compounds such as Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$, the depression of the perpendicular component of the melting field $H_{m}^{\perp}$ by larger parallel fields was interpreted as the consequence of the decomposition of the tilted vortex lattice into a combined lattice structure of Josephson Vortices (JVs) and Abrikosov-type pancake vortices (PVs) [@CrossLatPRL99]. For very small field components $H^{\perp}$ perpendicular to the layers, chain structures [@ChainReview] arising from the attractive interaction of PVs with JVs were directly visualized by Bitter decoration [@Bolle91; @Grig95], scanning Hall-probe [@ChainReview; @GrigNat01] and magneto-optical techniques [@VlaskoPRB02; @TokunagaPRB02]. At higher $H^{\perp} \sim H_{m}^{\perp}$, the contribution of the JVs to the free energy of the pancake vortex crystal results in the almost linear depression of $H_{m}^{\perp}$ as function of the parallel field [@CrossLatPRL99; @OoiPRL99; @KonczPhysC00; @MirkovPRL01]. This behavior in moderate $H^{\parallel}$ stops at a temperature dependent characteristic field $H^{\parallel}_{cr}$. Even though melting is still observed above $H_{cr}^{\parallel}$, the variation of $H_{m}^{\perp}$ with increasing $H^{\parallel}$ becomes much weaker [@KonczPhysC00; @MirkovPRL01]. Several controversial interpretations of this changing behavior were proposed, such as layer decoupling [@KonczPhysC00], a commensurate transition [@Savel01], and a matching effect [@TokunagaPRB02A].
In this Letter we focus on the high-temperature portion of the vortex phase diagram in single crystalline Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ in oblique fields, which can be established precisely using the well-defined discontinuity of the vortex density at the melting transition. We show that ($H_{m}^{\perp},H_{cr}^{\parallel})$ corresponds to a tricritical point in the vortex lattice phase diagram, where the melting crosses a novel transition from a composite lattice at low parallel fields, to another tilted lattice structure at *high $H^{\parallel}$. The experimental observation of large hysteresis suggests that this transition is strongly first order, consistent with recent predictions [@Koshelev2006]. The identification of the vortex ground state at high parallel field as a tilted lattice structure resolves the open problem of the apparent anisotropy factor $\gamma_{eff}$, and allows one to determine the enhancement of $H_{m}^{\perp}$ by magnetic coupling. We find the temperature dependence of $\gamma_{eff}$ to be consistent with previous observations [@SchillingPRB00; @Mirkovic2002] and in quantitative agreement with the proposed model.*
Experiments were performed on rectangular samples cut from Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ single crystals with different oxygen content [@MingLi]. The $c$-axis component of the local magnetic induction $B^{\perp}({\bf r})$ was measured by micro-Hall sensors placed on the central part of the sample. The 2D electron gas Hall sensors were fabricated in GaAlAs heterostructures and had $8\times
8$ $\mu$m$^{2}$ active area. Results are presented in Fig. \[Fig:DC-AC-loops\](a) as the local magnetization $H_{s}^{\perp} \equiv
B^{\perp}-H^{\perp}$. The local $dc$ magnetization of all crystals shows a sharp discontinuity, $\Delta B^{\perp}$, at the vortex melting transition, that was tracked as function of $H^{\parallel}$ at various fixed temperatures. The angle $\theta$ between the magnetic field and the crystalline $c$-axis was computer-controlled with 0.001$^\circ$ resolution, while the field magnitude could be swept up to 1 T using an electromagnet. Two types of magnetization loops were measured. In the first, the magnetization is traced as function of the $c$-axis field at constant $H^{\parallel}$; in the second, the magnetization is measured as function of $H^{\parallel}$ at constant $H^{\perp}$.
![(a) $dc$ local magnetization loops recorded on an as-grown Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\protect\delta}$ single crystal ($T_{c} = 88$ K) at 75 K, as function of the magnetic field component $H^{\perp}$ perpendicular to the superconducting layers, with the in-plane field $H_{\parallel}$ held constant. The inset shows a magnified view of the discontinuity at the vortex melting transition in $H^{\parallel} = 0$. (b) the in–phase (screening) component of the $ac$ response of the same crystal, recorded under the same conditions, with an $ac$ magnetic field of amplitude $h_{ac} = 0.8$ Oe and frequency $f = 11$ Hz applied along the $c$-axis. The melting transition shows up as a paramagnetic peak (see inset), the transition from combined to tilted vortex lattice is indicated by arrows. The $ac$ response is plotted as the transmittivity $T^{\prime} \equiv [B^{\prime}(f,T) - B(f,T
\ll T_{c})]/[B(f,T > T_{c}) - B(f,T \ll T_{c})]$. []{data-label="Fig:DC-AC-loops"}](Figure_1v2.eps){width="3.in"}
While the discontinuity in the $dc$ magnetization gives a clear identification of the melting field, another method [@MorozovPRL96], in which the magnitude $B(f,T)$ of the periodic part of the induction above the sample is measured at the frequency $f$ of an $ac$ ripple field applied perpendicularly to the sample plane, is more convenient and precise. The $ac$ response is represented as the transmittivity $T^{\prime}$, [*i.e.*]{} the in-phase component $B^{\prime}(f,T)$, normalized by the amplitude $h_{ac}$ of the $ac$ ripple [@Gilchrist93]. The steplike feature in the $dc$ magnetization loop at $H_{m}^{\perp}$ translates to a paramagnetic peak in the $ac$ response, shown in Fig. \[Fig:DC-AC-loops\](b) [@MorozovPRL96]. The magnitude of this peak depends on the ratio of $\Delta B^{\perp}$ to $h_{ac}$. The peak position is independent of both the amplitude and frequency of the $ac$ ripple. In the explored temperature range (above 50 K) and at low frequency (below 27 Hz), a true paramagnetic signal is measured. At higher frequencies or lower temperatures, flux pinning results in the partial shielding of the $ac$ field [@Indenbom96]. Nevertheless, a peak-like feature persists at melting.
Figure \[Fig:DC-AC-loops\](a) shows that at $T > 50$ K, the application of even a small magnetic field component parallel to the layers results in the drastic suppression of magnetic irreversibility. This is expected when the geometric barrier is at the origin of flux pinning [@ZeldovPRL94; @Morozov97]. Simultaneously, $H_{m}^{\perp}$ is depressed linearly with increasing $H^{\parallel}$. However, at a well-defined value $H^{\parallel}_{cr}$, the dependence of $H_{m}^{\perp}$ on in-plane field changes to a much slower, quadratic behavior that very well fits the anisotropic London model, $
H_{m}(\theta) = H_{m0} \slash ( \cos^{2}\theta + \sin^{2}\theta /
\gamma_{eff}^{2})^{1/2}$ [@BlatterPRL92]; *i.e. the perpendicular component of the melting field $$H_{m}^{\perp} = \sqrt{ H_{m0}^{\perp 2} -
\frac{H^{\parallel 2}}{\gamma_{eff}^{2}} }
\approx
H_{m0}^{\perp} \left( 1 - \frac{
H^{\parallel 2}}{ 2 \gamma_{eff}^{2} H_{m0}^{\perp 2}} \right).
\label{Bm_theta_AL}$$*
The characteristic field $H_{m0}$ and the effective anisotropy parameter $\gamma_{eff}$ will be defined below.
![Transmittivity $T^{\prime}$ of the same crystal as in Fig. \[Fig:DC-AC-loops\], recorded at $T = 70$ K and constant $H^{\perp} = 58$ Oe, as a function of $H^{\parallel}$ for various frequencies $f$ and amplitudes $h_{ac}$ of the $ac$ ripple field. A marked hysteresis of the $ac$ screening is observed. This hysteresis disappears when $h_{ac}$ is increased.[]{data-label="Fig:Hysteresis"}](Figure_2.eps){width="3.4in"}
In Fig. \[Fig:DC-AC-loops\](b), another feature in the in-phase component of the $ac$ response can be distinguished, at perpendicular fields $H^{\perp}$ somewhat smaller than $H_{m}^{\perp}$. This feature is brought out much more clearly in sweeps of the parallel field, shown in Fig. \[Fig:Hysteresis\]. There is an abrupt jump from lower to higher values of $T^{\prime}$ on increasing $H^{\parallel}$, that only appears for parallel fields $H^{\parallel}
\lesssim H_{cr}^{\parallel}$. The position of the jump does not depend on $ac$ frequency. At low amplitude of the $ac$ field, a pronounced hysteresis of $T^{\prime}$ is observed; this disappears if $h_{ac}$ is sufficiently increased.
The transmittivity $T^{\prime}$ is simply related to the magnitude of the shielding current flowing in the sample in response to the applied $ac$ magnetic field, a higher $T^{\prime}$ corresponding to a smaller current and less screening [@Gilchrist93]. In the present case, $dc$ magnetization loops point to the geometrical barrier [@ZeldovPRL94] as the main source of screening. However, increasing the $ac$ field frequency reduces the role of thermally activated depinning of vortices in the crystal bulk; as a consequence, a bulk screening current due to vortex pinning emerges [@Indenbom96]. At the frequencies of Fig. \[Fig:Hysteresis\], the step in $T^{\prime}$ is due to a discontinuous change of the magnitude of this bulk current at the well-defined in-plane field, $H^{\parallel}
\equiv H^{\parallel}_{ct}$. The location of $H^{\parallel}_{ct}$ does not depend on the frequency and $h_{ac}$ which indicates *a vortex phase transition in the bulk*, from a low $H^{\parallel}$-phase with higher pinning, to a high $H^{\parallel}$-phase with lower pinning. The hysteresis of the screening current indicates it to be first order. In Fig. \[Fig:PhaseDiagr\] we collect, for $T = 75$ K, the positions of the two first order transitions in a plot of $H^{\perp}$ versus $H^{\parallel}$. The usual melting field $H_{m}^{\perp}$ of the vortices, deduced from the paramagnetic peak in the transmittivity, shows the well-known linear decrease as function of $H^{\parallel}$ [@CrossLatPRL99; @OoiPRL99; @KonczPhysC00; @MirkovPRL01], up to the field component $H_{cr}^{\parallel}$. The field $H^{\perp}_{ct}$ of the first order transition revealed by the (irreversible) transmittivity rapidly increases with $H_{\parallel}$ and crosses the melting line at $H_{cr}^{\parallel}$ in a tricritical point. The same scenario is observed at all explored temperatures ($T > 50$ K), with temperature dependent values of $H_{cr}^{\parallel}$. The anisotropy factor $\gamma_{eff}$, extracted from the London model fit to the high–$H^{\parallel}$ part of the melting line, depends on temperature as well as on the oxygen content of the Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ crystals; it is depicted in Fig \[Fig:Gamma-T-HmJ\].
![Two field-component vortex-lattice phase diagram, with the first-order melting transition $H_{m}^{\perp}$ ($\circ$) determined from the paramagnetic peak in $T^{\prime}$, and the first-order transition to the tilted PV lattice at $B_{t}$ ($\bullet$), determined from the “glitch” in $T^{\prime}$ (Fig. \[Fig:DC-AC-loops\] b). The dashed line is a fit to the composite–to–tilted lattice transition, Eq. (\[eq:Bt\]) with $C =
0.030$; the continuous line is a fit of the high-field portion of the vortex lattice melting line to Eq. (\[Bm\_theta\_AL\]).[]{data-label="Fig:PhaseDiagr"}](Figure_3_new.eps){width="3.4in"}
The low-$H^{\parallel}$ portion of the phase diagram with almost linear $H_m^{\perp} (H^{\parallel})$ dependence has been interpreted as the region of crossing vortex lattices of JVs and PV stacks [@CrossLatPRL99]. A more accurate analysis shows that other “composite-lattice” configurations compete for the ground state in the parameter range of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$. These are the soliton lattice [@Koshelev2003], as well as the set of combined lattices composed of regularly spaced rows of *tilted pancake stacks, separated by $M$ rows of pancake stacks aligned with the $c$-axis. The latter type of lattice becomes favorable at smaller anisotropies and larger $H^{\parallel}$. Moreover, if $H^{\parallel}$ is sufficiently large and the material anisotropy is not extremely high, a *simple tilted lattice ($M = 0$) turns out to be the most favorable configuration. We interpret the experimentally observed transition as that from a composite to such a uniformly tilted lattice. A simple estimate for the in-plane field at which this transition is expected, $B^{\parallel}_{ct}$, can be obtained by comparing the ground state energies of the simplest ($M = 1$) composite lattice and of the uniformly tilted lattice, giving [@Koshelev2006] $$B^{\parallel}_{ct} \approx C \frac{\gamma}{\lambda_{ab}}
\left[ B^{\perp} \Phi_{0} \slash \ln
\left( \frac{1.55 \sqrt{B^{\perp}\Phi_{0}}}{s
B^{\parallel}_{ct}} \right)
\right]^{1/2}.
\label{eq:Bt}$$ Here $\Phi_{0}$ is the flux quantum, $\lambda_{ab}$ is the $ab$–plane penetration depth, $\gamma$ is the penetration depth ratio $\lambda_{c} / \lambda_{ab}$, and $s$ is the layer spacing. Eq. (\[eq:Bt\]) gives a very good fit to the experimental transition line, as illustrated in Fig. \[Fig:PhaseDiagr\].**
The anisotropic three-dimensional behavior (\[Bm\_theta\_AL\]) of $H_{m}^{\perp}$ for large in-plane field $B^{\parallel} >
B_{cr}^{\parallel}$ strongly supports this interpretation. The $ H_{m}^{\perp}( H^{\parallel} )$–dependence is the direct consequence of the vanishing contribution of the magnetic interaction between PVs to the vortex tilt stiffness in a highly inclined tilted vortex structure. The angular dependence of the melting field can be derived using a scaling transformation of coordinates, $\tilde{z}=\gamma
^{2/3}z;\;\tilde{r}_{\perp}=\gamma^{-1/3}r_{\perp}$, which reduces the larger part of the free energy to an isotropic form [@BlatterPRL92]. In scaled coordinates the magnetic field is given by $\tilde{B} =B\gamma^{2/3}\left(
\cos^{2}\theta+\sin^{2}\theta/\gamma^{2}\right)^{1/2}$, while the tilt angle $\tan \tilde{\theta}=\tan\theta/\gamma$. The Josephson tilt energy of a deformed vortex line (PV stack) in scaled coordinates, $$E_{J,t}=\int\frac{d\tilde{k}_{l}}{2\pi}\frac{\tilde{\varepsilon}_{1}(\tilde
{k}_{l})}{2}\tilde{k}_{l}^{2}|\delta\mathbf{\tilde{u}}(\tilde{k}_{l})|^{2},%$$ is determined by the effective line tension $\tilde{\varepsilon}_{1}(\tilde{k}_{l})=\tilde{\varepsilon}_{0}\ln\left(
1 \slash \tilde{r}_{\mathrm{cut}}\tilde{k}_{l} \right)$, valid when the wave vector along the line direction, $\tilde{k}_{l}$, is much larger than the vortex lattice zone boundary vector. Here, $\delta\mathbf{\tilde{u}}(\tilde{k}_{l})$ is the Fourier transform of the line deformation, and $\tilde{\varepsilon}_{0}=\varepsilon_{0}\gamma^{-2/3}$ with $\varepsilon_{0}\equiv\Phi_{0}^{2}/(4\pi\lambda_{ab})^{2}$. For near-perpendicular fields ($\tilde{\theta} \ll 1$) the core cut-off distance $\tilde{r}_{\mathrm{cut}}$ is determined by the so-called thermal vortex wandering length, $\tilde{r}_{\mathrm{cut}} \approx
\langle \mathbf{\tilde{u}}_{n,n+1}^{2} \rangle ^{1/2}
\equiv \langle ( \mathbf{\tilde{u}}_{n+1}-\mathbf{\tilde{u}}_{n}
)^{2} \rangle ^{1/2}$, where $\mathbf{u}_{n}$ is the position of the PV vortex in layer $n$ [@Koshelev98]. For a [*tilted*]{} vortex line, $\mathbf{\tilde{u}}_{n,n+1} =
s\tan\tilde{\theta}+\delta
\mathbf{\tilde{u}}_{n,n+1}$ consists of the average displacement as well as random (thermal) fluctuations meaning that the core cut-off $\tilde{r}_{\mathrm{cut}}^{2}
\approx\ s^{2}\tan^{2}\tilde{\theta}+\left\langle (\delta\mathbf{\tilde
{u}}_{n,n+1})^{2}\right\rangle $. The melting temperature is given by $T_{m}=A \sqrt{
\tilde{\varepsilon}_{1}(1/\tilde{a})\tilde{\varepsilon}_{0}}
\tilde{a}$, with $\tilde{a}\equiv (\Phi
_{0}/\tilde{B})^{1/2}$ and $A\approx0.1$ [@NordborgAEKPRB99]. Returning to real coordinates, we obtain$$T_{m}^{2}=A^{2}(\varepsilon_{0}s)^{2}\ln\left( \frac{C_{J}B_{sc}(\theta
)/B}{r_{0}^{2}+\tan^{2}\theta/\gamma^{2}}\right) \frac{B_{sc}(\theta)}{B}
\label{Tm_theta}%$$ where the numerical constant $C_{J}\approx 5$ can be estimated within the self-consistent harmonic approximation, $B_{sc}(\theta) =
(\Phi_{0}/\gamma^{2}s^{2})/\sqrt{\cos^{2}\theta
+\gamma^{-2}\sin^{2}\theta}$, and $r_{0}^{2} =\left\langle
(\delta\mathbf{\tilde{u}}_{n,n+1}
)^{2}\right\rangle /(\gamma s)^{2} \approx 2A [\Phi_{0}/s^{2}
\gamma^{2}B_{m}(0) ]^{1/2}$. Note that the angular-dependent core cutoff introduces an additional angular dependence of melting field: $T_{m}$ no longer depends only on the ratio $B_{sc}(\theta)/B$. In particular, a new angular scale appears given by $\tan\theta=\gamma r_{0}$. In the experimental angular range $\tan\theta\ll\gamma,\gamma
r_{0}$, we recover Eq. (\[Bm\_theta\_AL\]) with the apparent anisotropy $$\gamma_{eff} \approx \gamma\left( 1+\frac{10\sqrt{B_{m}(0)\gamma^{2}s^{2}%
/\Phi_{0}}}{\ln\left[ 68\sqrt{\Phi_{0}/(B_{m}(0) \gamma^{2}s^{2}%
)}\right] }\right) ^{-1/2} \label{app_anis}.%$$ We note several key points. First, the effective anisotropy $\gamma_{eff}$ is manifestly smaller than the intrinsic $\gamma$. It increases with temperature, and is in excellent agreement with the experimental data of Fig.\[Fig:Gamma-T-HmJ\], strongly suggesting that the modified core cut–off length originating from the tilting of the PV stacks determines the behavior of the melting line at high parallel fields. Very similar behavior has been observed in YBa$_{2}$Cu$_{3}$O$_{7-\delta}$ [@SchillingPRB00]. Next, the prefactor $H_{m0}^{\perp} = B_{m}(0)/\mu_{0}$ in Eq. (\[Bm\_theta\_AL\]) is to be interpreted as the hypothetical vortex melting field $H_{m,J}(\theta = 0)$ in the *absence of the magnetic coupling between PVs. The difference $\Delta H_{mag} =
H_{m}^{\perp}(\theta = 0) - H_{m,J}(0) \approx 0.15 H_{m}^{\perp}$ between the real (experimental) melting field and this prefactor represents the (remarkably modest) enhancement of the melting field due to magnetic coupling.*
![Temperature dependence of the apparent anisotropy $\gamma_{eff}$ extracted from the fits of the melting line to Eq. (\[Bm\_theta\_AL\]). The drawn line shows a fit to Eq. (\[app\_anis\]) with intrinsic $\gamma = 500$. []{data-label="Fig:Gamma-T-HmJ"}](Figure_4.eps){width="3.0in"}
Summarizing, we have established the existence of phase transition of the vortex lattice in single crystalline Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ in oblique fields. The transition has a strong first order character and intersects the usual first order vortex lattice melting line at a tricritical point $[H_{m}^{\perp}(T),H_{ct}^{\parallel}(T)]$. For fields parallel to the superconducting layers $H^{\parallel} < H_{cr}^{\parallel}$ the melting line shows the signature of a composite lattice. For $H^{\parallel} > H_{cr}^{\parallel}$, the melting line is fully consistent with that of a uniformly tilted lattice of PV stacks. We thus propose that the new first order transition takes place between the combined and the tilted vortex lattice. For low in-plane fields, the combined vortex lattice is stabilized by the magnetic interaction between PV’s in the same stack (vortex line). The enhancement of the melting line in the combined lattice regime is due to the contribution of this magnetic interaction to the vortex line tilt stiffness.
[99]{} E. Zeldov [*et al.*]{}, Nature **375**, 373 (1995)
A. E. Koshelev, Phys. Rev. Lett. **83**, 187 (1999).
S. Bending and M. J. W. Dodgson, J. Phys.: Condens. Matter **17**, R955 (2005).
C. A. Bolle, P. L. Gammel, D. G. Grier, C. A. Murray, D. J. Bishop, D.B. Mitzi, and A. Kapitulnik, Phys. Rev. Lett. **66**, 112 (1991).
I. V. Grigorieva, J. W. Steeds, G. Balakrishnan, and D. M. Paul, Phys. Rev. B **51**, 3765 (1995).
A. Grigorenko, S. Bending, T. Tamegai, S. Ooi, and M. Henini, Nature **414**, 728 (2001).
V. K. Vlasko-Vlasov, A. Koshelev, U. Welp, G. W. Crabtree, and K. Kadowaki, Phys. Rev. B **66**, 014523 (2002).
M. Tokunaga, M. Kobayashi, Y. Tokunaga, and T. Tamegai, Phys. Rev. B **66**, 060507 (R) (2002).
B.Schmidt,M. Konczykowski, N. Morozov, and E. Zeldov, Phys. Rev.B **55**, R8705 (1997).
S. Ooi, T.Shibauchi, N.Okuda, and T.Tamegai, Phys. Rev. Lett. **82**, 4308 (1999).
M. Konczykowski, C. J. van der Beek, M. V. Indenbom, E. Zeldov, Physica C, **341**, 1213 (2000).
J. Mirkovic, S. E. Savel’ev, E. Sugahara, and K. Kadowaki, Phys. Rev. Lett. **86**, 886 (2001).
S. E. Savel’ev, J. Mirkovic, K. Kadowaki, Phys. Rev. B **64**, 094521 (2001).
M. Tokunaga, M. Kishi, N. Kameda, K. Itaka, and T. Tamegai, Phys. Rev B **66**, 220501(R) (2002).
A.E. Koshelev, unpublished. A. Schilling, M. Willemin, C. Rossel, H. Keller, R.A. Fisher, N.E. Phillips, U. Welp, W.K. Kwok, R.J. Olsson, and G.W. Crabtree, Phys. Rev. B [**61**]{}, 3592 (2000).
J. Mirković, S.E. Savel’ev, E. Sugahara, and K. Kadowaki, Phys. Rev. B [**66**]{}, 132505 (2002).
M. Li, C.J. van der Beek, M. Konczykowski, A.A. Menovsky, and P.H. Kes, Phys. Rev. B [**66**]{}, 024502 (2002).
N. Morozov, E. Zeldov, D. Majer, and M. Konczykowski, Phys. Rev. B **54**, R3784 (1996).
M.V. Indenbom, C.J. van der Beek, V. Berseth, M. Konczykowski, N. Motohira, H. Berger and W. Benoit, J. Low Temp. Phys., [**105**]{}, 1117 (1996).
J. Gilchrist and M. Konczykowski, Physica [**C 212**]{}, 43 (1993).
E. Zeldov, A.I. Larkin, V.B. Geshkenbein, M. Konczykowski, D. Majer, B. Khaykovich, V.M. Vinokur, and H. Shtrikman, Phys. Rev. Lett. [**73**]{}, 1428 (1994).
N. Morozov, E. Zeldov, M.Konczykowski, and R. A. Doyle, Physica [**C 291**]{}, 113 (1997).
G. Blatter, V.B. Geshkenbein, and A.I. Larkin, Phys. Rev. Lett. **68**, 875 (1992).
[**12**]{},14 (1964)
A.E. Koshelev, Phys. Rev. [**68**]{}, 094520 (2003). A.E. Koshelev and V.M. Vinokur, Phys. Rev. B [**57**]{}, 8026 (1998); T.R. Goldin and B. Horovitz, Phys. Rev. B [**58**]{}, 9524 (1998).
A.E. Koshelev, H. Nordborg, Phys.Rev. **59**, 4358 (1999).
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---
abstract: 'We present a decomposition of the space of twisted arcs of a toric stack. As a consequence, we give a combinatorial description of the motivic integral associated to a torus-invariant divisor of a toric stack.'
address: 'Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA'
author:
- 'A. Stapledon'
bibliography:
- 'alan.bib'
title: Motivic Integration on Toric Stacks
---
Introduction
============
Let $N$ be a lattice of rank $d$ and set $N_{\mathbb{R}} = N \otimes_{\mathbb{Z}} \mathbb{R}$. Let $\Sigma$ be a simplicial, rational, $d$-dimensional fan in $N_{\mathbb{R}}$, and assume that the support $|\Sigma|$ of $\Sigma$ is convex. Denote the primitive integer generators of $\Sigma$ by $v_{1}, \ldots, v_{r}$ and fix positive integers $a_{1}, \ldots, a_{r}$. The data ${\mbox{\boldmath$\Sigma$}}= (N, \Sigma, \{ a_{i}v_{i} \} )$ is called a *stacky fan*. In [@BCSOrbifold], Borisov, Chen and Smith associated to ${\mbox{\boldmath$\Sigma$}}$ a smooth Deligne-Mumford stack $\mathcal{X} = \mathcal{X}({\mbox{\boldmath$\Sigma$}})$ over $\mathbb{C}$, called a *toric stack*, with coarse moduli space the toric variety $X = X(\Sigma)$ (see also [@FMNSmooth]). To any complex variety $Y$, one can associate its corresponding arc space $J_{\infty}(Y) := \operatorname{Hom}( \operatorname{Spec}\mathbb{C}[[t]], Y)$. The geometry of $J_{\infty}(Y)$ encodes a lot of information about the birational geometry of $Y$ and has been the subject of intensive study in recent times (see, for example, [@ELMContact; @EMInversion; @MusJet]). Kontsevich introduced the theory of *motivic integration* in [@KonMotivic], which assigns a measure to subsets of $J_{\infty}(Y)$ and allows one to compare invariants on birationally equivalent varieties. This theory has been developed over the last decade by many authors including Denef and Loeser [@DLMotivic; @DLGerms; @DLGeometry; @DLMotivic2] and Batyrev [@BatNon; @BDStrong].
Yasuda extended the theory of arc spaces and motivic integration to Deligne-Mumford stacks in [@YasTwisted] and [@YasMotivic]. More specifically, to any Deligne-Mumford stack $\mathcal{Y}$, he associated the space $|\mathcal{J}_{\infty}\mathcal{Y}|$ of *twisted arcs* of $\mathcal{Y}$ and defined an associated measure. The goal of this paper is to give an explicit description of the space of twisted arcs of the toric stack $\mathcal{X}$ above. We will prove a decomposition of $|\mathcal{J}_{\infty}\mathcal{X}|$ which allows us to compute motivic integrals on $\mathcal{X}$ and we will show how these motivic integrals relate to combinatorial invariants from the theory of lattice point enumeration of polyhedral complexes. We briefly recall Ishii’s decomposition of the arc space $J_{\infty}(X)$ of the toric variety $X$ in [@IshArc] (for more details, see Section \[arc\]). If $T$ denotes the torus of $X$ with corresponding arc space $J_{\infty}(T)$, then $J_{\infty}(T)$ has the structure of a group and acts on $J_{\infty}(X)$. If $D_{1}, \ldots, D_{r}$ denote the $T$-invariant divisors of $X$, then the open subset $J_{\infty}(X)' = J_{\infty}(X) \smallsetminus \cup_{i} J_{\infty}(D_{i})$ is invariant under the action of $J_{\infty}(T)$. Ishii gave a decomposition of $J_{\infty}(X)'$ into $J_{\infty}(T)$-orbits indexed by $|\Sigma| \cap N$ and described the orbit closures.
Our first main result is a stacky analogue of Ishii’s decomposition. We describe an action of $J_{\infty}(T)$ on $|\mathcal{J}_{\infty}\mathcal{X}|$, which restricts to an action on $|\mathcal{J}_{\infty}\mathcal{X}|' = |\mathcal{J}_{\infty}\mathcal{X}| \smallsetminus \cup_{i} |\mathcal{J}_{\infty}\mathcal{D}_{i}|$, where $\mathcal{D}_{1}, \ldots, \mathcal{D}_{r}$ are the $T$-invariant prime divisors of $\mathcal{X}$ (see Section \[arc\]). We define a twisted arc $\tilde{\gamma}_{v}$ for each element $v$ in $|\Sigma| \cap N$, and describe the $J_{\infty}(T)$-orbits of $|\mathcal{J}_{\infty}\mathcal{X}|'$ (Theorem \[decomp\]), as follows.
We have a decomposition of $|\mathcal{J}_{\infty}\mathcal{X}|'$ into $J_{\infty}(T)$-orbits $$|\mathcal{J}_{\infty}\mathcal{X}|' = \coprod_{v \in |\Sigma| \cap N} \tilde{\gamma}_{v} \cdot J_{\infty}(T).$$ Moreover, $\tilde{\gamma}_{w} \cdot J_{\infty}(T) \subseteq \overline{\tilde{\gamma}_{v} \cdot J_{\infty}(T)}$ if and only if $w - v = \sum_{\rho_{i} \subseteq \sigma} \lambda_{i}a_{i}v_{i}$ for some non-negative integers $\lambda_{i}$ and some cone $\sigma$ containing $v$ and $w$.
We now turn our attention to motivic integrals on $\mathcal{X}$. If $\mathcal{E} = \sum_{i = 1}^{r} b_{i} \mathcal{D}_{i}$ is a $T$-invariant $\mathbb{Q}$-divisor on $\mathcal{X}$, then the pair $(\mathcal{X}, \mathcal{E})$ is *Kawamata log terminal* if $b_{i} < 1$ for $i = 1, \ldots, r$. To any Kawamata log terminal pair $(\mathcal{X}, \mathcal{E})$, Yasuda associated the motivic integral $\Gamma(\mathcal{X},\mathcal{E})$ of $\mathcal{E}$ on $\mathcal{X}$. In the case when each $a_{i} = 1$, Theorem 3.41 in [@YasMotivic] implies that $\Gamma(\mathcal{X},\mathcal{E})$ is equal to Batyrev’s stringy invariant of a corresponding Kawamata log terminal pair of varieties [@BatNon]. We may view $\Gamma(\mathcal{X},\mathcal{E})$ either as an element of a Grothendieck ring of varieties [@YasTwisted] or as an element in a ring of convergent stacks [@YasMotivic]. For simplicity, we will specialise $\Gamma(\mathcal{X},\mathcal{E})$ and view it as a convergent power series in $\mathbb{Z}[(uv)^{1/N}][[ (uv)^{-1/N} ]]$, for some positive integer $N$, and refer the reader to section \[motivic\] for the relevant details.
In order to motivate and state our results, we recall some notions from the theory of lattice point enumeration of polyhedral complexes (see, for example, [@BarLattice]). Denote by $\psi: |\Sigma| \rightarrow \mathbb{R}$ the piecewise $\mathbb{Q}$-linear function satisfying $\psi(a_{i}v_{i}) = 1$ for $i = 1, \ldots ,r$, and consider the polyhedral complex $Q = \{ v \in |\Sigma| \, | \, \psi(v) \leq 1 \}$. If for each positive integer $m$, $f_{Q}(m)$ denotes the number of lattice points in $mQ$, then $f_{Q}(m)$ is a polynomial in $m$ of degree $d$, called the *Ehrhart polynomial* of $Q$. The generating series of $f_{Q}(m)$ can be written in the form $$1 + \sum_{m \geq 1} f_{Q}(m)t^{m} = \delta_{Q}(t)/ (1 - t)^{d + 1},$$ where $\delta_{Q}(t)$ is a polynomial of degree less than or equal to $d$, called the *Ehrhart $\delta$-polynomial* of $Q$. More generally, if $\mu: |\Sigma| \rightarrow \mathbb{R}_{\geq 0}$ is a piecewise linear function, one can consider the power series $$\delta_{Q}(t;\mu) := (1 - t)^{d + 1}(1 + \sum_{m \geq 1} \sum_{v \in mQ \cap N} t^{\mu(v) + m}),$$ so that $\delta_{Q}(t;\mu) = \delta_{Q}(t)$ when $\mu$ is identically zero. The Ehrhart $\delta$-polynomial of $Q$ and its associated generalisations have been studied extensively over the last forty years by many authors including Stanley [@StaHilbert2; @StaDecompositions; @StaHilbert1; @StaMonotonicity] and Hibi [@HibSome; @HibEhrhart; @HibLower; @HibStar].
Motivated precisely by computations of motivic integrals on toric stacks, the following variant of the above definition was recently introduced in [@YoWeightI]. If $\lambda: |\Sigma| \rightarrow \mathbb{R}$ is a piecewise $\mathbb{Q}$-linear function satisfying $\lambda(a_{i}v_{i}) > - 1$ for $i = 1, \ldots, r$, then the *weighted $\delta$-vector* $\delta^{\lambda}(t)$ is defined by $$\delta^{\lambda}(t) = (1 - t)^{d + 1}(1 + \sum_{m \geq 1} \sum_{v \in mQ \cap N} t^{\psi(v) - \lceil \psi(v) \rceil + \lambda(v) + m}).$$ Observe that if $\lambda$ is a piecewise linear function, then the coefficient of $t^{i}$ in $\delta_{Q}(t; \lambda)$ is equal to the sum of the coefficients of $t^{j}$ in $\delta^{\lambda}(t)$ for $i - 1 < j \leq i$. The weighted $\delta$-vector satisfies the following symmetry property.
With the notation above, the expression $\delta^{\lambda}(t)$ is a rational function in $\mathbb{Q}(t^{1/N})$, for some positive integer $N$. If $\Sigma$ is a complete fan, then $$\delta^{\lambda}(t) = t^{d} \delta^{\lambda}(t^{-1}).$$
The following result (Theorem \[tropo\]) may be viewed as a geometric realisation of weighted $\delta$-vectors.
Consider a Kawamata log terminal pair $(\mathcal{X}, \mathcal{E})$, where $\mathcal{E}$ is a $T$-invariant $\mathbb{Q}$-divisor on the toric stack $\mathcal{X}$. There exists a corresponding piecewise $\mathbb{Q}$-linear function $\lambda: |\Sigma| \rightarrow \mathbb{R}$ satisfying $\lambda(a_{i}v_{i}) > - 1$ for $i = 1, \ldots, r$, such that $$\Gamma(\mathcal{X}, \mathcal{E}) = (uv)^{d} \delta^{\lambda}((uv)^{-1}).$$ In particular, $\Gamma(\mathcal{X}, \mathcal{E})$ is a rational function in $\mathbb{Q}(t^{1/N})$, for some positive integer $N$. If $\Sigma$ is a complete fan, then $$\Gamma(\mathcal{X}, \mathcal{E})(u,v) = (uv)^{d} \Gamma(\mathcal{X}, \mathcal{E})(u^{-1}, v^{-1}) = \delta^{\lambda}(uv).$$ Moreover, every weighted $\delta$-vector has the form $\delta^{\lambda}(uv) = (uv)^{d}\Gamma(\mathcal{X}, \mathcal{E})(u^{-1}, v^{-1})$, for such a pair $(\mathcal{X}, \mathcal{E})$.
In the case when $\mathcal{E} = 0$, the invariant $\Gamma(\mathcal{X}, 0)$ is a polynomial in $\mathbb{Z}[ (uv)^{1/N} ]$, for some positive integer $N$, and the coefficient of $(uv)^{j}$ is equal to the dimension of the $2j^{\textrm{th}}$ orbifold cohomology group of $\mathcal{X}$ with compact support [@YasTwisted]. It follows from Poincaré duality for orbifold cohomology [@CRNew] and the above discussion that the coefficients of the Ehrhart $\delta$-polynomial of $Q$ are sums of dimensions of orbifold cohomology groups of $\mathcal{X}$. A detailed discussion of this result is provided in [@YoWeightI].
We conclude the introduction with an outline of the contents of the paper. In Section \[tstack\], we recall Borisov, Chen and Smith’s notion of a toric stack and give an explicit description of the local construction. We review Yasuda’s theory of twisted jets in Section \[jet\] and establish a concrete description of these spaces in the toric case. We use these results in Section \[arc\] to prove our decomposition of the space of twisted arcs of a toric stack. In Section \[contact\], we compute the contact order of a twisted arc along a torus-invariant divisor and, in Section \[motivic\], we use the results of the previous sections to compute certain motivic integrals on toric stacks. Finally, in Section \[transform\], we interpret Yasuda’s change of variables formula in the toric case to give a geometric proof of a combinatorial result in [@YoWeightI].
The author would like to thank his advisor Mircea Mustaţǎ for all his help. He would also like to thank Bill Fulton, Sam Payne and Kevin Tucker for some useful discussions. The author was supported by Mircea Mustaţǎ’s Packard Fellowship and by an Eleanor Sophia Wood travelling scholarship from the University of Sydney.
Toric Stacks {#tstack}
============
We fix the following notation throughout the paper. Let $N$ be a lattice of rank $d$ and set $N_{\mathbb{R}} = N \otimes_{\mathbb{Z}} \mathbb{R}$. Let $\Sigma$ be a simplicial, rational, $d$-dimensional fan in $N_{\mathbb{R}}$, and let $X = X(\Sigma)$ denote the corresponding toric variety with torus $T$. We assume that the support $|\Sigma|$ of $\Sigma$ in $N_{\mathbb{R}}$ is convex. Let $\rho_{1}, \ldots, \rho_{r}$ denote the rays of $\Sigma$, with primitive integer generators $v_{1}, \ldots, v_{r}$ in $N$ and corresponding $T$-invariant divisors $D_{1}, \ldots, D_{r}$. Fix elements $b_{1}, \ldots , b_{r}$ in $N$ such that $b_{i} = a_{i}v_{i}$ for some positive integer $a_{i}$. The data ${\mbox{\boldmath$\Sigma$}}= (N, \Sigma, \{ b_{i} \} )$ is called a *stacky fan*. We denote by $\psi: |\Sigma| \rightarrow \mathbb{R}$ the function that is $\mathbb{Q}$-linear on each cone of $\Sigma$ and satisfies $\psi(b_{i}) = 1$ for $i = 1, \ldots ,r$. For each non-zero cone $\tau$ of $\Sigma$, set $$\label{mcbox}
\operatorname{Box}({\mbox{\boldmath$\tau$}}) = \{ v \in N \mid v = \sum_{\rho_{i} \subseteq \tau} q_{i}b_{i} \textrm{ for some }
0 < q_{i} < 1 \}.$$ We set $\operatorname{Box}(\mbox{\boldmath$\{ 0 \}$}) = \{0\}$ and $\operatorname{Box}({\mbox{\boldmath$\Sigma$}}) = \cup_{\tau \in \Sigma} \operatorname{Box}({\mbox{\boldmath$\tau$}})$. We will always work over $\mathbb{C}$ and will often identify schemes with their $\mathbb{C}$-valued points.
Associated to a stacky fan, there is a smooth Deligne-Mumford toric stack $\mathcal{X} = \mathcal{X}({\mbox{\boldmath$\Sigma$}})$ over $\mathbb{C}$ with coarse moduli space $X$ [@BCSOrbifold]. Each cone $\sigma$ in $\Sigma$ corresponds to an open substack $\mathcal{X}({\mbox{\boldmath$\sigma$}})$ of $\mathcal{X}$. We identify $\mathcal{X}(\mathbf{ \{ 0 \}})$ with the torus $T$ of $X$. The open substacks $\{ \mathcal{X}({\mbox{\boldmath$\sigma$}}) \, | \, \dim \sigma = d \}$ give an open covering of $\mathcal{X}$. We will give an explicit description of $\mathcal{X}({\mbox{\boldmath$\sigma$}})$, for a fixed cone $\sigma$ of dimension $d$. If $N_{\sigma}$ denotes the sublattice of $N$ generated by $\{ b_{i} \, | \, \rho_{i} \subseteq \sigma \}$, then $N(\sigma) = N/N_{\sigma}$ is a finite group with elements in bijective correspondence with $\coprod_{\tau \subseteq \sigma}\operatorname{Box}({\mbox{\boldmath$\tau$}})$. Let $M_{\sigma}$ be the dual lattice of $N_{\sigma}$ and let $M$ be the dual lattice of $N$. If $\sigma'$ denotes the cone in $N_{\sigma}$ generated by $\{ b_{i} \, | \, \rho_{i} \subseteq \sigma \}$, with corresponding dual cone $(\sigma')^{\vee}$ in $M_{\sigma}$, then we will make the identifications $$\operatorname{Spec}\mathbb{C}[(\sigma')^{\vee} \cap M_{\sigma}] \cong \mathbb{A}^{d}, \: \operatorname{Hom}_{\mathbb{Z}}(M_{\sigma}, \mathbb{C}^{*}) \cong (\mathbb{C}^{*})^{d}.$$ If we regard $\mathbb{Q}/\mathbb{Z}$ as a subgroup of $\mathbb{C}^{*}$ by sending $p$ to $\exp (2 \pi \sqrt{-1}p)$, then we have a natural isomorphism $$\label{eqnZ}
N(\sigma) \cong \operatorname{Hom}_{\mathbb{Z}}( \operatorname{Hom}_{\mathbb{Z}}(N(\sigma), \mathbb{Q}/\mathbb{Z}), \mathbb{C}^{*}) =
\operatorname{Hom}_{\mathbb{Z}}( \operatorname{Ext}_{\mathbb{Z}}^{1}(N(\sigma), \mathbb{Z}), \mathbb{C}^{*}).$$ We apply the functor $\operatorname{Hom}_{\mathbb{Z}}( \, \: , \mathbb{Z})$ to the exact sequence $$0 \rightarrow N_{\sigma} \rightarrow N \rightarrow N(\sigma) \rightarrow 0,$$ to get $$0 \rightarrow M \rightarrow M_{\sigma} \rightarrow \operatorname{Ext}_{\mathbb{Z}}^{1}(N(\sigma),\mathbb{Z}) \rightarrow 0.$$ Applying $\operatorname{Hom}( \, \: , \mathbb{C^{*}})$ and the natural isomorphism (\[eqnZ\]), gives an injection $N(\sigma) \rightarrow (\mathbb{C}^{*})^{d}$. The natural action of $(\mathbb{C}^{*})^{d}$ on $\mathbb{A}^{d}$ induces an action of $N(\sigma)$ on $\mathbb{A}^{d}$. We identify $\mathcal{X}({\mbox{\boldmath$\sigma$}})$ with the quotient stack $[ \mathbb{A}^{d} / N(\sigma)]$, with corresponding coarse moduli space the open subscheme $U_{\sigma} = \mathbb{A}^{d} / N(\sigma)$ of $X$ [@BCSOrbifold].
Consider an element $g$ in $N(\sigma)$ of order $l$ corresponding to $v$ in $\operatorname{Box}({\mbox{\boldmath$\sigma(v)$}})$, where $\sigma(v)$ denotes the cone containing $v$ in its relative interior. We can write $v = \sum_{i = 1}^{d} q_{i} b_{i}$, for some $0 \leq q_{i} < 1$. Note that $q_{i} \neq 0$ if and only if $\rho_{i} \subseteq \sigma(v)$. If $x_{1}, \ldots, x_{d}$ are the coordinates on $\mathbb{A}^{d} = \operatorname{Spec}\mathbb{C}[(\sigma')^{\vee} \cap M_{\sigma}]$ and $\zeta_{l} = \exp(2\pi \sqrt{-1}/l )$, then one can verify that the action of $N(\sigma)$ on $\mathbb{A}^{d}$ is given by $$\label{actions}
g \cdot (x_{1}, \ldots, x_{d}) = (\zeta_{l}^{\, lq_{1}}x_{1}, \ldots , \zeta_{l}^{\,lq_{d}}x_{d}),$$ and the *age* of $g$ (see Subsection 7.1 [@AGVAlgebraic]) is equal to $$\label{age}
\operatorname{age}(g) = (1/l) \sum_{i = 1}^{d} lq_{i} = \psi(v).$$
Twisted Jets of Toric Stacks {#jet}
============================
We use the discussion of twisted jets of Deligne-Mumford stacks in [@YasMotivic] to give an explicit description of the toric case. More specifically, for any non-negative integer $n$, we describe the stack of twisted $n$-jets $\mathcal{J}_{n}\mathcal{X}$ of the toric stack $\mathcal{X}$. Fix an affine scheme $S = \operatorname{Spec}R$ over $\mathbb{C}$ and let $D_{n,S}$ denote the affine scheme $\operatorname{Spec}R[t]/(t^{n + 1})$. If we fix a positive integer $l$ and consider the group $\mu_{l}$ of $l$th roots of unity with generator $\zeta_{l} = \exp (2 \pi \sqrt{-1} /l)$, then $\mu_{l}$ acts on $D_{nl,S}$ via the morphism $p: D_{nl,S} \times \mu_{l} \rightarrow D_{nl,S}$, corresponding to the ring homomorphism $R[t]/(t^{nl + 1}) \rightarrow R[t]/(t^{nl + 1}) \otimes \mathbb{C}[x]/(x^{l} - 1), t \mapsto t \otimes x$. That is, $\mu_{l}$ acts on $D_{nl,S}$ by scaling $t$. If $\mathcal{D}_{n,S}^{l}$ denotes the quotient stack $[D_{nl,S} / \mu_{l} ]$, then we have morphisms $$D_{nl,S} \stackrel{\pi}{\rightarrow} \mathcal{D}_{n,S}^{l} \rightarrow D_{n,S},$$ such that $\pi$ is an atlas for $\mathcal{D}_{n,S}^{l}$ and $D_{n,S}$ is the coarse moduli space of $\mathcal{D}_{n,S}^{l}$. The composition of the two maps is the quotient of $D_{nl,S}$ by $\mu_{l}$ and corresponds to the ring homomorphism $R[t]/(t^{n + 1}) \rightarrow R[t]/(t^{nl + 1})$, $t \mapsto t^{l}$. The atlas $\pi$ corresponds to the object $\alpha$ of $\mathcal{D}_{n,S}^{l}$ over $D_{nl,S}$ $$\label{eqnat}
\xymatrix{ D_{nl,S} \times \mu_{l} \ar[d]^{pr_{1}} \ar[r]^p & D_{nl,S} \\
D_{nl,S} & }$$ and every object in $\mathcal{D}_{n,S}^{l}$ is locally a pullback of $\alpha$. Consider the automorphism $$\theta = \zeta_{l} \times \zeta_{l}^{-1}: D_{nl,S} \times \mu_{l} \rightarrow D_{nl,S} \times \mu_{l}$$ over $\zeta_{l}: D_{nl,S} \rightarrow D_{nl,S}$. Every automorphism of $\alpha$ is a power of $\theta$ and hence every object and automorphism in $\mathcal{D}_{n,S}^{l}$ is locally determined by a pullback of $\alpha$ and a power of $\theta$.
A *twisted $n$-jet of order $l$* of $\mathcal{X}$ over $S$ is a representable morphism $\mathcal{D}_{n,S}^{l} \rightarrow \mathcal{X}$. By the above discussion, a twisted jet is determined by the images of $\alpha$ and $\theta$. Yasuda defined the stack $\mathcal{J}_{n}^{l}\mathcal{X}$ of twisted $n$-jets of order $l$ of $\mathcal{X}$. An object of $\mathcal{J}_{n}^{l}\mathcal{X}$ over $S$ is a twisted $n$-jet $\gamma: \mathcal{D}_{n,S}^{l} \rightarrow \mathcal{X}$ of order $l$. Suppose $\gamma': \mathcal{D}_{n,T}^{l} \rightarrow \mathcal{X}$ is another twisted $n$-jet of order $l$. If $f: S \rightarrow T$ is a morphism, there is an induced morphism $f': \mathcal{D}_{n,S}^{l} \rightarrow \mathcal{D}_{n,T}^{l}$. A morphism in $\mathcal{J}_{n}^{l}\mathcal{X}$ from $\gamma$ to $\gamma'$ over $f: S \rightarrow T$ is a 2-morphism from $\gamma$ to $\gamma' \circ f'$. The stack $\mathcal{J}_{n}\mathcal{X}$ of twisted $n$-jets is the disjoint union of the stacks $\mathcal{J}_{n}^{l}\mathcal{X}$ as $l$ varies over the positive integers. Both $\mathcal{J}_{n}\mathcal{X}$ and the $\mathcal{J}_{n}^{l}\mathcal{X}$ are smooth Deligne-Mumford stacks (Theorem 2.9 [@YasMotivic]). The stack $\mathcal{J}_{0}\mathcal{X}$ is identified with the *inertia stack* $\mathcal{I}(\mathcal{X})$ of $\mathcal{X}$. That is, an object of $\mathcal{J}_{0}\mathcal{X}$ over $S$ is determined by a pair $(x, \alpha)$, where $x$ is an object of $\mathcal{X}$ over $S$ and $\alpha$ is an automorphism of $x$. We identify the stack $\mathcal{J}_{0}^{1}\mathcal{X}$ with $\mathcal{X}$. For any $m \geq n$ and for each $l > 0$, the truncation map $R[t]/(t^{ml + 1}) \rightarrow R[t]/(t^{nl + 1})$ induces a morphism $\mathcal{D}_{n,S}^{l} \rightarrow \mathcal{D}_{m,S}^{l}$. Via composition, we get a natural affine morphism (Theorem 2.9 [@YasMotivic]) $$\pi^{m}_{n} : \mathcal{J}_{m}\mathcal{X} \rightarrow \mathcal{J}_{n}\mathcal{X}.$$ The projective system $\{ \mathcal{J}_{n} \mathcal{X} \}_{n}$ has a projective limit with a projection morphism (p15 [@YasMotivic]) $$\mathcal{J}_{\infty}\mathcal{X} = \lim_{\leftarrow} \mathcal{J}_{n}\mathcal{X}.$$ $$\pi_{n}: \mathcal{J}_{\infty}\mathcal{X} \rightarrow \mathcal{J}_{n}\mathcal{X}.$$
The open covering $\{ \mathcal{X}({\mbox{\boldmath$\sigma$}}) \, | \, \dim \sigma = d \}$ of $\mathcal{X}$ induces an open covering $\{ \mathcal{J}_{n}\mathcal{X}({\mbox{\boldmath$\sigma$}}) \, | \, \dim \sigma = d \}$ of $\mathcal{J}_{n}\mathcal{X}$, for $0 \leq n \leq \infty$.
Recall that if $\mathcal{Y}$ is a stack over $\mathbb{C}$, then we can consider the set of *points* $|\mathcal{Y}|$ of $\mathcal{Y}$ over $\mathbb{C}$ [@LMBChamps]. Its elements are equivalence classes of morphisms from $\operatorname{Spec}\mathbb{C}$ to $\mathcal{Y}$. Two morphisms $\psi, \psi': \operatorname{Spec}\mathbb{C} \rightarrow \mathcal{Y}$ are equivalent if there is a $2$-morphism from $\psi$ to $\psi'$. It has a Zariski topology; for every open substack $\mathcal{Y'}$ of $\mathcal{Y}$, $|\mathcal{Y}'|$ is an open subset of $|\mathcal{Y}|$. If $\mathcal{Y}$ has a coarse moduli space $Y$, then $|\mathcal{Y}|$ is homeomorphic to $Y(\mathbb{C})$. We will often identify $Y$ with $Y(\mathbb{C})$.
Let $D_{\infty,\mathbb{C}} = \operatorname{Spec}\mathbb{C}[[t]]$ and $\mathcal{D}_{\infty,\mathbb{C}}^{l} = [D_{\infty,\mathbb{C}}/\mu_{l}]$. A *twisted arc* of order $l$ of $\mathcal{X}$ over $\mathbb{C}$ is a representable morphism from $\mathcal{D}_{\infty,\mathbb{C}}^{l}$ to $\mathcal{X}$. Two twisted arcs $\alpha, \alpha': \mathcal{D}_{\infty,\mathbb{C}}^{l} \rightarrow \mathcal{X}$ of order $l$ are equivalent if there is a $2$-morphism from $\alpha$ to $\alpha'$. The set of equivalence classes of twisted arcs over $\mathcal{X}$ is identified with $|\mathcal{J}_{\infty}\mathcal{X}|$ (p16 [@YasMotivic]), which we will call the *space of twisted arcs* of $\mathcal{X}$.
For $0 \leq n \leq \infty$ and a positive integer $l$, the scheme $D_{nl, \mathbb{C}}$ has a unique closed point. A (not necessarily representable) morphism $\gamma: \mathcal{D}_{n,\mathbb{C}}^{l} \rightarrow \mathcal{X}$ induces an object $\bar{\gamma}$ of $\mathcal{X}$ over $\mathbb{C}$, $$\bar{\gamma}: \operatorname{Spec}\mathbb{C} \rightarrow D_{nl, \mathbb{C}} \rightarrow \mathcal{D}_{n,\mathbb{C}}^{l} \rightarrow \mathcal{X}.$$ The automorphism group of the object in $\mathcal{D}_{n,\mathbb{C}}^{l}$ corresponding to the morphism $\operatorname{Spec}\mathbb{C} \rightarrow \mathcal{D}_{n,\mathbb{C}}^{l}$ is $\mu_{l}$. Hence we get a homomorphism $\phi: \mu_{l} \rightarrow
\operatorname{Aut}(\bar{\gamma})$. Since every automorphism in $\mathcal{D}_{n,\mathbb{C}}^{l}$ is locally induced by $\theta$, $\phi$ is injective if and only if $\operatorname{Aut}(\chi) \rightarrow \operatorname{Aut}(\gamma(\chi))$ is injective for all objects $\chi$ in $\mathcal{D}_{n,\mathbb{C}}^{l}$. This holds if and only if $\gamma$ is representable [@LMBChamps]. We conclude that $\gamma$ is representable if and only if $\phi$ is injective.
By considering coarse moduli spaces we have a commutative diagram $$\xymatrix{ \mathcal{D}_{n,\mathbb{C}}^{l} \ar[d] \ar[r]^\gamma & \mathcal{X} \ar[d] \\
D_{n,\mathbb{C}} \ar[r]^{\gamma'} & X .}$$ We will consider the $n$th *jet scheme* $J_{n}(X) =
\operatorname{Hom}(D_{n,\mathbb{C}}, X)$ of $X$ and identify jet schemes with their $\mathbb{C}$-valued points. When $n = \infty$, $J_{\infty}(X)$ is called the *arc space* of $X$. We have a map $$\label{smitten}
\tilde{\pi}_{n}: |\mathcal{J}_{n}\mathcal{X}| \rightarrow J_{n}(X)$$ $$\tilde{\pi}_{n}(\gamma) = \gamma'.$$
Fix a $d$-dimensional cone $\sigma$ of $\Sigma$ and consider the open substack $\mathcal{X}({\mbox{\boldmath$\sigma$}}) = [\mathbb{A}^{d}/N(\sigma)]$ of $\mathcal{X}$. A twisted $n$-jet $\mathcal{D}_{n, S}^{l} \rightarrow \mathcal{X}({\mbox{\boldmath$\sigma$}})$ can be lifted to a morphism between atlases, $D_{nl,S} \rightarrow \mathbb{A}^{d}$. Yasuda used these lifts to describe the stack $\mathcal{J}_{n}^{l}\mathcal{X}$ (Proposition 2.8 [@YasMotivic]). We will present his result and a sketch of the proof in our situation. We first fix some notation. The action of $\mu_{l}$ on $D_{nl, \mathbb{C}}$ extends to an action on $J_{nl}(\mathbb{A}^{d})$. The action of $N(\sigma)$ on $\mathbb{A}^{d}$ also extends to an action on $J_{nl}(\mathbb{A}^{d})$. If $g$ is an element of $N(\sigma)$ of order $l$, let $J_{nl}^{(g)}(\mathbb{A}^{d})$ be the closed subscheme of $J_{nl}(\mathbb{A}^{d})$ on which the actions of $\zeta_{l}$ in $\mu_{l}$ and $g$ in $N(\sigma)$ agree.
\[lemon\] For $0 \leq n \leq \infty$, we have a homeomorphism $$|\mathcal{J}_{n}\mathcal{X}({\mbox{\boldmath$\sigma$}})| \cong \coprod_{g \in N(\sigma)} J_{nl}^{(g)}(\mathbb{A}^{d})/N(\sigma).$$
We will only show that there is a bijection between the two sets. We have noted that a representable morphism $\gamma: \mathcal{D}_{n,\mathbb{C}}^{l} \rightarrow [\mathbb{A}^{d}/N(\sigma)]$ is determined by the images of $\alpha$ and $\theta$. These images have the form $$\xymatrix{ & & \mathbb{A}^{d} \\
D_{nl,\mathbb{C}} \times N(\sigma) \ar[urr]^{(v,\lambda) \mapsto \nu(v) \cdot \lambda} \ar[d]^{pr_{1}} \ar[r]_{\zeta_{l} \times g^{-1}} & D_{nl,\mathbb{C}} \times N(\sigma)
\ar[ur]_{(v,\lambda) \mapsto \nu(v) \cdot \lambda} \ar[d]^{pr_{1}} & \\
D_{nl,\mathbb{C}} \ar[r]^{\zeta_{l}} & D_{nl,\mathbb{C}} & }$$ for some $g$ in $N(\sigma)$ of order $l$ and some $nl$-jet $\nu: D_{nl,\mathbb{C}} \rightarrow \mathbb{A}^{d}$. Conversely, given such a diagram, we can construct a representable morphism. The diagram is determined by any choice of $g$ and $\nu$ satisfying $$\xymatrix{ D_{nl,\mathbb{C}} \ar[d]^{\zeta_{l}} \ar[r]^{\nu} & \mathbb{A}^{d} \ar[d]^{g} \\
D_{nl,\mathbb{C}} \ar[r]^{\nu} & \mathbb{A}^{d}.
}$$ That is, $\gamma$ is determined by the pair $(g, \nu)$, where $g$ has order $l$ and $\nu$ lies in $J_{nl}^{(g)}(\mathbb{A}^{d})$. Suppose $\gamma'$ is a twisted $n$-jet of order $l$ determined by the pair $(g', \nu')$. A $2$-morphism from $\gamma$ to $\gamma'$ is determined by a morphism $\beta: \gamma(\alpha) \rightarrow \gamma'(\alpha)$ in $[\mathbb{A}^{d}/N(\sigma)]$ over the identity morphism on $D_{nl,\mathbb{C}}$, such that the following diagram commutes $$\xymatrix{ \gamma(\alpha) \ar[d]^{\gamma(\theta)} \ar[r]^{\beta} & \gamma'(\alpha) \ar[d]^{\gamma'(\theta)} \\
\gamma(\alpha) \ar[r]^{\beta} & \gamma'(\alpha).
}$$ One verifies that the morphism $\beta$ is determined by an element $h$ in $N(\sigma)$ such that $\nu = \nu' \cdot h$ and that the diagram above is commutative if and only if $g = g'$. Hence the equivalence class of $\gamma$ in $|\mathcal{J}_{n}\mathcal{X}({\mbox{\boldmath$\sigma$}})|$ corresponds to a point in $J_{nl}^{(g)}(\mathbb{A}^{d})/N(\sigma)$. This gives our desired bijection.
\[doko\] Borisov, Chen and Smith gave a decomposition of $|\mathcal{I}\mathcal{X}|$ into connected components indexed by $\operatorname{Box}({\mbox{\boldmath$\Sigma$}})$ (Proposition 4.7 [@BCSOrbifold]). Given $v$ in $\operatorname{Box}({\mbox{\boldmath$\Sigma$}})$, let $\sigma(v)$ be the cone containing $v$ in its relative interior and let $\Sigma_{\sigma(v)}$ be the simplicial fan in $(N/N_{\sigma(v)})_{\mathbb{R}}$ with cones given by the projections of the cones in $\Sigma$ containing $\sigma(v)$. The associated toric variety $X(\Sigma_{\sigma(v)})$ is a $T$-invariant closed subvariety of $X$. The connected component of $|\mathcal{I}\mathcal{X}|$ corresponding to $v$ is homeomorphic to the simplicial toric variety $X(\Sigma_{\sigma(v)})$. By taking $n = 0$ in Lemma \[lemon\], we recover this decomposition for $|\mathcal{I}\mathcal{X}({\mbox{\boldmath$\sigma$}})|$.
Consider an element $g$ in $N(\sigma)$ of order $l$ corresponding to $v$ in $\operatorname{Box}({\mbox{\boldmath$\tau$}})$, for some cone $\tau$ contained in $\sigma$. We will give an explicit description of $J_{nl}^{(g)}(\mathbb{A}^{d})$. If we write $v = \sum_{i = 1}^{d} q_{i} b_{i}$, for some $0 \leq q_{i} < 1$, then recall from (\[actions\]) that the action of $g$ on $\mathbb{A}^{d}$ is given by $$g \cdot (x_{1}, \ldots, x_{d}) = (\zeta_{l}^{\,lq_{1}}x_{1}, \ldots , \zeta_{l}^{\,lq_{d}}x_{d}).$$ An element $\nu$ of $J_{nl}(\mathbb{A}^{d})$ can be written in the form $$\nu = ( \, \sum_{j= 0}^{nl} \alpha_{1,j}t^{j}\, , \ldots , \, \sum_{j= 0}^{nl} \alpha_{d,j}t^{j} \, ),$$ for some $\alpha_{i,j}$ in $\mathbb{C}$, $1 \leq i \leq d$, $0 \leq j \leq nl$. We have $$g \cdot \nu = ( \, \zeta_{l}^{\,lq_{1}} \sum_{j= 0}^{nl} \alpha_{1,j}t^{j}\, , \ldots , \, \zeta_{l}^{\,lq_{d}} \sum_{j= 0}^{nl} \alpha_{d,j}t^{j} \, )$$ $$\zeta_{l} \cdot \nu = ( \, \sum_{j= 0}^{nl} \alpha_{1,j}\zeta_{l}^{j}t^{j}\, , \ldots , \, \sum_{j= 0}^{nl}\alpha_{d,j} \zeta_{l}^{j}t^{j} \, ) .$$ Hence $v$ lies in $J_{nl}^{(g)}(\mathbb{A}^{d})$ if and only if $\alpha_{i,j} = 0$ whenever $j \not\equiv lq_{i} \, \left(\textrm{mod } l \right)$. We conclude that $$\label{weagles}
J_{nl}^{(g)}(\mathbb{A}^{d}) = \{ ( \, \sum_{j= 0}^{nl} \alpha_{1,j}t^{j}\, , \ldots , \, \sum_{j= 0}^{nl} \alpha_{d,j}t^{j} \, ) \mid
\alpha_{i,j} = 0 \textrm{ if } j \not\equiv lq_{i} \, \left(\textrm{mod } l \right) \}.$$
Twisted Arcs of Toric Stacks {#arc}
============================
In this section we describe an action of $J_{\infty}(T)$ on an open, dense subset of $|\mathcal{J}_{\infty}\mathcal{X}|$ and compute the corresponding orbits and orbit closures.
We first recall Ishii’s decomposition of the arc space of the toric variety $X$ [@IshArc]. Recall that $D_{1}, \ldots, D_{r}$ denote the $T$-invariant prime divisors of $X$ and let $J_{\infty}(X)' = J_{\infty}(X) \smallsetminus \cup_{i} J_{\infty}(D_{i})$. The action of $T$ on $X$ extends to an action of $J_{\infty}(T)$ on $J_{\infty}(X)'$ (Proposition 2.6 [@IshArc]). Given an arc $\gamma$ in $J_{\infty}(X)'$, we can find a $d$-dimensional cone $\sigma$ such that $\gamma: \operatorname{Spec}\mathbb{C}[[t]] \rightarrow U_{\sigma} \subseteq X$ corresponds to the ring homomorphism $\gamma^{\#}: \mathbb{C}[\sigma^{\vee} \cap M] \rightarrow \mathbb{C}[[t]]$. We have an induced semigroup morphism $\sigma^{\vee} \cap M \rightarrow \mathbb{N}$, $u \mapsto \operatorname{ord}_{t} \gamma^{\#}(\chi^{u})$, which extends to homomorphism from $M$ to $\mathbb{Z}$, necessarily of the form $\langle \: , v \rangle$, for some $v$ in $\sigma \cap N$. Consider the arc $$\gamma_{v} : \operatorname{Spec}\mathbb{C}[[t]] \rightarrow U_{\sigma} \subseteq X$$ corresponding to the ring homomorphism $$\gamma_{v}^{\#}: \mathbb{C}[\sigma^{\vee} \cap M] \rightarrow \mathbb{C}[[t]]$$ $$\chi^{u} \mapsto t^{\langle u , v \rangle}.$$ If $\phi$ denotes the arc in $J_{\infty}(T)$ corresponding to the ring homomorphism $\phi^{\#}: \mathbb{C}[\sigma^{\vee} \cap M] \rightarrow \mathbb{C}[[t]]$, $\chi^{u} \mapsto \gamma^{\#}(\chi^{u})/t^{\langle u , v \rangle}$, then $\gamma = \gamma_{v} \cdot \phi$ and both $v$ in $|\Sigma| \cap N$ and $\phi$ in $J_{\infty}(T)$ are uniquely determined. We have shown the first part of the following theorem.
\[Ishii\] We have a decomposition of $J_{\infty}(X)'$ into $J_{\infty}(T)$-orbits $$J_{\infty}(X)' = \coprod_{v \in |\Sigma| \cap N} \gamma_{v} \cdot J_{\infty}(T).$$ Moreover, $\gamma_{w} \cdot J_{\infty}(T) \subseteq \overline{\gamma_{v} \cdot J_{\infty}(T)}$ if and only if $w - v$ lies in some cone $\sigma$ containing $v$ and $w$.
We will give a similar decomposition of the space $|\mathcal{J}_{\infty}\mathcal{X}|$ of twisted arcs of $\mathcal{X}$. For $i = 1, \ldots, r$, we first describe a closed substack $\mathcal{D}_{i}$ of $\mathcal{X}$ with coarse moduli space $D_{i}$ [@BCSOrbifold]. Fix a maximal cone $\sigma$. If $\rho_{i}$ is not in $\sigma$ then $\mathcal{D}_{i}$ is disjoint from $\mathcal{X}({\mbox{\boldmath$\sigma$}})$. Suppose $\rho_{i}$ is a ray of $\sigma$ and consider the projection $p_{i}: N \rightarrow N(\rho_{i}) = N / N_{\rho_{i}}$, where $N_{\rho_{i}}$ is the sublattice of $N$ generated by $b_{i}$. If $\sigma_{i}$ denotes the cone $p_{i}(\sigma)$ in $N(\rho_{i})$, then $(N(\rho_{i}), \sigma_{i}, \{ p_{i}(b_{j}) \}_{\rho_{j} \subseteq \sigma} )$ is the stacky fan corresponding to $\mathcal{D}_{i} \cap \mathcal{X}({\mbox{\boldmath$\sigma$}})$[^1]. Note that $p_{i}$ induces an isomorphism between $N(\sigma) = N/N_{\sigma}$ and $N(\rho_{i})(\sigma_{i}) = N(\rho_{i}) / N(\rho_{i})_{\sigma_{i}}$. We have an inclusion of $\mathbb{A}^{d - 1}$ into $\mathbb{A}^{d}$ by setting the coordinate corresponding to $\rho_{i}$ to be zero. We conclude that $\mathcal{D}_{i} \cap \mathcal{X}({\mbox{\boldmath$\sigma$}}) \cong [\mathbb{A}^{d - 1}/ N(\sigma)]$ and the inclusion of $\mathbb{A}^{d - 1}$ into $\mathbb{A}^{d}$ induces the inclusion of $\mathcal{D}_{i} \cap \mathcal{X}({\mbox{\boldmath$\sigma$}})$ into $\mathcal{X}({\mbox{\boldmath$\sigma$}})$. Moreover, by Lemma \[lemon\], if $g$ is an element of $N(\sigma)$, then we have an induced inclusion $$\label{very}
J_{\infty}^{(g)}(\mathbb{A}^{d - 1})/N(\sigma) \hookrightarrow J_{\infty}^{(g)}(\mathbb{A}^{d})/N(\sigma),$$ corresponding to the closed inclusion $|\mathcal{J}_{\infty}(\mathcal{D}_{i} \cap \mathcal{X}({\mbox{\boldmath$\sigma$}}))| \hookrightarrow
|\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\sigma$}})|$. We define $|\mathcal{J}_{\infty}\mathcal{X}|'$ to be the open subset $$|\mathcal{J}_{\infty}\mathcal{X}|' = |\mathcal{J}_{\infty}\mathcal{X}| \smallsetminus \cup_{i = 1}^{r} |\mathcal{J}_{\infty}\mathcal{D}_{i}|.$$ Similarly, we consider $|\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\sigma$}})|'$ and let $J_{\infty}(\mathbb{A}^{d})'$ be the open locus of arcs of $\mathbb{A}^{d}$ that are not contained in a coordinate hyperplane. Setting $J_{\infty}^{(g)}(\mathbb{A}^{d})' = J_{\infty}^{(g)}(\mathbb{A}^{d}) \cap J_{\infty}(\mathbb{A}^{d})'$, it follows from Lemma \[lemon\] and (\[very\]) that $$\label{dash}
|\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\sigma$}})|' \cong \coprod_{g \in N(\sigma)} J_{\infty}^{(g)}(\mathbb{A}^{d})'/N(\sigma).$$ We will often make this identification.
For a fixed positive integer $l$, the open embedding of $T$ in $\mathcal{X}({\mbox{\boldmath$\sigma$}})$ extends to an open embedding of $J_{\infty}(T) \cong \mathcal{J}_{\infty}^{l}T$ in $\mathcal{J}_{\infty}^{l}\mathcal{X}({\mbox{\boldmath$\sigma$}})$. We obtain an action of $J_{\infty}(T)$ on $|\mathcal{J}_{\infty}^{l}\mathcal{X}({\mbox{\boldmath$\sigma$}})|$, which restricts to an action on $|\mathcal{J}_{\infty}^{l}\mathcal{X}({\mbox{\boldmath$\sigma$}})|'$. More specifically, we identify $J_{\infty}(T) \cong J_{\infty}^{(l)}(T)/N(\sigma)$, where $J_{\infty}^{(l)}(T)$ is the closed subscheme of $J_{\infty}(T)$ fixed by the action of $\mu_{l}$. In coordinates, $$J_{\infty}^{(l)}(T) = ( \, \sum_{j= 0}^{\infty} \beta_{1,j}t^{lj}\, , \ldots , \, \sum_{j= 0}^{\infty} \beta_{d,j}t^{lj} \, ) \mid \beta_{i,0} \neq 0 \textrm{ for }
i = 1, \ldots, d \}.$$ Fix an element $g$ in $N(\sigma)$ of order $l$ and recall from (\[weagles\]) that $$J_{\infty}^{(g)}(\mathbb{A}^{d}) = \{ ( \, \sum_{j= 0}^{\infty} \alpha_{1,j}t^{j}\, , \ldots , \, \sum_{j= 0}^{\infty} \alpha_{d,j}t^{j} \, ) \mid
\alpha_{i,j} = 0 \textrm{ if } j \not\equiv lq_{i} \, \left( \textrm{mod } l \right) \}.$$ The elements in $J_{\infty}^{(g)}(\mathbb{A}^{d})'$ also satisfy the property that for each $i = 1, \ldots , d$, there exists a non-negative integer $j$ such that $\alpha_{i,j} \neq 0$. Componentwise multiplication of power series gives an action, $$J_{\infty}^{(l)}(T)/N(\sigma) \times J_{\infty}^{(g)}(\mathbb{A}^{d})'/N(\sigma) \rightarrow J_{\infty}^{(g)}(\mathbb{A}^{d})'/N(\sigma).$$ By (\[dash\]), this induces an action of $J_{\infty}(T)$ on $|\mathcal{J}_{\infty}^{l}\mathcal{X}({\mbox{\boldmath$\sigma$}})|'$. By varying $l$, we obtain an action of $J_{\infty}(T)$ on $|\mathcal{J}_{\infty}\mathcal{X}|'$. We would like to describe the $J_{\infty}(T)$-orbits.
Let $w$ be an element of $\sigma \cap N$. There is a unique decomposition $w = v + \sum_{i = 1}^{d} \lambda_{i}b_{i}$, where $v$ lies in $\operatorname{Box}({\mbox{\boldmath$\tau$}})$, for some $\tau \subseteq \sigma$, and the $\lambda_{i}$ are non-negative integers. We will use the notation $\{ w \} = v$. Suppose $v$ corresponds to an element $g$ in $N(\sigma)$ of order $l$. If we write $v = \sum_{i = 1}^{d} q_{i} b_{i}$, for some $0 \leq q_{i} < 1$, then $w = \sum_{i = 1}^{d} w_{i} b_{i} = \sum_{i = 1}^{d} (\lambda_{i} + q_{i}) b_{i}$, where $w_{i} = \lambda_{i} + q_{i}$. We define $\tilde{\gamma}_{w} \in |\mathcal{J}_{\infty}^{l}\mathcal{X}({\mbox{\boldmath$\sigma$}})|' \subseteq |\mathcal{J}_{\infty}\mathcal{X}|'$ to be the equivalence class of twisted arcs corresponding to the element $$\label{MickyO}
( t^{ l(\lambda_{1} + q_{1}) }, \ldots , t^{ l(\lambda_{d} + q_{d}) } ) = ( t^{ lw_{1}}, \ldots , t^{ lw_{d}} )$$ of $J_{\infty}^{(g)}(\mathbb{A}^{d})'$, under the isomorphism (\[dash\]).
\[decomp\] We have a decomposition of $|\mathcal{J}_{\infty}\mathcal{X}|'$ into $J_{\infty}(T)$-orbits $$|\mathcal{J}_{\infty}\mathcal{X}|' = \coprod_{v \in |\Sigma| \cap N} \tilde{\gamma}_{v} \cdot J_{\infty}(T).$$ Moreover, $\tilde{\gamma}_{w} \cdot J_{\infty}(T) \subseteq \overline{\tilde{\gamma}_{v} \cdot J_{\infty}(T)}$ if and only if $w - v = \sum_{\rho_{i} \subseteq \sigma} \lambda_{i}b_{i}$ for some non-negative integers $\lambda_{i}$ and some cone $\sigma$ containing $v$ and $w$. With the notation of Theorem \[Ishii\] and (\[smitten\]), $$\tilde{\pi}_{\infty}: |\mathcal{J}_{\infty}\mathcal{X}|' \rightarrow J_{\infty}(X)'$$ is a $J_{\infty}(T)$-equivariant bijection satisfying $$\tilde{\pi}_{\infty}(\tilde{\gamma}_{v}) = \gamma_{v}.$$
Let $\sigma$ be a $d$-dimensional cone in $\Sigma$ and let $g$ be an element in $N(\sigma)$ of order $l$ corresponding to an element $v$ in $\operatorname{Box}({\mbox{\boldmath$\tau$}})$, for some $\tau \subseteq \sigma$. By (\[weagles\]), and with the notation of the previous discussion, we have a decomposition $$J_{\infty}^{(g)}(\mathbb{A}^{d})' = \coprod_{ \substack{w \in \sigma \cap N \\ \{w\} = v}} ( t^{ lw_{1}}, \ldots , t^{ lw_{d}} ) \cdot J_{\infty}^{(l)}(T),$$ and hence $$J_{\infty}^{(g)}(\mathbb{A}^{d})'/N(\sigma) \cong \coprod_{ \substack{w \in \sigma \cap N \\ \{w\} = v}} \tilde{\gamma}_{w} \cdot J_{\infty}(T).$$ Also, $\tilde{\gamma}_{w} \cdot J_{\infty}(T) \subseteq \overline{\tilde{\gamma}_{w'} \cdot J_{\infty}(T)}$ in $J_{\infty}^{(g)}(\mathbb{A}^{d})'/N(\sigma)$ if and only if $w_{i} \geq w_{i}'$ for $i = 1, \ldots, d$. We conclude that $$|\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\sigma$}})|' = \coprod_{w \in \sigma \cap N} \tilde{\gamma}_{w} \cdot J_{\infty}(T),$$ and $\tilde{\gamma}_{w} \cdot J_{\infty}(T) \subseteq \overline{\tilde{\gamma}_{w'} \cdot J_{\infty}(T)}$ in $|\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\sigma$}})|'$ if and only if $\{w\} = \{w'\}$ and $w_{i} \geq w_{i}'$ for $i = 1, \ldots, d$. This is equivalent to $w - w' = \sum_{i = 1}^{d} \lambda_{i}b_{i}$ for some non-negative integers $\lambda_{i}$. Since the open subsets $|\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\sigma$}})|'$ cover $|\mathcal{J}_{\infty}\mathcal{X}|'$ as $\sigma$ varies over all maximal cones, we obtain the desired decomposition and closure relations.
Consider the sublattice $N_{\sigma} \subseteq N$ and recall that $\sigma'$ is the cone in $N_{\sigma}$ generated by $b_{1}, \ldots, b_{d}$. If $H$ denotes the quotient of $N$ by the sublattice generated by $v_{1}, \ldots, v_{d}$, then we have a pairing $\langle \; , \, \rangle : N \times M_{\sigma} \rightarrow \mathbb{Q}$, $\langle v_{i}, u_{j} \rangle = \delta_{i,j}|H|$, where $\delta_{i,j} = 1$ if $i = j$ and $0$ otherwise. If $u_{1}, \ldots, u_{d}$ are the primitive integer generators of $\sigma^{\vee}$ in $M$, then $u_{1}/a_{1}|H|, \ldots, u_{d}/a_{d}|H|$ are the primitive integer generators of $(\sigma')^{\vee}$ in $M_{\sigma}$. We have made an identification throughout that $\operatorname{Spec}\mathbb{C}[(\sigma')^{\vee} \cap M_{\sigma}] \cong \mathbb{A}^{d}$. Consider an element $\gamma$ in $J_{\infty}^{(g)}(\mathbb{A}^{d})'$, for some $g$ in $N(\sigma)$ of order $l$, corresponding to an element of $|\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\sigma$}})|'$. Applying $\tilde{\pi}_{\infty}$ gives an arc in $U_{\sigma}$, which we denote by $\tilde{\pi}_{\infty}(\gamma)$. We have a commutative diagram $$\xymatrix{ \mathbb{C}[(\sigma')^{\vee} \cap M_{\sigma}] \ar[r]^{\: \; \: \gamma^{\#}} & \mathbb{C}[[t]] \\
\mathbb{C}[\sigma^{\vee} \cap M] \ar[r]^{\: \: \: \: \tilde{\pi}_{\infty}(\gamma)^{\#}} \ar[u] & \mathbb{C}[[t]] \ar[u]_{t \mapsto t^{l}}.
}$$ It follows that $\tilde{\pi}_{\infty}$ is $J_{\infty}(T)$-equivariant. Let $w$ be an element of $\sigma \cap N$ and consider the notation of (\[MickyO\]). With a slight abuse of notation, $(\tilde{\gamma}_{w})^{\#}(\chi^{u_{i}/a_{i}|H|}) = t^{lw_{i}}$, and we compute $$\tilde{\pi}_{\infty}(\tilde{\gamma}_{w})^{\#}(\chi^{u_{i}}) = ((t^{1/l})^{lw_{i}})^{a_{i}|H|} = t^{a_{i}w_{i}|H|} = t^{\langle u_{i}, w \rangle}.$$ It follows from the definition of $\gamma_{w}$ that $\tilde{\pi}_{\infty}(\tilde{\gamma}_{w}) = \gamma_{w}$.
The morphism from a Deligne-Mumford stack to its coarse moduli space is proper [@LMBChamps]. Hence the fact that the map $\tilde{\pi}_{\infty}: |\mathcal{J}_{\infty}\mathcal{X}|' \rightarrow J_{\infty}(X)'$ is bijective follows from Proposition 3.37 of [@YasMotivic].
\[ginvo\] We give a geometric description of a well-known involution $\iota$ on $|\Sigma| \cap N$ (see, for example, Section 2 [@YoWeightI]). If $\tau$ is a cone in $\Sigma$ and $v$ is a lattice point in $\operatorname{Box}({\mbox{\boldmath$\tau$}})$, then $v$ can be uniquely written in the form $v = \sum_{\rho_{i} \subseteq \tau} q_{i}b_{i}$, for some $0< q_{i} <1$. We define $\iota = \iota_{{\mbox{\boldmath$\Sigma$}}} : \operatorname{Box}({\mbox{\boldmath$\tau$}}) \rightarrow \operatorname{Box}({\mbox{\boldmath$\tau$}})$ by $\iota(v) = \sum_{\rho_{i} \subseteq \tau} (1 - q_{i})b_{i}$. By Remark \[doko\], the connected components of $|\mathcal{I}\mathcal{X}|$ are indexed by $\operatorname{Box}({\mbox{\boldmath$\Sigma$}})$. Recall that the elements of $|\mathcal{I}\mathcal{X}|$ are equivalence classes of pairs $(x, \alpha)$, where $x$ is an object of $\mathcal{X}$ over $\mathbb{C}$ and $\alpha$ is an automorphism of $x$. There is a natural involution on $|\mathcal{I}\mathcal{X}|$, taking a pair $(x, \alpha)$ to $(x, \alpha^{-1})$. Observe that $I$ induces the involution $\iota$ on the connected components of $|\mathcal{I}\mathcal{X}|$.
Every lattice point $w$ in $|\Sigma| \cap N$ can be uniquely written in the form $w = \{ w \} + \tilde{w}$, where $\{ w \}$ lies in $\operatorname{Box}({\mbox{\boldmath$\tau$}})$ for some $\tau \subseteq \sigma(w)$ and $\tilde{w}$ lies in $N_{\sigma(w)}$. Here $\sigma(w)$ is the cone of $\Sigma$ containing $w$ in its relative interior. The map $\iota$ extends to an involution on $|\Sigma| \cap N$, taking $w$ to $\iota(\{ w \}) + \tilde{w}$. Recall that we have a projection morphism $\pi: \mathcal{J}_{\infty}\mathcal{X} \rightarrow \mathcal{I}\mathcal{X} = \mathcal{J}_{0}\mathcal{X}$. We see that $I$ extends to a $J_{\infty}(T)$-equivariant involution $I : |\mathcal{J}_{\infty}\mathcal{X}|' \rightarrow |\mathcal{J}_{\infty}\mathcal{X}|'$ satisfying $I(\tilde{\gamma}_{v}) = \tilde{\gamma}_{\iota(v)}$.
For any non-negative integer $n$, we can consider $|\mathcal{J}_{n}\mathcal{X}|' = |\mathcal{J}_{n}\mathcal{X}| \smallsetminus \cup_{i = 1}^{r} |\mathcal{J}_{n}\mathcal{D}_{i}|$ and an action of $J_{n}(T)$ on $|\mathcal{J}_{n}\mathcal{X}|'$. Recall that we have projection morphisms $\pi_{n}: \mathcal{J}_{\infty}\mathcal{X} \rightarrow \mathcal{J}_{n}\mathcal{X}$. For each non-zero cone $\tau$ of $\Sigma$, let $$\overline{\operatorname{Box}}({\mbox{\boldmath$n\tau$}}) = \{ v \in N \mid v = \sum_{\rho_{i} \subseteq \tau} q_{i}b_{i} \textrm{ for some }
0 < q_{i} \leq n \}.$$ We set $\overline{\operatorname{Box}}(\mbox{\boldmath$n\{ 0 \}$}) = \{0\}$ and $\overline{\operatorname{Box}}({\mbox{\boldmath$n\Sigma$}}) = \cup_{\tau \in \Sigma} \overline{\operatorname{Box}}({\mbox{\boldmath$n\tau$}})$. It can be shown that there is a decomposition of $|\mathcal{J}_{n}\mathcal{X}|'$ into $J_{n}(T)$-orbits $$|\mathcal{J}_{n}\mathcal{X}|' = \coprod_{v \in \overline{\operatorname{Box}}({\mbox{\boldmath$n\Sigma$}})} \pi_{n}(\tilde{\gamma}_{v}) \cdot J_{n}(T).$$
Contact Order along a Divisor {#contact}
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In this section, we compute the contact order of a twisted arc along a $T$-invariant divisor on $\mathcal{X}$. Recall that for each maximal cone $\sigma$ in $\Sigma$, we have morphisms $$\mathbb{A}^{d} \stackrel{p}{\rightarrow} [\mathbb{A}^{d}/N(\sigma)] \cong \mathcal{X}({\mbox{\boldmath$\sigma$}}) \stackrel{q}{\rightarrow}
\mathbb{A}^{d}/N(\sigma) \cong U_{\sigma} ,$$ where $\mathbb{A}^{d}$ is the atlas of $\mathcal{X}({\mbox{\boldmath$\sigma$}})$ via $p$ and $U_{\sigma}$ is the coarse moduli space of $\mathcal{X}({\mbox{\boldmath$\sigma$}})$. For every ray $\rho_{i}$ in $\Sigma$, there is a corresponding divisor $\mathcal{D}_{i}$ of $\mathcal{X}({\mbox{\boldmath$\Sigma$}})$ (see Section \[arc\]). If $\rho_{i} \nsubseteq \sigma$ then $\mathcal{D}_{i}$ does not intersect $\mathcal{X}({\mbox{\boldmath$\sigma$}})$. If $\rho_{i} \subseteq \sigma$, then $\mathcal{D}_{i}$ corresponds to the divisor $\{ x_{i} = 0 \}$ on $\mathbb{A}^{d}$, with an appropriate choice of coordinates. We say that a $\mathbb{Q}$-divisor $\mathcal{E}$ on $\mathcal{X}$ is *$T$-invariant* if $\mathcal{E} = \sum_{i} \beta_{i} \mathcal{D}_{i}$, for some $\beta_{i} \in \mathbb{Q}$. There is a natural isomorphism of Picard groups over $\mathbb{Q}$ (Example 6.7 [@VisIntersection]) $$q^{*}: \operatorname{Pic}_{\mathbb{Q}} (X(\Sigma) ) \rightarrow \operatorname{Pic}_{\mathbb{Q}} (\mathcal{X}({\mbox{\boldmath$\Sigma$}}) )$$ $$D_{i} \mapsto q^{*}D_{i} = a_{i}\mathcal{D}_{i} .$$ The inverse map is induced by the pushforward functor $q_{*}: \operatorname{Pic}_{\mathbb{Q}} (\mathcal{X}({\mbox{\boldmath$\Sigma$}}) ) \rightarrow \operatorname{Pic}_{\mathbb{Q}} (X(\Sigma) )$. The canonical divisor on $\mathcal{X}$ is given by $K_{\mathcal{X}} = - \sum_{i} \mathcal{D}_{i}$, and so $q_{*}K_{\mathcal{X}} = - \sum_{i} a_{i}^{-1}D_{i}$. Recall that a $T$-invariant $\mathbb{Q}$-divisor $E = \sum_{\rho_{i} \in \Sigma} \alpha_{i} D_{i}$ on $X$ corresponds to a real-valued piecewise $\mathbb{Q}$-linear function $\psi_{E}: |\Sigma| \rightarrow \mathbb{R}$ satisfying $\psi_{E}(v_{i}) = -\alpha_{i}$ [@FulIntroduction]. Note that $\psi_{q_{*}K_{\mathcal{X}}}(b_{i}) = 1$ for $i = 1, \ldots,r$, and hence, with the notation of Section \[tstack\], $\psi = \psi_{q_{*}K_{\mathcal{X}}}$. Let $\mathcal{E} = \sum_{i} \beta_{i} \mathcal{D}_{i}$ be a $T$-invariant $\mathbb{Q}$-divisor on $\mathcal{X}$. Following Yasuda (Definition 4.17 [@YasMotivic]), we say that the pair $(\mathcal{X}, \mathcal{E})$ is *Kawamata log terminal* if $\beta_{i} < 1$ for $i = 1, \ldots, r$. Geometrically, this says that for each maximal cone $\sigma$, the representative of $\mathcal{E}$ in the atlas $\mathbb{A}^{d}$ of $\mathcal{X}({\mbox{\boldmath$\sigma$}}) = [\mathbb{A}^{d}/N(\sigma)]$ is supported on the coordinate axes with all coefficients less than $1$. Equivalently, the condition says that $\psi_{q_{*}\mathcal{E}}(b_{i}) > -1$ for $i = 1, \ldots,r$.
If $\mathcal{E} = \sum u_{j} \mathcal{E}_{j}$ is a $\mathbb{Q}$-divisor on $\mathcal{X}$, for some prime divisors $\mathcal{E}_{j}$ and rational numbers $u_{j}$, then Yasuda [@YasMotivic] defined a function $$\operatorname{ord}\mathcal{E}: |\mathcal{J}_{\infty} \mathcal{X}| \smallsetminus \cup_{j} |\mathcal{J}_{\infty} \mathcal{E}_{j}| \rightarrow \mathbb{Q},$$ $$\operatorname{ord}\mathcal{E} = \sum_{j} u_{j} \operatorname{ord}\mathcal{E}_{j}.$$ When $\mathcal{E}$ is a prime divisor, the function $\operatorname{ord}\mathcal{E}$ is defined as follows: if $\gamma: \mathcal{D}_{\infty,\mathbb{C}}^{l} \rightarrow \mathcal{X}$ is a representable morphism, then consider the composition of $\gamma$ with the atlas $\operatorname{Spec}\mathbb{C}[[t]] \rightarrow \mathcal{D}_{\infty,\mathbb{C}}^{l}$, and choose a lifting to an arc $\bar{\gamma}$ of an atlas $M$ of $\mathcal{X}$. If $m$ is the contact order of $\bar{\gamma}$ along the representative of $\mathcal{E}$ in $M$, then $\operatorname{ord}\mathcal{E}(\gamma) = m/l$. The following lemma describes the function $\operatorname{ord}\mathcal{E}$ in the toric case. We will use the notation of Theorem \[decomp\].
\[order\] If $\mathcal{E}$ is a $T$-invariant $\mathbb{Q}$-divisor on $\mathcal{X}$, then $$\operatorname{ord}\mathcal{E}( \tilde{\gamma}_{w} \cdot J_{\infty}(T) ) = -\psi_{q_{*}\mathcal{E}}(w).$$ In particular, $$\operatorname{ord}K_{\mathcal{X}}( \tilde{\gamma}_{w} \cdot J_{\infty}(T) ) = -\psi(w).$$
It will be enough to prove the result for $\mathcal{E} = \mathcal{D}_{i}$ and twisted arcs of the form $\tilde{\gamma}_{w}$. Consider the representable morphism $\tilde{\gamma}_{w}: \mathcal{D}_{\infty,\mathbb{C}}^{l} \rightarrow \mathcal{X}({\mbox{\boldmath$\sigma$}}) \subseteq \mathcal{X}$, for some lattice point $w$ in a maximal cone $\sigma$. With the notation of (\[MickyO\]), the composition of $\tilde{\gamma}_{w}$ with $\operatorname{Spec}\mathbb{C}[[t]] \rightarrow \mathcal{D}_{\infty,\mathbb{C}}^{l}$ lifts to an arc $( t^{ lw_{1}}, \ldots , t^{ lw_{d}} )$ in $J_{\infty}(\mathbb{A}^{d})$. If $\rho_{i} \nsubseteq \sigma$ then $\operatorname{ord}\mathcal{D}_{i} (\tilde{\gamma}_{w}) = 0 = -\psi_{a_{i}^{-1}D_{i}}(w)$. If $\rho_{i} \subseteq \sigma$ then the divisor $\mathcal{D}_{i}$ is represented by the divisor $\{ x_{i} = 0 \}$ in the atlas $\mathbb{A}^{d}$ and we conclude that $\operatorname{ord}\mathcal{D}_{i}( \tilde{\gamma}_{w}) = w_{i} = -\psi_{a_{i}^{-1}D_{i}}(w)$. The second statement follows since $\psi = \psi_{q_{*}K_{\mathcal{X}}}$.
Motivic Integration on Toric Stacks {#motivic}
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We consider motivic integration on a Deligne-Mumford stack as developed by Yasuda in [@YasMotivic]. We will compute motivic integrals associated to $T$-invariant divisors on $\mathcal{X}$ and show that they correspond to weighted $\delta$-vectors of an associated polyhedral complex.
Recall that to any complex algebraic variety $X$ of dimension $r$, we can associate its *Hodge polynomial* (see, for example, [@SriHodge]) $$E_{X}(u,v) = \sum_{i,j = 0}^{r} (-1)^{i + j}h_{i,j}u^{i}v^{j} \in \mathbb{Z}[u,v].$$ The Hodge polynomial is determined by the properties
1. $h_{i,j} = \dim H^{j}(X, \Omega_{X}^{i})$ if $X$ is smooth and projective,
2. $E_{X}(u,v) = E_{U}(u,v) + E_{X \smallsetminus U}(u,v)$ if $U$ is an open subvariety of $X$,
3. $E_{X \times Y}(u,v) = E_{X}(u,v)E_{Y}(u,v)$.
The second property means we can consider the Hodge polynomial of a constructible subset of a complex variety. For example, $E_{\mathbb{A}^{1}}(u,v) = E_{\mathbb{P}^{1}}(u,v) - E_{\{ \textrm{pt} \} }(u,v) = uv$ and hence $E_{\mathbb{A}^{n}}(u,v) = (uv)^{n}$. Similarly, $E_{(\mathbb{C}^{*})^{n}}(u,v) = (uv - 1)^{n}$. More generally, we can compute the Hodge polynomial of any toric variety. Recall that if $\triangle$ is a fan in a lattice $N'_{\mathbb{R}}$, for some lattice $N'$, then its associated $h$-vector $h_{\triangle}(t)$ is defined by $$h_{\triangle}(t) = \sum_{\tau \in \triangle} t^{\dim \tau} (1 - t)^{\operatorname{codim}\tau}.$$
\[kingston\] If $X = X(\triangle)$ is an $r$-dimensional toric variety associated to a fan $\triangle$, then $$E_{X}(u , v) = (uv)^{r}h_{\triangle}((uv)^{-1}).$$
We can write $X$ as a disjoint union of torus orbits indexed by the cones of $\triangle$ (see, for example, [@FulIntroduction]). The orbit corresponding to $\tau$ is isomorphic to $(\mathbb{C}^{*})^{\operatorname{codim}\tau}$. We compute, using the example above, $$E_{X}(u,v) = \sum_{\tau \in \triangle} E_{(\mathbb{C}^{*})^{\operatorname{codim}\tau}}(u,v) = \sum_{\tau \in \triangle} (uv - 1)^{\operatorname{codim}\tau} = (uv)^{r}h_{\triangle}((uv)^{-1}).$$
If $\triangle$ is an $r$-dimensional, simplicial fan with convex support, then $X = X(\triangle)$ has no odd cohomology and the coefficient of $(uv)^{i}$ in $E_{X}(u,v)$ is equal to the dimension of the $2i^{\textrm{th}}$ cohomology group of $X$ with compact support and rational coefficients. This follows from the above lemma, Lemma 4.1 of [@YoWeightI] and Poincaré duality.
Recall that we have projection morphisms $\pi_{n}: |\mathcal{J}_{\infty}\mathcal{X}| \rightarrow |\mathcal{J}_{n}\mathcal{X}|$, for each non-negative integer $n$. A subset $A \subseteq |\mathcal{J}_{\infty}\mathcal{X}|$ is a *cylinder* if $A = \pi_{n}^{-1}\pi_{n}(A)$ and $\pi_{n}(A)$ is a constructible subset, for some non-negative integer $n$. In this case, the measure $\mu_{\mathcal{X}}(A)$ of $A$ is defined to be $$\mu_{\mathcal{X}}(A) = E_{\pi_{n}(A)}(u,v)(uv)^{-nd} \in \mathbb{Z}[u,u^{-1},v,v^{-1}].$$ By Lemma 3.18 of [@YasMotivic], the right hand side is independent of the choice of $n$. The collection of cylinders is closed under taking finite unions and finite intersections and $\mu_{\mathcal{X}}$ defines a finite measure on $|\mathcal{J}_{\infty}\mathcal{X}|$. We will compute the measure of certain cylinders. Recall the decomposition of $|\mathcal{J}_{\infty}\mathcal{X}|'$ in Theorem \[decomp\]. Every $w$ in $|\Sigma| \cap N$ can be uniquely written in the form $w = \{ w \} + \tilde{w}$, where $\{ w \}$ lies in $\operatorname{Box}({\mbox{\boldmath$\tau$}})$ for some $\tau \subseteq \sigma(w)$ and $\tilde{w}$ lies in $N_{\sigma(w)}$. Recall that $\sigma(w)$ is the cone of $\Sigma$ containing $w$ in its relative interior and that $\psi: |\Sigma| \rightarrow \mathbb{R}$ is the piecewise $\mathbb{Q}$-linear function satisfying $\psi(b_{i}) = 1$ for $i = 1, \ldots, r$.
\[orbitz\] For any $w$ in $|\Sigma| \cap N$, the orbit $\tilde{\gamma}_{w} \cdot J_{\infty}(T)$ is a cylinder in $|\mathcal{J}_{\infty}\mathcal{X}|$ with measure $$\mu_{\mathcal{X}}(\tilde{\gamma}_{w} \cdot J_{\infty}(T)) = (uv - 1)^{d} (uv)^{-\psi(w) + \psi(\{w\}) - \dim \sigma(\{w\}) }.$$
Fix a $d$-dimensional cone $\sigma$ containing $w$ and consider the notation of (\[MickyO\]). The twisted arc $\tilde{\gamma}_{w}$ lies in $J_{\infty}^{(g)}\mathbb{A}^{d}/N(\sigma)$ and $\pi_{n}(\tilde{\gamma}_{w})$ lies in $J_{nl}^{(g)}\mathbb{A}^{d}/N(\sigma)$. If we consider the natural projection $\pi_{n}^{(g)}: J_{\infty}^{(g)}\mathbb{A}^{d} \rightarrow J_{nl}^{(g)}\mathbb{A}^{d}$, then $\pi_{n}^{(g)}( (t^{lw_{1}}, \ldots, t^{lw_{d}}) \cdot J_{\infty}^{(l)}(T))$ is equal to $$\{ ( \sum_{k = \lambda_{1}}^{n_{1}} \alpha_{1, l(k + q_{1})} t^{l(k + q_{1})}, \ldots,
\sum_{k = \lambda_{d}}^{n_{d}} \alpha_{d, l(k + q_{d})} t^{l(k + q_{d})} ) \mid \alpha_{i, w_{i}} \neq 0 \},$$ for $n \geq \max \{ \lambda_{1} + 1, \ldots, \lambda_{d} + 1 \}$. Here $n_{i} = n$ if $q_{i} =0$ and $n_{i} = n -1$ if $q_{i} \neq 0$. Note that $q_{i} \neq 0$ if and only if $\rho_{i} \subseteq \sigma(\{w\})$. Hence $(\pi_{n}^{(g)})^{-1}\pi_{n}^{(g)}( (t^{lw_{1}}, \ldots, t^{lw_{d}}) \cdot J_{\infty}^{(l)}(T)) = (t^{lw_{1}}, \ldots, t^{lw_{d}}) \cdot J_{\infty}^{(l)}(T)$, and $\pi_{n}^{(g)}( (t^{lw_{1}}, \ldots, t^{lw_{d}}) \cdot J_{\infty}^{(l)}(T))$ is isomorphic to $$\label{cmon}
(\mathbb{C}^{*})^{d} \times \mathbb{A}^{ nd - \sum_{i = 1}^{d}\lambda_{i} - \dim \sigma(\{w\}) }.$$ Note that (\[cmon\]) has a decomposition into a disjoint union of locally closed subspaces, each isomorphic to $(\mathbb{C}^{*})^{i}$ for some $i$, which is preserved after quotienting by the induced action of the finite group $N(\sigma)$. The result follows by considering the corresponding spaces and maps after quotienting by $N(\sigma)$, and observing that this doesn’t affect the Hodge polynomial of (\[cmon\]).
Let $F: |\mathcal{J}_{\infty}\mathcal{X}|' \rightarrow \mathbb{Q}$ be a $J_{\infty}(T)$-invariant function on the space of twisted arcs of $\mathcal{X}$ and let $A$ be a $J_{\infty}(T)$-invariant subset of $|\mathcal{J}_{\infty}\mathcal{X}|$. By Theorem \[decomp\], we have a decomposition $$A \cap |\mathcal{J}_{\infty}\mathcal{X}|' = \coprod_{\substack{ w \in |\Sigma| \cap N \\ \tilde{\gamma}_{w} \in A }} \tilde{\gamma}_{w} \cdot J_{\infty}(T).$$ We define $$\int_{A} (uv)^{F} d\mu_{\mathcal{X}} := \sum_{\substack{ w \in |\Sigma| \cap N \\ \tilde{\gamma}_{w} \in A }} \mu_{\mathcal{X}}(\tilde{\gamma}_{w} \cdot J_{\infty}(T)) (uv)^{F(\tilde{\gamma}_{w})},$$ when the right hand sum is a well-defined element in $\mathbb{Z}[(uv)^{1/N}][[(uv)^{-1/N}]]$, for some positive integer $N$. With the definitions of Yasuda, this is the *motivic integral* of $F$ over $A$.[^2]
Recall the involution $\iota: \operatorname{Box}({\mbox{\boldmath$\Sigma$}}) \rightarrow \operatorname{Box}({\mbox{\boldmath$\Sigma$}})$ (Remark (\[ginvo\])). Following [@YasMotivic], we define the *shift function* $s_{\mathcal{X}}: |\mathcal{J}_{\infty}\mathcal{X}|' \rightarrow \mathbb{Q}$ by $$s_{\mathcal{X}}(\tilde{\gamma}_{w} \cdot J_{\infty}(T)) = \dim \sigma(\{w\}) - \psi(\{ w \}) = \psi(\iota(\{w\})).$$ The shift function factors as $|\mathcal{J}_{\infty}\mathcal{X}|' \stackrel{\pi}{\rightarrow} |\mathcal{J}_{0}\mathcal{X}| = |\mathcal{I}\mathcal{X}|
\stackrel{\operatorname{sft}}{\rightarrow} \mathbb{Q}$, where the function $\operatorname{sft}$ is constant on the connected components of $|\mathcal{I} \mathcal{X}|$. Borisov, Chen and Smith [@BCSOrbifold] showed that the connected components of $|\mathcal{I}\mathcal{X}|$ are indexed by $\operatorname{Box}({\mbox{\boldmath$\Sigma$}})$ (c.f. Remark \[doko\]). The value of $\operatorname{sft}$ on the component of $|\mathcal{I} \mathcal{X}|$ corresponding to $g$ in $N(\sigma)$ is the age of $g^{-1}$ (see (\[age\]) and Remark \[ginvo\]).
Consider a Kawamata log terminal pair $(\mathcal{X}, \mathcal{E})$, where $\mathcal{E}$ is a $T$-invariant $\mathbb{Q}$-divisor on $\mathcal{X}$ (see Section \[contact\]). Following Yasuda (Definition 4.1 [@YasMotivic]), we consider the invariant $\Gamma(\mathcal{X}, \mathcal{E})$ defined by $$\Gamma(\mathcal{X}, \mathcal{E}) = \int_{|\mathcal{J}_{\infty}\mathcal{X}|} (uv)^{s_{\mathcal{X}} + \operatorname{ord}\mathcal{E}} d\mu_{\mathcal{X}}.$$ Yasuda showed that $\Gamma(\mathcal{X}, \mathcal{E})$ is a well-defined element in $\mathbb{Z}[(uv)^{1/N}][[(uv)^{-1/N}]]$, for some positive integer $N$ [@YasMotivic Proposition 4.15]. Our goal is to compute such invariants and give them a combinatorial interpretation. We first recall the notion of weighted $\delta$-vector from [@YoWeightI]. Consider the polyhedral complex $Q = \{ v \in |\Sigma| \, | \, \psi(v) \leq 1 \}$ and let $\lambda: |\Sigma| \rightarrow \mathbb{R}$ be a piecewise $\mathbb{Q}$-linear function satisfying $\lambda(b_{i}) > - 1$ for $i = 1, \ldots, r$. The *weighted $\delta$-vector* $\delta^{\lambda}(t)$ is defined by $$\delta^{\lambda}(t) = (1 - t)^{d + 1}(1 + \sum_{m \geq 1} \sum_{v \in mQ \cap N} t^{\psi(v) - \lceil \psi(v) \rceil + \lambda(v) + m}).$$ The following symmetry property was established in Corollary 2.13 of [@YoWeightI].
\[stu\] With the notation above, the expression $\delta^{\lambda}(t)$ is a rational function in $\mathbb{Q}(t^{1/N})$, for some positive integer $N$. If $\Sigma$ is a complete fan, then $$\delta^{\lambda}(t) = t^{d} \delta^{\lambda}(t^{-1}).$$
For each cone $\tau$ in $\Sigma$, let $N_{\tau}$ be the sublattice of $N$ generated by $\{ b_{i} \mid \rho_{i} \subseteq \tau \}$ and let $\Sigma_{\tau}$ be the simplicial fan in $(N/N_{\tau})_{\mathbb{R}}$ with cones given by the projections of the cones in $\Sigma$ containing $\tau$. We define $h_{\tau}^{\lambda}(t)$ to be the expression $$h_{\tau}^{\lambda}(t) = \sum_{\tau \subseteq \sigma}t^{\sum_{\rho_{i} \subseteq \sigma \backslash \tau} \lambda(b_{i})} t^{\dim \sigma - \dim \tau}(1 - t)^{\operatorname{codim}\sigma}
\prod_{\rho_{i} \subseteq \sigma \backslash \tau} (1 - t)/(1 - t^{\lambda(b_{i})+ 1}),$$ so that when $\lambda \equiv 0$, $h_{\tau}^{\lambda}(t)$ is the $h$-vector of $\Sigma_{\tau}$ (see, for example, [@StaEnumerative]). By expanding and collecting terms, we have $$\label{ditch}
t^{\operatorname{codim}\tau} h_{\tau}^{\lambda}(t^{-1}) =
(t - 1)^{\operatorname{codim}\tau} \sum_{\tau \subseteq \sigma} \prod_{\rho_{i} \subseteq \sigma \backslash \tau} 1 /(t^{\lambda(b_{i})+ 1} - 1).$$ The following expression for the weighted $\delta$-vector was established in Proposition 2.7 of [@YoWeightI], $$\label{mccoy}
\delta^{\lambda}(t) = \sum_{\tau \in \Sigma} h_{\tau}^{\lambda}(t) \sum_{ v \in \operatorname{Box}({\mbox{\boldmath$\tau$}})} t^{\psi(v) + \lambda(v)}
\prod_{\rho_{i} \subseteq \tau} (t -1)/(t^{\lambda(b_{i}) + 1} -1) .$$ After rearranging we see that $$\label{blue}
\delta^{\lambda}(t^{-1}) = \sum_{\tau \in \Sigma} h_{\tau}^{\lambda}(t^{-1}) \sum_{ v \in \operatorname{Box}({\mbox{\boldmath$\tau$}})} t^{- \psi(v) - \lambda(v) + \sum_{\rho_{i} \subseteq \tau}\lambda(b_{i}) }
\prod_{\rho_{i} \subseteq \tau} (t -1)/(t^{\lambda(b_{i}) + 1} -1) .$$
\[tropo\] If $(\mathcal{X}, \mathcal{E})$ is a Kawamata log terminal pair, where $\mathcal{E}$ is a $T$-invariant $\mathbb{Q}$-divisor on $\mathcal{X}$, then the corresponding piecewise $\mathbb{Q}$-linear function $\lambda = \psi_{q_{*}\mathcal{E}}$ satisfies $\lambda(b_{i}) > - 1$ for $i = 1, \ldots, r$, and $$\Gamma(\mathcal{X}, \mathcal{E}) = (uv)^{d} \delta^{\lambda}((uv)^{-1}).$$ In particular, $\Gamma(\mathcal{X}, \mathcal{E})$ is a rational function in $\mathbb{Q}(t^{1/N})$, for some positive integer $N$. If $\Sigma$ is a complete fan, then $$\Gamma(\mathcal{X}, \mathcal{E})(u,v) = (uv)^{d} \Gamma(\mathcal{X}, \mathcal{E})(u^{-1}, v^{-1}) = \delta^{\lambda}(uv).$$ Moreover, every weighted $\delta$-vector has the form $\delta^{\lambda}(uv) = (uv)^{d}\Gamma(\mathcal{X}, \mathcal{E})(u^{-1}, v^{-1})$, for some such pair $(\mathcal{X}, \mathcal{E})$.
We showed in Lemma \[order\] that $\operatorname{ord}\mathcal{E}( \tilde{\gamma}_{w} \cdot J_{\infty}(T) ) = -\lambda(w)$ and hence $(s_{\mathcal{X}} + \operatorname{ord}\mathcal{E})( \tilde{\gamma}_{w} \cdot J_{\infty}(T)) = \dim \sigma(\{w\}) - \psi(\{ w \}) -\lambda(w)$. Using Lemma \[orbitz\], we compute $$\Gamma(\mathcal{X}, \mathcal{E}) = (uv - 1)^{d} \sum_{w \in |\Sigma| \cap N} (uv)^{-\psi(w) -\lambda(w) }.$$ For each $w$ in $|\Sigma| \cap N$, we have a unique decomposition $$w = \{w\} + w' + \sum_{\rho_{i} \subseteq \sigma(w) \smallsetminus \sigma(\{w\})} b_{i},$$ where $w'$ is a non-negative linear combination of $\{ b_{i} \mid \rho_{i} \subseteq \sigma(w) \}$. Here $\sigma(w)$ is the cone containing $w$ in its relative interior. Using this decomposition, we compute the following expression for $\Gamma(\mathcal{X}, \mathcal{E})$, $$\sum_{\substack{ \tau \in \Sigma \\ v \in \operatorname{Box}({\mbox{\boldmath$\tau$}}) } } (uv - 1)^{d}(uv)^{-(\psi + \lambda)(v)}
\sum_{\tau \subseteq \sigma} (uv)^{- \sum_{\rho_{i} \subseteq \sigma \smallsetminus \tau} (\lambda(b_{i}) + 1)}
\prod_{\rho_{i} \subseteq \sigma} 1/( 1 - (uv)^{-\lambda(b_{i}) - 1} ).$$ Rearranging and using (\[ditch\]) gives $$\begin{aligned}
\Gamma(\mathcal{X}, \mathcal{E})& = \sum_{\tau \in \Sigma} (uv)^{\operatorname{codim}\tau}h_{\tau}^{\lambda}((uv)^{-1}) \times \\
&\sum_{ v \in \operatorname{Box}({\mbox{\boldmath$\tau$}})} (uv)^{\sum_{\rho_{i} \subseteq \tau}\lambda(b_{i}) + \dim \tau - \psi(v) - \lambda(v)}
\prod_{\rho_{i} \subseteq \tau} (uv -1)/((uv)^{\lambda(b_{i}) + 1} -1). \end{aligned}$$ Comparing with (\[blue\]) gives $\Gamma(\mathcal{X}, \mathcal{E}) = (uv)^{d} \delta^{\lambda}((uv)^{-1})$. The second statement follows from Theorem \[stu\].
If $\lambda: |\Sigma| \rightarrow \mathbb{R}$ is a piecewise $\mathbb{Q}$-linear function satisfying $\lambda(b_{i}) > - 1$ for $i = 1, \ldots, r$, then we may consider the corresponding $T$-invariant $\mathbb{Q}$-divisor $E$ on $X$. The pair $(\mathcal{X}, q^{*}E)$ is a Kawamata log terminal pair and, by the above argument, $\delta^{\lambda}(uv) = (uv)^{d} \Gamma(\mathcal{X}, q^{*}E)(u^{-1}, v^{-1})$. Hence every weighted $\delta$-vector corresponds to a motivic integral on $\mathcal{X}$.
By Theorem \[tropo\] and (\[blue\]), we obtain a formula for the invariant $\Gamma(\mathcal{X}, \mathcal{E})$. This formula is a stacky analogue of Batyrev’s formula for the motivic integral of a simple normal crossing divisor on a smooth variety (see [@VeyArc], [@BatNon]).
When $\mathcal{E} = 0$, $\Gamma(\mathcal{X}, 0)$ is a polynomial in $uv$ of degree $d$ and the coefficient of $(uv)^{j}$ is equal to the dimension of the $2j^{\textrm{th}}$ orbifold cohomology group of $\mathcal{X}$ with compact support [@YasTwisted]. When $\Sigma$ is complete, the symmetry $\Gamma(\mathcal{X}, 0)(u,v) = (uv)^{d} \Gamma(\mathcal{X}, 0)(u^{-1}, v^{-1})$ is a consequence of Poincaré duality for orbifold cohomology [@CRNew].
The Transformation Rule {#transform}
=======================
A morphism $f: \mathcal{Y} \rightarrow \mathcal{X}$ of smooth Deligne-Mumford stacks is *birational* if there exist open, dense substacks $\mathcal{Y}_{0}$ of $\mathcal{Y}$ and $\mathcal{X}_{0}$ of $\mathcal{X}$ such that $f$ induces an isomorphism $\mathcal{Y}_{0} \cong \mathcal{X}_{0}$ (Definition 3.36 [@YasMotivic]). If $f$ is proper and birational then Yasuda [@YasMotivic] proved a transformation rule relating motivic integrals on $\mathcal{X}$ to motivic integrals on $\mathcal{Y}$, generalising a classic result of Kontsevich for smooth varieties (see, for example, [@VeyArc]). The goal of this section is to interpret the transformation rule in our context in order to give a geometric proof of a combinatorial result in [@YoWeightI].
If ${\mbox{\boldmath$\triangle$}}= (N, \triangle, \{ \bar{b}_{i} \})$ and ${\mbox{\boldmath$\Sigma$}}= (N, \Sigma, \{ b_{i} \} )$ are stacky fans, we say that ${\mbox{\boldmath$\triangle$}}$ *refines* ${\mbox{\boldmath$\Sigma$}}$ if
1. The fan $\triangle$ refines $\Sigma$ in $N_{\mathbb{R}}$,
2. \[hoy\] For any ray $\bar{\rho}_{i}$ in $\triangle$, $\bar{b}_{i}$ is an integer combination of the lattice points $\{ b_{j} \mid \rho_{j} \subseteq \sigma \}$, where $\sigma$ is any cone of $\Sigma$ containing $\bar{\rho}_{i}$.
If ${\mbox{\boldmath$\triangle$}}$ *refines* ${\mbox{\boldmath$\Sigma$}}$, we have an induced morphism of toric stacks $f: \mathcal{X}({\mbox{\boldmath$\triangle$}}) \rightarrow \mathcal{X}({\mbox{\boldmath$\Sigma$}})$ (Remark 4.5 [@BCSOrbifold]) such that $f$ restricts to the identity map on the torus and hence is birational. Note that the corresponding map of coarse moduli spaces is proper [@FulIntroduction]. Since the morphism from a Deligne-Mumford stack to its coarse moduli space is proper [@KMQuotients], it follows that $f$ is proper [@LMBChamps]. We will give a local description of $f$. Let $\tau$ be a maximal cone in $\triangle$, contained in a maximal cone $\sigma$ of $\Sigma$. By Property (\[hoy\]), we have an inclusion of lattices $N_{\tau} \hookrightarrow N_{\sigma}$, inducing a homomorphism of groups $j: N(\tau) \rightarrow N(\sigma)$. Let $\tau'$ be the cone generated by $\{ \bar{b}_{i} \mid \bar{\rho}_{i} \subseteq \tau \}$ in $N_{\tau}$ and let $\sigma'$ be the cone generated by $\{ b_{i} \mid \rho_{i} \subseteq \sigma \}$ in $N_{\sigma}$. The inclusion $\tau' \cap N_{\tau} \hookrightarrow \sigma' \cap N_{\sigma}$ induces a $j$-equivariant map $\phi: \operatorname{Spec}\mathbb{C}[ \check{\tau}' \cap M_{\tau} ] \rightarrow \operatorname{Spec}\mathbb{C}[ (\sigma')^{\vee} \cap M_{\sigma} ]$. Taking stacky quotients of both sides yields the restriction of $f$ to $\mathcal{X}({\mbox{\boldmath$\tau$}})$, $f: \mathcal{X}({\mbox{\boldmath$\tau$}}) = [ \operatorname{Spec}\mathbb{C}[ \check{\tau}' \cap M_{\tau} ] / N(\tau) ] \rightarrow
\mathcal{X}({\mbox{\boldmath$\sigma$}}) = [ \operatorname{Spec}\mathbb{C}[ (\sigma')^{\vee} \cap M_{\sigma} ] / N(\sigma) ]$.
Given a morphism $g: \mathcal{Y} \rightarrow \mathcal{Z}$ of Deligne-Mumford stacks, Yasuda described a natural morphism $g_{\infty}: |\mathcal{J}_{\infty}\mathcal{Y}| \rightarrow |\mathcal{J}_{\infty}\mathcal{Z}|$ (Proposition 2.14 [@YasMotivic]). We outline his construction. Given a representable morphism $\gamma: \mathcal{D}^{l}_{\infty, \mathbb{C}} \rightarrow \mathcal{Y}$, we can consider the composition $\gamma': \mathcal{D}^{l}_{\infty, \mathbb{C}} \rightarrow \mathcal{Y} \rightarrow \mathcal{X}$. If $l'$ is a positive integer dividing $l$, then we have a group homomorphism $\mu_{l} \rightarrow \mu_{l'}$, $\zeta_{l} \mapsto (\zeta_{l})^{l/l'} = \zeta_{l'}$, and an equivariant map $\operatorname{Spec}\mathbb{C}[[t]] \rightarrow \operatorname{Spec}\mathbb{C}[[t]]$, $t \mapsto t^{l/l'}$. Taking stacky quotients of both sides gives a morphism $\mathcal{D}^{l}_{\infty, \mathbb{C}} \rightarrow \mathcal{D}^{l'}_{\infty, \mathbb{C}}$. By Lemma 2.15 of [@YasMotivic], there exists a unique positive integer $l'$ dividing $l$ so that $\gamma'$ factors (up to $2$-isomorphism) as $\mathcal{D}^{l}_{\infty, \mathbb{C}} \rightarrow \mathcal{D}^{l'}_{\infty, \mathbb{C}} \stackrel{\psi}{\rightarrow} \mathcal{X}$, where $\psi$ is representable, and then $g_{\infty}(\gamma) = \psi$.
Let $Y$ be the coarse moduli space of $\mathcal{Y}$ and let $Z$ be the coarse moduli space of $\mathcal{Z}$. Then $g: \mathcal{Y} \rightarrow \mathcal{Z}$ induces a morphism $g': Y \rightarrow Z$. Note that, by considering coarse moduli spaces, the morphism $\mathcal{D}^{l}_{\infty, \mathbb{C}} \rightarrow \mathcal{D}^{l'}_{\infty, \mathbb{C}}$ yields the identity morphism on $\operatorname{Spec}\mathbb{C}[[t]]$. It follows that we have a commutative diagram $$\label{Kirk}
\xymatrix{ |\mathcal{J}_{\infty}\mathcal{Y}| \ar[r]^{g_{\infty}} \ar[d]^{\tilde{\pi}_{\infty} } &
|\mathcal{J}_{\infty}\mathcal{Z}| \ar[d]^{ \tilde{\pi}_{\infty} } \\
J_{\infty}(Y) \ar[r]^{g_{\infty}'} & J_{\infty}(Z) .
}$$
Consider the notation of Theorem \[decomp\].
\[light\] If ${\mbox{\boldmath$\triangle$}}= (N, \triangle, \{ \bar{b}_{i} \})$ is a stacky fan refining ${\mbox{\boldmath$\Sigma$}}= (N, \Sigma, \{ b_{i} \})$, then the birational morphism $f: \mathcal{X}({\mbox{\boldmath$\triangle$}}) \rightarrow \mathcal{X}({\mbox{\boldmath$\Sigma$}})$ induces a $J_{\infty}(T)$-equivariant map $$f_{\infty}: |\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\triangle$}})|' \rightarrow |\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\Sigma$}})|'$$ $$f_{\infty}(\tilde{\gamma}_{v}) = \tilde{\gamma}_{v}.$$
By considering coarse moduli spaces, $f: \mathcal{X}({\mbox{\boldmath$\triangle$}}) \rightarrow \mathcal{X}({\mbox{\boldmath$\Sigma$}})$ gives rise to the toric morphism $f': X(\triangle) \rightarrow X(\Sigma)$, with induced map of arc spaces $f_{\infty}': J_{\infty}(X(\triangle))' \rightarrow J_{\infty}(X(\Sigma))'$. Consider the commutative diagram (\[Kirk\]). By Theorem \[decomp\], the maps $\tilde{\pi}_{\infty}$ are $J_{\infty}(T)$-equivariant bijections satisfying $\tilde{\pi}_{\infty}(\tilde{\gamma}_{v}) = \gamma_{v}$. Hence we only need to show that $f_{\infty}'$ is $J_{\infty}(T)$-equivariant and satisfies $f_{\infty}'(\gamma_{v}) = \gamma_{v}$. This fact is well-known but we recall a proof for the convenience of the reader. Suppose $v$ lies in a cone $\tau$ of $\triangle$ and let $\sigma$ be a cone in $\Sigma$ containing $\tau$. The arc $\gamma_{v}$ corresponds to the ring homomorphism $\mathbb{C}[\check{\tau} \cap M] \rightarrow \mathbb{C}[[t]]$, $\chi^{u} \mapsto t^{\langle u, v \rangle}$ and hence $f_{\infty}'(\gamma_{v})$ corresponds to the ring homomorphism $\mathbb{C}[\sigma^{\vee} \cap M] \hookrightarrow \mathbb{C}[\check{\tau} \cap M] \rightarrow \mathbb{C}[[t]]$, $\chi^{u} \mapsto \chi^{u} \mapsto t^{\langle u, v \rangle}$. We see that $f_{\infty}'$ is $J_{\infty}(T)$-equivariant and $f_{\infty}'(\gamma_{v}) = \gamma_{v}$.
We now state Yasuda’s transformation rule in the case of toric stacks.
\[yap\] Let ${\mbox{\boldmath$\triangle$}}= (N, \triangle, \{ \bar{b}_{i} \})$ be a stacky fan refining ${\mbox{\boldmath$\Sigma$}}= (N, \Sigma, \{ b_{i} \})$, with corresponding birational morphism $f: \mathcal{X}({\mbox{\boldmath$\triangle$}}) \rightarrow \mathcal{X}({\mbox{\boldmath$\Sigma$}})$. If $F: |\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\Sigma$}})|' \rightarrow \mathbb{Q}$ is a $J_{\infty}(T)$-invariant function and $A$ is a $J_{\infty}(T)$-invariant subset of $|\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\Sigma$}})|'$, then $$\int_{A} (uv)^{ s_{\mathcal{X}(\Sigma)} + F } d\mu_{\mathcal{X}}
= \int_{f_{\infty}^{-1}(A)} (uv)^{s_{\mathcal{X}(\triangle)} + F \circ f_{\infty}
- \operatorname{ord}K_{\mathcal{X}(\triangle)/\mathcal{X}(\Sigma)} } d\mu_{\mathcal{X}'},$$ where $K_{\mathcal{X}(\triangle)/\mathcal{X}(\Sigma)} = K_{\mathcal{X}(\triangle)} - f^{*}K_{\mathcal{X}(\Sigma)}$.
As a corollary, we deduce a geometric proof of the following result, which was proved using combinatorial methods in Proposition 2.13 of [@YoWeightI]. If ${\mbox{\boldmath$\Sigma$}}= (N, \Sigma, \{b_{i}\})$ is a stacky fan and $\lambda$ is a piecewise $\mathbb{Q}$-linear function on $|\Sigma|$, we fix the following notation. Let $\psi_{\Sigma} = \psi_{q_{*}K_{\mathcal{X}(\Sigma) } }$ be the piecewise $\mathbb{Q}$-linear function on $|\Sigma|$ satisfying $\psi_{\Sigma}(b_{i}) = 1$, and let $\delta^{\lambda}_{\Sigma}(t)$ denote the weighted $\delta$-vector corresponding to $\lambda$.
\[fastball\] Let $N$ be a lattice of rank $d$ and let ${\mbox{\boldmath$\Sigma$}}= (N, \Sigma, \{ b_{i} \})$ and ${\mbox{\boldmath$\triangle$}}= (N , \triangle, \{ b'_{j} \})$ be stacky fans such that $|\Sigma| = |\triangle|$. Let $\lambda$ be a piecewise $\mathbb{Q}$-linear function with respect to $\Sigma$ satisfying $\lambda(b_{i}) > -1$ for every $b_{i}$, and set $\lambda' = \lambda + \psi_{\Sigma} - \psi_{\triangle}$. If $\lambda'$ is piecewise $\mathbb{Q}$-linear with respect to $\triangle$ and satisfies $\lambda'(b'_{j}) > -1$ for every $b'_{j}$, then $\delta^{\lambda}_{\Sigma}(t) = \delta^{\lambda'}_{\triangle}(t)$.
Let $\tilde{\Sigma}$ be a common refinement of $\Sigma$ and $\triangle$. We can choose lattice points $\{\tilde{b}_{i}\}$ on the rays of $\tilde{\Sigma}$ so that $\tilde{{\mbox{\boldmath$\Sigma$}}} = (N, \tilde{\Sigma}, \{\tilde{b}_{i}'\})$ is a stacky fan refining ${\mbox{\boldmath$\Sigma$}}$ and ${\mbox{\boldmath$\triangle$}}$. Hence we can reduce to the case when ${\mbox{\boldmath$\triangle$}}$ refines ${\mbox{\boldmath$\Sigma$}}$. In this case, consider the corresponding birational morphism $f: \mathcal{X}({\mbox{\boldmath$\triangle$}}) \rightarrow \mathcal{X}({\mbox{\boldmath$\Sigma$}})$. By Lemma \[order\], there is a Kawamata log terminal pair $(\mathcal{X}({\mbox{\boldmath$\Sigma$}}), \mathcal{E})$ such that $\operatorname{ord}\mathcal{E}( \tilde{\gamma}_{w} \cdot J_{\infty}(T) ) = -\lambda(w)$. It follows from Lemma \[light\] that $(\operatorname{ord}\mathcal{E} \circ f_{\infty} )( \tilde{\gamma}_{w} \cdot J_{\infty}(T) ) = -\lambda(w)$ and Lemma \[order\] and Lemma \[light\] imply that $\operatorname{ord}K_{\mathcal{X}(\triangle)/\mathcal{X}(\Sigma)} ( \tilde{\gamma}_{w} \cdot J_{\infty}(T) ) = \psi_{\Sigma}(w) - \psi_{\triangle}(w)$. The result now follows from Theorem \[tropo\] and Theorem \[yap\], with $F = \operatorname{ord}\mathcal{E}$ and $A = |\mathcal{J}_{\infty}\mathcal{X}({\mbox{\boldmath$\Sigma$}})|$.
Remarks {#remedy}
=======
More generally, we can replace $N$ by a finitely generated abelian group of rank $d$. All the results go through with minor modifications. We mention the local construction of the toric stack [@BCSOrbifold]. Let $\bar{N}$ be the lattice given by the image of $N$ in $N_{\mathbb{R}}$ and for each $v$ in $N$, let $\bar{v}$ denote the image of $v$ in $\bar{N}$. Let $\Sigma$ be a complete, simplicial, rational fan in $N_{\mathbb{R}}$ and let $\bar{v}_{1}, \ldots, \bar{v}_{r}$ be the primitive integer generators of $\Sigma$ in $\bar{N}$. Fix elements $b_{1}, \ldots , b_{r}$ in $N$ such that $\bar{b}_{i} = a_{i}\bar{v}_{i}$, for some positive integer $a_{i}$. The data ${\mbox{\boldmath$\Sigma$}}= (N, \Sigma, \{ b_{i} \})$ is a *stacky fan*. For each maximal cone $\sigma$ of $\Sigma$, let $N_{\sigma}$ denote the subgroup of $N$ generated by $\{ b_{i} \, | \, \rho_{i}
\subseteq \sigma \}$. We obtain a homomorphism of finite groups $N(\sigma) = N/N_{\sigma} \rightarrow \bar{N}(\sigma) = \bar{N}/\bar{N}_{\sigma}$. Composing with the injection $\bar{N}(\sigma) \rightarrow (\mathbb{C}^{*})^{d}$ from Section \[tstack\] gives a homomorphism $N(\sigma) \rightarrow \bar{N}(\sigma) \rightarrow (\mathbb{C}^{*})^{d}$, and $\mathcal{X}({\mbox{\boldmath$\sigma$}}) = [\mathbb{A}^{d}/ N(\sigma)]$.
Using this more general setup, one can apply Theorem \[decomp\] to the $T$-invariant closed substacks of $\mathcal{X}$ to give a decomposition of $|\mathcal{J}_{\infty}\mathcal{X}|$ into $J_{\infty}(T)$-orbits.
[^1]: Note that $N(\rho_{i})$ may have torsion and so $\sigma_{i}$ is a cone in the image of $N(\rho_{i})$ in $N(\rho_{i})_{\mathbb{Q}}$. There are no difficulties in generalising to this situation. See Section \[remedy\] for a discussion of this issue. This is the level of generality used in [@BCSOrbifold].
[^2]: To check the definitions agree, we can replace $A$ by $A \cap |\mathcal{J}_{\infty}\mathcal{X}|'$, by Propositions 3.24 and 3.25 in [@YasMotivic]. Now apply the decomposition of Theorem \[decomp\].
|
---
abstract: 'We prove the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in $\mathbb{Z}^d$. Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.'
address:
- 'Sami Mustapha, Institut Mathématiques de Jussieu, Sorbonne Université , Tour 25 5e étage Boite 247. 4, place Jussieu F-75252 PARIS CEDEX 05. '
- 'Mohamed Sifi, Université de Tunis El Manar, Faculté des Sciences de Tunis, LR11ES11 Laboratoire d’Analyse Mathématique et Applications LR11ES11. 2092, Tunis, Tunisie. '
author:
- Sami Mustapha
- Mohamed Sifi
title: |
Discrete harmonic functions\
in Lipschitz domains
---
[^1]
[^2]
Introduction and main results
=============================
Random walks conditioned to live in domains $\mathcal{C}\subset\mathbb{Z}^d$ are of growing interest because of the range of their applications in enumerative combinatorics, in probability theory and in harmonic analysis (cf. [@B1], [@BM], [@DW], [@FIM], [@FR], [@R1]). Doob $h$-transforms, where $h$ is harmonic for the random walk, positive within $\mathcal{C}$ and vanishing on its boundary $\partial\mathcal{C}$, are used to perform such conditioning. It is therefore crucial to identify the set of all positive harmonic functions associated with a killed random walk.
General results for homogeneous random walks with non-zero drift killed at the boundary of a half-space or an orthant were obtained in [@I1], [@IR], [@KR2]. For random walks with zero drift, only few results are available [@BMS], [@DW], [@GS], [@R1], [@R2]. The first systematical result was obtained by K. Raschel, who introduced in [@R2] a new approach based on the investigation of a functional equation satisfied by the generating function of the values taken by the harmonic function. This approach allows him to establish the existence of positive harmonic functions for random walks with small steps and zero drift killed at the boundary of the quadrant $\mathbb{N}^2$. It should be also mentioned that [@R2] provides explicit expressions for these harmonic functions.
In a recent work Ignatiouk-Robert [@I2] investigated the properties of harmonic functions for random walks in via ladder heights. Applying her general results to random walk in a convex cone she deduced the uniqueness (up to a multiplicative constant) of the harmonic function constructed by Denisov and Wachtel in [@DW] under some moment condition on the jumps. Alternative constructions of this harmonic function are proposed by Denisov and Wachtel in [@DW1]. These new constructions allow them to remove quite restrictive extendability assumption imposed in [@DW]. In [@RT] Raschel and Tarrago studied the behavior of the Green function for random walks in convex cone which gives the uniqueness of the harmonic function (see also [@DuW].
Regarding spatially inhomogeneous random walks the problem is more difficult.\
Uniqueness of positive harmonic functions for random walks with symmetric spatially inhomogeneous increments, killed at the boundary of a half space, was established in [@M2] and more recently in the case of an orthant [@Bou].
The main purpose of the present paper is to extend the results of [@Bou] for the whole class of spatially inhomogeneous centered random walks satisfying finite span and ellipticity conditions and killed when leaving a globally Lipschitz unbounded domain in $\mathbb{Z}^d$.
Consider $\Gamma\subset \mathbb{Z}^d$ a finite subset of $\mathbb{Z}^d$ and let $\pi: \mathbb{Z}^d\times
\Gamma\rightarrow [0,1]$ such that $$\sum_{e\in\Gamma}\pi(x,e)=1,\quad \sum_{e\in\Gamma}\pi(x,e)e=0;\quad e\in\Gamma,\:
x\in\mathbb{Z}^d.$$ Then, we let $\{S(n),\, n\in \mathbb{N}\}=(S_n)_{n\in\mathbb{N}}$ be the Markov chain on $\mathbb{Z}^d$ defined by $$\mathbb{P}[S_{n+1}=x+e/S_n=x]=\pi(x,e);\,\,\quad e\in\Gamma,\:
x\in\mathbb{Z}^d,\: n=0,1,\ldots$$ $(S_n)_{n\in\mathbb{N}}$ is a centered random walk with bounded increments which becomes spatially homogeneous if we assume the probabilities $\pi(x,e)$ are independent of $x$. We shall assume that the set $\Gamma$ contains all unit vectors in $\mathbb{Z}^d$, i.e. all the vectors $e_k=(0, \ldots,0,1,0,\ldots0)\in
\mathbb{Z}^d$, where the $1$ is the $k$-th component. We shall impose to the random walk $(S_n)_{n\in\mathbb{N}}$ to satisfy the following uniform ellipticity condition: $$\label{0} \pi(x,e)\geq\alpha,\quad e\in\Gamma,\:
x\in\mathbb{Z}^d,$$ for some $\alpha>0$.
We shall denote by:
$\bullet\quad\mathcal{C}$ a globally Lipschitz domain of $\mathbb{Z}^d$ that is, a domain $\mathcal{C}=\mathcal{D}\cap
\mathbb{Z}^d$ where $$\mathcal{D}=\left\{(x_1,x')\in\mathbb{R}\times\mathbb{R}^{d-1};x_1>
\varphi(x')\right\}$$ for some Lipschitz function on $\mathbb{R}^{d-1}$ satisfying $$|\varphi(x')-\varphi(y')|\leq A|x'-y'|,\quad x',y'\in
\mathbb{R}^{d-1},$$ for some $A>0$, where $|.|$ denote the Euclidean norm. We shall assume that $\varphi(0)=0$.
$\bullet\quad\tau$ the first exit time from $\mathcal{C}$, i.e., $$\tau=\inf\{n=0,1,\ldots;\,\,S_n\notin\mathcal{C}\}.$$
$\bullet\quad G_x^y$, $x,y\in\mathcal{C}$, the Green function defined by $$G_x^y=\sum_{n\in \mathbb{N}}\mathbb{P}_x(S_n=y,
\tau>n).$$
We are interested in positive functions $h$ which are discrete harmonic for the random walk $(S_j)_{j\in\mathbb{N}}$ killed at the boundary of $\mathcal{C}$, i.e. in functions $h:\overline{\mathcal{C}}\rightarrow \mathbb{R}_+$ such that:
- For any $x\in \mathcal{C}$, $\,h(x)=\displaystyle \sum_{e\in\Gamma}\pi(x,e)h(x+e)$;
- If $x\in\partial\mathcal{C}$, then $h(x)=0$;
- If $x\in\mathcal{C}$, then $h(x)>0$;
where $\, \overline{\mathcal{C}}=\partial\mathcal{C}\cup\mathcal{C}$. The boundary of a set $A\subset \mathbb{Z}^d$ is defined by $$\partial A=\{x\in A^c, x=z+e\;\mbox{for some}\; z\in A\; \mbox{and}\; e\in \Gamma\}$$ and $\overline{A}=A\cup \partial A$. In terms of the first exit time of the random walk from $\mathcal{C}$, we have that $$h(x)=\mathbb{E}^x\left(h(S_1),\tau>1\right),\quad
x\in\mathcal{C}.$$
\[thm1\] Let $(S_n)_{n\in\mathbb{N}}$ be a centered random walk satisfying the above finite support and ellipticity conditions. Assume that $\mathcal{C}$ is a globally Lipschitz domain of $\mathbb{Z}^d$. Then, up to a multiplicative constant, there exists a unique positive function, harmonic for the random walk killed at the boundary.
The previous result has an important consequence on the Martin boundary theory attached to the random walk $(S_n)_{n\in\mathbb{N}}$ killed on the boundary of $\mathcal{C}$. Recall that for a transient Markov chain on a countable state space $E$, the Martin compactification of $E$ is the unique smallest compactification $E_M$ of the discrete set $E$ for which the Martin kernels $y\rightarrow \displaystyle k_y^x=G_y^x/G_y^{x_0}$ (where $x_0$ is a given reference state in $E$) extend continuously for all $x\in E$. The minimal Martin boundary $\partial_mE_M$ is the set of all those $\gamma\in\partial E_M$ for which the function $x\rightarrow k_\gamma^x$ is minimal harmonic. Recall that a harmonic function $h$ is minimal if $0\leq g\leq h$ with $g$ harmonic implies $g=ch$ with some $c>0$. By the Poisson-Martin boundary representation theorem, every nonnegative harmonic function $h$ can be written as $$h(x)=\int_{\partial_mE_M}k_\gamma^x\mu(d\gamma),$$ for a some positive Borel measure $\mu$ on $\partial_mE_M$ (cf. [@Dy], [@Ma], [@NY]).\
An immediate consequence of Theorem \[thm1\] is the following.
\[thm2\] For all transient random walks satisfying centering, finite support and ellipticity conditions and all global Lipschitz domains of $\mathbb{Z}^d$, the minimal Martin boundary is reduced to one point.
We conclude this introduction with some comments which may be helpful in placing the results of this paper in their proper perspective.
[*(i)*]{} The proof of Theorem \[thm1\] given in [@Bou] uses in a crucial way the parabolic Harnack principle. We noted in [@Bou] that a more satisfactory approach should dispense with parabolic information and restrict to elliptic tools. A way to get round the difficulties encountered in [@Bou] is to use a lower estimate for superharmonic extensions of discrete positive harmonic functions derived by Kuo and Trudinger in [@KT0]. This lower estimate encompasses three powerful ingredients: the Aleksandrov-Bakel’man-Pucci’s maximum principle, a barrier technique and a Calderón-Zygmund covering argument. Going trough the superharmonic extension gives an alternative to the use of [@Bou Lemma 2.5] and provides a purely elliptic derivation of [@Bou Proposition 2.6] . An advantage of this approach is that it allows us to relax the assumptions $0\in \Gamma$ and $\Gamma=-\Gamma$ made in [@Bou].
[*(ii)*]{} In case of homogeneous symmetric random walks on unbounded Lipschitz domains, the main results of this paper follows from [@GS]. Although the work of Gyrya and Saloff-Coste concerns diffusion on Dirichlet spaces, to derive the desired results for symmetric random walks, it suffices to consider the corresponding cable process (see [@BB §2]). Since the harmonic functions for cable process and the random walk on the corresponding graph are essentially the same one has all the desired results (namely Theorem \[thm1\], Theorem \[thm2\] and Theorem \[thm6\]).
[*(iii)*]{} Spatially inhomogeneous random walks can be considered as the discrete analogues of diffusions generated by second-order differential operators in nondivergence form. As in [@Bou], the main tools in this paper are discrete versions of Carleson estimate and boundary Harnack inequality (cf. [@BaBu], [@Bau], [@FS], [@FSY]).
[*(iv)*]{} We restrict ourselves in this paper to random walks in Lipschitz domains. However, the proofs given below should work for a larger class of domains, for instance uniform or inner uniform domains (cf. [@Ai]).
Proof of Theorem \[thm1\]
=========================
Harnack principle
-----------------
We say that a function $u:\overline{A}=A\cup\partial A\rightarrow \mathbb{R}$ is harmonic in $A\subset\mathbb{Z}^d$ if $Lu=0$ in $A$, where $L$ is the difference operator defined by $$Lu(x)=\sum_{e\in \Gamma}\pi(x,e)u(x+e)-u(x).$$ In addition to an obvious maximum principle, harmonic functions satisfy, when they are positive, a Harnack principle. For convenience this principle is formulated in balls. The discrete Euclidean ball of center $y\in\mathbb{Z}^d$ and radius $R\geq 1$ is denoted $B_R(y)$ and simply $B_R$ when $y$ is clearly understood. We shall also have to use cubes. The cube of center $y\in\mathbb{Z}^d$ and sides $2R$, parallel to the coordinate axes is denoted $Q_R(y)$ and simply $Q_R$ when $y$ is clear. The following theorem (see [@KT0] and [@KT01]) is a centered version of Harnack principle established by Lawler [@L] for random walks with symmetric bounded increments (as well homogeneous and inhomogeneous).
\[thm3\] [**(Harnack principle)**]{} Assume that $u$ is a nonnegative harmonic function associated to a random walk satisfying centering, finite support and uniform ellipticity conditions in a ball $B_{2R}(y)$. Then $$\max_{B_R(y)}u \leq C\,\displaystyle{\min_{B_R(y)}}\,u,$$ where $C=C(d,\alpha,\Gamma)>0$.
Carleson estimate
-----------------
The classical Carleson estimate [@C] asserts that a positive harmonic function vanishing on a portion of the boundary is bounded, up to a smaller portion, by the value at a fixed point in the domain with a multiplicative constant independent of the function.
\[thm5\] Assume that $u$ is a nonnegative harmonic function in $\mathcal{C}\cap B_{3R}(y)$. Assume that $u=0$ on $\partial\mathcal{C}\cap B_{2R}(y)$. Then $$\label{1}
\max\left\{u(x),\,\,x\in \mathcal{C}\cap B_{R}(y)\right\}\leq C\:
u(y+Re_1),\quad R\geq C,$$ where $C=C(d,\alpha,\Gamma,A)>0$ is independent of $y
, R$ and $u$.
The proof of Theorem \[thm1\] relies on the following Proposition.
\[prop1\] Let $y\in
\partial\mathcal{C}$ and $R$ large enough. Let $u$ be a nonnegative harmonic function in $\mathcal{C}\cap B_{3\sqrt{d}R}(y)$ which vanishes on $\partial\mathcal{C}\cap B_{2\sqrt{d}R}(y)$. Then $$\label{7}
\max\left\{u(x),\,\,\,x\:\in\:\overline{\mathcal{C}\cap B_{R}(y)}\right\}\leq
\rho \max\left\{u(x),\,\,\,x\:\in\:{\overline{\mathcal{C}\cap
B_{2\sqrt{d}R}(y)}}\right\},$$ with a constant $0<\rho=\rho(d,\alpha,\Gamma,A)<1$.
To prove we first observe that it suffices to show that $$\label{71}
\max\left\{u(x),\: x\in \overline{\mathcal{C}\cap Q_R(y)}\right\}\leq \rho \max \left\{u(x),\; x\in \overline{\mathcal{C}\cap Q_{2R}(y)}\right\}.$$ Without loss of generality, we assume $y=0$ and $\max \left\{u(x),\; x\in \overline{\mathcal{C}\cap Q_{2R}}\right\}=1$. Then considering the function $v: \overline{Q_{2R}}\rightarrow\mathbb{R}$ defined by $v=1-u$ in $\overline{\mathcal{C}\cap Q_{2R}}$ and $v=1$ on $\overline{Q_{2R}}\,\backslash\,\overline{\mathcal{C}\cap Q_{2R}}$, we see that reduces to the following lower estimate $$\label{72}
v(x)\geq \lambda=\lambda(d,\alpha,\Gamma, A)>0,\quad x\in \overline{Q_R}.$$ Since $v$ is superharmonic in $Q_{2R}$ (i.e $Lv\leq 0$ in $Q_{2R}$) we can use the estimate [@KT01 3.24] and deduce that $$\label{73}
\min_{\overline{Q_R}}v\geq \gamma\left(\frac{\left|\overline{Q_R}\cap \{v\geq 1\}\right|}{|\overline{Q_R}|}\right)^{\frac{\log\gamma}{\log \delta}}$$ where $0<\gamma,\delta<1$ are two positive constants depending on $d,\alpha$ and $\Gamma$ and where the notation $|S|$ is used to denote the cardinality of a subset $S\subset\mathbb{Z}^d$.
On the other hand, the fact that $\mathcal{C}$ is Lipschitz allows us to find a circular cone $\mathcal{C}'$ with vertex at the origin such that $\mathcal{C}'\subset \mathcal{C}^c$. It follows then that there exists a positive constant $\mu$ (depending on $A$) such that for $R$ large enough $$\label{74}
\left|\overline{Q_R}\cap \{v\geq 1\}\right|\geq\left| \overline{Q_R}\cap \mathcal{C}'\right|\geq \mu |\overline{Q_R}|.$$ We conclude from and that $$\min_{\overline{Q_R}}v\geq \gamma^{1+\frac{\log\mu}{\log\delta}},$$ which implies and completes the proof of .
[*Proof of Theorem \[thm5\].* ]{} To prove the Carleson estimate we first observe that the uniform ellipticity assumption implies that $u(\xi)
\leq Ce^{C\,|\xi-\zeta|}u(\zeta)$, $\xi, \zeta \in \mathcal{C}\cap
B_{3R}(y)$; where $C=C(d,\alpha,\Gamma,A)>0$. This local Harnack principle allows us to assume that the distance of $x$ from $\partial\mathcal{C}$ is sufficiently large. We shall denote by $\delta(x)$ ($x\:\in\:\mathcal{C}\cap B_{2R}(y)$) this distance and suppose that $\delta(x)\geq C$. The fact that $\mathcal{C}$ is Lipschitz combined with Harnack principle (Theorem \[thm3\]) imply that $$\label{8}
u(x)\leq C\, \left(\frac{R}{\delta(x)}\right)^\gamma
u(y+Re_1),\quad x\:\in\:\mathcal{C}\cap B_{2R}(y),$$ where $\gamma$ and $C$ are positive constants depending on $d,\alpha,\Gamma$ and $A$.
Let $x\in \mathcal{C}\cap {B_{2R}(y)}$, and let us assume that $$\label{9}
\delta(x) < \left(1-\left(\frac{1+\rho}{2}
\right)^{\frac{1}{\gamma}}\right)
\frac{2R-|x-y|}{8\sqrt{d}}$$ where $\rho$ is the constant obtained in (\[7\]) and $\gamma$ the exponent that appears in (\[8\]). Let $x_0\in \partial\mathcal{C}$ such that $|x-x_0|=\delta(x)$. It follows easily from (\[9\]) and the fact that $\delta(x)$ is sufficiently large that $\overline{B_{3\sqrt{d}\delta(x)}(x_0)}\subset B_{2R}(y)$. By Proposition \[prop1\] applied to the harmonic function $u$ in the domain $B_{{3}\sqrt{d}\delta(x)}(x_0)\cap\mathcal{C}$, we have $$\label{10}
u(x)\leq \max\left\{u(x'),\,\,\, x'\in
\mathcal{C}\cap\overline{B_{\frac{3}{2}\delta(x)}(x_0) }\right\}
\leq\rho \max\left\{u(x'),\,\,\, x'\in \mathcal{C}\cap
{\overline{B_{3\sqrt{d}\delta(x)}(x_0)}} \right\}
.$$ Let $z\in \mathcal{C}\cap
\overline{B_{3\sqrt{d}\delta(x)}(x_0)}$ satisfying $$u(z)=\max\left\{u(x'),\,\,\, x'\in \mathcal{C}\cap\overline{B_{3\sqrt{d}\delta(x)}(x_0)
} \right\}.$$ We have $$(2R-|x-y|)\leq (2R-|z-y|)+8\sqrt{d}\delta(x).$$ Hence, thanks to $$(2R-|x-y|)\leq\left(\frac{1+\rho}{2}\right)^{-\frac{1}{\gamma}}(2R-|z-y|).$$ It follows that $$(2R-|x-y|)^\gamma u(x)\leq\left(\frac{1+\rho}{2}\right)^{-1}(2R-|z-y|)^\gamma
u(x)
,$$\
and therefore, by $$\begin{aligned}
\label{11}
(2R-|x-y|)^\gamma u(x)&\leq& \frac{2\rho}{1+\rho} (2R-|z-y|)^\gamma
u(z)\\ &\leq& \nonumber \theta_0\max_{x'\,\in {\mathcal{C}\cap
B_{2R}(y)}}(2R-|x'-y|)^\gamma u(x')\end{aligned}$$ where $$\theta_0=\displaystyle\frac{2\rho}{1+\rho}<1.$$
It remains to consider the case where
$$\label{12}
\delta(x)\geq \left(1-\left(\frac{1+\rho}{2}\right)^{\frac{1}{\gamma}}\right)
\frac{2R-|x-y|}{8\sqrt{d}}.$$
In follows from (\[12\]) that $$(2R-|x-y|)^\gamma u(x)\leq\varepsilon_0^{-\gamma}\delta(x)^\gamma u(x)
\leq \varepsilon_0^{-\gamma}\max_{x'\,\in\mathcal{C}\cap
B_{2R}(y)}\delta(x')^\gamma u(x')$$ where $8\sqrt d \varepsilon_0=\displaystyle
1-\left(\frac{1+\rho}{2}\right)^{\frac{1}{\gamma}}$ and, thanks to , $$\label{13}
(2R-|x-y|)^\gamma u(x)\leq C\,\varepsilon_0^{-\gamma}R^\gamma
u(y+Re_1).$$
Putting together and and taking the supremum over $\mathcal{C}\cap B_{2R}(y)$, we deduce that $$\max_{x\,\in {\mathcal{C}\cap
B_{2R}(y)}}(2R-|x-y|)^\gamma u(x) \leq \theta_0 \max_{x\,\in
{\mathcal{C}\cap B_{2R}(y)}}(2R-|x-y|)^\gamma u(x)
+C\,\varepsilon_0^{-\gamma}R^\gamma
u(y+Re_1) .$$ Using the fact that $(2R-|x-y|)\approx R$ for $x\,\in\mathcal{C}\cap B_{R}(y)$ we deduce the estimate .$\hfill\Box$
Boundary Harnack principle
--------------------------
Carleson estimate can be extended to the ratio $u/v$ of positive harmonic functions.
\[thm6\] [**(Boundary Harnack principle)**]{} Let $y\in\partial\mathcal{C}$ and $K>0$ large enough. Assume that $u$ and $v$ are two nonnegative harmonic functions in $\mathcal{C}\cap
B_{3KR}(y)$. Assume that $u,\, v =0$ on $\partial\mathcal{C}\cap B_{2KR}(y)$. Then $$\label{14}
\max_{x\,\in\:\mathcal{C}\cap B_R(y)}\frac{u(x)}{v(x)}\leq C\:
\frac{u(y+Re_1)}{v(y+Re_1)},\quad R\geq C,$$ where $C=C(d,\alpha,\Gamma,A,K)>0$.
The above formulation of the boundary Harnack principle follows the classical formulation but the proof of which will be given below shows that the assumption $v=0$ on $\partial\mathcal{C}\cap B_{2R}(y)$ is not needed so that constitutes a special case of .
The estimate is an immediate consequence of the lower estimate contained in the following lemma.
For $y\in \partial \mathcal{C}$ and $R\geq r\geq 1$, we shall denote by $$\begin{aligned}
\mathcal{D}_{R,r}(y)&=&B_R(y)\cap \{x\in \mathcal{C},\delta(x)>r\}; \\
\mathcal{C}_{R,r}(y)&=&\left(B_R(y)\cap \mathcal{C}\right) \,\setminus\,\mathcal{D}_{R,r}(y).\end{aligned}$$ For $R\geq r\geq 1$, the boundary of $\mathcal{C}_{R,r}$ is the union of three sets: the “bottom” $\partial\mathcal{C}_{R,r}\cap \mathcal{C}^c$, the “lateral side” $\partial\mathcal{C}_{R,r}\cap\{x\in\mathcal{C},\:0\leq \delta(x)\leq r\}$ and the “top” $\partial\mathcal{C}_{R,r}\cap \mathcal{D}_{R,r}$.
\[lem2\] There exists a constant $K_0>0$ such that for all $K\geq K_0$ and for all $y\in \partial\mathcal{C}, $ $r\geq 1$, $$\label{15}
\min_{x\in \mathcal{C}\cap B_r(y)}\frac{\mathbb{P}_x
\left[S(\tau_{\mathcal{C}_{Kr,r}(y)})\in \mathcal{D}_{Kr,r}(y)
\right]}{\mathbb{P}_x \left[S(\tau_{\mathcal{C}_{Kr,r}(y)})\in
\left(\partial\mathcal{C}_{Kr,r}(y)\cap\mathcal{C}\right)\,\setminus\,
\mathcal{D}_{Kr,r}(y) \right]}\geq 1$$ where $\tau_{\mathcal{C}_{Kr,r}(y)}$ denotes the exit time from $\mathcal{C}_{Kr,r}(y)$.
[ *Proof of Theorem \[thm6\].*]{} In order to derive estimate from , we first observe that it is always possible to assume that $u(y+Re_1)=v(y+Re_1)=1$. For a large $R$, Carleson estimate implies that the function $u$ is dominated by a positive constant $c_0$ in the region $B_{KR}(y)\cap \mathcal{C}$. This constant $c_0$ can be chosen so that by Harnack principle the lower estimate $v\geq \frac{1}{c_0}$ holds on $\mathcal{D}_{KR,R}(y)$. Let $v_0=c_0v$ and $u_0=\displaystyle \frac{u}{c_0}-v_0$. Let $x\in
B_{R}(y)\cap \mathcal{C}$. We have: $$\begin{aligned}
u_0(x) &\leq &
\mathbb{P}_x\left[S(\tau_{\mathcal{C}_{KR,R}(y)})\in
\left(\partial\mathcal{C}_{KR,R}(y)\cap\mathcal{C}\right)\,\setminus\, \mathcal{D}_{KR,R}(y)\right]\\
&\leq & \mathbb{P}_x\left[S(\tau_{\mathcal{C}_{KR,R}(y)})\in
\mathcal{D}_{KR,R}(y)\right]\\
& \leq & v_0(x),\end{aligned}$$ where the second inequality follows from . We deduce then that $$\frac{u(x)}{c_0}-c_0v(x)\leq c_0 v(x),
\quad x\in B_{R}(y)\cap \mathcal{C}.$$ So that $$\frac{u(x)}{v(x)}\leq 2 c_0^2,\quad x\in B_{R}(y)\cap \mathcal{C},$$ which completes the proof of . $\hfill\Box$\
[*Proof of Lemma \[lem2\].*]{} To prove estimate it suffices to show that if $u,\, v :
\overline{\mathcal{C}_{Kr,r}(y)}\rightarrow \mathbb{R}$ (where $y\in
\partial\mathcal{C}$, $r\geq 1$ are fixed) satisfy $$\label{16}
\left\{
\begin{array}{cc}
\displaystyle u(x)=\sum_{e\in \Gamma}\pi(x,e)u(x+e), & x\in\mathcal{C}_{Kr,r}(y) \\
u(x)\geq 0 & \mbox{in} \quad {\overline{\mathcal{C}_{Kr,r}(y)}} \\
u(x)\geq 1 &\mbox{ on} \quad \partial\mathcal{C}_{Kr,r}(y)\cap \mathcal{D}_{Kr,r}(y)
\end{array}\right.$$ $$\label{17}
\left\{\begin{array}{ccc} &\displaystyle v(x)=
\sum_{e\in \Gamma}\pi(x,e)v(x+e),& \quad x\in\mathcal{C}_{Kr,r}(y)\\
&v(x)\leq 1 & \quad \mbox{in} \quad
{\overline{\mathcal{C}_{Kr,r}(y)}}\\
& v(x)\leq 0 & \quad \mbox{ on} \quad
\partial\mathcal{C}_{Kr,r}(y)\cap \mathcal{D}_{Kr,r}(y)
\end{array}\right.$$ then we have $$\label{18}
v(x)\leq u(x), \quad x\in B_r(y)\cap\mathcal{C}$$ provided that $K\geq K_0$ is large enough.
First we prove that under the function $u$ satisfies $$\label{19}
u(x) \geq 2 \alpha \left( \frac{ \delta(x)}{r}\right)^\beta,
\,\,\,\,\, x \in B_r(y)\cap \mathcal{C},$$ for appropriate constants $\alpha, \, \beta >0.$
The proof of relies on the following construction.
We assume $K$ large enough and we define $ \tilde u : {\overline {
B_{Mr}(y) \cap \mathcal{C}}} \longrightarrow \mathbb{R} $ by $$\left\{
\begin{array}{cc}
\displaystyle \tilde u(x)=\sum_{e\in \Gamma}\pi(x,e)
\tilde u(x+e), & x\in B_{Mr}(y) \cap \mathcal{C} \\
\tilde u = \min(u,1) & \mbox{on} \quad
\partial \left( B_{Mr}(y) \cap \mathcal{C} \right) \,\backslash\,
{ \mathcal{D}_{Kr,r}(y)} \\
\tilde u = 1 &\mbox{on} \quad
\partial \left( B_{Mr}(y) \cap \mathcal{C} \right)
\cap
{ \mathcal{D}_{Kr,r}(y)}
\end{array}\right.$$ where $0<M \leq K $ is chosen so that $\tilde y = y +(M-1) re_1$ satisfies $$\label{20}
\delta( \tilde y) \geq 10 r.$$
Let $\mathcal{U} = \left( B_{Mr}(y) \cap \mathcal{C} \right) \cap
B_{2r}(\tilde{y})$ and $w: \overline{B_{2r}(\tilde{y})}
\longrightarrow \mathbb{R} $ be defined by $$w(x) = \tilde u (x), \,\,\,\, x\in \overline{\mathcal{U}};\; w(x)=1,\quad x \in \mathcal{U}^c\cap \overline{B_{2r}(\tilde{y})}.$$ It is easy to see that $w$ is superharmonic. Let $\tilde{z}=y+(M-2)re_1$. By the same argument used in the proof of the lower estimate combined with Harnack principle, we see that $w(\tilde{z})$ satisfies a lower estimate $w(\tilde{z})\geq c$. It follows then that $ \tilde{u}(\tilde{z})\geq c$.
Since $u\geq 1$ on $\partial\mathcal{C}_{Kr,r}(y)\cap \mathcal{D}_{Kr,r}(y)$, we deduce by the maximum principle that $u\geq \tilde{u}$ on $B_r(y)\cap\mathcal{C}$. Combining with Harnack inequality we de deduce .
It follows from that if $x\in
\overline{\mathcal{C}_{r,r}(y)}\,\backslash\,
\mathcal{C}_{r,r/K}(y)$ then we have $$\label{22}
u(x)\geq 2\alpha K^{-\beta}.$$
Let us now prove that there exists $N>0$ such that $$\label{23}
v(x)\leq e^{-NK}, \quad x \in \overline{\mathcal{C}_{r,r}(y)}.$$ Let $j=1,\ldots, \lfloor\frac{K-1}{2}\rfloor$ and let $x_j\in
\partial\mathcal{C}_{(2j-1)r,r}(y)$ be such that $$v(x_j)=\max\left\{v(x),\quad x \in
\overline{\mathcal{C}_{(2j-1)r,r}(y)}\right\}.$$ Let $\mathcal{U}_j=
B_{2r}(x_j)\cap {\mathcal{C}_{(2j+1)r,r}(y)}$ and $\tau_{\mathcal{U}_j}$ be the exit time from $\mathcal{U}_j$. By the same argument used in the proof of we see that $$\mathbb{P}_{x_j}\left[S(\tau_{\mathcal{U}_j})\in \mathcal{D}_{Kr,r}(y)
\right]\geq c>0.$$ Using (in particular, the fact that $v\leq0$ on $\partial\mathcal{C}_{Kr,r}(y)\cap \mathcal{D}_{Kr,r}(y)$) we deduce then that $$v(x_j)\leq\theta\max_{
\overline{\mathcal{U}_j}}v,$$ where $0<\theta<1$. Hence $$\max\left\{ v(x),\quad x\in
\overline{\mathcal{C}_{(2j-1)r,r}(y)}\right\}\leq\theta \max\left\{
v(x),\quad x\in \overline{\mathcal{C}_{(2j+1)r,r}(y)}\right\}.$$ Iterating this estimate we obtain $$\max\left\{ v(x),\quad x\in
\overline{\mathcal{C}_{r,r}(y)}\right\}\leq\theta^{\lfloor\frac{K-1}{2}\rfloor} \max\left\{ v(x),\quad
x\in \overline{\mathcal{C}_{Kr,r}(y)}\right\}\leq e^{-NK},$$ which proves . It follows from that $$\label{231}
v\leq\alpha
K^{-\beta}\quad \mbox{in}\quad \overline{\mathcal{C}_{r,r}(y)}$$ provided that $K$ is large enough.
From the previous considerations it follows that $$u_1=\frac{K^\beta}{2\alpha}u\geq 0 \quad \mbox{ in}\quad
\overline{\mathcal{C}_{r,r/K}(y)}$$ with $$u_1\geq1 \quad
\mbox{in}\quad \partial
\mathcal{C}_{r,r/K}(y)\cap \mathcal{D}_{r,r/K}(y)$$ thanks to and, thanks to , $$v_1=\frac{K^\beta}{2\alpha}(2v-u)\leq \frac{K^\beta}{\alpha}v\leq 1
\quad \mbox{in}\quad\overline{\mathcal{C}_{r,r/K}(y)},$$ with $$v_1\leq 0 \quad \mbox{ on}\quad \partial\mathcal{C}_{r,r/K}(y)\cap\mathcal{D}_{r,r/K}(y).$$ In particular, we have $$u_1-v_1= \frac{K^\beta}{\alpha}(u-v) \geq 0 \quad \mbox{ on}\quad
\overline{\mathcal{C}_{r,r}(y)\,\backslash\,\mathcal{C}_{r,r/K}(y)}.$$ It follows that $u_1$, $v_1$ satisfy the same assumptions as $u$, $v$ with $r$ replaced by $r/K$. We can then iterate and define $u_i$, $v_i$ such that $$u_i-v_i=\left(\frac{K^\beta}{\alpha}\right)^i(u-v) \geq 0
\quad \mbox{ on}\quad \overline{\mathcal{C}_{r/K^i,r/K^i}(y)\,\backslash\, \mathcal{C}_{r/K^i,r/K^{i+1}}(y)}$$ $i=1,2\ldots$. We deduce then that $$\label{24}
u-v\geq 0 \quad \mbox{ on}\quad
S(y)=\bigcup_{i\geq
0}\overline{\mathcal{C}_{r/K^i,r/K^i}(y)\,\backslash\,
\mathcal{C}_{r/K^i,r/K^{i+1}}(y)}.$$ Let $x\in
B_r(y)$ and $\tilde{x}\in\partial\mathcal{C}$ satisfying $\delta(x)=|x-\tilde{x}|$. Then $$\mathcal{C}_{Kr,r}(\tilde{x})\subset
\mathcal{C}_{(K+2)r,r}(y).$$ Replacing $K$ by $K+2$ in the previous considerations we deduce that $u\geq v$ on $S(\tilde{x})$ that contains $x$. This shows that $u(x)\geq v(x)$ and completes the proof of . $\hfill \Box$\
\
[*Proof of Theorem \[thm1\].*]{} The proof is the same as [@Bou]. We observe that instead of Carleson estimate, we can simply use the estimate $$u(\xi)\leq C\, e^{C|\xi-\zeta|}u(\zeta),\quad \xi,\zeta\in \mathcal{C},$$ which follows from uniform ellipticity. The advantage of this estimate is that it works for all connected infinite domains, and not just for domains satisfying Carleson estimate. It is already enough to run the diagonal process argument used in [@Bou].
As in [@Bou] the uniqueness can be deduced by essentially the same method as in [@Ai Proof of Theorem 3] and [@An Lemma 6.2]. $\hfill\Box$
[00]{} H. *Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain,* J. Math. Soc. Japan, **53** (2001), 119-145. A. Ancona, *Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien*, Ann. Inst. Fourier (Grenoble) 28 (1978) 169-213. M. T. Barlow and R. F. Bass, *Stability of parabolic Harnack inequalities,* Trans. Amer. Math. Soc. **356** (2004) 1501-1533. R. F. Bass and K. Burdzy, *The boundary Harnack principle for nondivergence form elliptic operators,* J. London Math. Soc. **50** (1994) 157-169. P. E. Bauman, *Positive solutions of elliptic equations in nondivergence form and their adjoints,* Ark. Math. **22** (1984) 536-565. N. Ben Salem, S. Mustapha and M. Sifi, *Potential Theoretic Tools and Randon Walks*, ESAIM: Proccedings and Surveys, to appear. P. Biane, *Quantum random walk on the dual of $SU(n)$*, Probab. Theory and Related Fields **89** (1991) 117-129. A. Bouaziz, S. Mustapha and M. Sifi, *Discrete harmonic functions on an orthant in $\mathbb{Z}^{d}$*, Electron. Commun. Probab. **20** (2015) 1-13. M. Bousquet-Mélou and M. Mishna, *Walks with small steps in the quarter plane*, Contemp. Math. **520** (2010) 1-40. L. Carleson, *On the existence of boundary values for harmonic functions in several variables,* Ark. Mat. **4** (1962) 393-399. D. Denisov and V. Wachtel, *Random Walks in Cones*, Ann. Probab. **43** (2015) 992-1044. D. Denisov and V. Wachtel, *Alternative constructions of a harmonic function for a random walk in a cone*, Preprint 2018, arXiv:1805.01437. E. B. Dynkin, *The boundary theory of Markov processes (discrete case)*, Uspehi Mat. Nauk **24** (1969) 3-42. J. Duraj and V. Wachtel, *Green function of a random walk in a cone*, Preprint 2018, arXiv:1807.07360. E. B. Fabes and M. V. Safonov, *Behavior near the boundary of positive solutions of second order parabolic equations,* J. Fourier Anal. Appl. **3** (1997) 871-882. E. B. Fabes, M. V. Safonov and Y. Yuan, *Behavior near the boundary of positive solutions of second order parabolic equations, II*, Trans. Amer. Math. Soc. **351** (1999) 4947-4961. G. Fayolle, R. Iasnogorodski and V. Malyshev, *Random walks in the quarter-plane*, Springer-Verlag, Berlin, 1999. G. Fayolle and K. Rashel, *Random walks in the quarter plane with zero drift: an explicit criterion for the finiteness of the associated group*, Markov Processes and Related Fields **17** (2011) 619-636. P. Gyrya and L. Saloff-Coste, *Neumann and Dirichelet Heat Kernels in Inner Uniform Domains,* Volume **336**, Astérisque, 2011. I. Ignatiouk-Robert, *Martin boundary of a killed random walk on a half-space,* J. Theoret. Probab. **21** (2008) 35-68. I. Ignatiouk-Robert, *Harmonic functions of random walks in a semigroup via ladder heights*, Preprint 2018, arXiv:1803.05682. I. Ignatiouk-Robert and C. Lorée, *Martin boundary of a killed random walk on a quadrant*, Ann. Probab. **38** (2010) 1106-1142. H. J. Kuo and N. S. Trudinger, *Linear differential elliptic difference inequalities with random coefficients,* Math. Comp. 55 (1990), no. 191, 37–53. Kuo, Hung-Ju; Trudinger, Neil S, *Positive difference operators on general meshes,* Duke Math. J. 83 (1996), no. 2, 415-433. I. Kurkova and K. Raschel, *Random walks in $\mathbb{Z}_+^2$ with non-zero drift absorbed at the axes,* Bulletin de la Société Mathématique de France 139 (2011), 341-387. G. F. Lawler, *Estimates for differences and Harnack inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous, increments*, Proc. London Math. Soc. **63** (1991) 552-568. R. S. Martin, *Minimal positive harmonic functions*, Trans. Amer. Math. Soc. **49** (1941) 137-172. S. Mustapha, *Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments*, Bernoulli **13** (2007) 131-147. Ney, P. and Spitzer, F. (1966). *The Martin boundary for random walk*, Trans. Amer. Math. Soc. **121** 116-132. K. Raschel, *Green functions for killed random walks in the Weyl chamber of ${\rm Sp}(4)$*, Ann. Inst. Henri Poincaré Probab. Stat. **47** (2011) 1001-1019. K. Raschel, *Random walks in the quarter plane, discrete harmonic functions and conformal mappings, with an appendix by Sandro Franceschi*, Stochastic Processes and their Applications **124** (2014) 3147-3178. K. Raschel and P. Tarrago, *Martin boundary of random walks in convex cones*, Preprint 2018, arXiv:1803.09253.
[^1]: 2010 *Mathematics Subject Classification*: Primary 60G50, 31C35; Secondary 60G40, 30F10.
[^2]: *Key words and phrases*: Random walk in Lipchitz domain, Discrete harmonic function, Martin boundary.
|
=-1cm
$^{(1)}$ [ CERN-CH-1211 Geneva 23, Switzerland\
and\
Universidad Autónoma de Madrid\
Cantoblanco, Madrid 28049, Spain\
]{} $^{(2)}$ [ Laboratorie de Physique Theorique et Hautes Energies\
Université de Paris-sud, Bat. 211, Orsay 91405, France]{}
ABSTRACT
Low energy effects of generic extensions of the Standard Model can be comprehensively parametrized in terms of higher dimensional effective operators. After the success of all the recent precission tests on the Standard Model, we argue that any sensible description of these extensions at the Z-scale must be stable under higher order quantum corrections. The imposition of $SU(2)_L \times U(1)_Y$ gauge invariance seems to be the simplest and most natural way to fulfill this requirement. With this assumption, all the possible deviations from the standard triple gauge boson vertices can be consistently parametrized in terms of a finite set of gauge invariant operators. We deal here with those operators that do not give any tree level effect on present experimental observables and constrain them by computing their effects at the one-loop level. We conclude that for a light Higgs boson, the direct measurement at LEP200 can improve present bounds on these “blind directions”, while for a heavy Higgs it is most unlikely to provide any new information.
Introduction {#intro}
============
Half a dozen of the Standard Model (SM) predictions are already tested to first order in their quantum corrections[^1]. For these quantum corrections to be finite, subtle cancellations occur, which require the algebraic relations imposed by the theory’s gauge invariance on the various couplings to be exactly satisfied. In this situation, the natural hypothesis is to assume that theories beyond the SM must be such that they generate an effective theory, which preserves at least the gauge symmetry $SU(2) \times U(1)$ over the symmetry breaking scale $v$. Other assumptions need to prove their “quantum consistency”. As emphasized in [@deru1; @deru3], this has often been overlooked in the literature, leading to overly optimistic expectations concerning the “new physics” sensitivity of future machines.
The leading low energy effects of a generic “meta-theory” at a scale $\Lambda \,(\geq v)$ can be comprehensively and systematically parametrized in terms of a linear combination of higher dimensional operators, constructed out of the light fields. If the low energy limit of this “meta-theory” has the symmetries and light fields of the minimal SM (it takes care of its own ultraviolet divergences), the higher dimensional operators must also preserve those symmetries. Besides, even if the low energy effective theory at the Z-scale is not the renormalizable linear SM (this is for instance the case of a theory with a strongly interacting symmetry breaking sector), the successes of the minimal SM still imply that $\Lambda$ cannot be independent of $v$ and that the pattern of the symmetry breaking must be $SU(2)_L \times SU(2)_C
\rightarrow SU(2)$, at least[^2] to a precission of 1%.
In [@deru1] the effects of “new-physics” on the structure and strength of the triple gauge-boson vertices (TGVs) were systematically analysed in an effective Lagrangian parametrization. The basis of this method is discussed in sections 2 and 3. It was found that all the operators affecting the TGVs can be expressed as a linear combination of six independent ones, which were chosen to be those giving tree level effects on present observables. All the rest can be expressed in terms of these by means of the equations of motion. However, some combinations of the basis operators are such that their tree level effects on present observables exactly cancel. These are the so called “blind directions”. Even when there is no known symmetry or dynamical reason why a meta-theory would be so contrived as to generate a low effective Lagrangian pointing exclusively in these directions, in order to leave out any theoretical prejudice, we do not exclude this possibility. In this paper, we constrain them directly from present data through their one-loop effects. In section 4, we present the results of our computation, which has also been partially carried out in [@chori]. In section 5, we compare LEP-1 and LEP-2 sensitivities. We conclude in section 6 that the better chances for LEP-2 concerning an anomalous correction to the triple gauge boson vertices in these directions, come from a relatively light Higgs (as expected also for many other reasons), while for a heavy Higgs LEP-1 constraints are already considerably better.
Effective Lagrangian Parametrization
=====================================
Recently, there has been some controversy in relation with the “uses and abuses” of Effective Lagrangian parametrizations in the search for non- standard effects in present and future experiments [@reac]. Questions like what are the symmetries one must impose on an Effective Lagrangian or whether the latter can be used in higher orders of perturbation theory without loosing their predictive power, seem to be misunderstood even though the principles of this approach were settled long ago [@fermi; @classics]. We briefly review the ideas behind the Effective Lagrangian parametrization of New Physics and how they can consistently be used in higher orders of perturbation theory.
The Operator Product Expansion [@wilson] is an example of the factorization property of renormalizable theories, by which, at a given energy scale, the effects on physical observables of the much higher energy modes, can be factorized in the effective couplings of the light ones. In any renormalizable theory in which the typical scale of some fields (given for example by their mass) becomes large, the effect of this heavy sector on Feynman’s amplitudes between light states can be expanded in perturbation theory as a sum of local operators of the light fields (multiplied by the appropiate dimensional coefficients which depend on the heavy scale)[^3].
Suppose that at very high energy we have a renormalizable theory in which the masses of some fields are much higher than the energy of our experiment (if the masses are generated by spontaneous symmetry breaking we assume that the vev for that sector is large). If the light sector is symmetric under a given gauge subgroup (then it is renormalizable), obviously the full theory must also be symmetric and necessarily the operators of the large mass expansion also preserve this gauge symmetry[^4]. In neglecting the contributions vanishing as $\Lambda
\rightarrow \infty$, we are left with local operators of light fields of dimension $d\leq4$, whose coefficients may contain positive powers of $\Lambda$ times logarithms. However, as long as they preserve the simmetries of the light sector, they just produce a finite renormalization of the light couplings and all the non-vanishing dependence on $\Lambda$ is physically unobservable. The heavy fields decouple [@appelcar]. This is in fact the case in many physically interesting situations where the new physics comes from extended gauge groups or larger symmetries. We will refer to this situation as the decoupling case.
An effective Lagrangian parametrization is natural here, because the renormalizability of the light theory provides a power counting which controls the non-renormalizable interactions coming from the heavy sector. If we want to consider the effects up to some given order in 1/$\Lambda$, we can truncate the expansion at that order and the theory so obtained takes care of its own ultraviolet divergences [@polchi].
Thus, in parametrizing as generally as possible the leading effects of any decoupling new physics, we will consider a combination of light operators of the smallest possible dimension greater than 4,
which preserve the gauge symmetries of the light sector:
$$\begin{aligned}
{\cal L}_{eff} = {\cal L}_{light} + \sum_{j} {\alpha}_{j}
\frac {O_j^{d_j}}{\Lambda^{d_j-4}}.\end{aligned}$$
This is the most general large mass expansion of a renormalizable meta-theory whose low energy limit is ${\cal L}_{light}$. The quantum corrections of this lagrangian are well defined order by order in $1/\Lambda$, which implies, in particular, that there is no physical divergence coming out at the one loop level.
On the contrary, if the light theory is non-renormalizable, there must be some physical cutoff related to the heavy scale. The effects of the heavy fields cannot decouple, as they must compensate the tower of divergent light counterterms needed to make the full theory finite. Even though these effects can also be written as a sum of local operators of the light fields, this is in general of no use since there are now contributions with positive powers of $\Lambda$ that can not be renormalized in ${\cal L}_{light}$ and, a priori, it is not clear that there is any useful expansion to sum the leading effects in $\Lambda$, in terms of a finite set of couplings. The quantum corrections in this case contain non-renormalizable divergences and, contrary to what is claimed in [@london], this has a clear physical meaning: the cutoff can not be sent to infinity, furthermore, there must be some natural relation between the light and heavy scales. One example of the fact that these divergences are physical is the Minimal Standard Model. Suppose that an elementary Higgs do exist, but its mass is larger than $M_Z$ and $M_W$. Then, at LEP energies, we can integrate it out and work in the resulting effective theory, which is a non-linear sigma model. The non-renormalizable divergences that appear in the perturbative computation of quantum corrections in the effective theory, turn out to be the corrections in $M_H$ computed in the full renormalizable theory [@appel], they do not disappear! In fact, it is through these divergences that one is able to estimate the natural scale of the couplings in the effective theory (contrary to the case of a decoupling sector, for which there is no naturality argument to estimate $\Lambda$ and it can only be determined experimentally).
Anomalous Triple Gauge Boson Vertices
======================================
The literature abounds in estimates of the LEP-2 sensitivity to the structure of the gauge boson vertex, based on a general Lorentz invariant parametrization of that vertex which does not preserve the gauge symmetry [@chorisa]. As we have seen, the effect of these new interactions is to break the renormalizability of the light theory (as the gauge symmetry is lost) and consequently its predictivity. This leads to very optimistic expectations concerning the “new physics” sensitivity of future machines, but makes it difficult to understand how it can be that half a dozen of the predictions of the SM are already succesfully tested to first order in their quantum corrections [@deru2].
In [@deru1] a general Effective Lagrangian parametrization of novel effects in the structure of the TGVs was considered. Two main possibilities were analysed:
1\) The case of Decoupling type of New Physics. In this case the hypothesis is that the Minimal Standard Model is the correct theory to describe physical phenomena at the Z scale. Novel effects would then come from extra particles, larger gauge symmetries, compositeness... characterized by a mass scale $\Lambda$ distinct from the scale $v$. In particular, this means that the unspecified heavy objects are assumed not to acquire their masses from the standard machinery of symmetry breaking, though they may well be involved in the mechanisms that trigger it. The leading effects were shown to come from d=6 operators, which by the previous reasoning must be SU(2)xU(1) invariant. In [@deru1], the basis operators and the blind direction $O_W \equiv i \vec{W^{\nu}_{\mu}}
\times \vec{W^{\lambda}_{\nu}} \cdot \vec{W^{\mu}_{\lambda}}$ were studied in detail. Here we deal with the remaining blind-directions. In the decoupling case, we have two more of them:
$$\begin{aligned}
O_{B\Phi} & \equiv & i B^{\mu\nu} {( D_{\mu} \Phi )}^{\dag}
D_{\nu} \Phi,\\
O_{W\Phi} & \equiv & i \vec{W^{\mu\nu}} {( D_{\mu} \Phi )}^{\dag}
\vec{\sigma} D_{\nu} \Phi.\end{aligned}$$
where $W^a_{\mu\nu} \equiv \partial_{\mu} W_{\nu}^a -
\partial_{\nu} W_{\mu}^a - g \epsilon_{abc} W_{\mu}^b W_{\nu}^c$; and $B_{\mu\nu} \equiv \partial_{\mu} B_{\nu} - \partial_{\nu}
B_{\mu}$.
2\) The case of a strongly interacting symmetry breaking sector does not fit in the previous picture, because the light theory is in this case a non-linear sigma model which is not perturbatively renormalizable. We assume that the interactions responsible for the generation of intermediate vector boson masses have a global $SU(2)_L \times SU(2)_C$ symmetry, with $SU(2)_C$ the accidental custodial symmetry [@silvi] of the standard potential of scalar doublets. This is the only natural situation given the experimental fact that $\rho
\simeq 1$ to a precision of a percent. Without any further assumption, the most general Lagrangian is that of a nonlinearly realized $SU(2)_L
\times SU(2)_C$ which breaks to $SU(2)$. After switching on the gauge fields, this implies the $SU(2)_L \times U(1)$ gauge symmetry [@coli].
As shown by Weinberg [@wein], a loop expansion in this theory is equivalent to a momentum expansion, so that only a finite number of operators are needed to describe the physical phenomena at low energies (this number increasing with the order in the momentum expansion). There is a natural dimension-full parameter which suppresses these non-renormalizable terms ($\Lambda = 4 \pi v$) [@georgi] and the effective Lagrangian is just a Taylor expansion in $U$ , $\frac{D}{\Lambda}$, $\frac{W_{\mu\nu}}{\Lambda^2}$ and $\frac{B_{\mu\nu}}{\Lambda^2}$, where $U$ is the unitary matrix describing the longitudinal gauge-boson degrees of freedom, transforming as a $(2,2)$ of $SU(2)_L \times SU(2)_C$. The covariant derivative acting on $U$ is the usual one:
$$\begin{aligned}
D_{\mu} U \equiv \partial_{\mu} U + i g \frac{\vec{\sigma}}{2}
\vec{W_{\mu}} U - i g U \frac{\sigma_3}{2} B_{\mu}.\end{aligned}$$
It is important to remark that in the gauge sector, the loop expansion is not necessarily a low-energy expansion.
The leading non-standard effects on TGVs come from operators of $d_{\chi}$=4. After reducing the list with the use of the equations of motion, we are left [@deru1] with three independent ones and three blind directions. Here, we will consider the effect of these blind directions[^5]
$$\begin{aligned}
{\cal L}_2 & \equiv & i g' \beta_2 B^{\mu\nu} Tr \{ T [ (D_{\mu}U)
U^{\dag},(D_{\nu}U)U^{\dag}] \},\\
{\cal L}_3 & \equiv & i g \beta_3 Tr \{ \vec{W_{\mu\nu}} \vec{\sigma}
[(D_{\mu}U) U^{\dag},(D_{\nu}U)U^{\dag}] \},\\
{\cal L}_9 & \equiv & i g \beta_9 Tr \{ T W^{\mu\nu}\} Tr \{
T [ (D_{\mu}U)U^{\dag},(D_{\nu}U)U^{\dag}] \}.\end{aligned}$$
where $T \equiv U \sigma_3 U^{\dag}$. Those operators containing $T$ are not custodial preserving. Even when we assume that the symmetry breaking sector is symmetric under $SU(2)_C$, the coupling to hypercharge breaks this symmetry and operators containing $T$ can appear, although they always have a factor $g'$.
It turns out that in practice, the effects of these operators are proportional to the effects of equivalent operators in the linear realization with the identification $M_H \sim \Lambda$ (the linear model with a Higgs scalar is a regulator for the nonlinear model [@appel]):
$$\begin{aligned}
{\cal L}_2 & \Rightarrow & \frac{8 g' \beta_2}{v^2} O_{B\Phi}{|}_{(\Phi
\rightarrow v)},\\
{\cal L}_3 & \Rightarrow & \frac{8 g \beta_3}{v^2} O_{W\Phi}{|}_{(\Phi
\rightarrow v)},\\
{\cal L}_9 & \Rightarrow & \frac{-32 i g \beta_9}{v^4} ( \Phi^{\dag} W^{\mu\nu}
\Phi)(D_\mu \Phi)^{\dag}D_\nu\Phi {|}_{(\Phi \rightarrow v)}.\end{aligned}$$
We see that the first two operators on the right-hand side are two of the three blind directions of the decoupling case, while the third one is an operator of d=8, which was not leading in that case. This shows the different power counting of the two parametrizations.
In fact, at any given dimension, one expects more operators in the non-decoupling case. If we consider for example the Standard Model with a higgs in the intermediate region (not so heavy as to render the theory non-renormalizable, but heavier than the energies of LEP-1 and LEP-2 experiments), we can work in both the linear and non-linear realizations. When we integrate the higgs out, all the contributions that appear in the linear case as finite corrections in $M_H$, should appear after its integration as new effective operators. The linear treatment would be more adequate then, because there are fewer degrees of freedom.
Present Data Constraints
========================
Following [@deru1], we will use $\alpha$, $G_F$ and $M_z$ as the input parameters for the minimal Standard Model, because they are the most accurately measured quantities. Then, for constraining the previous operators from present data we will use as observables the W mass, the leptonic and hadronic widths of the Z, the forward-backward asymmetry in leptonic Z decays, the $\tau$ polarization $P_\tau$ and the ratio of inclusive neutral to charged-current neutrino cross sections on aproximately isoscalar targets $R_\nu$:
$$\begin{aligned}
M_W & = & 80.13 \pm 0.31 GeV \cite{mw} \nonumber \\
\Gamma_l & = & 83.52 \pm 0.33 MeV \cite{nash} \nonumber \\
\Gamma_h & = & 1742 \pm 8 MeV \cite{nash} \nonumber \\
A_{FB}^l & = & 0.0157 \pm 0.003 \cite{nash} \nonumber \\
P_\tau & = & -.140 \pm 0.024 \cite{nash} \nonumber \\
R_\nu & = & 0.308 \pm 0.002 \cite{rnu}\end{aligned}$$
To the experimental error in $R_\nu$ we have added a theoretical error of $1 \%$ to reflect the theoretical uncertainty associated with the charm threshold in the charged currents, discussed and estimated in [@26]. We adopt the safe recipe of adding linearly the theoretical uncertainties to the experimental errors. We shall choose to perform our analysis at the 2 $\sigma$ level, the inputs to our limits on novel effects are then twice the quoted errors.
None of these observables are affected at tree level by the insertion of blind operators, their effects start at the one-loop level. Our calculation has been done in a generic $\xi$-gauge, and we have been able to check the cancellation of the $\xi$-dependence in physical observables. We have used two regularization procedures, dimensional regularization and a simple momentum cutoff. Both give the same results with the identification[^6] of quadratic divergences with poles at $d=2$ and logarithmic ones with poles at $d=4$. As discussed before, in the decoupling case all divergences but logarithms must get renormalized in the couplings of the SM, so they must also cancel in physical observables. The coefficients of the logarithmic divergences are the coefficients of the renormalization group logarithms that appear in the running of the effective Lagrangian coefficients from the scale $\Lambda$ to the Z-scale. Our computation is valid up to these logarithms, not including constant terms. In the nonlinear case, however, the leading contribution comes from quadratic divergences (poles in $d=2$) that do not cancel. These clearly correspond to the terms in $M_H^2$ of the linear case. There are also some leading contributions coming from finite terms in $M_H^2$.
We express the effect of the insertion of blind operators in terms of the following dimensionless parameters:
$$\begin{aligned}
{\delta}_{B \Phi} & \equiv & \frac {g s}{c} \frac{{M_W}^2}
{(4 \pi)^2 \Lambda^2}
{\alpha}_{B \Phi}, \\
{\delta}_{W{\Phi}} & \equiv & \frac {g s^2}{c^2} \frac {{M_W}^2}
{(4\pi)^2{\Lambda}^2}
{\alpha}_{W\Phi} , \\
\delta_9 & \equiv & -8 \frac{s^2}{c^2} \frac{g^4 \beta_9}{(4 \pi)^2}.\end{aligned}$$
where, from now on, $s=\sin {\theta_W}$ and $c=\cos {\theta_W}$.
The quantum effects of these operators appear either as boson self-energies $\Pi^{\gamma\gamma}$, $\Pi^{\gamma Z}$, $\Pi^{ZZ}$ and $\Pi^{WW}$, or as corrections to the $Z f f$ vertex ($\delta c_L^f$) and the $W l \nu$ vertex ($\delta g_{Wl\nu}$). We collect all this quantum corrections in the appendix A. In terms of these objects, the shifts induced in the renormalization parameters are:
$$\begin{aligned}
\frac {\Delta\alpha}{\alpha} & = & \frac{{\Pi}_{\gamma\gamma}}
{q^2} {|}_{q^2 = 0} ,\\
\frac {\Delta{M_Z}^2}{{M_Z}^2} & = & \frac{{\Pi}_{ZZ}(q^2)}{{M_Z}^2}
{|}_{q^2 = {M_Z}^2} ,\\
\frac {\Delta G_F}{G_F} & = & [\frac{2\delta g_{Wl\nu}}{g_{Wl\nu}}
- \frac {{\Pi}_{WW}(q^2)}{{M_W}^2} ] {|}_{q^2 = 0}.\end{aligned}$$
All the physical observables can now be expressed in terms of the preceeding objects.
We parametrize the shifts in the widths in terms of $\delta \gamma_f$, $\delta \kappa_f$, which also contain the main contribution from the standard radiative corrections [^7]:
$$\begin{aligned}
\delta{\gamma}_f & = & -\frac{\Delta G_F}{G_F}- \frac{\Delta {M_Z}^2}
{{M_Z}^2} + Re( \frac{{\Pi}_{ZZ}(q^2)-{\Pi}_{ZZ}({M_Z}^2)}{q^2-{M_Z}^2})
+ 2 \frac{ \delta {c_L}^f}{({c_L}^f-{c_R}^f)},\end{aligned}$$
$$\begin{aligned}
\delta{\kappa}_f & = & - \frac{c^2}{(c^2-s^2)}(\frac{\Delta\alpha}
{\alpha}-\frac{\Delta{M_Z}^2}{{M_Z}^2}-\frac{\Delta G_F}{G_F})
- \frac {c}{s} \frac{Re({\Pi}_{\gamma Z} (q^2))}{q^2}
- \frac{ \delta {c_L}^f}{({c_L}^f-{c_R}^f)}.\end{aligned}$$
The observables we will use to constrain the blind operators are:
$$\begin{aligned}
\frac {\delta {M_W}^2}{{M_W}^2} & = & \frac{{\Pi}_{WW}({M_W}^2)}{{M_W}^2}
- \frac{\Delta {M_Z}^2}{{M_Z}^2}
+ \frac{s^2}{(c^2-s^2)}(\frac{\Delta\alpha}
{\alpha}-\frac{\Delta {M_Z}^2}{{M_Z}^2}-\frac{\Delta G_F}{G_F}),\end{aligned}$$
$$\begin{aligned}
\frac{\delta{\Gamma}_l}{\Gamma_l} & = & \delta{\gamma}_l(M_Z^2)-0.250
\delta{\kappa}_l (M_Z^2),\end{aligned}$$
$$\begin{aligned}
\frac{\delta{\Gamma}_h}{\Gamma_h} & = & \delta{\gamma}_q (M_Z^2)-0.318
\delta{\kappa}_q (M_Z^2),\end{aligned}$$
$$\begin{aligned}
\frac{\delta {A_{FB}^l}}{A_{FB}^l} & = & 4 \frac{s^2 \delta \kappa^l}
{g_v} \;
\frac{{g_a}^2-g_v^2}{g_a^2+g_v^2},\end{aligned}$$
$$\begin{aligned}
\frac{\delta P_{\tau}}{P_{\tau}} & = & -2 \; \frac{s^2
\delta \kappa^l}{g_v} \; \frac{(g_a - g_v)^2}{g_v ^2 + g_a ^2},\end{aligned}$$
$$\begin{aligned}
\frac{\delta R_{\nu}}{R_{\nu}} & = & \delta \gamma_q (0) + \delta
\gamma_\nu (0) + 2
[ \frac{ \delta \kappa_u(0)(c_L^u c_R^u + \frac{1}{3} {c_R^u}^2)
+ \delta \kappa_d(0) (c_L^d c_R^d + \frac{1}{3} {c_R^d}^2) }{
{c_L^u}^2 +{c_L^d}^2 +\frac{1}{3} ( {c_R^u}^2 + {c_R^d}^2)}].\end{aligned}$$
The standard one-loop contributions to these observables depend sensitively on $m_t$ and weakly on $M_H$. In [@deru1] the only blind direction considered did not give any extra effect on $M_H$ and the uncertainty due to the dependence on this parameter was summed linearly with the experimental error. Our operators, on the contrary, have a strong dependence on $M_H$ at the one loop level, so we will consider separately the cases of a light Higgs ($M_H = 50
GeV$) and a heavy Higgs ($M_H \sim 1 TeV$).
We assume no acccidental cancellations between the contributions of the various operators to the different observables and extract from existing experiments the combined allowed domains in the ($m_t$,${\delta}_i$) planes, whose projections are the 95.5 % confidence level (2 $\sigma$) intervals on the individual variables, for the limiting values of $M_H$. The bounds we find are weaker for a light Higgs and become more restrictive as $M_H$ grows, due to the quadratic dependence on $M_H$ of the loop effects. For $O_{B\Phi}$, the constraints become more restrictive as $M_H$ increases in the whole region from $M_H = 50$ GeV to 1 TeV, while for $O_{W\Phi}$ this behaviour starts only after $M_H \simeq 260$ GeV. In the region between 50 and 260 GeV the constraints are weaker for a heavier Higgs due to cancellations between terms in $M_H^2$ and the other quantum corrections.
- [Case of a light Higgs boson ]{}
In the following table we show the $2 \sigma$ contraints on $\delta_{B\Phi}$ and $\delta_{W\Phi}$, as a function of $M_H$ for any value of $m_t$:
$M_H = 50 GeV$ $-1.5 \cdot {10}^{-4} \leq \delta_{B\Phi} \leq 3.7 \cdot {10}^{-4}$ $-1.8 \cdot {10}^{-4} \leq \delta_{W\Phi} \leq 3.8 \cdot 10^{-4}$
----------------- --------------------------------------------------------------------- ----------------------------------------------------------------------
$M_H = 100 GeV$ $-1.6 \cdot {10}^{-4} \leq \delta_{B\Phi} \leq 3.0 \cdot {10}^{-4}$ $-2.0 \cdot {10}^{-4} \leq \delta_{W\Phi} \leq 4.6 \cdot {10}^{-4}$
$M_H = 260 GeV$ $-8.0 \cdot {10}^{-5} \leq \delta_{B\Phi} \leq 1.8 \cdot {10}^{-4}$ $-2.8 \cdot {10}^{-4} \leq \delta_{W\Phi} \leq 1.38 \cdot {10}^{-3}$
$M_H = 500 GeV$ $-4.0 \cdot {10}^{-5} \leq \delta_{B\Phi} \leq 1.1 \cdot {10}^{-4}$ $-1.4 \cdot {10}^{-4} \leq \delta_{W\Phi} \leq 4.9 \cdot {10}^{-4}$
These blind operators also generate new couplings involving scalars not present at tree level in the standard model, like the $Z H_0 \gamma$ vertex. We will translate the present experimental limits on the decay $Z \rightarrow H_0 \gamma$ into new constraints on the $\delta 's$ for a light Higgs ($M_H \leq M_Z$). Let $A= A_0 + A_{B \Phi} +A_{W \Phi}$ be the amplitude for $Z \rightarrow H_0\gamma$ decay. The corresponding width is
$$\begin{aligned}
\Gamma(Z \rightarrow H_0 \gamma) = |A|^2 \frac{E^3_{\gamma}}{12 \pi}.\end{aligned}$$
The standard $A_0$ is dominated by the triangle graph with intermediate W’s and is equal to [@50] :
$$\begin{aligned}
A_0 \simeq - \frac{e \alpha}{4 \pi sin^2 \theta M_W} [ 4.56+0.25
(\frac{M_H}{M_W})^2],\end{aligned}$$
practically independent of $M_H$ for $M_H \leq M_Z$. With the same normalization:
$$\begin{aligned}
A_{B\Phi} = - \frac{\alpha_{B\Phi} M_W}{\Lambda^2} ,\\
A_{W\Phi} = \frac{\alpha_{W\Phi} M_W}{\Lambda^2}.\end{aligned}$$
The ratio $\Gamma(Z \rightarrow H_0\gamma)/\Gamma_0(Z \rightarrow H_0\gamma
)$ varies from $\sim 0$ to $\sim 4.$ in the interval $|\delta_{B(W)\Phi}|
\leq 3.8 \cdot 10^{-5}$, indicating a strong sensitivity to the new physics. The trouble is that current bounds on the branching ratio B for $Z \rightarrow H_0
\gamma$ are not overly restrictive. From L3 result [@L3] $B \leq 10^{-
3}$ (for $48 \leq M_H \leq 86 GeV$) we get the following constraints for $M_H = 50 GeV$:
$$\begin{aligned}
|\delta_{B\Phi}| \leq 2.4 \cdot 10^{-4},\\
|\delta_{W\Phi}| \leq 2.4 \cdot 10 ^{-4}.\end{aligned}$$
It is remarkable that these constraints, which are completely independent of the preceeding ones, are of the same order of magnitude and do not depend on $m_t$. Also the dependence on $M_H$ is weaker (of course, only inside the range $M_H \leq M_Z$).
- [Case of a heavy Higgs ($M_H \sim 3 TeV$)]{}
In the case of a heavy Higgs, we take $\Lambda \simeq M_H \simeq 4 \pi v
\; (\sim 3 TeV)$, which corresponds to the natural situation in the non-linear case. For any value of $m_t$:
$$\begin{aligned}
-9.4 \cdot 10^{-3} & \leq \beta_2 \leq & 2.2 \cdot 10^{-2},\\
-1.5 \cdot 10^{-2} & \leq \beta_3 \leq & 3.9 \cdot 10^{-2},\\
-1.1 \cdot 10^{-2} & \leq \beta_9 \leq & 4.7 \cdot 10^{-3}.\end{aligned}$$
These latter constraints are much more restrictive due to the existence of quadratic divergences. Obviously, these leading contributions can be renormalized in the couplings of some other non-blind operators of the same dimension. However, for this chiral expansion to be natural, we expect that the divergent part coming from the loop contribution is (but for additional powers of the couplings $g$ or $g'$) of the same order of the renormalized coupling [@appel]. Thus, we constrain these chiral operators indirectly by constraining the counterterms they necessarily generate, which are non-blind to present observables. On naturality grounds, we expect these results to be a correct estimation of the order of magnitude. It is straightforward to check that the constraints on $\beta 's$ we have just derived are typically a factor $g^2$ worse than the constraints obtained in [@deru1] for non-blind operators.
In [@chori], only the situation $\delta_{B\Phi} = \delta_{W\Phi} = \delta$ was studied. This corresponds to the effect of the operator $L_9$ in Gasser-Leutwyler’s notation (GL). In order to compare our numerical results with theirs, we have also obtained the bounds form present data in this situation. We consider the case of a light Higgs ($M_H = 60 GeV$ and $\Lambda = 300 GeV$):
$$\begin{aligned}
-2.4 \cdot {10}^{-4} \leq \delta \leq 8.0 \cdot {10}^{-4},\end{aligned}$$
while their constraints translate into $$\begin{aligned}
-3.0 \cdot {10}^{-4} \leq \delta \leq 1.5 \cdot {10}^{-3}.\end{aligned}$$
We find similar results for other values of $M_H$ and $\Lambda$. To our understanding the differences between our results and those in [@chori] come from the fact that they use Altarelli-Barbieri $\epsilon's$ as present data constraints instead of directly measured observables, with the unavoidable propagation of errors. Besides, they partially lose the advantage of the different dependence on $m_t$ of the single observables.
In a recent reference [@vanderbij], it is claimed that there are quartic contributions to $\delta \rho$ for all the operators except those that break $SU(2)_C$ via a minimal coupling to hypercharge [@87]. These quartic divergences appear at next order in the coupling (they are $\sim {\delta}^2 $), as expected from simple power counting [@appel], and there is no physical reason to impose their cancellation on naturality grounds. Rather, if one considers at this order all the counterterms of lower chiral dimension, these quartic divergences disappear from physical observables. Besides, the minimal coupling to hypercharge is not even realized in the standard model with a heavy Higgs boson. Operators with non minimal coupling do appear [@appel] in this case, although they always contain a factor $g'$.
LEP200 Sensitivity
==================
Now, we turn to study the sentitivity of the future LEP-200 experiment to these operators. There, $\sqrt{s} \simeq 200GeV$ and the channel $e^{+}e_{-}
\rightarrow W^{+}W_{-}$ is opened, so any anomaly in the self-coupling of vector bosons will contribute at the tree level. We use the standard notation for the trilinear couplings:
$$\begin{aligned}
{\cal L}_0^{(3)}(V)= - i e g_V [( W_{\mu\nu}^{\dag} W^{\mu} -
W_{\mu\nu} W^{\dag \mu}) V^{\nu} + \kappa_V W_{\mu}^{\dag} W_{\nu}
V^{\mu\nu}] - i e g_V \frac{\lambda_V}{{M_W}^2}[V^{\mu\nu}
W^{\dag}_{\nu\rho} W^{\rho}_{\mu}],\end{aligned}$$
where $W_{\mu\nu} = \partial_{\mu} W_{\nu} - \partial_{\nu} W_{\mu}$ and V = $ \gamma$, Z. Blind operators produce tree level shifts on the couplings $\kappa_V$ and $g_V$.
A very sensitive direct test concerns the differential cross section $ d \sigma / d cos \theta_+$, with $\theta_+$ the $e^+ W^+$ scattering angle. The possible non-standard effects asssociated with our blind directions will come through shifts in the quantities $\kappa_Z ,\,\kappa_\gamma,\,g_Z,\,g_\gamma,\,M_W,\,c_L^e$ and $c_R^e$. For LEP-200 we will consider only the tree-level shifts and neglect the one-loop effects in $M_W$, $c_L^e$ and $c_R^e$ as well as the standard radiative corrections.
$$\begin{aligned}
\delta \kappa_\gamma & = & \lambda_{B\Phi} + \lambda_{W\Phi} -\frac{1}{2}
\lambda_9 ,\\
\delta \kappa_Z & = & - \frac{s^2}{c^2} ( \lambda_{B\Phi} + \lambda_{W\Phi} )
- \frac{1}{2} \lambda_9 ,\\
\delta g_\gamma & = & 0 ,\; \; \; \; \; \\
\delta g_Z & = & \frac{1}{c^2} \lambda_{W \Phi} g_Z.\end{aligned}$$
where we have defined:
$$\begin{aligned}
\lambda_{B \Phi} \equiv \frac{g c}{4 s}
\frac{v^2 \alpha_{B\Phi}}{\Lambda^2},\; \; \; \; \; \;
\lambda_{W \Phi} \equiv \frac{g}{4}
\frac{v^2 \alpha_{W \Phi}}{\Lambda^2},\; \; \; \; \; \; \;
\lambda_9 \equiv -8 g^2 \beta_9.\end{aligned}$$
We have Monte-Carlo generated $10^4$ W-pairs at $\sqrt{s} = 200 GeV$, which is a generous estimation of LEP2 statistics, and performed $\chi^2$ tests of significance of deviations from the standard differential cros-section at various values of $\lambda's$. The errors considered are only statistical.
In Figs.1, we show the biggest allowed domains on the planes $(m_t, \delta )$ ( at $M_H = 50GeV$ for $O_{B\Phi}$ and at $M_H = 260GeV$ for $O_{W\Phi}$) together with the expected LEP200 constraints, for $\Lambda = 1 TeV$. For the operator $O_{B\Phi}$, LEP200 will improve the present upper bound in a factor 2-3, at most. However, the sensitivity of LEP200 to $O_{W\Phi}$ is $\sim 5$ times better, and consequently there is an improvement of an order of magnitude with respect to present bounds in this case.
As we explained before, the worst constraints from present data on this operator are obtained for $M_H \sim 260 GeV$, where the allowed domain in the $(m_t, \delta_{W\Phi})$ plane extends to very large values of $m_t$. In fact, the upper limit on $\delta_{W\Phi}$ grows more than a factor 4 from the one obtained at $M_H = 50 GeV$, but it can only be saturated if $m_t$ turns out to be $\sim 350 GeV$. In this situation, the bound for $\delta_{W\Phi}$ from present data can be read from Fig. 1:
$$\begin{aligned}
-2.8\cdot 10^{-4} \leq \delta_{W\Phi} \leq 1.38 \cdot 10^{-3},\end{aligned}$$
to be compared with the expected sensitivity of LEP200, also shown in Fig. 1: $$\begin{aligned}
-3.7 \cdot 10^{-5} \leq \delta_{W\Phi} \leq 3.1 \cdot 10^{-5}.\end{aligned}$$
If $m_t$ is lighter than $200 GeV$, the present upper bound would go down to $\delta_{W\Phi} \leq 4.0 \cdot 10^{-4}$ and, for greater values, present data constrain $\delta_{W\Phi}$ to be in a band of width $\sim 4 \cdot 10^{-4}$ whose central value grows linearly with $m_t^2$. The allowed domain from LEP200 slightly intersects this region and, in this sense, future constraints will complement present ones and not supersede them (giving a new bound on the top mass in case LEP200 fails to detect a non-standard TGV).
The results for the case of a heavy Higgs are gathered in Figs. 2. They show the comparison between the $\chi^2$-test limits from LEP200 and present constraints for the three operators of (8)-(10). These results are very similar to that of non-blind operators [@deru1]. In particular, the bounds on $\beta_2 (\sim L_9^R)$ and $\beta_3 (\sim L_9^L)$ at LEP1, are much better than those that can be obtained from their tree level effects in LEP200 and CDF, although not competitive with those from SSC and LHC [@valencia][^8].
As we have explained in the previous section, we constrain these operators indirectly by constraining the counterterms they generate, which are non-blind to present observables. In other words, present constraints imply that the sensitivity of the experiments LEP200 and CDF to the TGVs will not be enough to measure a value of $L_9$ coupling of the order which is natural for a strongly interacting symmetry breaking sector [@espriu].
It is interesting to study whether a rise in energy to $E_{cm} = 500 GeV$ at NLC, will give much better constraints compared to LEP200’s. We have generated 25000 events at $E_b = 250 GeV$ (which corresponds to an integrated luminosity of $10 fb^{-1}$) and the bounds we obtain for the $\beta's$ are the following:
$$\begin{aligned}
-1.8 \cdot 10^{-2} \leq \beta_2 \leq 7.2 \cdot 10^{-3} \\
-7.2 \cdot 10^{-3} \leq \beta_3 \leq 5.5 \cdot 10^{-2}.\end{aligned}$$
NLC sensitivity to the blind directions is a factor $\sim 5$ better than that of LEP200, and competitive with present constraints in the case of a heavy Higgs. Obviously, the information obtained from direct measurement has less uncertainty than that obtained from the loop effects, so if nature has chosen any of these blind directions to “deform” the standard TGV’s, NLC will certainly improve our present knowledge.
Conclusions
===========
Our computation of the one-loop effects of the “blind operators” completes the analysis started in [@deru1] of the bounds on non-standard triple gauge boson vertices from present experimental data. We have argued, yet once again, the necessity of imposing the gauge symmetry on the effective Lagragian, for this approach to be stable under higher order perturbations. Present experiments are already sensitive to radiative corrections, consequently, any meaningful search for possible new physics by means of an effective Lagrangian parametrization must be such that these corrections are well defined.
We have considered two main possibilities for departures from the original version of the standard model. The first is that in which the effective theory at the Z-scale is correctly described by the minimal standard model with a relatively light Higgs and the new physics appears as larger symmetries or extended gauge groups. Secondly, we also considered the case of an spontaneous symmetry breaking sector involving some sort of strongly coupled dynamics, where the elementary scalars may play no role at all. If the Higgs is not found within the range up to $O(1 TeV)$, this possibility seems more likely.
The one-loop corrections due to these operators depend quadratically on $M_H$ (or equivalently, on the cutoff in the case of a strongly interacting symmetry breaking sector). We have studied separately the limiting cases of a heavy and a light Higgs. We conclude that LEP200 is more sensitive to any new physics pointing in these blind directions if the Higgs is light: for $O_{B\Phi}$, present bounds are only a factor 2-3 worse than our conservative estimation of future LEP200 sensitivity, while for $O_{W\Phi}$ this factor grows to an order of magnitude.
On the contrary, in the case of a heavy Higgs and considering the natural situation $M_H \sim 4 \pi v$, present constraints are already considerably better than those that can be obtained in LEP200. On naturality grounds, we have argued that the quadratic dependence on the cutoff is physically relevant in the non-linear realization, implying that the size of the counterterm must be that of the quadratic terms in $\Lambda$, within an order of magnitude.
Even though it is not natural to expect that an extension of the standard model will point exclusively in these blind directions, present data already constrain considerably this possibility and, still in those situations where present bounds are weaker, the allowed regions on the planes $(m_t,
\delta's)$ from present and future experiments are independent to a large extent.
Although our estimation of LEP200 sensitivity is quite optimistic, this analysis can certainly be refined. The measurement of helicity amplitudes, for instance, is expected to increase the sensitivity by a factor 2. Also, the measurement of $M_W$ with much better precission, can improve the indirect constraints.
Finally we have also estimated the sensitivity of NLC to these blind directions and found an increase of a factor $\sim 5$ with respect to LEP200.
Acknowledgments
===============
We thank Álvaro De Rújula, Andy Cohen, Belén Gavela, Juanjo Gómez Cadenas, Takeo Inami, Elisabeth Jenkins, Aneesh Manohar, Eduard Masso, Olivier Péne and Juan Terrón for encouragement and/or illuminating discussions. We are indebted to Álvaro De Rújula, Belén Gavela and Olivier Péne for careful reading of the manuscript. We also acknowledge the Ministerio de Educación y Ciencia and the Universidad Autónoma de Madrid for financial support during the completion of this work.
Appendix A
==========
- [Self-energies:]{}
$$\begin{aligned}
\Pi_{\gamma\gamma} & = & - 2 c^2 ( {\delta}_{B \Phi}
+ {\delta}_{W{\Phi}} - \frac{1}{2}{\delta_9} )
[ {\Lambda}^2- (\frac {q^2}{6} + 3 {M_W}^2){\log( \frac{\Lambda^2}{M_W^2} )}] \frac{q^2}{M_W^2}\end{aligned}$$
$$\begin{aligned}
\Pi_{\gamma Z} & = & \frac {c}{s} [ ( s^2 {\delta}_{B{\Phi}}
- c^2 {\delta}_{W{\Phi}} + \frac{c^2-s^2}{4} \delta_9)
(2 \frac {\Lambda^2}{{M_W}^2}
- (\frac {q^2}{3{M_W}^2} + 6){\log( \frac{\Lambda^2}{M_W^2} )}) \nonumber \\
& & + [ \frac {9}{4}( {\delta}_{B{\Phi}}
+ {\delta}_{W{\Phi}} -\frac{1}{2} \delta_9 )
- \frac {s^2}{4 c^2}( {\delta}_{B{\Phi}}
- {\delta}_{W{\Phi}}-\frac{1}{2}
\delta_9) - \frac {3}{2} \xi {\delta}_{W{\Phi}}]{\log( \frac{\Lambda^2}{M_W^2} )}] q^2 \nonumber\\
& & + \frac{c}{s} ( \delta_{W\Phi}
-\delta_{B\Phi} +\frac{\delta_9}{2}) \frac{M_H^2}{4 M_W^2}
( {\log( \frac{M_H^2}{M_W^2} )}-\frac{1}{2} ) q^2\end{aligned}$$
$$\begin{aligned}
\Pi_{ZZ} & = & ( \frac{c^4}{s^2} \delta_{W{\Phi}}
+ s^2 \delta_{B{\Phi}} + \frac{c^2}{2} \delta_9 )
( - 2 \Lambda^2 + (q^2/3){\log( \frac{\Lambda^2}{M_W^2} )}) \frac{q^2}{M_W^2} \nonumber \\
& + & ( \frac{c^2}{s^2} {\delta}_{W{\Phi}}
+ {\delta}_{B{\Phi}} + \frac{c^2}{2 s^2} \delta_9)
[ [ q^2 ( \frac{1}{6c^2}+ 1-6c^2- \frac{M_H^2}{2 M_W^2}) \nonumber\\
& + & \frac{3}{2c^2} ( M_H^2 +M_Z^2-\frac{q^2}{3} )] {\log( \frac{\Lambda^2}{M_W^2} )}+ \frac{1}{2} (q^2- 3 M_Z^2) \frac{M_H^2}{M_W^2}
( {\log( \frac{M_H^2}{M_W^2} )}- \frac{1}{2} ) +\frac{3}{c^2} \Lambda^2 ] \nonumber\\
& + & \frac{c^2}{s^2} \delta_{W\Phi} [ q^2
(3 +12 c^2 - \frac{2}{3 c^2} - 3 \xi ) + ( 3 (\xi+1) M_Z^2
- \frac{q^2}{c^2})] {\log( \frac{\Lambda^2}{M_W^2} )}-\frac{1}{ s^2} \delta_9 \Lambda^2\end{aligned}$$
$$\begin{aligned}
\Pi_{WW} & = & ( \delta_{B{\Phi}} + \frac {c^2}{s^2}
\delta_{W{\Phi}})
[ - \frac{q^2}{3} + 3 ( \Lambda^2 + (\frac {M_Z^2}{2}
-\frac {3}{2} M_W^2 - \frac {q^2}{6}){\log( \frac{\Lambda^2}{M_W^2} )}) ] \nonumber \\
& + & \frac {c^2}{s^2} \delta_{W{\Phi}}
[ [ ((M_W^2 ( \frac {1}{2c^2}+ \frac {28}{3}- 3 \xi) + \frac{q^2}{3}
- \frac {M_H^2}{2}){\log( \frac{\Lambda^2}{M_W^2} )}-2 {\Lambda}^2) \frac{q^2}{M_W^2} \nonumber\\
& + & 3 ( (\xi+6){M_W}^2- \frac {3}{2}{M_Z}^2
+ \frac {{M_H}^2}{2}){\log( \frac{\Lambda^2}{M_W^2} )}] + \frac{1}{2} ( q^2 - 3 M_W^2 )
\frac{M_H^2}{M_W^2} ( {\log( \frac{M_H^2}{M_W^2} )}- \frac{1}{2} ) ] \nonumber\\
& + & \frac{c^2}{s^2} \delta_9 [
- \frac{\Lambda^2}{2} +
( \frac{3q^2}{4} \xi -\frac{13q^2}{12}
+\frac{q^2}{4}
+ \frac{M_Z^2}{4} +\frac{5}{4} M_W^2- \frac{3}{4} \xi M_W^2){\log( \frac{\Lambda^2}{M_W^2} )})]\end{aligned}$$
- [Vertex corrections:]{}
$$\begin{aligned}
\delta {c_L}^f & = & \frac {3}{2} \frac{c^2}{s^2} (\xi+1)
( {c_L}^f-{c_R}^f) \delta_{W\Phi} {\log( \frac{\Lambda^2}{M_W^2} )}\end{aligned}$$
$$\begin{aligned}
\frac{\delta g_{Wl \nu}}{g_{Wl \nu}} & = & \frac {3}{4} \frac {c^2}{s^2}
(\xi+1) [(\delta_{W \Phi}- \frac{\delta_9}{2})
({c_L}^{\nu}-{c_L}^l+c_R^f) + \frac{1}{c^2} \delta_{W \Phi}
({c_L}^{\nu}-{c_L}^l)] {\log( \frac{\Lambda^2}{M_W^2} )}\end{aligned}$$
[10]{}
A. De Rújula, M.B. Gavela,P.Hernández,E.Massó, CERN-TH.6272/91-FTUAM 91-31. To appear in Nuclear Physics B. A. De Rújula, M.B. Gavela,O. Péne and F.J. Vegas, Nucl. Phys. B355 (1991) 549. M.S. Chanowitz, M.Golden and H. Georgi, Phys. Rev. D36 (1987)1490. C.P. Burguess and D.London, McGill-92/04, UdeM-LPN-TH-83. K. Hagiwara, et al. MAD/PH 690 (1992). C.P. Burgess and D.London, McGill-92/04,/05, UdeM-LPN-TH-83,84. M.B. Einhorn and J. Wudka, NSF-ITP-92-01,UCRHEP-T88,UM-TH-92-02. E. Fermi, Nuovo Cimento 11 (1934) 1; Z. Phys. 88(1934) 161. K. Wilson, Phys. Rev. D2(1970)1438,E. Witten, Nucl. Phys. B104(1976) 445 S. Weinberg, Physica 96A (1979)327, Phys. Lett. 91B (1980)51., J. Polchinski, Nucl. Phys. B231(1984)269, T. Appelquist and J. Carazzone, Phys. Rev. D11 (1975) 2856. K.G. Wilson, Phys.Rev.179 (1969)1499. Y. Kazama and YP. Yao. Phys. Rev. D25(1982) 1605. S. Weinberg Phys. Lett. 91B(1980)51. T. Appelquist and J. Carazzone, Phys. Rev. D11(1975) 2856. J. Polchinski, Nucl. Phys. B231 (1984) 269. T. Appelquist and C. Bernard. Phys. Rev. D22(1980) 200. A. C. Longhitano, Phys. Rev. D22(1980)1166 and Nucl. Phys. B188(1981)118., T. Appelquist in Gauge Theories and Experiments at High Energies, Ed. by K.C.Brower and D.G. Sutherland. Published by the Scottish University Summer School in Physics, St. Andrews(1980). Gounaris et al. BI-TP 91/40, G.Couture et al. OCIP/C-91-6. See \[1\] for many earlier references. A. De Rújula, CERN-TH.6374/92. P. Sikivie et al. Nucl. Phys. B173(1980)189. S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969)2239, C.G. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969)2247. S. Weinberg, Physica 96A (1979)327 J. Gasser and H. Leutwyler, Annals of Physics 155 (1984) 142 and Nucl. Phys. B250 (1985) 93. A. Manohar and H. Georgi, Nucl. Phys. B234(1984)189. J. Alliti et al. UA2 Col, Phys. Lett. B241(1990)150. F. Abe et al. CDF Col. , Phys. Rev. D43(1991)2070. J.Nash. Electroweak Results form LEP 1991. Talk given at XXVIIth Rencontres de Moriond on Electroweak Interactions and Unified Theories, Les Arcs, France, March 1992. H. Abramowicz et al. CDHS collaboration. Phys. Rev. Lett. 57(1986) 298. J. V. Allaby et al, CHARM collaboration. Phys. Lett. B177(1986)446. See P.Langaker. Pennsylvannia Report UPR-0435T(1990). R.N. Cahn, M.S. Chanowitz and N. Fleishon, Phys. Lett. 82B (1979) 112. L3 collaboration, CERN Preprint PPE 91-29 (1991). J.J. van der Bij, Quartic divergences and vector boson self couplings. To appear in Phys.Lett.B. J.J. Van der Bij, Phys. Rev. D, 35 1088(1987). J. Bagger, et al. FERMILAB-PUB-92/75-T, JHU-TIPAC-920009. G. Valencia, FERMILAB-CONF-92/246-T. D. Espriu and M.J. Herrero, Nucl. Phys. B373 (1992) 117.
Figs. 1 Allowed $2 \sigma$ contours in the $(\delta_i, m_t)$ planes from present data and $M_H = 50 GeV$/$M_H = 260GeV$ for $O_{B\Phi}$/ $O_{W\Phi}$. The dashed domains subtends the values of $\delta_i's$ that can not be distinguished from zero at LEP200 at the $2 \sigma$ level.
Figs. 2 $\chi^2$ test fo significance of the effect of $\lambda_i's
\neq 0$ on $d \sigma /d cos \theta_+$. The horizontal line shows the $2 \sigma$ sensitivity for $10^4$ W-pairs at $\sqrt{s} = 200 GeV$, the projections along the vertical arrows delimit the interval of $\lambda_i's$ inside which a LEP-2 measurement would test the hypothesis $\lambda_i's \neq 0$ with less than $2 \sigma$ significance. Vertical bands encompasses the values fo $\lambda_i's$ currently allowed by the lower-energy tests, for $M_H \sim 4 \pi v$.
[^1]: Quantitatively only the running of QED and QCD couplings has been significatively tested up to now, although including also the proper weak corrections one obtains a better agreement with experimental data. We acknowledge discussions on this point with L. Maiani and L. Okun.
[^2]: This pattern is not the most general [@chano], but it doesn’t seem very sensible at present to adopt the point of view of Burguess and London [@london] of substituting the hypothesis of a general symmetry principle ($SU(2)_C$) by an unnatural set of fine tunnings.
[^3]: Some care is needed when considering a theory in which masses are generated by spontaneous symmetry breaking, for in this case masses are given by some dimensionless couplings $\lambda$ times the vacuum expectation value (vev) of a scalar field. When the mass is large due to a large Yukawa coupling, while the vev is light, the large mass expansion does not exist in general [@kazama].
[^4]: In fact the operators must be BRS-invariant, so in principle, there could be operators involving ghost fields. However, as shown in [@weinb] the Faddeev-Popov procedure can be done in two steps, firstly for the “heavy subgroup” when integrating the heavy sector and secondly for the “light” subgroup when defining the effective theory. This way no extra interactions appear in the “light” ghost sector.
[^5]: In Gasser-Leutwyler notation [@gasser], the $L_9$ operator corresponds to a combination of these operators $(\beta_2 = \beta_3 )$.
[^6]: A common misconception is the statement that there are no quadratic divergences in dimensional regularization. There are, and they correspond to poles at d=2.
[^7]: The radiatively-corrected standard predictions on $\Gamma_f$ can be cast in the form: $\Gamma_f \simeq \frac{G_F M_Z^3}{3 \pi \sqrt{2}} N_c (1 + \delta
\gamma_f) ([c_f^R]^2+[c_f^L]^2)$, with $c_f^R = -(1+\delta \kappa_f) Q_f sin^2 \theta_Z$ and $c_f^L = T_3^f + c_f^R$.
[^8]: In terms of $L_9^R$ and $L_9^L$, our constraints translate into $-13.9 \leq \L_9^R \leq 6.0$ and $-24.6 \leq \L_9^L \leq 9.5$.
|
---
abstract: 'The $S=1$ antiferromagnetic Heisenberg model on a Kagome lattice is studied using the density-matrix renormalization group method. To identify the ground state, we take four kinds of clusters into account; periodic, cylindrical, and two open ones. The hexagonal singlet solid (HSS) and triangular valence bond solid (TVBS) states are artificially generated by modulating edge shapes of the open clusters. We find that the energy par sites of the HSS state is $e_0=E_0/N =-1.41095$, which is readily lower than that of the TVBS state ($e_0=-1.3912 \pm 0.0025$). This agrees well with those of the cylindrical ($e_0=-1.40988$) and periodic ($e_0=-1.409 \pm 0.005$) clusters, where no assumption as to the ordering is posed. Thus, we conclude that the HSS picture is consistent to describe the ground state of the $S=1$ Kagome Heisenberg model. This is further confirmed by finding non-symmetry-breaking state in the calculations of the dimer-dimer correlation functions as well as the entanglement entropy of cylindrical clusters. Finally, we estimate the single-triplet energy gap: The HSS ground state has $\Delta=0.1919$, while the TVBS excited state has a larger one $\Delta=0.2797$.'
author:
- Satoshi Nishimoto
- Masaaki Nakamura
title: 'Non-Symmetry-Breaking Ground State of the $S=1$ Heisenberg Model on the Kagome lattice'
---
For a long time, frustrated spin systems have been fascinating subjects of research for discovering new physics [@Moessner-R]. Among them a system attracting the most attention in recent years is Kagome antiferromagnetic Heisenberg (KAH) model, and a lot of experimental and theoretical studies on the Kagome system have been carried out, assisted by the improvement of research techniques.
In the $S=1/2$ KAH system, one of the most striking finding is the fact that this ground state is characterized as a $Z_2$ spin liquid [@Yan11; @Jiang12]. The experimental realization of spin liquid has been also demonstrated in the herbertsmithite ZnCu$_3$(OD)$_6$C$_{l2}$ [@Han]. Theoretically, it is still debated whether the ground state is gapless [@Iqbal-B-S-P] or gapped with very small energy gap [@Yan11; @Jiang08; @Depenbrock12; @Nishimoto-S-H]. Another notable feature is the appearance of a series of plateaus in the magnetization process [@Nishimoto-S-H; @Capponi13]. Of particular interest is a possible $Z_3$ spin liquid plateau at 1/9 magnetization [@Nishimoto-S-H]. Thus the $S=1/2$ KAH model exhibits a variety of phases, although it is a simple Heisenberg system consisting only of the nearest-neighbor exchange couplings.
We here turn our attention on an $S=1$ version of the KAH model. It would be a natural continuation of the study on the KAH system. The Hamiltonian is written as $$H=J\sum_{\langle ij \rangle} {\bf S}_i \cdot {\bf S}_j
\label{ham}$$ where ${\bf S}_i$ is a spin-one operator at site $i$, and the summation is taken for nearest neighbours $\braket{ij}$. $J$ is the antiferromagnetic exchange integral and we set $J=1$ hereafter. From the theoretical point of view, a very interesting point is an extension of valence-bond solid (VBS) picture [@Affleck-KLT] to two-dimension. Firstly, the triangular VBS (TVBS) state was suggested as a ground state by the perturbation expansion around the complete TVBS limit [@Asakawa]. After that, according to the analysis based on the exact diagonalization and variational method, it was claimed that hexagonal-singlet solid (HSS) ground state is realized [@Hida] (see Fig. \[states\]). Moreover, a resonating AKLT loop (RAL) state has been recently suggested as an alternative candidate [@Li14]. The ground state of the $S=1$ KAH model is still an open issue. So far, several materials have been synthesized as possible realizations of the $S=1$ KAH system, e.g., $m$-MPYNN$\cdot$BF$_4$ [@Wada; @Matsushita], KV$_3$Ge$_2$O$_9$ [@Hara], \[C$_6$N$_2$H$_8$\]\[NH$_4$\]$_2$\[Ni$_3$F$_6$(SO$_4$)$_2$\] [@Behera], and NaV$_3$(OH)$_6$(SO$_4$)$_2$ [@Papoutsakis]. Further experimental observations are strongly expected.
![Schematic illustrations of valence-bond solid (VBS) state in the $S=1$ Kagome antiferromagnetic Heisenberg model: (a) the hexagonal-singlet solid (HSS) and (b) the triangular VBS states. Blue (red) line indicates a spin-singlet formation in six (two) $S=1/2$ variables.[]{data-label="states"}](states)
![Lattices used for the DMRG calculations: (a) A cylindrical cluster, denoted as XC6-3, where periodic (open) boundary conditions is applied in the vertical (horizontal) direction. (b) A PBC cluster with $N=36$ sites. Other PBC clusters are shown in the Supplemental material. (c),(d) Open clusters for obtaining the TVBS and HSS states, respectively.[]{data-label="lattice"}](lattice.pdf)
In this Letter, the ground state of the $S=1$ KAH model is determined. We use the density-matrix renormalization group (DMRG) method, which enables us to study large-size clusters. For this aim, we exploit four kinds of clusters shown in Fig. \[lattice\]. Up to about 10000 density-matrix eigenstates are kept in the renormalization procedure and the discarded weight is below 10$^{-4}$ even for the most difficult case, namely, 36-site cluster under the periodic boundary conditions (PBC). First, by calculating the dimer-dimer correlation functions and entanglement entropy with the cylindrical clusters, we find that the translational symmetry is not broken in the ground state.It is further confirmed with the PBC cluster. Next, we [*intentionally*]{} produce the HSS and TVBS states by modulating the edge condition of open clusters (see below for details). It enables us to compare their energies directly. The lowest-state energy calculated with the HSS cluster is $E_0/N \equiv e_0 =-1.41095$ and it is in good agreement with those of the cylindrical ($e_0=-1.40988$) and PBC ($e_0=-1.409 \pm 0.005$) clusters. It is quite reasonable because the HSS state is a non-symmetry-breaking one. Moreover, it is striking that the energy estimated with the TVBS cluster ($e_0=-1.3912 \pm 0.0025$) is decidedly higher than the others. Thus, we verify the HSS state to be the ground state of the $S=1$ KAH model. Finally, we estimate the single-triplet gap. The ground state, i.e., the HSS state, has a gap $\Delta=0.1919$; while, the TVBS state as an excited eigenstate has a larger gap $\Delta=0.2797$.
Let us start with the cylindrical cluster \[Fig. \[lattice\](a)\]. We here use a type of cylinder denoted as XC6-3, the notation of which was defined in previous works on the $S=1/2$ KAH system [@Yan11; @Depenbrock12]. The reason for choosing it is related to the shape of both edges. In general, an open edge exerts a critical influence on the formation of plaquette or bond singlets. If the edge consists of either triangles or hexagons, the TVBS or HSS state may be artificially favoured in our system (\[ham\]). Such a signature was also identified in the $S=1/2$ KAH model [@Yan11; @Gong13]. This problem should be avoided by choosing a XC$n$-($n/2$) or YC$n$ type of cylinder, where both triangles and hexagons are [*equally*]{} arranged at the open edges (see Supplement). In other words, the TVBS and HSS states could be intentionally stabilized by modulating the shape of open edges in a (small) cluster. This [*technique*]{} is used in the latter part of this Letter. For instance, the same technique was used to detect the plaquette VBS state in the $J_1$-$J_2$ honeycomb Heisenberg model [@Ganesh13].
In order to check the configuration of singlet valence bonds, we calculate the dimer-dimer correlation functions defined by $$C_{(i,j)(k,l)}=
4[\braket{({\bf S}_i \cdot {\bf S}_j)({\bf S}_k \cdot {\bf S}_l)}
-\braket{{\bf S}_i \cdot {\bf S}_j}\braket{{\bf S}_k \cdot {\bf S}_l}].$$ In Fig. \[dimerdimer\](a) the values of the dimer-dimer correlation for each bond are shown with fixing one reference dimer bond. Blue and red links denote positive and negative correlations. It seems that the patterns for the sign of correlations does not exhibit any spatial periodicity. In addition, the correlation decays exponentially with distance of two bonds, as plotted in Fig. \[dimerdimer\](b). It clearly indicates the absence of symmetry-breaking order associated with dimer formations. A similar feature is also observed in the PBC clusters \[Fig. \[dimerdimer\](d)\] Moreover, to make sure, we examine the von Neumann entanglement entropy (EE) $S_L(l)=-{\rm Tr}_l \rho_l \log \rho_l$, where $\rho_l={\rm Tr}_{L-l}\rho$ is the reduced density matrix of the subsystem and $\rho$ is the full density matrix of the whole system [@Oshikawa06; @Jiang12]. We plot the value of $S_{\frac{L}{2}}$ as a function of the circumference of the cylinder $L_y$. The values should follow a relation $S(L_y)=\alpha L_y - \gamma$, where $\alpha$ is a constant and $\gamma = \ln D$ is the topological entropy with dimension $D$ [@Kitaev06; @Levin06]. By the fitting of our data with the equation, we obtain $\gamma=0.0014$ in the $L_y\to 0$ limit. This suggests that the system is in a topologically trivial phase, i.e., a unique ground state, and it is consistent to the results of the dimer-dimer correlation functions.
Next, we consider the TVBS state. As mentioned above, an ordered state like the symmetry-breaking TVBS state can be forcibly stabilized as the lowest state in a small cluster by taking a proper edge condition. However, we have to take notice the following two points to determine if the lowest state is really the ground state when this artificial technique is applied: (i) the ordering survives in the thermodynamic limit, (ii) the energy of the ordered state remains lowest among all eigenstates in the thermodynamic limit. If it is not the case, a level crossing with the true ground state occurs at some larger cluster. One possible realization of the TVBS cluster is shown in Fig. \[lattice\](c). For this cluster we present the values of the nearest-neighbour spin-spin correlations $\braket{-{\bf S}_i \cdot {\bf
S}_j}$ in Fig. \[dimerdimer\](e). A link with larger (smaller) value than that of the next bonds is coloured in red (blue). A TVBS configuration is obviously seen. Furthermore, with increasing the system size, the spin gap is smoothly scaled to a finite value $\Delta=0.2797$ at $1/L\to 0$ (see the inset of Fig. \[dimerdimer\](f)). The spin gap is evaluated as the energy difference between the lowest singlet and the first excited triplet states. Hence, the TVBS state is confirmed to survive in the thermodynamic limit. We also make certain that this state is indeed of the TVBS with triangular three-dimer formations. For this purpose, we modulate the exchange couplings as $(1+\delta)J$ and $(1-\delta)J$ for red and blue bonds in Fig. \[dimerdimer\](e), respectively ($0 \le \delta \le 1$). Figure \[dimerdimer\](f) shows the spin gap as a function of $\delta$. In the isolated triangle limit ($\delta=1$), the lowest state is exactly described by a direct product of triangular Haldane-gap dimers with three $S=1$ spins; as $\delta$ decreases, the fluctuations between the triangles are increased and the original Hamiltonian is recovered at $\delta=0$. It is worth noting the spin gap is scaled perfectly by that assuming an isolated triangle, $\Delta=2\delta$, for a wide range of $\delta(>0.5)$. This means that the triangular Haldane-gap dimers are quite robust against the inter-triangle fluctuations and the product state may be a good approximation of the TVBS state even at smaller $\delta$. We can further see that no gap closing point exists from $\delta=1$ to $\delta=0$, suggesting that the product state is intermittently connected to the TVBS state in the original model (\[ham\]). Therefore, we can confirm that the TVBS state indeed exists as an eigenstate of the $S=1$ KAH model.
![Extrapolation scheme of the lowest-lying-state energy as a function of $1/L_y$ for the cylindrical cluster and of $1/\sqrt{N}$ for the TVBS, HSS, and PBC clusters. For the cylindrical cluster the $L_x \to \infty$ limits are already taken.[]{data-label="gsenergy"}](gsenergy)
For identifying the true ground state, it would be a natural step to compare the lowest-state energies calculated with different clusters. In Fig. \[gsenergy\] the finite-size scaling analysis of the energy per site for each cluster is presented. For the TVBS and PBC clusters, the energy is plotted with $1/\sqrt{N}$ as usually assumed for two-dimensional systems. For the cylindrical cluster, we first take the $1/L_x\to 0$ limit followed by the $1/L_y\to 0$ limit. Only the scaling with $1/L_y$ is shown in Fig. \[gsenergy\]. We estimate the energy for the TVBS cluster in two different ways; one is simply the total energy divided by the total number of sites ($e_0(N)=E_0(N)/N$) and the other is an average of the neighbouring spin-spin correlations ($e_0(N)=2\overline{\langle {\bf S}_i {\bf S}_j \rangle}$, where the factor 2 comes from the ratio of the number of sites and bonds in the thermodynamic limit). Since one of them is extrapolated from the higher energy side with decreasing $1/L_y$ and the other from the lower side, as seen in Fig. \[gsenergy\], this should make the scaling analysis more reliable. The extrapolated values from the both ways agree very well in the thermodynamic limit and it is estimated as $e_0=-1.3912 \pm 0.0025$. The energies for the cylindrical and PBC clusters seem to converge rather faster with the system sizes, and they are $e_0=-1.40988$ and $e_0=-1.409 \pm 0.005$, respectively, in the thermodynamic limit. Clearly, the energy of the TVBS state is high in number ($\Delta e_0 =0.02$) compared to those for the other two clusters. We thus argue that the TVBS state exists as an eigenstate but it is not the ground state.
In a similar way as the TVBS state, we can also stabilize the HSS state in a small cluster. One possible realization is shown in Fig. \[lattice\](d), where the hexagons are placed at the corners of open cluster. Note that the outer $S=1$ spins are replaced by $S=1/2$ spins not to hold extra free spins when all hexagons form singlet plaquettes. By doing this we can easily detect the first excited triplet state by one spin-flip from the lowest state. For the HSS cluster we estimate the energy as an averaged value of the neighbouring spin-spin correlations between the $S=1$ spins, namely, the outer bonds are excluded. The size-scaling is shown in Fig. \[gsenergy\]. The extrapolated value to the thermodynamic limit is $e_0 =-1.41095$. It agrees to those of the cylindrical and PBC clusters within the error bars. Since the HSS state is a non-symmetry-breaking one, it is utterly reasonable.
Although it is not possible to study a RAL state with our method, our estimation of the energy of the HSS state is readily lower than the variational energy of the pure RAL state $e_0=-1.383$ [@Li14]. Therefore, we conclude that the HSS state is the ground state of the $S=1$ KAH model. It has been suggested that the TVBS state is the ground state when a certain amount of the second- and third-neighbour antiferromagnetic exchange interactions are taken into account [@Cai09]. It is likely to happen because a frustration induced by the second-neighbour interactions naively lifts up the energy of the HSS state.
![Extrapolation scheme of the spin gap as a function of $1/L_y$ for the cylindrical cluster, of $1/L$ for the TVBS, HSS, and of $1/\sqrt{N}$ for the PBC clusters. For the cylindrical cluster the limit for $L_x \to \infty$ are already taken. []{data-label="gap"}](gap)
Finally, we study the spin gap. In Fig. \[gap\] the finite-size-scaling analyses of the spin gap for the used four kinds of clusters are performed. The extrapolated values to the thermodynamic limit are $\Delta=0.183$, $0.17 \pm 0.03$, and $0.172$ for the cylindrical, PBC, and HSS clusters, respectively. As expected from the above discussion, the three numbers are very close to each other. We thus determine the spin gap of the HSS state, which is the ground state, is $\Delta=0.178 \pm 0.05$. It is considerably smaller than that of the TVBS state, i.e., $\Delta=0.2797$. This may imply that the triangular Haldane-gap dimers in the TVBS state is more robust than the hexagonal singlet in the HSS state. Hence, in the TVBS state the spins are strongly screened and the energy gain from the quantum fluctuations between the triangles might be small. Here we shall comment on the spin gap observed in a related material $m$-MPYNN$\cdot$BF$_4$. From the fitting of measured susceptibility with $(\Delta/k_B T)\exp(-\Delta/k_B T)$ at low temperature $T$, $\Delta$ was obtained as $\sim 0.2$K [@Wada; @Matsushita]. This system can be mapped to the $S=1$ KAH model with $J=0.65-0.95$ K. By our analysis the spin gap is estimated as $\Delta=0.178J \approx
0.12-0.17$K. This value seems to be reasonably close to the experimental one.
In summary, we have studied the $S=1$ KAH model using the DMRG technique. Four kinds of clusters have been used to determine the ground state. We have succeeded in extracting the ordered HSS and TVBS states by taking unique open clusters. It enabled us to extrapolate their low-lying energies individually to the thermodynamic limit. As a result, we have found that the ground state of the $S=1$ KAH model is the HSS state and the TVBS state is an excited one. However, their lowest-lying-state energies are very close, and the near degeneracy seems to make it more difficult to detect the true ground state. The dimer-dimer correlation functions and the entanglement entropy for the cylindrical and PBC clusters, where no assumptions for ordering are posed, suggest a non-symmetry-breaking ground state. It also supports the HSS ground state. The singlet-triplet gap of the TVBS state is larger than that of the HSS state. It means that the triangular singlet dimers are more robust than the hexagonal singlet, and the spin are strongly screened. It may prevent lowering of the energy derived by quantum fluctuations between triangular singlet dimers.
We acknowledge Max-Planck-Institut für Festkörperforschung where a part of the numerical calculations has been done.
[*Note added*]{} — In preparing this manuscript we noticed two preprints on the DMRG study of the $S=1$ KAH model [@Changlani14; @Liu14]. Although they argued different ground state from our conclusion, the ground-state energy in their calculations agrees with ours very well.
[99]{}
R. Moessner and A. P. Ramirez, Physics Today, pp. 24, February 2006.
S. Yan, D.A. Huse, and S.R. White, Science [**332**]{}, 1173 (2011).
H.-C. Jiang, Z. Wang, and L. Balents, Nature Physics [**8**]{}, 902 (2012).
T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J.A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Nature [**492**]{}, 406 (2012).
Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B [**87**]{}, 060405(R) (2013); Y. Iqbal, D. Poilblanc, and F. Becca, Phys. Rev. B [**89**]{}, 020407(R) (2014).
S. Depenbrock, I. P. McCulloch, and U. Schollwöck, , 067201 (2012).
H. C. Jiang, Z. Y. Weng, and D. N. Sheng, , 117203 (2008).
S. Nishimoto, N. Shibata, and C. Hotta, Nature Comm. [**4**]{}, 2287 (2013).
S. Capponi, O. Derzhko, A. Honecker, A. M. Läuchli, and J. Richter, Phys. Rev. B [**88**]{}, 144416(R) (2013).
I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. [**59**]{}, 799 (1987); Commun. Math. Phys. [**115**]{}, 477 (1988).
H. Asakawa and M. Suzuki, Physica A [**198**]{}, 210 (1993).
K. Hida, J. Phys. Soc. Jpn. [**69**]{}, 4003 (2000); [**70**]{}, 3673 (2001).
W. Li, S. Yang, M. Cheng, Z.-X.Liu, and H.-H. Tu, , 174411 (2014)
N. Wada, T. Kobayashi, H. Yano, T. Okuno, A. Yamaguchi, and K. Awaga, J. Phys. Soc. Jpn. [**66**]{}, 961 (1997).
T. Matsushita, N. Hamaguchi, K. Shimizu, N. Wada, W. Fujita, K. Awaga, A. Yamaguchi, and H. Ishimoto, J. Phys. Soc. Jpn. [**79**]{}, 093701 (1997).
S. Hara, H. Sato, and Y. Narumi, J. Phys. Soc. Jpn.[**81**]{}, 073707 (2012).
J. N. Behera and C. N. R. Rao, J. Am. Chem. Soc. [**128**]{}, 9334 (2006).
D. Papoutsakis, D. Grohol, and D.G. Nocera, J. Am. Chem. Soc. [**124**]{}, 2647 (2002).
S. Gong, W. Zhu, and D. N. Sheng, arXiv:1312.4519v1.
R. Ganesh, J. van den Brink, and S. Nishimoto, Phys. Rev. Lett. [**110**]{}, 127203 (2013).
M. Oshikawa and T. Senthil, , 060601 (2006).
A. Kitaev, and J. Preskill, , 110404 (2006).
M. Levin and X. G. Wen, , 110405 (2006).
Z. Cai, S. Chen, and Y. Wang, J. Phys.: Condens. Matter [**21**]{}, 456009 (2009).
H. J. Changlani and A. M. Läuchli, arXiv:1406.4767v1.
T. Liu, W. Li, A. Weichselbaum, J. von Delft, and G. Su, arXiv:1406.5905v1.
cylindrical clusters
====================
In this Letter, we follow the notation of cylindrical cluster in Ref. 2. A cylinder is labeled as XC$n$ (YC$n$) when the circumference is along or close in orientation to the $x$($y$)-axis, where $n$ is the circumference in unit of the lattice spacing. If the lattice is connected with a shift $d$ units in the periodic direction, the number of shift is added to the label such as XC$n$-$d$.
In Fig. \[suppl\_fig1\] several kinds of the cylindrical cluster are illustrated. Some arbitrariness remains on the shape of open edges. For example, the lattices (a) and (b) are both labeled by the XC8 cylinders but the shapes of the open edges are different. One of them has hexagons and the other has triangles at the open edges. Then, there is a possibility that the hexagonal singlet or triangular Haldane-gap dimers is artificially favoured. If we choose the XC$n$-($n/2$) type of cylinder, the hexagons and triangulars may be [*equally*]{} placed at the open edges, as shown in Fig. \[suppl\_fig1\](c). Also, a kind of the YC$n$ cylinder \[Fig. \[suppl\_fig1\](d)\] has the similar edges. Another kind of the YC$n$ cylinder like in Fig. \[suppl\_fig1\](e) has triangles at the open edges, and however, it could be useful cluster to detect the TVBS state directly as a translation-symmetry-breaking state along the OBC direction.
![(a),(b) Kinds of XC8 cylinders. (c) A kind of XC8-4 cylinder. (d),(e) Kinds of YC6 cylinders. []{data-label="suppl_fig1"}](XCYC.pdf)
periodic clusters
=================
In Fig. \[suppl\_fig2\] the periodic clusters used in our numerical calculations are shown. They are taken as isotropic as possible. As shown in the main text, the $N=36$ periodic cluster is completely isotropic.
![Periodic clusters used in our numerical calculations. []{data-label="suppl_fig2"}](PBCclusters.pdf)
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---
abstract: 'We adapt a nonlinear version of Peetre’s theorem on local operators in order to investigate representatives of nonlinear generalized functions occurring in the theory of full Colombeau algebras.'
author:
- 'E.A. Nigsch[^1]'
title: |
On a nonlinear Peetre’s theorem\
in full Colombeau algebras
---
**MSC2010 classification:** 46F30
**Keywords:** nonlinear Peetre’s theorem; local function; Colombeau algebra
Preliminaries
=============
Algebras of of nonlinear generalized functions in the sense of J.F. Colombeau [@Biagioni; @ColNew; @ColElem; @GKOS; @MOBook] provide a way to define a meaningful multiplication of arbitrary distributions while at the same time products of smooth functions and the partial derivatives of distribution theory are preserved. This is the best one can obtain in light of L. Schartz’ impossibility result [@Schwartz].
A certain variant of these algebras, namely those which are termed *full* Colombeau algebras, have been gaining more and more importance recently through their role in the development of a coordinate-invariant formulation of nonlinear generalized function algebras suitable for singular differential geometry and nonlinear problems in a geometrical context. We recall that in general, Colombeau algebras are given as quotients of certain basic spaces containing the representatives of generalized functions. In successive steps, these basic spaces have been modified and enlarged in order for the resulting algebras to accomodate certain desired properties [@found; @global; @bigone; @papernew]. At one point in this development, the sheaf property could only be obtained in the quotient by imposing so-called *locality conditions* on the elements of the basic space.
The object of this article is to study representatives of nonlinear generalized functions on an open subset $\Omega \subseteq {\mathbb{R}}^n$ which are given by smooth mappings $$R \colon C^\infty(\Omega, {\mathcal{D}}(\Omega)) \to C^\infty(\Omega)$$ which satisfy the most general of these locality conditions, i.e., which are *local* (Definition \[def\_local\]). Adapting arguments of J. Slovák from [@zbMATH04036782] we obtain a characterization of locality in simpler terms, i.e., $R(\vec\varphi)(x)$ does not depend on the germ of $\vec\varphi$ at $x$ but only on its jet of infinite order at $x$ (Theorem \[thm\_peetre\]). Furthermore, we examine in which sense such mappings $R$ have locally finite order (Theorem \[thm\_5\]). While any distribution is of finite order locally, no comparable statement exists for Colombeau algebras so far; our results are a first step in this direction.
Let use introduce some notation. Throughout this article we will work on open subsets $\Omega_1 \subseteq {\mathbb{R}}^n$ and $\Omega_2 \subseteq {\mathbb{R}}^{n'}$ with $n,n' \in {\mathbb{N}}$ fixed. We employ the usual multiindex notation ${\partial}^\alpha$, $\alpha!$, ${\left|\alpha\right|}$ etc. with differentiation indices $\alpha \in {\mathbb{N}}_0^n$. Given an open subset $\Omega \subseteq {\mathbb{R}}^n$ and a locally convex space ${\mathbb{E}}$, the space $C^\infty(\Omega, {\mathbb{E}})$ is endowed with its standard topology, which is that of uniform convergence on compact sets in all derivatives separately. For a function $f$ we denote by $j^r f(x)$ the $r$-jet of $f$ at $x$, i.e., the family $({\partial}^\alpha f(x))_{{\left|\alpha\right|} \le r}$, where also $r = \infty$ is allowed. The interior of a set $B$ is denoted by $B^\circ$. Note that for smooth functions $f(x,y)$ of two variables we will also write $f(x)(y)$, justified by the exponential law [@KM 3.12, p. 30].
The formulation of Theorem \[thm\_5\] requires a notion of smoothness for mappings between arbitrary locally convex spaces. The setting we use for this is that of convenient calculus [@KM], i.e., a mapping $f \colon {\mathbb{E}}\to {\mathbb{F}}$ between two locally convex spaces is said to be smooth in this sense if it maps each smoothly parametrized curve into ${\mathbb{E}}$ to a smoothly parametrized curve into ${\mathbb{F}}$, i.e., for all $c \in C^\infty({\mathbb{R}}, {\mathbb{E}})$ we have $f \circ c \in C^\infty({\mathbb{R}}, {\mathbb{F}})$.
For convenience we cite the extension theorem of Whitney [@zbMATH03934056 1.5.6, p. 31] which will be heavily used below.
\[whitney\] Let $\Omega$ be an open subset of ${\mathbb{R}}^n$ and $X$ a closed subset of $\Omega$. Given a continuous function $f^\alpha$ on $X$ for each $\alpha \in {\mathbb{N}}_0^n$, there exists a function $f \in C^\infty(\Omega)$ with ${\partial}^\alpha f|_X = f^\alpha$ for all $\alpha \in {\mathbb{N}}_0^n$ if and only if for all integers $m \ge 0$ and all compact subsets $K \subseteq X$ we have $$\label{whitcond}
f^\alpha(y) = \sum_{{\left|\beta\right|} \le m} \frac{1}{\beta!} f^{\alpha + \beta} (x) (y-x)^\beta + o ( {\left\lVerty-x\right\rVert}^m)$$ uniformly for $x,y \in K$ as ${\left\lVerty-x\right\rVert} \to 0$.
Main Results
============
We first recall the definition of locality for elements of the basic space $$C^\infty(C^\infty(\Omega, {\mathcal{D}}(\Omega)), C^\infty(\Omega))$$ given in [@papernew]. While only the case $\Omega_1 = \Omega_2$ was considered there, we use a slightly more general formulation which will be needed below.
\[def\_local\]A mapping $R \colon C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2)) \to C^\infty(\Omega_1)$ is called *local* if for all $x \in \Omega_1$ and all $\vec\varphi, \vec\psi \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ the following implication holds: $$\vec\varphi|_U = \vec\psi|_U\textrm{ for some open neighborhood $U$ of }x \Longrightarrow R(\vec\varphi)(x) = R(\vec\psi)(x).$$
Our first result is the following.
\[thm\_peetre\] A mapping $R \colon C^\infty( \Omega_1, {\mathcal{D}}(\Omega_2)) \to C^\infty(\Omega_1)$ is local if and only if for every $\vec\varphi \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ and every point $x \in \Omega_1$, $R(\vec\varphi)(x)$ depends on the $\infty$-jet $j^\infty(\vec\varphi)(x)$ only, i.e., if for all $x \in \Omega_1$ and $\vec\varphi, \vec\psi \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ the equality $j^\infty \vec\varphi (x) = j^\infty \vec\psi (x)$ implies $R(\vec\varphi)(x) = R(\vec\psi)(x)$.
The proof imitates that of [@zbMATH04036782 Theorem 1, p. 274] but is adapted in order to incorporate the additional variable $y$ of the smoothing kernels $\vec\varphi(x)(y)$.
Suppose we are given $\vec \varphi, \vec\psi \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ such that $({\partial}_x^\alpha \vec\varphi)(x) = ({\partial}_x^\alpha \vec\psi)(x)$ for some fixed $x \in \Omega_1$ and all $\alpha \in {\mathbb{N}}_0^n$. Choose an open neighborhood $W$ of $x$ which is convex and relatively compact in $\Omega_1$, as well as compact sets $K,L \subseteq \Omega_2$ with $K \subseteq L^\circ$ such that $\operatorname{supp}\vec\varphi(a) \cup \operatorname{supp}\vec\psi(a) \subseteq K$ for all $a \in W$.
Next, we construct a sequence $(x_k)_{k \in {\mathbb{N}}}$ in $W$ and an open neighborhood $U_k$ of each $x_k$ with $\overline{U}_k \subseteq W$ such that for all $k$ the following conditions hold: $$\begin{gathered}
{\left\lVerta-x\right\rVert} < {\left\lVertb-x\right\rVert}/2 \quad \forall a \in \overline{U}_{k+1}, b \in \overline{U}_k \label{cond1} \\
{\left|({\partial}_x^\alpha {\partial}_y^\beta \vec\varphi)(a,\xi) - ({\partial}_x^\alpha {\partial}_y^\beta \vec\psi)(a,\xi)\right|} \le \frac{1}{k} {\left\lVerta-x\right\rVert}^m \label{cond2} \\
\nonumber\forall a \in \overline{U}_k, \xi \in L, {\left|\alpha\right|} + {\left|\beta\right|} + m \le k.\end{gathered}$$ It suffices to show that holds for any fixed $\alpha, \beta \in {\mathbb{N}}_0^n$ and $m \in {\mathbb{N}}$ uniformly for all $\xi \in L$ if ${\left\lVerta-x\right\rVert}$ is small enough. By Taylor’s theorem we have for any $f=f(x,y) \in C^\infty(\Omega_1 \times \Omega_2)$, $m \in {\mathbb{N}}_0$, $a \in W$ and $\xi \in L$: $$\begin{aligned}
f(a,\xi) &= \sum_{{\left|\gamma\right|} < m} \frac{ ({\partial}_x^\gamma f)(x,\xi)}{\gamma!} (a-x)^\gamma \\
&\quad + m \sum_{{\left|\gamma\right|} = m} \frac{ (a-x)^\gamma }{\gamma!} \int_0^1 (1-t)^{m-1} ({\partial}_x^\gamma f) ( x + t(a-x), \xi) \,{\mathrm{d}}t.\end{aligned}$$ Replacing $f$ by ${\partial}^\alpha_x {\partial}^\beta_y \vec\varphi - {\partial}^\alpha_x {\partial}^\beta_y \vec\psi$ we see that $$\begin{gathered}
({\partial}^\alpha_x {\partial}^\beta_y \vec\varphi - {\partial}^\alpha_x {\partial}^\beta_y \vec\psi)(a,\xi) = \\
m \sum_{{\left|\gamma\right|} = m} \frac{ (a-x)^\gamma }{\gamma!} \int_0^1 (1-t)^{m-1} ( {\partial}^{\alpha+\gamma}_x {\partial}^\beta_y \vec\varphi - {\partial}^{\alpha+\gamma}_x {\partial}^\beta_y \vec\psi ) ( x + t(a-x), \xi) \,{\mathrm{d}}t \\
= o({\left\lVerta-x\right\rVert}^m) \qquad \textrm{as }{\left\lVerta-x\right\rVert} \to 0\end{gathered}$$ uniformly for $(a,\xi) \in W \times L$. In fact, $({\partial}^{\alpha+\gamma}_x {\partial}^\beta_y \vec\varphi - {\partial}^{\alpha+\gamma}_x {\partial}^\beta_y \vec\psi)(a, \xi)$ vanishes for $a=x$ by assumption and is uniformly continuous on the compact set $\overline{W} \times L$, hence the integrand converges to zero uniformly for $\xi \in L$ as $a$ and hence $x + t(a-x)$ approaches $x$.
Note that implies $$\label{cond3}
{\left\lVerta-x\right\rVert} < 2 {\left\lVerta-b\right\rVert}\quad \forall a \in \overline{U}_k, b \in \overline{U}_j, k \ne j.$$ In fact, for $k>j$ we have $2 {\left\lVerta-x\right\rVert} < {\left\lVertb-x\right\rVert} \le {\left\lVerta-b\right\rVert} + {\left\lVerta-x\right\rVert}$ and for $k<j$ we have ${\left\lVerta-x\right\rVert} \le {\left\lVerta-b\right\rVert} + {\left\lVertb-x\right\rVert} < {\left\lVerta-b\right\rVert} + {\left\lVerta-x\right\rVert}/2$. Moreover, $x_k \to x$ for $k \to \infty$.
With $A {\coloneqq}\{x\} \cup \bigcup_{k} \overline{U}_k$, which is a compact subset of $W$, we define a family of continuous functions $h^{\alpha,\beta}$ on $A \times L$ with $\alpha,\beta \in {\mathbb{N}}_0^n$ by $$\label{blubber}
h^{\alpha,\beta}(a,\xi) {\coloneqq}\left\{
\begin{aligned}
({\partial}_x^\alpha {\partial}_y^\beta \vec\varphi)(a,\xi)\qquad &a = x \textrm{ or }a \in \overline{U}_{2k} \textrm{ for some }k, \\
({\partial}_x^\alpha {\partial}_y^\beta \vec\psi)(a,\xi)\qquad &a \in \overline{U}_{2k+1} \textrm{ for some }k.
\end{aligned}
\right.$$ In order to apply Whitney’s theorem to this family we have to verify that $$\label{verify}
h^{\alpha,\beta}(b,\eta) = \sum_{{\left|(\gamma,\lambda)\right|} \le m } \frac{ h^{\alpha + \gamma, \beta + \lambda} (a,\xi) } { (\gamma,\lambda)! } (b-a)^\gamma (\eta-\xi)^\lambda + o ( {\left\lVert (b-a,\eta-\xi) \right\rVert}^m )$$ uniformly for $(b,\eta)$ and $(a,\xi)$ in $A \times L$ as ${\left\lVert (b-a,\eta-\xi) \right\rVert} \to 0$. This follows easily from Taylor’s theorem, and .
Consequently, there is a function $\tilde h \in C^\infty({\mathbb{R}}^n \times {\mathbb{R}}^{n'})$ whose derivatives on $A \times L$ are given by ${\partial}_x^\alpha{\partial}_y^\beta \tilde h = h^{\alpha,\beta}$. Choosing $\rho \in {\mathcal{D}}(\Omega_2)$ such that $\rho \equiv 1$ in an open neighborhood of $K$ and $\operatorname{supp}\rho \subseteq L$, set $h(a)(\xi) {\coloneqq}\tilde h(a,\xi) \cdot \rho(\xi)$ for $a \in \Omega_1$ and $\xi \in \Omega_2$. Then $h \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ and $$\begin{gathered}
h|_{U_{2k}} = \vec\varphi|_{U_{2k}},\quad h|_{U_{2k+1}} = \vec\psi|_{U_{2k+1}}\qquad \forall k \in {\mathbb{N}}, \\
({\partial}_x^\alpha h)(x) = ({\partial}_x^\alpha\vec\varphi)(x) = ({\partial}_x^\alpha\vec\psi)(x)\qquad \forall \alpha \in {\mathbb{N}}_0^n.\end{gathered}$$ The claim of the theorem then follows by $$\begin{aligned}
R(\vec\varphi)(x) &= \lim_{k \to \infty} R(\vec\varphi)(x_{2k}) = \lim_{k \to \infty} R(h)(x_{2k}) \\
&= \lim_{k \to \infty} R(h)(x_{2k+1}) = \lim_{k \to \infty} R(\vec\psi)(x_{2k+1}) = R(\vec\psi)(x).\qedhere\end{aligned}$$
In order to show that $R(\vec\varphi)(x)$ locally depends only on *finitely many* derivatives of $\vec\varphi(x)$ in a certain sense, we will employ the following lemma, paralleling [@zbMATH04036782 Lemma 1, p. 276].
\[thelemma\] Let $R \colon C^\infty ( \Omega_1, {\mathcal{D}}(\Omega_2) ) \to C^\infty(\Omega_1)$ be local and suppose we are given $f \in C^\infty( \Omega_1, {\mathcal{D}}(\Omega_2))$, $x_0 \in \Omega_1$ and $K \subseteq \Omega_2$ compact with $\operatorname{supp}f (x_0) \subseteq K$.
Define ${\varepsilon}\colon {\mathbb{R}}^n \to {\mathbb{R}}$ by $${\varepsilon}(x) {\coloneqq}\left\{ \begin{aligned}
& \exp ( -1 / {\left\lVertx - x_0\right\rVert}) \qquad & x &\ne x_0, \\
&0 & \qquad x &= x_0.
\end{aligned}
\right.$$
Then there exist a neighborhood $V$ of $x_0$ in $\Omega_1$ and $r \in {\mathbb{N}}$ such that for any $x \in V$ and $g_1, g_2 \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ satisfying
1. $\operatorname{supp}g_i (y) \subseteq K$ for $y$ in a neighborhood of $x$ and $i=1,2$,
2. $\sup_{\xi \in \Omega_2} {\left| {\partial}_x^\alpha {\partial}_y^\beta ( g_i-f)(x)(\xi) \right|} \le {\varepsilon}(x)$ for $i=1,2$ and $0 \le {\left|\alpha\right|} + {\left|\beta\right|} \le r$
we have the implication $j^r g_1 (x) = j^r g_2(x) \Longrightarrow (Rg_1)(x) = (Rg_2)(x)$.
Let $R$, $f$, $x_0$ and $K$ be as stated and suppose that the claim does not hold. Then we can find a sequence $x_k \to x_0$ and, for each $k \in {\mathbb{N}}$, functions $f_k, g_k \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ satisfying $$\begin{aligned}
& \operatorname{supp}f_k(y) \cup \operatorname{supp}g_k(y) \subseteq K \textrm{ for $y$ in a neighborhood of $x_k$}, \\
& \sup_{\xi \in \Omega_2} {\left| {\partial}_x^\alpha {\partial}_y^\beta ( f_k-f)(x_k)(\xi) \right|} \le {\varepsilon}(x_k),\textrm{ and} \label{dienstag1} \\
& \sup_{\xi \in \Omega_2} {\left| {\partial}_x^\alpha {\partial}_y^\beta ( g_k-f)(x_k)(\xi) \right|} \le {\varepsilon}(x_k) \label{dienstag2} \end{aligned}$$ for $0 \le {\left|\alpha\right|} + {\left|\beta\right|} \le k$ such that $$\label{cond7}
j^k f_k(x_k) = j^k g_k(x_k), \quad (Rf_k)(x_k) \ne (Rg_k)(x_k).$$ Taking suitable subsequences we may assume that $${\left\lVertx_{k+1} - x_0\right\rVert} \le {\left\lVertx_k - x_0\right\rVert}/2$$ for all $k \in {\mathbb{N}}$ and that all $x_k$ are contained in an open neighborhood $W$ of $x_0$ which is relatively compact in $\Omega_1$ and convex. Furthermore, we can assume that either $x_k \ne x_0$ or $x_k = x_0$ holds for all $k \in {\mathbb{N}}$.
In the first case, choose points $y_k \in W$ with $x_0 \ne y_k \ne x_j$ for all $k,j \in {\mathbb{N}}$ such that $$\begin{aligned}
{\left\lVerty_k - x_k\right\rVert} & \le \frac{1}{k} {\left\lVertx_k-x_0\right\rVert}, \label{cond4} \\
\sup_{\xi \in \Omega} {\left| {\partial}_x^\alpha {\partial}_y^\beta (f_k-f)(y_k)(\xi) \right|} & \le 2{\varepsilon}(x_k)\quad (0 \le {\left|\alpha\right|} + {\left|\beta\right|} \le k), \label{cond5} \\
{\left|(Rg_k)(x_k) - (R f_k)(y_k) \right|} & \ge k {\left\lVertx_k - y_k\right\rVert}^{1/k}, \label{cond6} \\
\operatorname{supp}f_k(y_k) & \subseteq K. \label{cond6b}\end{aligned}$$ Such points $y_k$ can be chosen if each of these finitely many conditions holds for $y_k$ in some neighborhood of $x_k$. Conditions and obviously are without problems. For condition with fixed $\alpha$ and $\beta$ we note that ${\partial}_x^\alpha{\partial}_y^\beta (f_k-f)(W)$ is relatively compact (i.e., bounded) in ${\mathcal{D}}(\Omega_2)$. Hence, there exists a compact set $B_k \subseteq \Omega_2$ such that $\operatorname{supp}{\partial}_x^\alpha {\partial}_y^\beta (f_k - f)(W) \subseteq B_k$. In particular, ${\partial}_x^\alpha {\partial}_y^\beta (f_k-f)$ is uniformly continuous in $\overline{W} \times B_k$ and requirement is satisfied for $y_k$ in a small enough neighborhood of $x_k$. Finally, for we first note that by there is $\delta>0$ such that ${\left|(R f_k)(y_k) - (R g_k)(x_k ) \right|} \ge \delta$ for $y_k$ in a small neighborhood of $x_k$ by continuity of $R f_k$. Moreover, we have $$k {\left\lVertx_k - y_k\right\rVert}^{1/k} \le \delta \Longleftrightarrow {\left\lVertx_k - y_k\right\rVert} \le (\delta/k)^k$$ which gives for $y_k$ in a small enough neighborhood of $x_k$.
Next, we want to construct a function $h \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ such that $$({\partial}_x^\alpha{\partial}_y^\beta h)(x)(\xi) = \left\{
\begin{aligned}
& ({\partial}_x^\alpha{\partial}_y^\beta g_k)(x)(\xi) & \qquad &x=x_k \textrm{ for some }k \in {\mathbb{N}},\\
& ({\partial}_x^\alpha{\partial}_y^\beta f_k)(x)(\xi) & \qquad &x=y_k \textrm{ for some }k \in {\mathbb{N}},\\
& ({\partial}_x^\alpha{\partial}_y^\beta f)(x)(\xi) & \qquad &x = x_0
\end{aligned}
\right.\label{rhs}$$ for all $\alpha,\beta \in {\mathbb{N}}_0^n$ and $\xi \in \Omega_2$. For this purpose we apply Whitney’s extension theorem to the family $h^{\alpha,\beta}(x)(\xi)$ defined by the right hand side of for $(x, \xi)$ in the compact set $A \times L$ where $A {\coloneqq}\{x_0\} \cup \{ x_k : k \in {\mathbb{N}}\} \cup \{ y_k : k \in {\mathbb{N}}\}$ and the compact set $L \subseteq \Omega_2$ is chosen such that $K \subseteq L^\circ$. Again, we have to verify , which is straightforward using Taylor’s formula in combination with , and .
Emplying a cut-off function as in the proof of Theorem \[thm\_peetre\], we obtain $h \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ as desired. Theorem \[thm\_peetre\] now implies $${\left| (Rh)(x_k) - (Rh)(y_k) \right|} = {\left| (R g_k) (x_k) - (R f_k) (y_k) \right|} \ge k {\left\lVertx_k-y_k\right\rVert}^{1/k}.$$ For large $k$ this gives a contradiction because $Rh$ is smooth and a fortiori locally Hölder continuous.
In the other case, i.e., $x_k = x_0$ for all $k$, our assumptions imply that $$\begin{gathered}
{\partial}_x^\alpha {\partial}_y^\beta f_k(x_0)(\xi) = {\partial}_x^\alpha {\partial}_y^\beta g_k(x_0)(\xi) = {\partial}_x^\alpha{\partial}_y^\beta f(x_0)(\xi) \label{blahber1} \\
(R f_k)(x_0) \ne (R g_k)(x_0) \label{blahber}\end{gathered}$$ for all $k \in {\mathbb{N}}$, $\xi \in \Omega_2$ and ${\left|\alpha\right|} + {\left|\beta\right|} \le k$.
Either $(Rf_k)(x_0)$ or $(Rg_k)(x_0)$ must be different from $(Rf)(x_0)$ for infinitely many values of $k$, hence without loss of generality we can assume that $(R f_k)(x_0) \ne (R f)(x_0)$. As in the previous case, we then choose a sequence $y_k \to x_0$ in an open convex neighborhood $W$ of $x_0$ which is relatively compact in $\Omega_1$ such that $$\begin{aligned}
{\left| (Rf_k)(y_k) - (R f)(x_0) \right|} & \ge k {\left\lVert y_k - x_0 \right\rVert}^{1/k}, \\
{\left\lVerty_{k+1} - x_0\right\rVert} & < {\left\lVerty_k - x_0\right\rVert}/2,\textrm{ and} \\
\sup_{\xi \in \Omega_2} {\left| {\partial}_x^\alpha {\partial}_y^\beta ( f_k - f)(y_k)(\xi)\right|} & \le \frac{1}{k} {\left\lVerty_k - x_0\right\rVert}^m \label{blahblu}\end{aligned}$$ for all ${\left|\alpha\right|} + {\left|\beta\right|} + m \le k$. is obtained using Taylor’s theorem as in the proof of Theorem \[thm\_peetre\]. Again using Whitney’s extension theorem together with a cut-off function in ${\mathcal{D}}(\Omega_2)$, we can construct a mapping $h \in C^\infty( \Omega_1, {\mathcal{D}}(\Omega_2))$ satisfying $$j^\infty h (y_k) = j^\infty f_k(y_k), \quad j^\infty h(x_0) = j^\infty f(x_0).$$ To summarize, by Theorem \[thm\_peetre\] we obtain $${\left| (R h)(y_k) - (R h)(x_0) \right|} = {\left| (R f_k)(y_k) - (R f) (x_0) \right|} \ge k {\left\lVerty_k - x_0\right\rVert}^{1/k}$$ in contradiction to Hölder continuity of $Rh$, which concludes the proof.
With this in place we are able to show the following (cf. [@zbMATH04036782 Theorem 3, p. 278]):
\[thm\_5\]Let $R \colon C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2)) \to C^\infty(\Omega_1)$ be local and smooth and suppose we are given $f \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$, $x_0 \in \Omega_1$ and a compact subset $K \subseteq \Omega_2$ such that $\operatorname{supp}f(x_0) \subseteq K$. Then there are $r \in {\mathbb{N}}$, a neighborhood $V$ of $x_0$ and $\kappa>0$ such that for all $x \in V$ and $g_1, g_2 \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ with
1. $\operatorname{supp}g_i(y) \subseteq K$ for $y$ in a neighborhood of $x$ and $i=1,2$,
2. $\sup_{\xi \in \Omega_2} {\left| {\partial}_x^\alpha {\partial}_y^\beta ( g_i - f)(x)(\xi)\right|} \le \kappa$ for $i=1,2$ and $0 \le {\left|\alpha\right|} + {\left|\beta\right|} \le r$,
the condition $j^r g_1(x) = j^r g_2(x)$ implies $(Rg_1)(x) = (Rg_2)(x)$.
Fix $R$, $f$, $x_0$ and $K$ as stated and assume the claim does not hold. With $r(k) {\coloneqq}2^{-k}$ there exists a sequence $x_k \to x_0$ and $f_k, g_k \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ with $$\begin{aligned}
& \operatorname{supp}f_k(y) \cup \operatorname{supp}g_k(y) \subseteq K \textrm{ for $y$ in a neighborhood of $x_k$}, \\
& \sup_{\xi \in \Omega_2} {\left| {\partial}_x^\alpha {\partial}_y^\beta ( f_k - f)(x_k)(\xi)\right|} \le e^{-r(k)},\textrm{ and} \\
& \sup_{\xi \in \Omega_2} {\left|{\partial}_x^\alpha {\partial}_y^\beta ( g_k - f)(x_k)(\xi)\right|} \le e^{-r(k)} \end{aligned}$$ for $0 \le {\left|\alpha\right|} + {\left|\beta\right|} \le k$ such that $$\label{cond8}
j^k g_k(x_k) = j^k f_k(x_k),\quad (R g_k)(x_k) \ne (R f_k)(x_k).$$ We may assume that ${\left\lVertx_{k+1} - x_0\right\rVert} \le {\left\lVert x_k - x_0 \right\rVert}/2$. We then construct $s \in C^\infty({\mathbb{R}}\times \Omega_1, {\mathcal{D}}(\Omega_2))$ such that $$\begin{aligned}
({\partial}_x^\alpha {\partial}_y^\beta s)(2^{-k}, x_k, \xi ) &= ({\partial}_x^\alpha {\partial}_y^\beta f_k)(x_k)(\xi), \\
({\partial}_x^\alpha {\partial}_y^\beta s)(0, x_0, \xi) &= ({\partial}_x^\alpha {\partial}_y^\beta f)(x_0)(\xi)\end{aligned}$$ for all $k \in {\mathbb{N}}$, $\xi \in \Omega_2$ and $\alpha,\beta \in {\mathbb{N}}_0^n$. Note that $A {\coloneqq}\{ ( 0, x_0 ) \} \cup \{ ( 2^{-k}, x_k) : k \in {\mathbb{N}}\}$ is compact in ${\mathbb{R}}\times {\mathbb{R}}^n$. The function $s$ is obtained by applying Whitney’s theorem to the family of functions $s^{l,\alpha,\beta}$ (with $l \in {\mathbb{N}}_0$ and $\alpha,\beta \in {\mathbb{N}}_0^n$) defined on $A \times L$, where $L \subseteq \Omega_2$ is any compact set such that $K \subseteq L^\circ$, by $$s^{l, \alpha, \beta}(t,x,\xi) {\coloneqq}\left\{
\begin{aligned}
& ({\partial}_x^\alpha {\partial}_y^\beta f_k)(x_k)(\xi) & &l=0 \textrm{ and } (t,x) = (2^{-k}, x_k)\textrm{ for some }k, \\
& ({\partial}_x^\alpha {\partial}_y^\beta f)(x_0)(\xi) & &l=0 \textrm{ and } (t,x) = (0, x_0), \\
&0 \quad& &l \ne 0.
\end{aligned}
\right.$$ The requirements for Whitney’s theorem then are easily verified and we obtain $\tilde s \in C^\infty({\mathbb{R}}\times {\mathbb{R}}^n \times {\mathbb{R}}^{n'})$ which, by multiplying it with a suitable cut-off function in ${\mathcal{D}}(\Omega_2)$ as before, gives $s$ as desired. Next, we define a map $$\begin{aligned}
\widetilde R \colon C^\infty( {\mathbb{R}}\times \Omega_1, {\mathcal{D}}(\Omega_2)) &\to C^\infty({\mathbb{R}}\times \Omega_1) \\
(\widetilde R h)(t,x) &{\coloneqq}R(h_t)(x)\end{aligned}$$ where $h_t \in C^\infty(\Omega_1, {\mathcal{D}}(\Omega_2))$ is given by $h_t(x) {\coloneqq}h(t,x)$. Obviously, $\widetilde R$ is local in the sense of Definition \[def\_local\]. Now $(R f_k)(x_k) = R(s(2^{-k}, .))(x_k)$ holds because ${\partial}_x^\alpha f_k(x_k, \xi) = {\partial}_x^\alpha s(2^{-k}, x_k, \xi)$ for all $\alpha,\beta \in {\mathbb{N}}_0^n$ and $\xi \in \Omega_2$ by the construction of $s$ above. Furthermore, $R(s(2^{-k},.))(x_k) = (\widetilde R s)(2^{-k}, x_k)$ by the definition of $\widetilde R$. We now define $\tilde g_k \in C^\infty({\mathbb{R}}\times \Omega_1, {\mathcal{D}}(\Omega_2))$ by $\tilde g_k(t,x,\xi) = g_k ( x, \xi)$ and see that $$({\partial}_t^l {\partial}_x^\alpha s)(2^{-k}, x_k, \xi) = ({\partial}_t^l {\partial}_x^\alpha \tilde g_k)(2^{-k}, x_k, \xi)$$ for $0 \le l + {\left|\alpha\right|} \le k$. Hence, $(\widetilde R s)(2^{-k}, x_k) = (\widetilde R \tilde g_k)(2^{-k}, x_k)$ holds for large values of $k$ by Lemma \[thelemma\]. Finally, $(\widetilde R \tilde g_k)(2^{-k}, x_k) = (Rg_k)(x_k)$. To summarize, we obtain $(Rf_k)(x_k) = (Rg_k)(x_k)$ for large $k$, which contradicts and concludes the proof.
Conclusion
==========
We have seen in Theorem \[thm\_peetre\] that $R(\vec\varphi)(x_0)$ does not depend on the entire germ of $\vec\varphi$ at $x_0$, but only on its $\infty$-jet. Moreover, the statement of Theorem \[thm\_5\] may be reworded as follows: if $R$ is smooth and local and we are given $\vec\varphi$ and $x_0$, there is a neighborhood of $(\vec\varphi,x_0)$ and a natural number $r$ such that that for all $(\vec\psi, x)$ in this neighborhood, the value of $R(\vec\psi)(x)$ depends only on the $r$-jet of $\vec\psi$ at $x$.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was supported by the Austrian Science Fund (FWF) project P23714.
[10]{}
H. A. Biagioni. [*A nonlinear theory of generalized functions*]{}. 2nd ed. Berlin: Springer-Verlag, 1990. [isbn]{}: 978-3-540-52408-3. J. F. Colombeau. [*New generalized functions and multiplication of distributions*]{}. North-Holland Mathematics Studies 84. Amsterdam: North-Holland Publishing Co., 1984. [isbn]{}: 978-0-444-86830-5. J. F. Colombeau. [*Elementary introduction to new generalized functions.*]{} North-Holland Mathematics Studies 113. Amsterdam: North-Holland Publishing Co., 1985. [isbn]{}: 0-444-87756-8. M. Grosser, E. Farkas, M. Kunzinger, and R. Steinbauer. “On the foundations of nonlinear generalized functions I, II”. [*Mem. Am. Math. Soc.*]{} 729 (2001). [issn:]{} 0065-9266. [doi:]{} 10.1090/memo/0729. M. Grosser, M. Kunzinger, M. Oberguggenberger, and R. Steinbauer. [*Geometric theory of generalized functions with applications to general relativity.*]{} Mathematics and its Applications 537. Dordrecht: Kluwer Academic Publishers, 2001. [isbn:]{} 1-4020-0145-2. M. Grosser, M. Kunzinger, R. Steinbauer, and J. A. Vickers. “A Global Theory of Algebras of Generalized Functions”. [*Adv. Math.*]{} 166.1 (2002), pp. 50–72. [issn:]{} 0001-8708. [doi:]{} 10.1006/aima.2001.2018. A. Kriegl and P. W. Michor. [*The convenient setting of global analysis.*]{} Mathematical Surveys and Monographs 53. Providence, RI: American Mathematical Society, 1997. [isbn:]{} 0-8218-0780-3. R. Narasimhan. [*Analysis on real and complex manifolds.*]{} Vol. 35. North-Holland Mathematical Library. Reprint of the 1973 edition. North-Holland Publishing Co., Amsterdam, 1985. [isbn:]{} 0-444-87776-2. E. A. Nigsch. “The functional analytic foundation of Colombeau algebras”. [*J. Math. Anal. Appl.*]{} 421.1 (2015), pp. 415–435. [doi:]{} 10.1016/j.jmaa.2014.07.014. E. A. Nigsch. “Nonlinear generalized sections of vector bundles”. [*J. Math. Anal. Appl.*]{} 440 (2016), pp. 183–219. [issn:]{} 0022-247X. [doi:]{} 10.1016/j.jmaa.2016.03.022. M. Oberguggenberger. [*Multiplication of Distributions and Applications to Partial Differential Equations.*]{} Pitman Research Notes in Mathematics 259. Harlow, U.K.: Longman, 1992. [isbn:]{} 978-0-582-08733-0. L. Schwartz. “Sur l’impossibilité de la multiplication des distributions”. *Comptes Rendus de l’Académie des Sciences* 239 (1954), pp. 847–848. J. Slovák. “Peetre theorem for nonlinear operators.” [*Ann. Global Anal. Geom.*]{} 6.3 (1988), pp. 273–283. [issn:]{} 0232-704X. [doi:]{} 10.1007/BF00054575.
[^1]: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria. e-mail: <eduard.nigsch@univie.ac.at>
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abstract: 'In this work, we present a work in progress towards an efficient and economical computational module which interfaces between Cauchy and characteristic evolution codes. Our goal is to provide a standardized waveform extraction tool for the numerical relativity community which will allow CCE to be readily applied to a generic Cauchy code. The tool provides a means of unambiguous comparison between the waveforms generated by evolution codes based upon different formulations of the Einstein equations and different numerical approximation.'
author:
- 'M. C. Babiuc${}^{1}$, J. Winicour${}^{2,3}$ and Y. Zlochower${}^{4}$'
title: Binary Black Hole Waveform Extraction at Null Infinity
---
Introduction
============
The study of gravitational waves will brighten unexplored features of our universe that are otherwise invisible to conventional astronomy and will increase our knowledge about the very nature of time and space [@shultz]. Gravitational wave detectors are already operating, and results from the first LIGO and Virgo collaboration were recently published in Nature [@collab]. The signal predicted by numerical relativity will provide a template bank used for filtering the noise, indispensable to the success of gravitational wave detectors such as LIGO, Virgo, and LISA. The current sensitivity levels of the detectors will be improved substantially in next-generation detection estimated by 2015. Although existing simulations are sufficiently accurate for populating the parameter space in current searches of ground-based detectors, the new generation of advanced detectors will be 10-15 times more sensitive by 2015.
Ideally, the emitted gravitational wave signature should be extracted at spatial or null infinity. However, most present codes impose artificial, finite outer boundaries and are performing the waveform extraction at finite radius. This method introduces systematic errors, especially for higher modes, which is the main obstacle in reaching the desired accuracy. With the exception of Pretorius [@pretorius], who uses coordinates that compactify spatial infinity, all the other codes use a computational domain with a finite outer boundary and Sommerfeld-like approximate outer boundary conditions must be imposed, which introduce errors in the computation of gravitational waves. The choice of proper boundary conditions is complicated by gauge freedom and constraint preservation [@lehnmor]. The emitted gravitational wave signature is calculated at finite distance, using either the Newman-Penrose Weyl scalar $\psi_4$ [@NP], or the odd and even parity functions $Q_+$, $Q_x$ in the Zerilli-Moncrief formalism [@moncrief]. The strain $h$ of the wave used in detection is obtained performing one time integration from the Zerilli-Moncrief multipoles, and two time integrations from the Newman-Penrose curvature. The waveform is affected by gauge ambiguities which are magnified by the integration [@lindblom].
Cauchy-characteristic extraction (CCE) [@cce], which is one of the pieces of the CCM strategy [@livccm], offers a means to avoid the error introduced by extraction at a finite world-tube. In CCE, the inner world-tube data supplied by the Cauchy evolution is used as boundary data for a characteristic evolution to future null infinity $\cal I^+$, where the waveform can be unambiguously computed by geometric methods. This characteristic initial-boundary value problem based upon a timelike world-tube [@tam] has been implemented as a mature evolution code, the PITT null code [@isaac; @highp], which incorporates a Penrose compactification of the space-time. By itself, CCE does not use the characteristic evolution to inject outer boundary data for the Cauchy evolution, which can be a source of instability in full CCM.
The PITT code has been tested to be second order convergent in a wide range of testbeds extending from the perturbative regime [@babiuc05] to highly nonlinear single black hole spacetimes [@highp]. However, in cases which require high resolution, such as the inspiral of matter into a black hole, the error in CCE has been a troublesome factor in the postprocessing phase [@partbh]. This has motivated a recent project [@strat] to increase the accuracy of the PITT code. Other results achieved with previous versions of the PITT have been recently reported [@reis1; @reis2]. Recently, the code underwent major improvements and corrections to previous versions, to improve accuracy and convergence [@xtract].
Here we test this improved version of CCE on a realistic application involving a Cauchy evolution of the inspiral and merger of two equal mass non-spinning black holes. We use the same code specifications described in [@strat] except that the accuracy of angular derivatives has been increased to a 4th order finite difference approximation. The results presented here are a work in progress towards our goal to develop CCE as a reliable and accurate waveform extraction tool for the numerical relativity community. This paper addresses the first two objectives:
- To create a robust and flexible interface between a binary black-hole Cauchy evolution code and a characteristic code for wave extraction at infinity.
- To prove the robustness of the interface by performing precise computations of gravitational waveforms at infinity from binary black-hole, using this Cauchy-characteristic extraction approach.
We construct an interface that takes the Cartesian data from a Cauchy evolution and converts it into boundary data on a spherical grid for the characteristic evolution. The data are evolved to future null infinity, where it is used to compute the gravitational waveform. The flexibility of the interface is due to implementation of a spectral decomposition of data. This implementation has been tested with a realistic application involving a binary black-hole inspiral. In Sec. \[sec:formalism\] we review the formalism underlying CCE, including enough details of the patching, evolution and extraction, to make clear the difficulties underlying the calculation of an accurate waveform at ${\cal I}^+$. In Sec. \[sec:initdata\] we briefly describe the initial data for Cauchy and characteristic evolution. In Sec. \[sec:interface\], we present the details of the CCE interface which allows the data from a Cauchy evolution to be used as boundary data on an inner worltube for a characteristic evolution to ${\cal I}^+$, where the waveform is extracted. In Sec. \[sec:results\], we test the CCE interface by extracting the waveform from a Cauchy evolution of a binary black-hole inspiral and merger, and by comparing it to the waveform obtained by other standard method in current practice.
The CCE Formalism {#sec:formalism}
=================
Cauchy-Characteristic Patching {#sec:cform}
------------------------------
Characteristic data are provided by the Cauchy evolution on a world-tube ${\cal WT}$, free initial data being given on the initial null hypersurface ${\cal N_I}$, which sets the metric on the entire initial cone (fig. 1).
![Cauchy and characteristic evolution are patched in the vicinity of a world-tube ${\cal WT}$, embedded in Cauchy evolution[]{data-label="fig:Patching"}](Figure01){width="6cm"}
The metric data from a Cauchy evolution are interpolated onto a timelike inner world-tube to extract the boundary data for the characteristic evolution. The characteristic evolution is embedded into the Cauchy evolution and is extending to future null infinity ${\cal I}^+$, where the waveform can be unambiguously computed using the geometric methods developed by Bondi et al [@bondi], Sachs [@sachsr] and Penrose [@Penrose]. The extraction process involves carrying out the complicated Jacobian transformation between the Cartesian coordinates used in the Cauchy evolution and the spherical null coordinates used in the characteristic evolution (the full details are given in [@ccm].)
Characteristic Evolution {#sec:cform}
------------------------
The characteristic formalism is based upon a family of outgoing null hypersurfaces, emanating from some inner world-tube, which extend to infinity where they foliate ${\cal I}^+$ into spherical slices. We let $u$ label these hypersurfaces, $x^A$ $(A=2,3)$ be angular coordinates which label the null rays and $r$ be a surface area coordinate. (fig. 2).
![Ongoing null hypersurfaces emanating from the world-tube and extending to ${\cal I}^+$[]{data-label="fig:Timelike"}](Figure02){width="6cm"}
In the resulting $x^\alpha=(u,r,x^A)$ coordinates, the metric takes the Bondi-Sachs form [@bondi; @sachsr] $$\begin{aligned}
ds^2 & = & -\left(e^{2\beta}\frac{V}{r} -r^2h_{AB}U^AU^B\right)du^2
-2e^{2\beta}dudr \nonumber \\
& -& 2r^2 h_{AB}U^Bdudx^A
+ r^2h_{AB}dx^Adx^B,
\label{eq:bmet}\end{aligned}$$ where $h_{AB}$ is the Bondi-Sachs conformal 2-metric with $h^{AB}h_{BC}=\delta^A_C$. The code introduces an auxiliary unit sphere metric $q_{AB}$, with associated complex dyad $q_A$ satisfying $ q_{AB} =\frac{1}{2}\left(q_A \bar q_B+\bar q_Aq_B\right)$. For a general Bondi-Sachs metric, the full nonlinear $h_{AB}$ is uniquely determined by the dyad component $J=h_{AB}q^Aq^B/2$, since the other dyad component $K=h_{AB}q^A \bar q^B /2$ is constrained by the determinant condition $1=K^2-J\bar J$. The spherically symmetric case characterized by $J=0$. We introduce the spin-weighted fields $U=U^Aq_A$ and $Q=Q_Aq^A$, where $$Q_A = r^2 e^{-2\,\beta} h_{AB} U^B_{,r}.$$ as well as the (complex differential) operators $\eth$ and $\bar \eth$. Refer to [@eth; @cce] for further details regarding numerical implementation. The auxiliary variables $$\nu =\eth J \, , \quad B=\eth \beta \, , \quad k=\eth K
\label{eq:aux}$$ are also introduced to eliminate all second angular derivatives. In certain applications this has been found to give rise to increased accuracy by suppressing short wavelength error [@gomezfo].
In this formalism, the Einstein equations $G_{\mu\nu}=0$ decompose into hypersurface equations, evolution equations and conservation conditions on the inner world-tube. As described in more detail in [@newt; @nullinf], the hypersurface equations take the form $$\begin{aligned}
\beta_{,r} &=& N_\beta,
\label{eq:beta} \\
U_{,r} &=& r^{-2}e^{2\beta}Q +N_U,
\label{eq:wua} \\
(r^2 Q)_{,r} &=& -r^2 (\bar \eth J + \eth K)_{,r}
+2r^4\eth \left(r^{-2}\beta\right)_{,r} + N_Q,
\label{eq:wq} \\
V_{,r} &=& \frac{1}{2} e^{2\beta}{\cal R}
- e^{\beta} \eth \bar \eth e^{\beta}
+ \frac{1}{4} r^{-2} \left(r^4
\left(\eth \bar U +\bar \eth U \right)
\right)_{,r} + N_V,
\label{eq:ww}\end{aligned}$$ where [@eth] $${\cal R} =2 K - \eth \bar \eth K + \frac{1}{2}(\bar \eth^2 J + \eth^2 \bar J)
+\frac{1}{4K}(\bar \eth \bar J \eth J - \bar \eth J \eth \bar J)
\label{eq:calR}$$ is the curvature scalar of the 2-metric $h_{AB}$. Those equations have a hierarchical structure in $[J,\beta,Q,U,V]$ such that the right hand sides, e..g. $N_\beta[J]$ only depend upon previous variables and their derivatives intrinsic to the hypersurface.
The evolution equation takes the form $$\begin{aligned}
2 \left(rJ\right)_{,ur}
- \left(r^{-1}V\left(rJ\right)_{,r}\right)_{,r} =
-r^{-1} \left(r^2\eth U\right)_{,r}
+ 2 r^{-1} e^{\beta} \eth^2 e^{\beta}- \left(r^{-1} V \right)_{,r} J
+ N_J,
\label{eq:wev}\end{aligned}$$ where, $N_\beta$, $N_U$, $N_Q$, $N_V$ and $N_J$ are nonlinear terms which vanish for spherical symmetry. Expressions for these terms as complex spin-weighted fields and a discussion of the conservation conditions are given in [@cce].
The characteristic Einstein equations are evolved in a domain between an inner radial boundary at the interior world-tube, and an outer boundary at future null infinity. The characteristic evolution code implements this formalism as an explicit finite difference scheme, based upon the compactified radial coordinate $$\xi=\frac{r}{R_E +r}
\label{eq:compx}$$ so that ${\xi}=1$ at ${\cal I}^+$. Here $R_E$ is a parameter based upon the extraction world-tube, which in the CCE module is chosen as the radius of the extraction world-tube, as determined by $R^2=\delta_{ij}x^i
x^j$ in terms of the Cartesian coordiates $x^i$ used in the Cauchy evolution code. The boundary data for $J$, $\beta$, $U$, $Q$, and $V$ on the world-tube supply the integration constants for a radial numerical integration of the hypersurface Einstein equations. The finite difference scheme for integrating the hypersurface and evolution equations is based on the marching equation for a spherically symmetric scalar field $\Phi$: $$\Phi_{\bf N}-\Phi_{\bf W}-\Phi_{\bf E}+\Phi_{\bf S} = -{1\over 2}\int_{\Sigma} \left ({V \over r} \right)_{,r} {\Phi \over r} du dr
\label{eq:march}$$ where the point N is the “new” point in the evolution scheme, and $V$ is defined by the spherically symmetric version of the Bondi-Sachs metric given above. The evolution scheme in the full gravitational case used to determine the metric at the next point on the null hypersurfaces is modeled after this example (see [@highp; @gomezfo] for details).
![The null parallelogram WSEN used to determine the field values at point N, as described by (\[eq:march\]).[]{data-label="fig:Null"}](Figure03){width="6cm"}
Gravitational Radiation Calculation {#sec:wave}
-----------------------------------
The theoretical derivation of the waveform at infinity is carried out in terms of an inverse surface-area coordinate $\ell=1/r$, where $\ell=0$ at ${\cal I}^+$. In the resulting $x^\mu=(u,\ell,x^A)$ coordinates, the physical space-time metric $g_{\mu\nu}$ (\[eq:bmet\]) has the conformal compactification $\hat g_{\mu\nu}=\ell^{2} g_{\mu\nu}$, where $\hat g_{\mu\nu}$ is smooth at ${\cal I}^+$ and takes the form [@tam] $$\hat g_{\mu\nu}dx^\mu dx^\nu=
-\left(e^{2\beta}V \ell^3 -h_{AB}U^AU^B\right)du^2
+2e^{2\beta}dud\ell -2 h_{AB}U^Bdudx^A + h_{AB}dx^Adx^B.
\label{eq:lmet}$$ As described in [@strat], the Bondi news function $N(u,x^A)$ and the Newman-Penrose Weyl tensor component $\Psi(u,x^A)=\lim_{r\rightarrow \infty} r \psi_4$ which describe the waveform are both determined by the asymptotic limit at ${\cal I}^+$ of the tensor field $$\hat \Sigma_{\mu\nu} = \frac{1}{\ell}(\hat \nabla_\mu\hat \nabla_\nu -\frac{1}{4}\hat g_{\mu\nu} \hat \nabla^\alpha\hat \nabla_\alpha)\ell.
\label{eq:Sigma}$$ constructed from the leading coefficients in an expansion of the metric in powers of $\ell$ $$\begin{aligned}
h_{AB}&= &H_{AB}+\ell c_{AB}+O(\ell^2),
\\
\beta&=&H+ O(\ell^2) ,
\\
U^A&=& L^A+2\ell e^{2H} H^{AB}D_B H+O(\ell^2) ,
\\
\ell^2 V&=& D_A L^A+\ell (e^{2H}{\cal R}/2 +D_A D^A e^{2H})+O(\ell^2),\end{aligned}$$ where ${\cal R}$ and $D_A$ are the 2-dimensional curvature scalar and covariant derivative associated with $H_{AB}$.
The expansion coefficients $H$, $H_{AB}$, $c_{AB}$ and $L^A$ (all functions of $u$ and $x^A$) completely determine the radiation field. Before the gravitational radiation is calculated from the metric in the neighborhood of ${\cal I}^+$, it is necessary to determine the conformal factor $\omega$ relating $H_{AB}$ to a unit sphere metric $Q_{AB}$, i.e. to an inertial conformal Bondi frame [@tam] satisfying $$Q_{AB}=\omega^2H_{AB}.
\label{eq:unsph}$$ The news function $N(u,x^A)$ is directly computed by the code in terms of the computational coordinates $(u,x^A)$, as opposed to the inertial coordinates $(\tilde u,y^A)$ on ${\cal I}^+$ corresponding to an idealized distant observatory. The transformation to inertial coordinates proceeds first by introducing the conformally rescaled metric $\tilde g_{\mu\nu} = \omega^2 \hat g_{\mu\nu}$ in which the cross-sections of ${\cal I}^+$ have unit sphere geometry, in accord with (\[eq:unsph\]). Then the rescaled null vector $\tilde n^\mu =
\omega^{-1} \hat n^\mu$ is the generator of time translations on ${\cal
I}^+$, i.e. $\tilde n^\mu \partial_\mu = \partial_{\tilde u}$. The inertial coordinates thus satisfy the propagation equations $$\hat n^\mu \partial_\mu \tilde u = \omega \, , \quad
\hat n^\mu \partial_\mu y^A =0,
\label{eq:inertialc}$$ where $\hat n^\mu\partial_\mu =e^{-2H}(\partial_u + L^A
\partial_{x^A})$ in terms of the computational coordinates. The inertial coordinates are obtained by integrating (\[eq:inertialc\]), thus establishing a second pair of stereographic grid patches corresponding to $y^A$. Then the news function is transformed into $N(\tilde u, y^A)$.
The Bondi news function $N$ is given by (\[eq:news\]), $$N={1\over 4}e^{-2i \delta}\omega^{-2}e^{-2H}F^A F^B
\{(\partial_u+{\pounds_L})c_{AB}-{1\over 2}c_{AB} D_C L^C
+2\omega D_A[\omega^{-2}D_B(\omega e^{2H})]\},
\label{eq:news}$$ where $\pounds_L$ is the Lie derivative with respect to $L^A$. The Newman-Penrose Weyl tensor component $\Psi$ is given by (\[eq:psia\]) $$\Psi=\frac{1}{2} \omega^{-3}e^{-2i\delta}
\hat n^\mu F^A F^B \bigg(
\partial_\mu \hat \Sigma_{AB}
-\partial_A \hat \Sigma_{\mu B}
- \hat \Gamma^\alpha_{\mu B}\hat \Sigma_{A \alpha}
+\hat \Gamma^\alpha_{A B}\hat \Sigma_{\mu\alpha}
\bigg)|_{\cal I^+} .
\label{eq:psia}$$ In the inertial Bondi coordinates, the expression for the news function (\[eq:news\]) reduces to the simple form $$N={1\over 4}{\cal Q}^A {\cal Q}^B \partial_u c_{AB},
\label{eq:inews}$$ and (\[eq:psia\]) reduces to the single term $$\Psi = \frac {1}{4} Q^A Q^B\partial_u^2 c_{AB} = \partial_u^2
\partial_l J|_{{\cal I}^+} .$$ This is related to the expression for the news function in inertial Bondi coordinates by $$\Psi =\partial_u N.
\label{eq:PsiNu}$$ Equation (\[eq:PsiNu\]) holds true in the linearized approximation of the Einstein equations. In the nonlinear case, the full expression for news and $\Psi$ must be used in the code. This introduces additional challenges to numerical accuracy due to high order angular derivatives of $\omega$ and large number of terms.
Initial Data {#sec:initdata}
============
Initial Cauchy Data {#sec:initCauchy}
-------------------
For the Cauchy evolution we used the LazEv code [@Campanelli:2005dd; @Zlochower:2005bj] along with the Cactus framework [@cactus_web] and Carpet [@Schnetter:2003rb] mesh refinement driver. LazEV is an eighth-order-accurate finite-difference code based upon the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation [@bssn1; @bssn2] of Einstein’s equations, which deals with the internal singularities by the moving puncture approach [@Campanelli:2005dd; @Baker:2005vv]. Our simulation used 9 levels of refinement with finest resolution of $h= M/53.76$, and outer Cauchy boundary at $400M$. The initial data consisted of a close quasicircular black-hole binary with orbital frequency $M\Omega = 0.050$, leading to more than a complete orbit before merger (see [@Campanelli:2006uy]). We output the metric data on the extraction world-tube every $\Delta t = M/32$.
Initial Characteristic data {#sec:initchar}
---------------------------
The initial data for the characteristic evolution consist of the values of $J$ on the initial hypersurface $u=0$. One way of supressing incoming radiation in the data would be to set the Newman-Penrose Weyl tensor component $\Psi_0=0$ on the initial null hypersurface. For a perturbation of the Schwarzschild metric, this condition implies no incoming radiation in the linearized approximation. However, in order to avoid shocks arising from incompatibility with the Cauchy data on the extraction world tube $\xi=\xi_E$ (with $\xi$ given by \[eq:compx\]), we also need to require that $J$ and $\partial_\xi J$ are continuous. In the linearized approximation, the condition that $\Psi_0=0$ implies that $\partial^2_\xi J=0$. The combination of those requirements leads to $J=J|_{\xi_E} + (\partial_\xi J)|_{\xi_E} (\xi-\xi_E)$, which would imply that $J\ne 0$ at ${\cal I}^+$. For technical simplicity we avoid this complication by initializing $J$ according to $$J = J|_{\xi_E}\frac {(\xi-1)}{(\xi_E-1)},
\label{eq:initJ}$$ which matches the Cauchy data and the derivatives at $\xi=\xi_E$ and is consistent with asymptotic flatness. Since this choice of $J$ vanishes at infinity, the initial slice of ${\cal I}^+$ has a unit sphere metric so that the conformal factor has the simple initialization $\omega(0,p,q) =1$.
Given the initial data (\[eq:initJ\]), this leads to complete knowledge of the metric on the initial null cone. Then (\[eq:wev\]) gives an expression for $J_{,ur}$, which is used to determine $J$ on the “next” null cone, so that the process can be repeated to yield the complete metric throughout the domain, which extends to ${\cal I}^+$.
Computational interface {#sec:interface}
=======================
We have designed an interface that takes Cartesian grid data from a Cauchy evolution and converts it into boundary data for characteristic evolution on a spherical grid extending to ${\cal I}^+$. We treat each component $g_{\mu\nu}(t,x^i)$ of the Cauchy metric as a scalar function in the $x^i$ Cartesian coordinates which are used in the $3+1$ evolution.
In order to make the interface as flexible as possible for future development as a community tool for waveform extraction, we have based it upon a spectral decomposition of the Cauchy data in the region between two world tubes or radii $R=R_1$ and $R=R_2$, where $R=\sqrt{\delta_{ij}x^i x^j}$ is the Cartesian coordinate radius. Then at a given time $t=T$, we decompose $g_{\mu\nu}(T,x^i)$ in terms of Tchebychev polynomials of the second kind $U_k(R)$ and spherical harmonics $Y_{l m}(\theta,\phi)$, where $(\theta,\phi)$ are related to $x^i/R$ in the standard way. The Tchebychev polynomials are conventionally defined as functions $U_k(\tau)$ on the interval $-1 \le
\tau \le 1$. Here we map them to the interval $R_1 \le R \le R_2$ by the transformation $$\tau (R)=\frac{2R-R_1 -R_2}{R_2 -R_1} .$$ where the extraction shell thickness is determined by the number $k_{Max}$ of Tchebychev polynomials used. (In tests of binary black holes with mass M we use a relatively small range $R_2 -R_1=10M$, a larger value of $k_{Max}$ would be needed if the range were expanded). Thus, for $R_1<R<R_2$, we expand $$g_{\mu\nu} (T,x^i) = \sum_{k l m} C_{\mu \nu [k l m]}
U_k (R) Y_{l m}(\theta,\phi).$$
For the applications to waveform extraction given in this paper, it is sufficient to consider $l \le l_{Max}$, where $l_{Max} =6$, and $k\le k_{Max}$, where $k_{Max} =6$. The coefficients $C_{\mu \nu [klm]}$ then allow us to reconstruct a spherical harmonic decomposition of each component of the Cauchy metric on the extraction world-tube $R=R_E$, i.e. $$g_{\mu\nu [l m]}(T,R_E)
= \sum_{k} C_{\mu \nu [k l m]} U_k (R_E) .$$ This decomposition is carried out at a sequence of Cauchy time steps $T_N=T_0 +N\Delta T$, where $\Delta T$ is chosen to be much smaller than the characteristic time scale of the problem but, for purposes of economy, larger than the time step used for the Cauchy evolution. A fifth-order polynomial interpolation is carried out locally over the $T_N$ to provide characteristic boundary data at any time $t$ in analytic form.
The extraction module also requires the derivatives $\partial_t
g_{\mu\nu}$ and $\partial_R g_{\mu\nu}$ at the extraction world-tube. The $t$-derivative is constructed by a fourth-order-accurate finite-difference stencil using the surrounding Cauchy times $t=T_N$. The $R$-derivative is obtained analytically, at each time level $T_N$, by differentiation of the Tchebychev polynomials.
The spherical harmonic interpolator from the Cartesian to the spherical coordinates is part of the extraction module, but its resolution is controlled by the Cauchy evolution.
The stereographic coordinates $x^A=(q,p)$ used to label the outgoing null rays in the Bondi metric are matched to the spherical coordinates $(\theta,\phi)$ induced by the Cartesian Cauchy coordinates on the extraction worldtube by a standard transformation, using the conventions in [@eth]. The value of the surface-area coordinate $r$ in the Bondi-Sachs metric is obtained on the extraction world-tube from the 2-determinant of the Cartesian metric on the surfaces $t=T_N,R=R_E$. As a result the radius of the Bondi coordinate $r\ne const$ on the extraction world-tube. The metric has to be calculated at a common value of the surface coordinate $r$, because the original Cauchy extraction was at constant R. In order to make this calculation possible, the transformation from Cartesian coordinates $(t, x^i)$ to Bondi-Sachs coordinates $(u,r,x^A)$ is carried out via an intermediate Sachs coordinate system $(u,\lambda,x^a)$ [@sachsr] where $\lambda$ is an affine parameter along the outgoing null rays. The affine freedom allows us to set $\lambda=0$ on the extraction world-tube $R=R_E$. After carrying out the Jacobian transformation from $(t,x^i)$ to $(u,\lambda,x^A)$, the Cartesian metric and its first derivatives at the extraction world-tube provide a first-order Taylor expansion in $\lambda$ (about $\lambda =0$) of the null metric in Sachs coordinates. The corresponding Taylor expansion of the metric in Bondi-Sachs coordinates then follows from the computed value of $r$ and $\partial_\lambda r$ at $\lambda=0$, which are obtained from the 2-determinant of the Cartesian metric. In order to obtain a first-order Taylor expansion for the Bondi metric variable $\beta$, the hypersurface equation (\[eq:beta\]) must be used to evaluate $\partial_r \beta$ at the extraction world-tube. All other metric variables are then initialized consistent with second order accuracy. Taylor expansions are also needed to start up the radial integration equations for the auxiliary variables (\[eq:aux\]) used to convert angular derivatives to first-order form. These expansions are obtained from applying the $\eth$-operator to the Taylor expansion of the underlying metric. This is a complicated process because the $\eth$ operator intrinsic to the $\lambda=0$ extraction world-tube is not the same as the $\eth$ operator intrinsic to the $r=const$ Bondi spheres (see [@xtract] for a discussion). The low order intermediate Taylor expansions limit the accuracy of the result. A new approach that avoids entirely the use of the Taylor expansion and gives better accuracy is presented in [@xtract]. In the original approach, used for this paper, the resulting Taylor expansion of the evolution variables is used to fill the points of the Bond-Sachs grid to start the integration of the characteristic hypersurface and evolution equations (\[eq:beta\]) - (\[eq:wev\]). The integration proceeds from the extraction world tube to ${\cal I}^+$ on a radial grid based upon the compactified $x$-coordinate (\[eq:compx\]).
Domain of dependence considerations place a constraint between the characteristic time step $\Delta u$ and the size of the characteristic grid analogous to the CFL condition for the Cauchy evolution. For an estimate, consider the Minkowski space case with the conformally rescaled metric $$ds^2=-\frac{(1-\xi)^2}{R_E^2} du^2 -\frac {2}{R_E} du d\xi
+ q_{AB} dx^A dx^B$$ where the unit sphere metric takes the form $$q_{AB} dx^A dx^B = \frac{4}{1+p^2 +q^2} (dp^2 + dq^2).$$ The past light cone is determined by $$\frac {du}{R_E} = \frac {-d\xi
- \sqrt {d\xi^2+(1-\xi)^2 q_{AB} dx^A dx^B}} {(1-\xi)^2}.$$ For typical characteristic grid parameters, $\Delta p = \Delta q =
\Delta \xi /4$, the resulting restriction is $$\frac {|\Delta u|}{R_E} < 8 \Delta \xi
\label{eq:cfl}$$ For a Cauchy simulation of a binary black-hole system of total mass $M$ with timestep $\Delta t =M/32$ (sufficient to describe the typical frequencies of a binary system), (\[eq:cfl\]) leads to $$\frac {M}{256 R_E} < \Delta \xi ,$$ for the choice of characteristic timestep $\Delta u = \Delta t$. The corresponding number of radial gridpoints must roughly satisfy $N_\xi < 128
R_E/M$. This places no limit of practical concern on the resolution of the characteristic evolution even for the small extraction radius $R_E =20 M$. Thus, for purposes of CCE, there are no demanding CFL restrictions.
The interface was debugged and calibrated using the analytic Schwarzschild metric in Kerr-Schild coordinates $(t,x^i)$, $$g_{\mu\nu} = \eta_{\mu\nu}+\frac{2m}{r}k_\mu k_\nu ,$$ where $k_\mu=(-1,x^i/r)$.
Results {#sec:results}
=======
We present results for the characteristic extracted waveform either in terms of $\Psi$, related to the Bondi news by $\Psi=\partial_u N$ in the linearized regime, or, when comparing to the perturbative waveform, in terms of the Newman-Penrose component $\psi_4$. The relationship between the Cauchy and the characteristic waveforms is: $(R-2M)\psi_4=-2\bar \Psi$. We decompose the signal in $l=10$ spherical harmonic modes but, for illustrative purposes, we concentrate on the dominant $(2,2)$ and sub-dominant $(4,4)$ modes. The Cauchy data were given at the extraction radii $R~=~20M,~50M,~100M$. The relationship between the Cauchy radius $R$ and the characteristic world-tube radius $R_E$ is: $R_E/R=1 + 1/R +1/(4R^2)$. The characteristic extraction module was run with the following specifications: angular gridpoints = radial gridpoints = $60,~120,~240$, and timestep $\Delta u~=~8 \Delta t,~4 \Delta t,~2 \Delta t$, where $\Delta t=M/32$. The test was run until $t/M=385$, using $4^{th}$-order accurate angular derivatives, on stereographic patches with circular boundaries and angular dissipation $\epsilon_{Jx} = 0.001$ (see [@strat] for details on how the angular dissipation is added to the evolution equation \[eq:wev\]). The results are shown for the highest resolution. Table \[bbhconvwt\] gives the convergence rates for the world-tube variables obtained with a small extraction radius $R_E=20M$ at a time corresponding to the peak of the signal ($t\approx 200M$). The rates are given for the real and the imaginary part. All quantities are very close to second order convergent, including $J_{,x}$, which is the term which determines the waveform.
$Variable$ $ Rate_{Re} $ $ Rate_{Im} $
------------ --------------- ---------------
$\beta$ $2.02$ $2.01$
$J$ $2.03$ $2.00$
$J_{,x}$ $2.04$ $1.99$
$Q$ $2.02$ $2.04$
$U$ $2.02$ $2.02$
$W$ $2.01$ $2.04$
: Convergence rates of the $l=2,m=2$ mode for the metric variables measured near the peak of the signal ($t\approx200/M$) at the world-tube, for an extraction radius $R=20M$.
\[bbhconvwt\]
We are reporting only first-order convergence rates at future null infinity $\cal I^+$ for the Bondi News $B$ and the Weyl complonent $\Psi$, but the error is relatively small ($0.5\%$ during the late inspiral). From an extraction point of view, these errors are smaller than the error in the Cauchy code data and are of little concern. The data are not convergent at early time when high frequencies dominate the error. For a thorough analysis of the causes for the first-order accurate results and major improvements to the code see [@xtract].
Figure \[fig:Comp22\] compares the imaginary and real parts of the $(l, m) = (2, 2)$ mode of the Cauchy $\psi_4$ with the complex conjugate of the $(l, m) = (2,-2)$ mode of the characteristic $\Psi$. We obtain very good amplitude match. Also, we observe improved phase agreement as the extraction radius is increased (from $50M$ to $100M$), because the phase error in $\psi_4$ is reduced with increased extraction radius.
Figure \[fig:Comp44\] compares the imaginary and real parts of the $(l, m) = (4, 4)$ mode of the Cauchy $\psi_4$ with the complex conjugate of the $(l, m) = (4,-4)$ mode of the characteristic $\Psi$. Here we see two effects. First the improved phase agreement as $R_E\to \infty$, but also an attenuation of the amplitude due to dissipation of higher-order modes. Also, the noise is apparent for the $(4,4)$ mode.
Figure \[fig:CompAP\] compares the amplitudes and the phases between the absolute value of the $(l, m) = (2, 2)$ mode of the Cauchy $\psi_4$ extracted at $R=50$, and the absolute value of the $(l, m) = (2,-2)$ mode of the characteristic $\Psi$ for the same extraction radius, at the highest resolution ($N=200$). The difference in amplitude is relatively small, maximum $0.17\%$ of the Cauchy $\psi_4$ amplitude in the wave zone.
Figure \[fig:CompRad\] compares the real part of the $(l, m) = (2,-2)$ modes of the characteristic $\Psi$ extracted at three different extraction radii: $R=20M$, $R=50M$ and $R=100M$. The waveform extracted at $R=20$ has the biggest amplitude, and a very small attenuation of the signal with the radius is observed.
![A plot that compares the phase in the $(l=2,m=2)$ mode of the Cauchy $\psi_4$ as calculated using the Null code and the Cauchy code. All plots were translated so that the time of the maximum in the amplitude agree.[]{data-label="fig:Comp22"}](Figure04){width="12cm"}
![A plot that compares the phase in the $(l=4,m=4)$ mode of the Cauchy $\psi_4$ as calculated using the Null code and the Cauchy code. All plots were translated so that the time of the maximum in the amplitude agree.[]{data-label="fig:Comp44"}](Figure05){width="12cm"}
![A plot of that compares the amplitudes and the phases between the $(l, m) = (2, 2)$ mode of the Cauchy $\psi_4$ extracted at $R=50$, and the $(l, m) = (2,-2)$ mode of the characteristic $\Psi$ for the same extraction radius.[]{data-label="fig:CompAP"}](Figure06){width="12cm"}
![A plot of that compares the real part of the $(l, m) = (2,-2)$ mode for the characteristic $\Psi$ extracted at three different radius (world-tubes). The waveforms are translated such that the maximum of the amplitude corresponds to t/M=0[]{data-label="fig:CompRad"}](Figure07){width="12cm"}
Conclusion
==========
We have presented here a method for interfacing outer boundary data from a Cauchy evolution with inner boundary data for a characteristic evolution so that the waveform can be accurately extracted at infinity. We have demonstrated how the PITT null code can be interfaced with the LazEv code, which is a finite-difference BSSN code, to produced calibrated waveforms from a binary black-hole inspiral. The extraction interface has been implemented as a thorn in the Einstein Computational Toolkit [@einsteintool] In this paper we are reporting only preliminary results (see [@xtract] improvements). Although we are aware of the deficiencies in the characteristic waveform extraction tool presented here, there is pressing interest from several numerical relativity groups to apply the tool to extract waveforms from binary black-hole inspirals.
[34]{} B.S. Sathyaprakash, and B. F. Shultz, [*Living Rev. Rel.*]{}, [**12**]{} 2 (2009), arXiv:0903.0338 The LIGO Scientific Collaboration and the Virgo Collaboration, [*Nature*]{}, [**Vol. 460**]{} 990 (2009) F. Pretorius, [*Class. Quant. Grav*]{}, [**22**]{} 425 (2005), gr-qc/0407110 L. Lehner, and O. M. Moreschi, [*Phys. Rev. D*]{}, [**76**]{} 124040 (2007), arXiv:0706.1319 E. T. Newman, and R. Penrose, [*J. Math. Phys.*]{}, [**3**]{} 566 (1962) V. Moncrief, [*Ann. Phys*]{}, [**88**]{} 323-342 (1974) L. Lindblom, B. J. Owen, D. A. Brown, N.T. Bishop, R. Gómez, L. Lehner, and J. Winicour, [*Phys. Rev. D*]{}, [**54**]{} 6153 (1996). Winicour, J. (2005), [*Living Rev. Relativity*]{} [**8**]{}, 10. L.A. Tamburino and J. Winicour, [*Phys. Rev.*]{} [**150**]{} 1039, 1966. R.A. Isaacson, J.S. Welling and J. Winicour, [*J. Math. Phys.*]{} [**24**]{} 1824 (1983). N. T. Bishop, R. Gómez, L. Lehner, M. Maharaj and J. Winicour, [*Phys. Rev. D*]{},[**56**]{} 6298 (1997). M. Babiuc, B. Szilágyi, I. Hawke, and Y. Zlochower, [*Class. Quantum Grav.*]{} [**22**]{} 5089 (2005) N. T. Bishop, R. Gómez, S. Husa, L. Lehner, J. Winicour [*Phys.Rev. D*]{} [**68**]{}, 084015 (2003) M.C. Babiuc, N.T. Bishop, B. Szilágyi and J. Winicour, [*Phys. Rev. D*]{}, [**79**]{} 084011 (2009). C. Reisswig, N. T. Bishop, D. Pollney abd B. Szilágyi, [*Phys. Rev. Lett.*]{}, [**95**]{}, 221101 (2009). C. Reisswig, N. T. Bishop, D. Pollney abd B. Szilágyi, [*Class. Quantum Grav.*]{} [**27**]{}, 075014 (2010). M.C. Babiuc, B. Szilágyi, J. Winicour and Y. Zlochower \[arXiv:1011.4223\] H. Bondi, M.J.G. van der Burg and A.W.K. Metzner, [*Proc. R. Soc. A*]{} [**269**]{} 21 (1962). R.K. Sachs, [*Proc. R. Soc. A*]{} [**270**]{} 103 (1962). R. Penrose, [*Phys. Rev. Letters*]{}, [**10**]{} 66 (1963). N.T. Bishop, R. Gómez, L. Lehner, B. Szilágyi, J. Winicour and R. A. Isaacson, [*Black Holes, Gravitational Radiation and the Universe*]{}, Kluwer Academic Publishers, Dordrecht, 1998 R. Gómez, L. Lehner, P. Papadopoulos and J. Winicour, [*Class. Quantum Grav.*]{} [**14**]{} 977, 1997. R. G[ó]{}mez, [*Phys. Rev. D*]{} [**64**]{}, 1–8 (2001). J. Winicour, [*J. Math. Phys.*]{} [**24**]{} 1193 (1983). J. Winicour, [*J. Math. Phys.*]{} [**25**]{} 2506 (1984). M. Campanelli, C. O. Lousto, P. Marronetti and Y. Zlochower, Phys. Rev. Lett. [**96**]{}, 111101 (2006) \[arXiv:gr-qc/0511048\]. Y. Zlochower, J. G. Baker, M. Campanelli and C. O. Lousto, Phys. Rev. D [**72**]{}, 024021 (2005) \[arXiv:gr-qc/0505055\]. . E. Schnetter, S. H. Hawley and I. Hawke, Class. Quant. Grav. [**21**]{}, 1465 (2004) \[arXiv:gr-qc/0310042\]. M. Shibata and T. Nakamura, [*Phys. Rev. D*]{} [**52**]{}, 5428 (1995). T. Baumgarte and S. L. Shapiro, [*Phys. Rev. D*]{} [**59**]{}, 024007 (1999). J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, [*Phys.Rev.Lett.*]{} [**96**]{}, 111102 (2006). M. Campanelli, C. O. Lousto and Y. Zlochower, Phys. Rev. D [**74**]{}, 041501 (2006) \[arXiv:gr-qc/0604012\]. .
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---
author:
- |
[Xinyang Zhang]{}$^\ast$ $^\star$ $^\ast$ $^\dagger$ $^\ddagger$ $^\ast$\
$^\ast$Pennsylvania State University $^\star$University of California Irvine\
$^\dagger$Zhejiang University and Alibaba-ZJU Joint Institute of Frontier Technologies\
$^\ddagger$Hong Kong Polytechnic University
bibliography:
- 'main.bib'
title: Interpretable Deep Learning under Fire
---
Acknowledgments {#acknowledgments .unnumbered}
===============
This material is based upon work supported by the National Science Foundation under Grant No. 1846151 and 1910546. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Shouling Ji is partially supported by NSFC under No. 61772466 and U1836202, the Zhejiang Provincial Natural Science Foundation for Distinguished Young Scholars under No. LR19F020003, and the Provincial Key Research and Development Program of Zhejiang, China under No. 2017C01055. Xiapu Luo is partially supported by Hong Kong RGC Project (No. PolyU 152279/16E, CityU C1008-16G).
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---
abstract: 'We study the interplay between the Zeeman field and spin-orbit coupling (SOC) in harmonically trapped Fermi gases loaded into a two-dimensional single-band tight-binding optical lattice. Using the Bogoliubov-de Gennes theory, we find that the Zeeman field combined with a Rashba SOC gives rise to $(i)$ Fulde-Ferrell-like superfluidity and $(ii)$ skyrmion-like polarization textures near the edges of the system. We also discussed the effects of interaction, temperature, SOC anisotropy and Zeeman field anisotropy on the superfluid ground state and polarization textures.'
author:
- 'M. Iskin'
title: 'Spin-orbit coupling induced Fulde-Ferrell-Larkin-Ovchinnikov-like Cooper pairing and skyrmion-like polarization textures in trapped optical lattices'
---
Introduction {#sec:intro}
============
The possibility of simulating non-Abelian artificial gauge fields with quantum Bose and Fermi gases in atomic systems has become one of the forefront research directions in the atomic and molecular physics community [@nistsoc; @chinasocb; @chinasocf; @mitsoc; @engels; @fu; @williams; @zhaireview; @galitskireview], primarily due to its direct connection to the topological phases of matter that have extensively been studied in the condensed-matter community in recent years [@volovik; @hasan; @sczhang; @wen]. In particular, the exciting possibility of the creation and observation of Majorana bound states in topological insulators, superconductors and superfluids is at the heart of topological quantum computation [@tqc]. These quasiparticles can be created at the boundaries (edges) of non-Abelian topological phases, and they allow for non-local storage of quantum information that is protected from local perturbations by the bulk gap. Motivated by these theoretical proposals, spin-orbit coupled Fermi gases have recently been created and detected near the quantum degeneracy limit by three groups [@chinasocf; @mitsoc; @fu; @williams]. While the Shanxi group in China studied the spin dynamics and momentum distribution asymmetry in the equilibrium state as hallmarks of the spin-orbit coupling [@chinasocf; @fu], and the MIT group used a more direct approach and analyzed the energy-momentum dispersion, spin-orbit gap and spin composition of the quantum states [@mitsoc], the NIST group has very recently identified a Feshbach resonance via its associated atomic loss feature [@williams]. Thus, assuming that sufficiently low temperatures are experimentally attainable in the near future, the physics of Majorana bound states can be studied in the clean and controllable environment uniquely offered by the atomic systems [@iskin; @liu].
Following the success of these initial experiments [@nistsoc; @chinasocb; @chinasocf; @mitsoc; @engels; @fu; @williams] (see also the recent reviews [@zhaireview; @galitskireview]), there has been growing theoretical interest in studying the one-, two-, few-, and many-body properties of spin-orbit coupled Fermi gases. For instance, the stability and phase diagrams have been studied for finite and thermodynamic systems as functions of the interaction strength, SOC strength, population imbalance, Zeeman fields, SOC anisotropy, Zeeman field anisotropy, temperature, etc. in one, two and three dimensions (see e.g. [@zheng; @vjshenoy2b; @dong2b; @iskinZ; @zhangyi; @huZ; @yiZ; @puZ; @seoZ]). There also appeared some recent works on the normal-state properties of repulsive Fermi gases with short-range interactions in the upper branch of the spectrum [@wmliu], which share similarities with repulsive electron gases with long-range Coulomb interactions [@ashrafi]. The amount of knowledge gained in these recent works is overwhelming, and here we briefly quote the most recent ones that are concentrated on the possibility of creation and observation of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) [@FF; @LO] type spatially-modulated non-uniform superfluid phases under in-plane Zeeman field [@zheng; @vjshenoy2b; @dong2b; @iskinZ; @zhangyi; @huZ; @yiZ; @puZ; @seoZ]. In sharp contrast to the out-of-plane Zeeman field, these works have shown that the non-uniform FFLO-like phases are energetically more favored than the uniform BCS-like phases in the case of an in-plane Zeeman field. It is important to note that the FFLO-type phases in spin-orbit coupled Fermi gases are stabilized mainly by the asymmetry of the Fermi surfaces in momentum space, and this mechanism is in contrast with that of the condensed-matter ones where they are stabilized by the symmetric Zeeman mismatch in momentum space.
Since all of these results are obtained through ansatz-based non-self-consistent momentum-space calculations [@zheng; @vjshenoy2b; @dong2b; @iskinZ; @zhangyi; @huZ; @yiZ; @puZ; @seoZ], one of our main objectives here is to investigate the stability of FFLO-like phases by solving the BdG equations in a self-consistent fashion. For this purpose, we study the interplay between the Zeeman field and SOC in two-dimensional Fermi gases [@kohl; @vogt; @sommer] loaded into a single-band tight-binding optical lattice. Our primary finding is that while the ground states of spin-orbit coupled systems may have weak Fulde-Ferrell (FF) [@FF] type non-uniform superfluid characters (i.e. phase modulations) but not a Larkin-Ovchinnikov (LO) [@LO] one (i.e. amplitude modulations) under the out-of-plane Zeeman field, the FF character of the superfluids is stronger for the Rashba SOC under in-plane Zeeman field. The FF-type phase oscillations are most prominent along the direction that is perpendicular to the Zeeman field. Therefore, our self-consistent real-space BdG results support recent findings on the thermodynamic continuum systems that are ansatz-based momentum-space calculations [@zheng; @vjshenoy2b; @dong2b; @iskinZ; @zhangyi; @huZ; @yiZ; @puZ; @seoZ]. We also comment on the effects of interaction, temperature, SOC anisotropy and Zeeman field anisotropy on the FFLO-like pairing and ground state of the system, and note that since the superfluid order parameters modulate only towards the edges of the system, where the densities of fermions are low and the magnitudes of the order parameters are small, it may be difficult to detect these modulations in atomic systems at finite temperatures.
Furthermore, our secondary finding is that any non-zero combination of the Zeeman field and Rashba SOC induces not only an easy-axis polarization along the direction of the Zeeman field everywhere in the system but also a spatially-modulated (ring-shaped in magnitude) transverse polarization near the edges. This is in sharp contrast with the trapped systems with no-SOC (and also with the thermodynamic systems with SOC) where only an easy-axis polarization can be induced beyond a threshold Zeeman field. We show that the induced polarization textures are skyrmion-like [@volovik; @skyrme] finite-size effects, broadened by the trapping potential, and that their microscopic origin can be traced back to the counter-flow of spontaneous spin currents in the case of Rashba SOC. The skyrmion particles were originally proposed in late 1950s by the nuclear physicist T. Skyrme as a model for baryons [@skyrme], and they were first observed in condensed-matter physics with quantum-Hall ferromagnets as a result of the interplay between the Zeeman field and Coulomb interactions [@sondhi; @barrett]. Note that similar skyrmion-like spin textures were previously predicted in atomic physics for rotating spinor BEC [@khawaja; @mueller] and spin-orbit coupled BEC [@sinha; @wu; @rama], where skyrmions are spontaneously produced by SOC in the latter case without rotation. We also argue that the transverse polarization textures may be used to probe and characterize the topological phase transitions and the associated Majorana bound states in finite spin-orbit coupled Fermi gases, and comment on the effects of interaction, temperature, SOC anisotropy and Zeeman field anisotropy on the polarization textures.
The rest of this paper is organized as follows. In Sec. \[sec:bdg\], first we introduce the mean-field Hamiltonian and then derive the self-consistency equations for the superfluid order parameter, total number of fermions, and out-of- and in-plane spin polarizations within the BdG framework. We numerically solve the resultant equations and discuss the obtained results in Sec. \[sec:numerics\]. Finally, the conclusions of this paper are briefly summarized in Sec. \[sec:conclusions\].
Bogoliubov-de Gennes theory {#sec:bdg}
===========================
The results mentioned above are obtained within the self-consistent BdG theory in real-space as discussed next. First of all, we describe the spin-orbit coupled Fermi gases loaded into a two-dimensional single-band tight-binding optical lattice by the grand-canonical mean-field Hamiltonian, $$\begin{aligned}
\label{eqn:ham}
H &= \sum_i \left(-t \sum_{\mathbf{\widehat{e}}}
C_{i+\mathbf{\widehat{e}}}^\dagger \phi_{i+\mathbf{\widehat{e}},i} C_i
+ \Delta_i c_{\uparrow i}^\dagger c_{\downarrow i}^\dagger + H.c.\right) \\
& - \sum_{i, \sigma} \left[ \left(\mu + s_\sigma h_z- V_i\right) c_{\sigma i}^\dagger c_{\sigma i}
+ (h_x -i s_\sigma h_y) c_{\sigma i}^\dagger c_{-\sigma i} \right] \nonumber,\end{aligned}$$ where the operator $c_{\sigma i}^\dagger$ ($c_{\sigma i}$) creates (annihilates) a pseudo-spin $\sigma = \{ \uparrow, \downarrow\}$ fermion at lattice site $i$, the spinor $C_i^\dagger = (c_{\uparrow i}^\dagger, c_{\downarrow i}^\dagger)$ denotes the fermion operators collectively, $
\mathbf{\widehat{e}} = \{\mathbf{\widehat{x}}, \mathbf{\widehat{y}}\}
$ allows only nearest-neighbor hopping with amplitude $t$, and $H.c.$ is the Hermitian conjugate. For a generic non-Abelian gauge field $\mathbf{A}=(\alpha \sigma_y, -\beta\sigma_x)$, where $\sigma_e$ is the Pauli matrix and $\{\alpha, \beta\} \ge 0$ are independent parameters characterizing both the strength and the symmetry of the SOC, the $\uparrow$ and $\downarrow$ fermions gain $
\phi_{i+\mathbf{\widehat{x}}, i} = e^{-i\alpha \sigma_y}
$ phase factors for hopping in the positive $\mathbf{\widehat{x}}$ direction and $
\phi_{i+\mathbf{\widehat{y}}, i} = e^{i\beta \sigma_x}
$ phase factors for hopping in the positive $\mathbf{\widehat{y}}$ direction. In addition, the complex number $\Delta_i$ is the local mean-field superfluid order parameter (to be specified below), $\mu$ is the chemical potential, $s_\uparrow = - s_\downarrow = 1$, $\mathbf{h} \equiv (h_x, h_y, h_z)$ is the Zeeman field, and $V_i = V_0 r_i^2$ is the harmonic confining potential where the distance $r_i$ of site $i$ is measured from the center of the lattice.
Using the Bogoliubov transformation, the mean-field Hamiltonian given in Eq. (\[eqn:ham\]) for a two-dimensional $L \times L$ square lattice can be compactly written as a $4L^2 \times 4L^2$ matrix-eigenvalue problem [@edoko], $$\begin{aligned}
\label{eqn:bdg.matrix}
\sum_{j} \left( \begin{array}{cccc}
T_{\uparrow \uparrow} & T_{\uparrow \downarrow} & 0 & \Delta \\
T_{\downarrow \uparrow} & T_{\downarrow \downarrow} & -\Delta & 0 \\
0 & -\Delta^* & -T_{\uparrow \uparrow}^* & -T_{\uparrow \downarrow}^* \\
\Delta^*& 0 & -T_{\downarrow \uparrow}^* & -T_{\downarrow \downarrow}^*
\end{array} \right)_{ij}
&
\left( \begin{array}{c}
u_{nj}^\uparrow \\
u_{nj}^\downarrow \\
v_{nj}^\uparrow \\
v_{nj}^\downarrow
\end{array} \right)
= \varepsilon_n
\left( \begin{array}{c}
u_{ni}^\uparrow \\
u_{ni}^\downarrow \\
v_{ni}^\uparrow \\
v_{ni}^\downarrow
\end{array} \right),\end{aligned}$$ where $u_{ni}^\sigma$ and $v_{ni}^\sigma$ are the components of the $n$th quasiparticle wave function at site $i$, and $\varepsilon_n \ge 0$ is the corresponding energy eigenvalue. Here, the offsite hopping and onsite energy terms are compactly written as $$\begin{aligned}
T_{\sigma \sigma'}^{ij} = - t_{\sigma \sigma'}^{ij} -
& \left[ \left(\mu + s_\sigma h_z - V_i\right) \delta_{s_\sigma s_{\sigma'}} \right. \nonumber \\
& \left. + \left(h_x - i s_\sigma h_y\right) \delta_{s_\sigma,-s_{\sigma'}}
\right] \delta_{ij},\end{aligned}$$ where $\delta_{ij}$ is the Kronecker delta. The non-vanishing nearest-neighbor hopping elements are $t_{\sigma\sigma}^{i,i+\mathbf{\widehat{x}}} = t \cos \alpha$ and $t_{\uparrow\downarrow}^{i,i+\mathbf{\widehat{x}}} = - t_{\downarrow\uparrow}^{i,i+\mathbf{\widehat{x}}} = -t \sin \alpha$ for the positive $\mathbf{\widehat{x}}$ direction, and $t_{\sigma\sigma}^{i,i+\mathbf{\widehat{y}}} = t \cos \beta$ and $t_{\uparrow\downarrow}^{i,i+\mathbf{\widehat{y}}} = t_{\downarrow\uparrow}^{i,i+\mathbf{\widehat{y}}} = i t \sin \beta$ for the positive $\mathbf{\widehat{y}}$ direction. Note that the hopping in the negative directions are simply the Hermitian conjugates, and also that the angles $\alpha$ and $\beta$ determine, respectively, the relative strength between the spin-conserving particle hopping and spin-flipping SOC terms in the $\mathbf{\widehat{x}}$ and $\mathbf{\widehat{y}}$ directions.
In this paper, we consider only the onsite interactions for which the off-diagonal couplings are $\Delta_{ij} = \Delta_i \delta_{ij}$ diagonal in the site index. Therefore, Eq. (\[eqn:bdg.matrix\]) needs to be solved simultaneously with $
\Delta_i = g \langle c_{\uparrow i} c_{\downarrow i} \rangle,
$ where $g \ge 0$ is the strength of the onsite interaction between $\uparrow$ and $\downarrow$ fermions, and $\langle \cdots \rangle$ is a thermal average. In addition, we use $\mu$ to fix the total number of fermions $N = \sum_i n_i$, where $0 \le n_i = \sum_\sigma \langle c_{\sigma i}^\dagger c_{\sigma i} \rangle \le 2$ gives the local fermion filling. Once the self-consistent solutions are obtained for the wave functions and the energy spectrum, it is a straightforward task to calculate any of the desired observables. For instance, we are interested in the local polarization vector $\mathbf{p_i} \equiv (p_{i x}, p_{i y}, p_{i z})$, the components of which follow from the expectation values of the Pauli spin matrices, i.e. $p_{i \nu} = \langle C_i^\dagger \sigma_\nu C_i \rangle$, and are given by $p_{i x} = 2\textrm{Re} \langle c_{\uparrow i}^\dagger c_{\downarrow i} \rangle$, $p_{i y} = 2\textrm{Im} \langle c_{\uparrow i}^\dagger c_{\downarrow i} \rangle$ and $p_{i z} = \sum_\sigma s_\sigma \langle c_{\sigma i}^\dagger c_{\sigma i} \rangle$. Thus, we need the following averages for our purposes $$\begin{aligned}
\label{eqn:op}
\langle c_{\uparrow i} c_{\downarrow i} \rangle &= \sum_n \Big[ (v_{n i}^\uparrow)^* u_{n i}^ \downarrow f(\varepsilon_n) + u_{n i}^\uparrow (v_{n i}^ \downarrow)^* f(-\varepsilon_n) \Big], \\
\label{eqn:inplane}
\langle c_{\uparrow i}^\dagger c_{\downarrow i} \rangle &= \sum_{n} \Big[ (u_{n i}^\uparrow)^* u_{n i}^\downarrow f(\varepsilon_n) + v_{n i}^\uparrow (v_{n i}^\downarrow)^* f(-\varepsilon_n) \Big], \\
\label{eqn:outofplane}
\langle c_{\sigma i}^\dagger c_{\sigma i} \rangle &= \sum_{n} \Big[ |u_{n i}^\sigma|^2 f(\varepsilon_n) + |v_{n i}^\sigma|^2 f(-\varepsilon_n) \Big],\end{aligned}$$ where $f(x)=1/(e^{x/T}+1)$ is the Fermi-Dirac distribution function, $T$ is the temperature and the Boltzmann constant $k_B$ is set to unity. We also define the total polarization components as $P_\nu = \sum_i p_{i \nu}/N$, where $\nu \equiv\{x, y, z\}$. Equations (\[eqn:bdg.matrix\])-(\[eqn:outofplane\]) correspond to the generalization of the BdG equations to the case of spin-orbit coupled Fermi gases on optical lattices.
Numerical results {#sec:numerics}
=================
Having established the BdG formalism, next we present our numerical solutions for the ground-state phases, which are performed on a $41a \times 41a$ square lattice with $N = 150$ fermions in total, where $a$ is the lattice spacing. We take $V_0 = 0.01t$ as the strength of the trapping potential, and discuss both the Rashba-type symmetric ($\alpha = \beta$) and asymmetric ($\alpha \ne \beta$) SOC fields. Note that the experimentally more relevant equal Rashba-Dresselhaus (ERD) SOC [@nistsoc; @chinasocb; @chinasocf; @mitsoc; @engels; @fu; @williams; @zhaireview; @galitskireview] can be obtained by setting $\alpha = 0$. The effects of higher fermion numbers and finite temperature are also briefly mentioned towards the end of the paper.
Before we discuss the aforementioned FFLO-like pairing and polarization textures, we make three important remarks. First, in the absence of a SOC, i.e. when $\alpha = \beta = 0$, we know that a sufficiently strong Zeeman field $\mathbf{h}$ (the threshold of which depends on $g$) can polarize the system along the easy-axis ($\mathbf{\widehat{h}}$) direction. Second, in the absence of a Zeeman field, i.e. when $\mathbf{h} \equiv (0,0,0)$, we also know that the system is trivially unpolarized no matter what the SOC is. Third, while any combination of Zeeman field (no matter how weak the field is) and SOC in a thermodynamic system may produce a uniform polarization along the easy-axis direction, it does not induce any polarization in the transverse direction, i.e. perpendicular to $\mathbf{\widehat{h}}$. With these remarks in mind, next we show that any non-zero Zeeman field can induce intricate polarization textures near the edges of finite-size spin-orbit coupled Fermi gases under various circumstances.
Out-of-plane Zeeman field {#sec:out}
-------------------------
Let us first consider an out-of-plane Zeeman field $\mathbf{h} \equiv (0, 0, h_z \ne 0)$, which is perpendicular to our square lattice. The magnitudes and phases of typical ground-state order parameters are illustrated for the Rashba and ERD-like SOCs in Figs. \[fig:outop\](a) and \[fig:outop\](b), respectively, for $h_z = 0.5t$. The phases of the order parameters clearly show the $C_4$ and $C_2$ symmetries of the Hamiltonian for the Rashba and ERD-like SOCs, respectively. These figures suggest that the ERD-like SOC has a stronger FF-type non-uniform superfluid character where the phase of the order parameter has a much larger spatial modulation. Note that the phases have both angular and radial oscillations towards the edges of the system where the densities of the fermions are low and the magnitudes of the order parameters are small. Therefore, it may be difficult to detect these modulations in atomic systems at finite $T$. However, since the spatial profiles of the magnitudes of the order parameters do not have any zeros (nodes), our results do not feature any LO-type non-uniform superfluidity. We emphasize that these effects become weaker and weaker with decreasing $h_z$ in such a way that all of the local phases of the order parameters vanish as $h_z \to 0$.
The corresponding ground-state polarization textures are illustrated in Fig. \[fig:out\], where we show two-dimensional vector maps of the transverse polarizations $(-p_{i x}, -p_{i y})$ together with color maps of the easy-axis polarizations $p_{i z}$. Here, we set $g = 3t$ but emphasize that setting it to 0 does not lead to any significant change in the results. These figures again clearly show the $C_4$ and $C_2$ symmetries of the Hamiltonian for the Rashba and ERD-like SOCs, respectively. First of all, the not-so-interesting $p_{i z}$ is finite everywhere in the trap with its maximum value at the center of the system in both figures, and it gradually decreases to zero towards the edges. The case of Rashba SOC is shown in Fig. \[fig:out\](a), where we find that $p_{i x} \ne 0$ and $p_{i y} \ne 0$ in general, except for the center of the trap. In the ERD-like case when $\alpha \to 0$ but $\beta = \pi/4$, we see in Fig. \[fig:out\](b) that while $p_{i x} \to 0$ everywhere in the system, $p_{i y}$ remains mostly unchanged. Therefore, in the ERD case when $\alpha = 0$, a domain-wall is formed on the $x$ axis where $p_{i x} = p_{i y} = 0$, and such a limiting behavior can be extracted from Fig. \[fig:out\](b). Similarly, when $\beta = 0$ but $\alpha \ne 0$, a domain-wall forms on the $y$ axis where $p_{i x} = p_{i y} = 0$ (not shown). Thus, we conclude that a non-zero out-of-plane Zeeman field no matter how small it is (not shown) induces spatially modulated transverse polarizations only along those directions where there is SOC. In most cases, the ratio of the transverse to the easy-axis polarizations are around $\%5-\%10$. However, we emphasize that while the total easy-axis polarizations are $P_z \approx 0.3$ and $P_z \approx 0.36$ in Figs. \[fig:out\](a) and \[fig:out\](b), respectively, the total transverse polarizations vanish, i.e. $P_x = 0 = P_y$, as one may expect.
Note that the transverse polarizations change sign in space, and that their magnitudes $\sqrt{p_{i x}^2 + p_{i y}^2}$ show ring-shaped structures. In Fig. \[fig:out\](a), in addition to the broad ring ranging mostly between $7a \lesssim r_i \lesssim 12a$, there is also a narrow one with a very weak peak around $r_i \approx 14a$. Similarly, in Fig. \[fig:out\](b), there are two incomplete rings especially along the $y$ axis, and they both have comparable peaks around $r_i \approx \{10.5a, 14.5a\}$. We also find that increasing the size of the square lattice pushes the ring-shaped structures further away from the center, and reducing the strength of the trapping potential intensifies them around a narrower region near the edges, eventually both leaving no transverse polarization near the center. In the box-potential limit when $V_0 \to 0$, we find that while the continuum- and edge-state contributions to the transverse polarizations are competing with each other for low $h_z$ values, the latter contribution gets stronger with increasing $h_z$, and eventually dominates beyond the $h_z$ threshold for the creation of zero-energy (Majorana) edge-bound states. These findings suggest that the transverse polarizations observed here are finite-size edge effects, broadened by the trapping potential, and also that the working mechanism is similar to the one that is responsible for the creation of edge-bound Majorana states. We note that similar spin textures are referred to as skyrmions in the contexts of rotating spinor BEC [@khawaja; @mueller] and spin-orbit coupled BEC without rotation [@sinha; @wu; @rama]. In particular, we especially note the great similarity between Fig. \[fig:out\](a) presented here and Fig. 5(a) of Ref [@rama].
To understand the microscopic origin of these textures, next we employ the local-density approximation and analyze the single-particle excitation spectrum of the local system. The spectrum of the local Hamiltonian in momentum space involves two quasihole and two quasiparticle branches that are given by $$\begin{aligned}
\label{eqn:ek}
E_{\mathbf{k} i, \pm}^2 &= \xi_{\mathbf{k} i}^2 + h_z^2 + |s_{\mathbf{k}}|^2 + |\Delta_i|^2 \nonumber \\
& \pm 2\sqrt{h_z^2(\xi_{\mathbf{k},i}^2 + |\Delta_i|^2) + \xi_{\mathbf{k} i}^2 |s_{\mathbf{k}}|^2},\end{aligned}$$ where $
\xi_{\mathbf{k} i} = -2t[\cos\alpha \cos(k_x a) + \cos\beta \cos(k_y a)] - \mu_i
$ is the shifted kinetic energy and $
|s_{\mathbf{k}}|^2 = 4t^2[\sin^2\alpha \sin^2(k_x a) + \sin^2\beta \sin^2(k_y a)]
$ is the SOC contribution. Here, the local chemical potential $\mu_ i = \mu - V_i$ includes the trapping potential. We immediately see that the minus branches can become gapless at some $\mathbf{k}$-space points, i.e. $E_{\mathbf{k_0} i, -}^2 = 0$, and therefore the location of zero-energy states are determined by the following conditions: ($i$) $|s_{\mathbf{k_0}}| = 0$ and ($ii$) $h_z = \sqrt{\xi_{\mathbf{k_0} i}^2 + |\Delta_i|^2}$. While the Rashba SOC satisfies the former condition at four points $\mathbf{k_0} \equiv \{(0, 0)$; $(0, \pi)$; $(\pi, 0)$; $(\pi, \pi)$}, the ERD SOC satisfies it only at two points $\mathbf{k_0} \equiv \{(k_x, 0)$; $(k_x, \pi)$}.
It is clear that the latter condition ($ii$) is easier to satisfy towards the edges of the system when $|\Delta_i| \to 0$, and since the transverse polarizations are found to be very similar for $g = 3t$ and $g = 0$, we may set $|\Delta_i| = 0$ for our purpose. When this is the case, the condition ($ii$) becomes $
\mu_i = \pm h_z \pm 2t(\cos \alpha \pm \cos \beta)
$ for the Rashba SOC, and $
\mu_i = \pm h_z - 2t[\cos(k_x a) \pm \cos \beta]
$ for the ERD SOC, where all $\pm$ combinations are possible and $|\cos(k_x a)| \le 1$. For the parameters of Figs. \[fig:out\](a) and \[fig:out\](b), where $\mu \approx -1.76 t$, these conditions are satisfied at two distances $
r_i \approx \{ 7.5a, 12.5a \}
$ and $
r_i \approx \{ 10.7a, 14.6a \},
$ respectively, which are very close to our numerical results given above. Thus, we conclude that the microscopic origin of the transverse polarizations can be traced back to the changes in the momentum-space topology of the single-particle excitation spectrum of the local system.
In the case of Rashba SOC, we can also interpret these textures as a direct consequence of counter-flow of spontaneously-induced spin currents [@edoko]. This is because the Rashba SOC gives rise to an effective momentum-dependent in-plane magnetic field in the direction that is perpendicular to the in-plane momentum. Since the induced spin currents are circulating along the trap edges, i.e. the in-plane momentum is in the azimuthal direction, the induced in-plane spin texture is in the radial direction. The relative contribution between the radially-outward and -inward helicity bands depends on the local chemical potential $\mu_i$, and this competition produces the spatial structure of the spin textures such as the one illustrated in Fig. \[fig:out\](a). Note that the time-reversal symmetry of the spins must be broken via e.g. the Zeeman field in order to have a non-zero polarization in any particular direction.
In-plane Zeeman field {#sec:in}
---------------------
Next, we consider an in-plane Zeeman field $\mathbf{h} \equiv (0, h_y \ne 0, 0)$, which lies in the $\mathbf{\widehat{y}}$ direction parallel to our square lattice. The magnitudes and phases of typical ground-state order parameters are illustrated for the Rashba and ERD-like SOCs in Figs. \[fig:inop\](a) and \[fig:inop\](b), respectively, for $h_y = 0.5t$. In sharp contrast to the out-of-plane Zeeman case discussed above, this comparison clearly shows that the Rashba SOC has a much stronger FF-type non-uniform superfluid character in the in-plane Zeeman case, without again featuring any LO-type order parameter node. Note again that the phases have both angular and radial oscillations towards the edges of the system, and the FF-type oscillations are most prominent along the $x$ direction, i.e. perpendicular to the direction of the Zeeman field. We emphasize that these effects become weaker and weaker with decreasing $h_y$ in such a way that all of the local phases vanish as $h_y \to 0$.
The corresponding ground-state polarization textures are illustrated in Fig. \[fig:in\], where we show two-dimensional vector maps of the transverse polarizations $(p_{i x}, p_{i z})$ together with color maps of the easy-axis polarizations $p_{i y}$. Here, we set $g = 3t$ but setting $g$ to 0 again does not lead to any significant change in the results. First of all, the not-so-interesting $p_{i y}$ is finite everywhere in the trap with its maximum value near the center in both figures, and it gradually decreases to zero towards the edges. The case of Rashba SOC is shown in Fig. \[fig:in\](a), where we find that $p_{i x} \ne 0$ and $p_{i z} \ne 0$ in general, except for a domain-wall on the $x$ axis where $p_{i x} = p_{i z} = 0$. In the ERD-like SOC when $\alpha \to 0$ but $\beta \ne 0$, we see in Fig. \[fig:in\](b) that while $p_{i x} \to 0$ everywhere in the system, $p_{i z}$ remains mostly unchanged. Therefore, similar to the Rashba case, the ERD case also has a domain-wall that is formed on the $x$ axis where $p_{i x} = p_{i z} = 0$, and such a limiting behavior can be extracted from Fig. \[fig:in\](b). On the other hand, when $\beta = 0$ but $\alpha \ne 0$, there is not any transverse polarization in the entire system, i.e. $p_{i x} = p_{i z} = 0$ for every $i$ (not shown). Thus, we conclude that, when a non-zero in-plane Zeeman field (no matter how small it is) is not perpendicular to the direction of the SOC, a spatially modulated polarization is induced in the transverse direction. In most cases, the ratio of the transverse to the easy-axis polarizations are around $\%5-\%15$. However, we emphasize that while the total easy-axis polarizations are $P_y \approx 0.37$ and $P_y \approx 0.36$ in Figs. \[fig:in\](a) and \[fig:in\](b), respectively, the total transverse polarizations again vanish, i.e. $P_x = 0 = P_z$.
Similar to the out-of-plane Zeeman case, we note that the magnitudes of the transverse polarizations $\sqrt{p_{i x}^2 + p_{i z}^2}$ show ring-shaped structures in both Figs. \[fig:in\](a) and \[fig:in\](b), where the peaks occur, respectively, at $r_i \approx \{10.5a, 13.5a\}$ and $r_i \approx \{10.5a, 14.5a\}$ away from the center of the trap especially along the $y$ axis. The microscopic origin of these structures can again be traced back to the changes in the momentum-space topology of the single-particle excitation spectrum.
Topological phase transitions {#sec:top}
-----------------------------
As elaborated above, although gap closings occur only at a few $\mathbf{k_0}$ points in the lowest quasiparticle and highest quasihole bands, these gapless excitation points are sufficient to induce intricate polarization textures in real space, at and around the boundary between phases with locally different momentum-space topology. We emphasize that since the symmetry of the order parameters of the local phases have the same $s$-wave symmetry across the boundary, the transition is topological. Such topological changes are known as Lifshitz-type transition in the literature, and they are extensively discussed in the context of nodal, e.g. $p$-wave, superfluids and superconductors [@volovik]. In thermodynamic systems, while the primary signatures of Lifshitz transitions are seen in the momentum distribution and single-particle spectral function, some thermodynamic quantities, e.g. atomic compressibility and spin susceptibility, also show anomalies at the transition boundary.
It is also worth noting that both spinless $p_x \pm ip_y$ (chiral) superfluids [@volovik] and spin-orbit coupled Fermi gases under Zeeman field [@iskin; @liu] can host Majorana bound states near the edges of the system. More specifically, these bound states can only exist at the phase boundary between a topologically non-trivial and a trivial phase, the classification of which is based on the value of the topological charges, i.e. Chern numbers [@volovik]. In contrast to the spinless chiral superfluids, our numerical results suggest that the induced transverse polarizations can be used as a probe to characterize the topological phase transitions and the associated Majorana bound states in finite spin-orbit coupled Fermi gases. This is in accordance with a recent work on one-dimensional quantum wires with strong Rashba and Dresselhaus SOC, where it is shown that the Majorana polarization can be used as an order parameter to characterize the topological transition between the trivial system and the system exhibiting Majorana bound modes [@simon].
We also remark that spin textures may also occur in $p$-wave superfluids (as well as in all other systems with vector order parameters) as topological defects, i.e. a coreless vortex may exist as a spin texture. For instance, such textures were recently observed in superfluid $^3$He [@he-skyrm], in good agreement with the early predictions [@merminho; @at]. Furthermore, in the condensed-matter literature, these two-dimensional topological defects were characterized depending on how the local spin changes from the center of the defect to its boundary. Assuming that the local spin $\mathbf{p_i} = p_i \mathbf{\widehat{z}}$ is perpendicular to the system at the center of the defect, the topological object is called (a) an Anderson-Toulouse spin texture or a baby skyrmion if the spin continuously rotates through an angle $\pi$ towards the boundary and anti-aligns with respect to the center [@at], or (b) a Mermin-Ho spin texture (meron) or a half-skyrmion if the spin continuously rotates through an angle $\pi/2$ and aligns with the plane of the system [@merminho]. Note that the polarization textures presented in this work do not belong to these classes, and are unique to trapped Fermi gases with SOC.
Having discussed the low-filling Fermi gases at zero temperature, next we briefly comment on the effects of finite $T$ and high fillings. First, although these topological transitions are quantum in their nature, signatures of them can still be observed at finite $T$, where the observables are smeared out due to thermal effects. In particular, we find for the parameters of Figs. \[fig:out\] and \[fig:in\] that the maximum magnitudes of the transverse polarizations reduce, respectively, to $50\%$ and $10\%$ at $T = 0.1t$ and $T = 0.2t$. Second, due to the particle-hole symmetry of the parent Hamiltonian around half filling, in addition to the ring-shaped transverse polarizations induced near the edges of the system, additional ring-shaped structures are further induced near the center of the trap when the center is close to a band insulator. Therefore, the transverse polarizations show multiple ring-shaped structures in high-filling lattice systems. Having discussed the numerical results, next we conclude the paper with a brief summary of our main findings.
Conclusions {#sec:conclusions}
===========
In this paper, we studied the interplay between the Zeeman field, SOC, FFLO pairing and polarization textures in harmonically trapped two-dimensional Fermi gases, loaded into a single-band tight-binding optical lattice. The trapping potential, SOC and Zeeman field are taken self-consistently into account via the real-space mean-field BdG theory, and two of our main findings can be summarized as follows.
First, we showed that while the ground states of the spin-orbit coupled systems in general have weak FF-type non-uniform superfluid characters but not an LO one under the out-of-plane Zeeman field, the FF character of the superfluids is stronger for the Rashba SOC under in-plane Zeeman field. The FF-type phase oscillations are also most prominent along the direction that is perpendicular to the Zeeman field. Therefore, our self-consistent results on a finite lattice support recent findings on the thermodynamic continuum systems that are ansatz-based non-self-consistent momentum-space calculations [@zheng; @vjshenoy2b; @dong2b; @iskinZ; @zhangyi; @huZ; @yiZ; @puZ; @seoZ]. We also discussed the effects of interaction, temperature, SOC anisotropy and Zeeman field anisotropy on the FFLO-like pairing and ground state of the system, and noted that since the superfluid order parameters modulate only towards the edges, it may be difficult to detect these modulations in atomic systems at finite $T$.
Second, in sharp contrast to the no-SOC case where only an easy-axis polarization is possible beyond a threshold Zeeman field, we showed that any non-zero combination of the Zeeman field and Rashba SOC induces not only an easy-axis polarization everywhere in the system but also a spatially-modulated transverse one near the edges. We found that the induced polarization textures are skyrmion-like finite-size effects, which are very similar to the spin textures that were previously predicted for rotating spinor BEC [@khawaja; @mueller] and spin-orbit coupled BEC without rotation [@sinha; @wu; @rama]. We also argued that the transverse polarizations can be used to probe and characterize the topological phase transitions and the associated Majorana bound states in finite spin-orbit coupled Fermi gases, and briefly discussed the possibility of observing these effects in atomic systems.
Finally, we emphasize that while all of these results are obtained using an optical lattice model, they are equally applicable to continuum systems in the low-filling limit. We preferred the lattice description mainly because of its easier numerical implementation and versatility, e.g. self-consistent inclusion of the trapping potential, and anisotropic SOC and/or Zeeman field do not require any additional cost in numerics. However, since the lattice model allows particle fillings up to unity, but with a particle-hole symmetry around half filling, it leads to richer finite-size effects compared to the continuum model away from the low-filling limit.
Acknowledgments {#sec:ack}
===============
This work is supported by the Marie Curie IRG Grant No. FP7-PEOPLE-IRG-2010-268239, TÜB$\dot{\mathrm{I}}$TAK Career Grant No. 3501-110T839, and TÜBA-GEB$\dot{\mathrm{I}}$P. The author thanks E. Doko, V. B. Shenoy, and A. L. Suba[ş]{}[i]{} for discussions.
[99]{}
Y.-J. Lin, Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Nature (London) **471**, 83 (2011). S. Chen, J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, Y. Deng, H. Zhai, and J.-W. Pan, Phys. Rev. Lett. **109**, 115301 (2012). P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. **109**, 095301 (2012). L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. **109**, 095302 (2012). C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels, arXiv:1301.0658 (2013). Z. Fu, L. Huang, Z. Meng, P. Wang, X.-J. Liu, H. Pu, H. Hu, and J. Zhang, arXiv:1303.2212 (2013). R. A. Williams, M. C. Beeler, L. J. LeBlanc, K. Jiménez-García, and I. B. Spielman, arXiv:1306.1965 (2013).
H. Zhai, Int. J. Mod. Phys. B **26**, 1230001 (2012). V. Galitski and I. B. Spielman, Nature **494**, 49 (2013).
G. Volovik, *The universe in a helium droplet*, Oxford (2003). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. **82**, 3045 (2010). X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. **83**, 1057 (2011). X.-G. Wen, arXiv:1210.1281 (2012).
C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. das Sarma, Rev. Mod. Phys. **80**, 1083 (2008).
M. Iskin, Phys. Rev. A. **85**, 013622 (2012); and Phys. Rev. A **86**, 065601 (2012). X.-J. Liu, L. Jiang, H. Pu, and H. Hu, Phys. Rev. A **85**, 021603(R)(2012); X.-J. Liu and H. Hu, Phys. Rev. A **85**, 033622 (2012).
Z. Zheng, M. Gong, X. Zou, C. Zhang, and G.-C. Guo, Phys. Rev. A **87**, 031602(R) (2013). V. B. Shenoy, arXiv:1211.1831 (2012). L. Dong, L. Jiang, H. Hu, and H. Pu, Phys. Rev. A **87**, 043616 (2013). M. Iskin and A. L. Subaşi, arXiv:1211.4020 (2012) (to appear in PRA). F. Wu, G.-C. Guo, W. Zhang, and W. Yi, Phys. Rev. Lett. **110**, 110401 (2013). X.-J. Liu and H. Hu, arXiv:1302.0553 (2013). X.-F. Zhou, G.-C. Guo, W. Zhang, and W. Yi, Phys. Rev. A **87**, 063606 (2013). L. Dong, L. Jiang, and H. Pu, arXiv:1302.1189 (2013). K. Seo, L. Han, and C. A. R. Sá de Melo, arXiv:1301.1353 (2013).
X.-L. Yu, S.-S. Zhang, and W.-M. Liu, Phys. Rev. A **87**, 043633 (2013). A. Ashrafi, E. I. Rashba, and D. L. Maslov, arXiv:1306.1165.
P. Fulde and R. A. Ferrell, Phys. Rev.**135**, A550 (1964). A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. **47**, 1136 (1964); and Sov. Phys. JETP **20**, 762 (1965).
M. Feld, B. Frohlich, E. Vogt, M. Koschorreck, and M. Köhl, Nature **480**, 75 (2011). E. Vogt, M. Feld, B. Frohlich, D. Pertot, M. Koschorreck, and M. Köhl; Phys. Rev. Lett. **108**, 070404 (2012). A. T. Sommer, L. W. Cheuk, M. J.-H. Ku, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. **108**, 045302 (2012).
T. H. R. Skyrme, Proc. R. Soc. A 260, **127** (1961); and Nucl. Phys. **31**, 556 (1962).
S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Phys. Rev. B **47**, 16419 (1993). S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, Phys. Rev. Lett. **74**, 5112 (1995).
U. Al Khawaja and H. T. C. Stoof, Phys. Rev. A **64**, 043612 (2001). E. J. Mueller, Phys. Rev. A **69**, 033606 (2004). S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. **107**, 270401 (2011). C. Wu, I. Mondragon-Shem, and X.-F. Zhou, Chin. Phys. Lett. **28**, 097102 (2011). B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D. Drummond, and H. Hu, Phys. Rev. A **85**, 023606 (2012).
E. Doko, A. L. Suba[ş]{}[i]{}, and M. Iskin, Phys. Rev. A **85**, 053634 (2012).
D. Sticlet, C. Benna, and P. Simon, Phys. Rev. Lett. **108**, 096802 (2012).
R. Blaauwgeers, V. B. Eltsov, M. Krusius, J. J. Ruohio, R. Schanen, and G. E. Volovik, Nature **404**, 471 (2000). N. D. Mermin and T. L. Ho, Phys. Rev. Lett. **36**, 594 (1976). P. W. Anderson and G. Toulouse, Phys. Rev. Lett. **38**, 508 (1977).
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---
abstract: 'We classify by numerical invariants the finite subgroups $H$ of a primary abelian group $G$ for which every homomorphism or monomorphism of $H$ into $G$, or every endomorphism of $H$, extends to an endomorphism of $G$. We apply these results to show that for finitely generated subgroups of general abelian groups, the extendibility of monomorphisms implies the extendibility of all homomorphisms.'
address:
- '(Breaz) “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu 1, 400084 Cluj-Napoca, Romania'
- '(Călugăreanu) “Babeş-Bolyai” University, Faculty of Mathematics and Computer Science, Str. Mihail Kogălniceanu 1, 400084 Cluj-Napoca, Romania'
- '(Schultz) School of Mathematics and Statistics, The University of Western Australia, Nedlands, 6009, Australia'
author:
- 'Simion Breaz, Grigore Călugăreanu and Phill Schultz'
title: Subgroups which admit extensions of homomorphisms
---
[^1]
Introduction
============
We begin by characterizing in module theoretic terms the extension properties described above. The problem of characterizing the subgroups satisfying these properties forms part of the more general question of characterizing special classes of submodules of a module. For example, Birkhoff observed in [@B34] that although for some rings $R$ (and in particular for $R={\mathbb Z}$) finitely generated modules can be completely characterized using numerical invariants, in general it is difficult to describe even cyclic modules as submodules of a given module. A complex study in this direction was initiated by Ringel and Schmidmeier [@RS08] for artinian algebras, and it was continued by several authors who show that in many cases the category of submodules is wild(cf. the introduction of [@RS06]). For the case of abelian groups, we mention the studies realized in [@A00] and [@RW99].
Let $R$ be a unital ring and $M$ an $R$–module, and let ${\operatorname{Sub}}(M)$ be the set of all submodules of $M$.
If $N\in{\operatorname{Sub}}(M)$, and $\iota:\, N\to M$ is the inclusion map, by applying the contravariant functors ${\operatorname{Hom}}(-,M)$ and ${\operatorname{Hom}}(-,N)$, we obtain a commutative diagram: $$\begin{CD} {\operatorname{Hom}}(M,N)@> \iota_M^*>>{\operatorname{End}}(M)\\
@V {\mathit{res}}_N VV @VV {\mathit{res}}_M V\\
{\operatorname{End}}(N) @>>\iota_N^*> {\operatorname{Hom}}(N,M)\end{CD},$$ where $\iota_X^*:\,{\operatorname{Hom}}(X,N) \to
{\operatorname{Hom}}(X,M)$ are the induced inclusion maps and ${\mathit{res}}_X:\,{\operatorname{Hom}}(M,X)\to{\operatorname{Hom}}(N,X)$ the induced restriction maps.
Several module theoretic properties that appear in the literature in other guises can be described in terms of this diagram: $\iota_M^*$ is an isomorphism if and only if $N=M$; if ${\mathit{res}}_M$ is an isomorphism then $M$ is a localization of $N$ ([@Du04]); ${\mathit{res}}_N$ is an isomorphism if and only if $N$ is a direct summand with a unique complement; ${\mathit{res}}_M$ factors through $\iota_N^*$ if and only if $N$ is fully invariant in $M$; for a given $N$, ${\mathit{res}}_M$ and ${\operatorname{im}}^*_N$ have the same image for all $M$ if and only if $N$ is rigid [@DG96].
Now consider, for a given module $M$, the following sets of submodules: $$\begin{aligned}
{\mathcal {S}}(M)&=\{ N\in {\operatorname{Sub}}(M):\, {\mathit{res}}_N \textrm{ is epic}\};\\
{\mathcal Q}(M)&=\{ N\in {\operatorname{Sub}}(M):\, {\mathit{res}}_M \textrm{ is epic}\};\\
{\mathcal W}(M)&=\{ N\in {\operatorname{Sub}}(M):\, {\operatorname{im}}({\mathit{res}}_M)={\operatorname{im}}(\iota^*_N)\};\\
{\mathcal {P}}(M)&=\{N\in{\operatorname{Sub}}(M): {\operatorname{im}}({\mathit{res}}_M)\supseteq{\operatorname{Mon}}(N,M)\}\end{aligned}$$ where ${\operatorname{Mon}}(N,M)$ is the set of monomorphisms of $N$ into $M$.
It is easy to see that ${\mathcal {S}}(M)$ is the set of all direct summands of $M$, ${\mathcal Q}(M)$ is the set of all submodules $N$ of $M$ such that all homomorphisms of $N$ into $M$ can be extended to endomorphisms of $M$, ${\mathcal W}(M)$ is the class of all submodules $N$ of $M$ such that all endomorphisms of $N$ can be extended to endomorphisms of $M$. and ${\mathcal {P}}(M)$ is the class of all submodules $N$ of $M$ for which all monomorphisms of $N$ into $M$ can be extended to endomorphisms of $M$.
Therefore, if one of these classes coincides with the set of all submodules of $M$, i.e., $\mathcal{X}(M)={\operatorname{Sub}}(M)$ for $\mathcal{X}=\mathcal{S}$, ${\mathcal Q}$, ${\mathcal W}$, or ${\mathcal {P}}$ then $M$ is semi-simple, respectively quasi-injective ([@Fa67]), weakly-injective ([@mis]) or pseudo-injective ([@JS67]). Using known results about the structure of these modules, it is easy to see that in general ${\mathcal {S}}(M)\subseteq {\mathcal Q}(M) \subseteq
{\mathcal {P}}(M)\subseteq {\operatorname{Sub}}(M)$ and all inclusions can be strict. With one exception (the inclusion ${\mathcal Q}(M)\subseteq {\mathcal {P}}(M)$) this strictness can be demonstrated in the class of abelian groups, using structure theorems in [@kil] and [@mis]. For ${\mathcal Q}(M)\subsetneqq {\mathcal {P}}(M)$, there are pseudo-injective modules which are not quasi-injective (see [@JS67] or [@T75]). However, it is proved in [@sin] that over principal ideal domains quasi-injective and pseudo-injective modules coincide, and we do not know if there exists an abelian group $G$ such that ${\mathcal Q}(G)\neq {\mathcal {P}}(G)$.
Recently Er, Singh and Srivastava [@ESS13] showed that pseudo–injective modules are precisely those modules which are invariant under automorphisms of their injective hulls. This description should be compared with the well known characterization of quasi–injective modules as those modules which are fully invariant, i.e., invariant under endomorphisms of their injective hulls.
The object of the paper is to characterize finitely generated subgroups which lie in these classes for the case of primary abelian groups. Let $G$ be a $p$-group. Cyclic subgroups in ${\mathcal Q}(G)$ are described in Theorem \[3.14\], while finitely generated subgroups in ${\mathcal Q}(G)$ are characterized in Theorem \[finite-Q(G)\]. Using these results we prove in Theorem \[Pf=Qf\] that for all abelian groups $G$ (not only for quasi-injective or pseudo-injective abelian groups) we have ${\mathcal Q}_f(G)={\mathcal {P}}_f(G)$, where $\mathcal{X}_f(G)$ denotes the set of all finitely generated subgroups in $\mathcal{X}(G)$. Theorem \[finite-W(G)\] gives us information about finitely generated subgroups in ${\mathcal W}(G)$.
All groups in this paper are abelian. Unless specifically noted we use the standard notation of [@fuc1] and [@fuc2].
*With the exception of the final section, in the rest of this paper we assume that $p$ is a fixed prime and $G$ a $p$-group.*
If $G$ is bounded, we denote by $\exp(G)$ the least positive integer $k$ such that $p^k G=0$. If $G$ is not bounded then $\exp(G)=\infty$. The exponent of an element $x\in G$, denotes the positive integer $\exp(x)$ such that the order of $x$ is $p^{\exp(x)}$. If $x\in G$, then $h(x)$ denotes the height of $x$. A group $G$ is *homogeneous* if it is a direct sum of isomorphic quasi-cyclic groups, i.e. $G\cong {\mathbb Z}(p^k)^{(\lambda)}$ where $k$ is a positive integer or $\infty$ and $\lambda$ is a cardinal.
It is well known that the semi–simple groups are the direct sums of elementary groups. Moreover, quasi-injective (primary) groups are exactly the homogeneous groups.
Cyclic subgroups in ${\mathcal Q}(G)$ {#sect-cyclic}
=====================================
In this section, we consider the question: which cyclic subgroups of $G$ are in ${\mathcal Q}(G)$ or in ${\mathcal {P}}(G)$?
If $G$ is divisible, then for all $x\in G,\ {\langle}x{\rangle}\in{\mathcal Q}(G)$, so we need consider only non–divisible $G$. We recall that $G^1=\bigcap_{n>0}p^nG$ denotes [*the first Ulm subgroup*]{} of $G$.
\[cyclic2\] Let $G$ be non–divisible and $\langle
x\rangle\in{\mathcal {P}}(G)$.
1. If $G$ has an unbounded basic subgroup then $\langle
x\rangle\cap G^1=\{0\}$.
2. If $G=B\oplus D$ with $B$ bounded and $D$ divisible then either
1. $\exp(x)\leq\exp(B)$ and $\langle x\rangle\cap D=0$ or
2. $\exp(x)>\exp(B)$ and $x=b+d$ with $b\in B$, $d\in D$ such that $\exp(B)=\exp(b)$ and $\exp(d)>\exp(B)$.
\(1) If $G$ has an unbounded basic subgroup then $G$ has a cyclic direct summand $\langle a\rangle$ of order $\geq {\operatorname{ord}}(x)$. Hence there is a monomorphism $\langle x\rangle\to \langle
a\rangle$ which can be extended to an endomorphism of $G$. It follows that the heights of all non-zero elements of $\langle
x\rangle$ are finite.
\(2) Suppose $G=B\oplus D$ with $\exp(B)=n$ and $D$ divisible.
\(a) If $\exp(x)\leq n$ then as in (1) $G$ has a cyclic direct summand of order $\geq {\operatorname{ord}}(x)$ and non-zero elements from $\langle x\rangle$ have finite height.
\(b) Suppose that $\exp(x)> n$, and $x=b+d$ with $b\in B$ and $d\in
D$. Then $\exp(d)>n\geq \exp(b)$. Let $y\in B$ have exponent $n$. Then $\exp(y+d)=\exp(d)=\exp(x)$, and there exists $f\in{\operatorname{End}}( G)$ such that $f(x)=y+d$. Since $h(y)=0$, it follows that $h(x)=0$, hence $\exp(b)=n$.
\[sufficient\] If $G$ and $x$ are as in Lemma [\[cyclic2\](2)(b)]{} then $\langle x\rangle\in{\mathcal Q}(G)$.
Since $\exp(b)=\exp(B),\ \langle b\rangle$ is a direct summand of $B$.
Let $f\in{\operatorname{Hom}}({\langle}x{\rangle},G)$ with $f(x)=a+e$, where $a\in B$ and $e\in D$. Since ${\langle}b{\rangle}$ is a summand of $B$ and $\exp(b)\geq\exp(a)$, the map $f_1$ defined by $f_1(b)=a$ extends to $g_1\in {\operatorname{Hom}}(B,G)$. Since $D$ is injective and $\exp(d)=\exp(x)\geq\exp(e)$, the map $f_2 $ defined by $f_2(d)=e$ extends to $g_2\in{\operatorname{Hom}}(D,G)$. Hence $g=g_1+g_2$ is an extension of $f$ to ${\operatorname{End}}(G)$.
It remains to find intrinsic criteria for cyclic groups satisfying the conditions of Lemma \[cyclic2\] (1) and (2) (a) to be in ${\mathcal Q}(G)$. We consider therefore cyclic groups ${\langle}x{\rangle}$ containing no elements of infinite height in $G$.
Recall ([@fuc2 Section 65]) that for $n\in{\mathbb N}$, an *Ulm sequence of length $n$* is a strictly increasing infinite sequence $U=(h_0,h_1,\dots,h_{n-1},\infty,\dots)$ with each $h_i$ an ordinal, under the conventions that each ordinal $h_i<\infty,\ \infty< \infty$ and the constant sequence $(\mathbf{\infty})$ is the unique Ulm sequence of length 0. The set of Ulm sequences is well–ordered pointwise with maximum $(\mathbf \infty)$, no minimum but infimum ${\mathbb N}=(0,1,\dots,
n,n+1,\dots)$. This means in particular that if $U\leq V$ where $U$ has length $n$ and $V$ has length $m$, then $n\geq m$. An Ulm sequence $U$ is called *finite* if all its non–infinity entries are finite. In particular, $(\infty)$ is a finite Ulm sequence.
By Lemma \[cyclic2\] we have:
\[pseudo\] If ${\langle}x{\rangle}\in{\mathcal {P}}(G)$ then $U(x)$ is finite.
We say that the Ulm sequence *$U$ has a gap before $k$* if $h_k
>h_{k-1}+1$, where $h_{-1}$ denotes by definition the integer $-1$. The gap before $n$, where $n$ is the length of $U$, is called *the trivial gap*.
Let $x \in G$ with $\exp(x)=n$. Then $x$ determines an Ulm sequence of length $n$ by $U(x)=(h(x),\,h(px),\dots,h(p^{n-1}x),\,\infty,\dots)$. It is clear from this definition that $U (x)$ is finite if and only if ${\langle}x{\rangle}\cap G^1=\{0\},\ h(p^kx)=\infty$ if and only if $p^kx\in
D$, the divisible part of $G,\ U(x)=(0,1,\dots,n-1,\infty,\dots)$ if and only if ${\langle}x{\rangle}$ is a summand of exponent $n$ and for $x,\ y\in G,\ U(x+y)\geq\min\{U(x),\, U(y)\}$. Finally, note that by [@fuc2 Lemma 65.3], if $h(x)=0$ and $U(x)$ has the first non–trivial gap before $k$, then $G$ has a direct summand of exponent $k$.
By [@fuc2 Lemma 65.5 and Exercise 6] we have:
\[65.5\] Let $G$ be a group and $x\in G$ such that $\langle
x\rangle\cap G^1=0$.
1. The following are equivalent:
1. $\langle x\rangle\in{\mathcal Q}(G)$;
2. if $y\in G$ such that $\exp(x)\geq \exp(y)$ then $U(x)\leq
U(y)$.
2. The following are equivalent:
1. $\langle x\rangle\in{\mathcal {P}}(G)$;
2. if $y\in G$ such that $\exp(x)= \exp(y)$ then $U(x)\leq
U(y)$.
Using this result we can characterize cyclic groups ${\langle}x{\rangle}$ with no elements of infinite height in ${\mathcal Q}(G)$ as follows:
\[3.14\] Let $G$ be a group and $x\in G$ an element of exponent $n$ such that ${\langle}x{\rangle}\cap G^1=\{0\}$. The following are equivalent:
1. $\langle x\rangle\in{\mathcal Q}(G)$;
2. $\langle x\rangle\in{\mathcal {P}}(G)$;
3. $U(x)$ has at most one non-trivial gap and if a gap occurs before the index $k\geq 0$ and $h(p^{k}x)=k+\ell$, then $G$ has no cyclic summands of exponents between $k+1$ and $n+\ell-1$.
(1)$\Rightarrow$(2) This is obvious.
(2)$\Rightarrow$(3) Let $x\in G$ such that $\langle
x\rangle\in{\mathcal {P}}(G)$. If $U(x)$ has no non-trivial gaps then $\langle x\rangle$ is a direct summand of $G$. Therefore we can assume that $U(x)$ has at least one non-trivial gap.
Suppose that $U(x)=(h_0,\dots,h_{n-1},\infty,\dots)$ has at least two non-trivial gaps. Since all height $h_i$ are integers, we can apply [@fuc2 Lemma 65.4], and it follows that there is a direct summand $C=\langle c_1\rangle\oplus \dots \oplus \langle
c_t\rangle$ of $G$ and a strictly increasing chain of positive integers $0<k_1<k_2<\dots<k_t$ such that
1. $t\geq 3$,
2. $\exp(c_1)<\exp(c_2)<\dots<\exp(c_t)=k_t+n$, and
3. $x=p^{k_1}c_1+p^{k_2}c_2+\dots +p^{k_t}c_t$.
We observe that the exponent of $$y=p^{k_1}c_1+p^{k_2-1}c_2+p^{k_3}c_3\dots +p^{k_t}c_t$$ is $n$. But $$h(p^{\exp(c_1)-k_1}y)=\exp(c_1)-k_1+k_2-1<\exp(c_1)-k_1+k_2=h(p^{\exp(c_1)-k_1}x),$$ hence $U(y)\ngeq U(x)$, a contradiction.
Therefore $U(x)$ has exactly one non-trivial gap. Let $k$ be the index such that $U(x)$ has a gap before $k$. Hence $h(p^kx)=k+\ell$ with $\ell>0$.
Suppose that $\langle z\rangle$ is a direct summand of $G$ of exponent $n\leq e\leq n+\ell-1$. If $v=p^{e-n} z$ then $p^{k}v\neq
0$ since $n>k$. Moreover, $$h(p^k v)=e-n+k\leq n+\ell-1-n+k=k+\ell-1<h(p^k x).$$ Therefore $U(v)\ngeqq U(x)$, but $\exp(v)=\exp(x)$, a contradiction.
Suppose that $\langle z\rangle$ is a direct summand of $G$ of exponent $k+1\leq e\leq n-1$. We observe that $v=x+z$ is of exponent $n$. But $h(p^kx)>k=h(p^kz)$, hence $$h(p^kv)=h(p^kx+p^kz)=k<h(p^kx),$$ and it follows that $U(v)\ngeqq
U(x)$. This leads to a contradiction and the proof is complete.
(3)$\Rightarrow$(1) Let $x$ be as in (3). If $U(x)$ has no nontrivial gaps then $\langle x\rangle$ is a direct summand of $G$.
Suppose that $U(x)$ has a gap before the index $k$, and we fix an element $y$ of exponent $e\leq \exp(x)$. We will prove that $U(x)\leq U(y)$.
We consider the Ulm sequence $U(y)=(r_0,\dots,r_{e-1},\infty,\dots)$.
**Case I: $r_{e-1}$ is finite.** In order to prove that $U(x)\leq U(y)$, since $U(x)$ has only one gap and this occurs before $k$, it is enough to prove that $h(p^k
x)\leq h(p^k y)$.
Suppose by contradiction that $h(p^k x)> h(p^k y)$. As in [@fuc2 Lemma 65.4], if $n_1,\dots,n_t$ are the positive indexes before the gaps occur and we set $r_{n_i}=n_i+k_{i+1}$ and $k_1=r_0$ then we have cyclic direct summands of exponent $n_i+k_i$, with $i=1,\dots,t$.
If $k=0$ then we have no cyclic direct summands of exponent $1,\dots,n+\ell-1$. Then every element $y$ of exponent $\leq n$ must have height $\geq \ell$.
If $k>0$, let $n_j\leq k$ be the largest index $n_i\leq k$. Then $h(p^k y)=k+k_{j+1}<k+ \ell$ and $G$ has a direct summand of exponent $n_{j+1}+k_{j+1}$. Since $k<n_{j+1}\leq n$, we obtain that $G$ has a cyclic direct summand of exponent $e$ with $k<e<n+\ell$, a contradiction.
**Case II: $r_{e-1}$ is infinite.** Let $u=h(p^{n-1}x)$. If $B=\bigoplus_{i>0}B_i$ is a basic subgroup of $G$, where $B_i$ are homogeneous subgroups of exponent $i$, we consider the direct decomposition $G=B_1\oplus\dots \oplus B_u\oplus p^u G$ and write $y=y_1+\dots +y_u+y^*$ with $y_i\in B_i$ for all $i=1,\dots,u$ and $y^*\in p^u G$. It is obvious that $U(x)\leq U(y^*)$. Moreover, $y-y^*$ satisfies Case I and $\exp(y-y^*)<\exp(x)$. Therefore $U(x)\leq\min\{U(y-y^*),U(y^*)\}$, and it follows that $U(x)\leq
U(y)$.
Let ${\langle}x{\rangle}\in{\mathcal Q}(G)$. If $U(x)$ has no non–trivial gap then ${\langle}x{\rangle}\in{\mathcal {S}}(G)$. If $U(x)$ has a non–trivial gap at $k$, then $x\in H$, a summand of $G$, where $H$ is cyclic if $k=0$ and finite of rank $2$ otherwise.
It is easy to see that ${\mathcal Q}(G)$ is closed with respect direct summands.
The set ${\mathcal Q}(G)$ is not closed under direct sums, even in the case that $G$ is a finite $p$–group.
Let $x,\ y\in G$ such that $U(x)$ has a single non–trivial gap before index $k$ and $h(p^kx)=k+\ell$ wih $\ell>1$. Let ${\langle}y{\rangle}$ be cyclic of exponent $k+1$. Then ${\langle}x{\rangle}$ and ${\langle}y{\rangle}\in{\mathcal Q}(G)$ but ${\langle}x{\rangle}\oplus{\langle}y{\rangle}\not\in{\mathcal Q}(G)$.
Finite subgroups in ${\mathcal Q}(G)$ {#finite}
=====================================
We now extend the results of Section \[sect-cyclic\] from cyclic to finite subgroups. The main result of this section is that a finite subgroup $H$ of a group $G$ is in ${\mathcal Q}(G)$ if and only if it is a valuated direct sum of cyclic subgroups from ${\mathcal Q}(G)$.
Recall from [@HRW77] that if $H\subseteq G$, the *valuation of $H$ induced by heights in $G$* is defined by $v(x)=h(x)$, the height of $x$ in $G$, for all $x\in H$ and $H=K\oplus L$ is a *valuated direct sum* if $v(k+\ell)=\min\{v(k), v(\ell)\}$ for all $k\in K$ and $\ell\in
L$. In the following results, the valuation of $H$ is always that induced by heights in $G$. Consequently, $f\in{\operatorname{Hom}}(H,G)$ does not decrease valuations if and only if $f$ does not decrease Ulm sequences in $G$.
\[valuated\] Let $G$ be a group and let $K,\ L\leq {\mathcal Q}(G) $ with $K\cap L=0$. If $K\oplus L\in{\mathcal W}(G)$ then $K\oplus L$ is a valuated direct sum.
Suppose the direct sum $K\oplus L$ is not valuated. Then there exists $(k,\ell)\in K\oplus L$ such that $h^G((k,\ell))>\min\{h^G(k),\,h^G(\ell)\}$. For example, say $h^G((k,\ell))>h^G(k)$. Let $f\in{\operatorname{End}}(K\oplus L)$ be the natural projection onto $K$. Then $h^G(f(k,\ell))=h^G(k)<h^G((k,\ell))$ so $f$ cannot be extended to ${\operatorname{End}}(G)$, a contradiction.
\[finite-1\] Let $G$ be a group, $K$ a [pure]{} subgroup of $G$ such that $K$ is a direct sum of cyclic groups and $G/K$ is divisible, and $H\leq K$ a finite subgroup. If $f:\,H\to G$ is a homomorphism, the following are equivalent:
1. $f$ can be extended to an endomorphism of $G$;
2. $f$ can be extended to a homomorphism $\overline{f}:K\to
G$;
3. $f$ can be extended to a homomorphism $\overline{f}:K\to G$ such that $\overline{f}(K)$ is bounded.
(1)$\Rightarrow$(2) is obvious.
(2)$\Rightarrow$(3) We extend $f$ to a homomorphism $g:K\to G$. Since $H$ is finite and $K$ is a direct sum of cyclic groups, there is a finite direct summand $L$ of $K$ such that $H\leq L$. If $L\oplus M=K$, we define $\overline{f}:K\to G$ by $\overline{f}(x+y)=g(x)$ for all $x\in L$ and $y\in M$.
(3)$\Rightarrow$(1) Let $\overline{f}$ be as in (3). We have to extend $\overline{f}$ to an endomorphism of $G$. Let $k>0$ be an integer such that $\overline{f}(K)$ is bounded by $p^k$. Since $G/K$ is divisible, it is easy to see that $G=K+p^kG$, hence for every $x\in G$ there are $y\in K$ and $z\in G$ such that $x=y+p^kz$. [Since $K$ is pure in $G$]{}, it is not hard to see that the map $g:G\to G$, $g(x)=\overline{f}(y)$ is well defined, and it represents an endomorphism of $G$ which extends $f$.
\[finite-1-Q-W\] Let $G$ be a group and let $K$ be a pure subgroup of $G$ such that $K$ is a direct sum of cyclic groups with $G/K$ is divisible. Let $H$ be a finite subgroup of $K$.
1. The following are equivalent:
- $H\in{\mathcal Q}(G)$;
- every homomorphism $f:H\to G$ can be extended to a homomorphism $\overline{f}:K\to G$;
- every homomorphism $f:H\to G$ can be extended to a homomorphism $\overline{f}:K\to G$ such that $\overline{f}(K)$ is bounded.
2. The following are equivalent:
- $H\in{\mathcal W}(G)$;
- every endomorphism $f:H\to H$ can be extended to a homomorphism $\overline{f}:K\to G$;
- every endomorphism $f:H\to H$ can be extended to a homomorphism $\overline{f}:K\to G$ such that $\overline{f}(K)$ is bounded.
We are now able to characterize finite subgroups of $G$ in ${\mathcal Q}(G)$, respectively in ${\mathcal W}(G)$.
\[finite-Q(G)\] Let $G$ be a group and let $H=\bigoplus_{i=1}^n H_i$ be a finite subgroup such that all $H_i$ are cyclic groups. The following are equivalent:
1. $H\in{\mathcal Q}(G)$;
2. 1. $H_i\in{\mathcal Q}(G)$ for all $i=1,\dots,n$,
2. $H=\bigoplus_{i=1}^n H_i$ is a valuated direct sum of cyclic groups.
(1)$\Rightarrow$(2) This follows from Lemma \[valuated\].
(2)$\Rightarrow$(1) **Case I: $G$ has an unbounded basic subgroup.** Then $H_i\cap G^1=0$ for every $i\in \{1,\dots,n\}$. Since the direct sum $\bigoplus_{i=1}^n H_i$ is valuated, it follows that $H\cap G^1=0$, hence there is a basic subgroup $B\leq
G$ such that $H\leq B$. By Lemma \[finite-1\], it is enough to prove that every homomorphism $f:H\to G$ can be extended to a homomorphism $f':B\to G$.
We consider a homomorphism $f:H\to G$. If $x\in B$ we denote by $h^B(x)$ the height of $x$ calculated in $B$ and by $h(x)$ the height of $x$ as an element of $G$.
Then the restrictions $f|_{H_i}$ can be extended to endomorphisms of $G$, and it follows that $h(x_i)\leq h(f(x_i))$ for all $i$ and all $x_i\in H_i$.
Let $x=x_1+\dots+x_n\in H$ with $x_i\in H_i$ for all $i$. We observe that $$\begin{aligned}
h^B(x)\leq h(x)&=\min\{h(x_i)\mid
i=1,\dots,n\}\leq \min\{h(f(x_i))\mid i=1,\dots,n\}\\ &\leq
h(f(x_1)+\dots+f(x_n))=h(f(x)).\end{aligned}$$ Since $B/H$ is a direct sum of cyclic groups, and $H$ is a nice subgroup of $B$ as a consequence of [@fuc2 79(b)], we can apply [@fuc2 Corollary 81.4] to conclude that there is a homomorphism $f':B\to G$ which extends $f$, and the proof is complete.
**Case II: $G=B\oplus D$ with $B$ bounded and $D$ divisible.** Let $X=\{i\in\{1,\dots,n\}\mid H_i\cap G^1=0\}$ and $Y=\{1,\dots,n\}\setminus X$. Since the direct sum $\bigoplus_{i=1}^n H_i$ is valuated, it follows that $\bigoplus_{i\in X} H_i\cap G^1=0$, hence we can suppose $\bigoplus_{i\in X} H_i\leq B$.
For every $i\in Y$ we fix a generator $h_i$ for $H_i$, and write $h_i=b_i+d_i$ with $\langle b_i\rangle$ a direct summand of $B$, $d_i\in D$. Since $H_i\cap G^1\neq 0$, it follows that $\exp(B)=\exp(b_i)<\exp(d_i)=\exp(h_i)$ by Lemma \[cyclic2\].
We claim that $\sum_{i\in Y}\langle b_i\rangle=\bigoplus_{i\in
Y}\langle b_i\rangle$. In order to prove this, suppose by contradiction that there exist an index $j\in Y$ and a non-zero element $0\neq k_j b_j=\sum_{i\in Y\setminus \{j\}}k_i b_i$. Then $k_j h_j- \sum_{i\in Y\setminus \{j\}}k_i h_i$ is of infinite height, hence the sum $\bigoplus_{i\in Y} H_i$ is not valuated since $k_jh_j$ is of finite height. This contradicts our hypothesis, so the claim is true. Moreover, $\bigoplus_{i\in
Y}\langle b_i\rangle$ is a direct summand of $B$ as a bounded pure subgroup, and using [@fuc1 Exercise 9.8] we conclude that it is an absolute direct summand of $B$.
Using a similar argument, if we suppose that $(\bigoplus_{i\in
Y}\langle b_i\rangle)\cap (\bigoplus_{i\in X}H_i)\neq 0$ we obtain that $\bigoplus_{i=1}^{n}H_i$ is not a valuated direct sum, a contradiction. Hence $$\textstyle{(\bigoplus_{i\in Y}\langle
b_i\rangle)\cap (\bigoplus_{i\in X}H_i) =0,}$$ and it follows that there is a direct complement $C$ of $(\bigoplus_{i\in Y}\langle
b_i\rangle)$ in $B$ such that $\bigoplus_{i\in X}H_i\leq C$.
Moreover, since the sum $\bigoplus_{i\in Y}H_i$ is direct and $H_i[p]=\langle d_i\rangle[p]$ for all $i\in Y$, it follows that the sum $\sum_{i\in Y}\langle d_i\rangle$ is a direct sum. Hence we can find infinite quasi-cyclic subgroups $D_i$, $i\in Y$, such that $D=(\bigoplus_{i\in Y}D_i)\oplus D'$ and $d_i\in D_i$ for all $i\in Y$.
Let $f:H\to G$ be a homomorphism. Using the same argument as in the first case we observe that every homomorphism $\bigoplus_{i\in
X} H_i\to G$ can be extended to a homomorphism $f_0:C\to G$ (note that the valuation induced on $\bigoplus_{i\in X} H_i$ by $C$ is the same as the valuation induced by $G$).
For all $i\in I$, we have $\exp(f(h_i))\leq\exp(h_i)$, hence $f(h_i)=a_i+z_i$ with $a_i\in B$ and $z_i\in D$ such that $\exp(z_i)\leq\exp(d_i)$. Therefore there exist homomorphisms $f'_i:\langle b_i\rangle \to G$ such that $f'_i(b_i)=a_i$ and $f''_i:D_i\to D$ such that $f''_i(d_i)=z_i$. Since $$\textstyle{G=C\oplus (\bigoplus_{i\in Y}\langle
b_i\rangle)\oplus (\bigoplus_{i\in Y}D_i)\oplus D',}$$ the homomorphisms $f_0$, $f'_i$ and $f''_i$, $i\in Y$, induce an endomorphism $\overline{f}:G\to G$, and it is easy to see that $\overline{f}$ extends $f$.
Cyclic valuated groups are characterized using invariants in [@HRW77 Theorem 3]. Therefore this result together with Theorem \[finite-Q(G)\] and Theorem \[3.14\] give us a characterization by invariants for subgroups in ${\mathcal Q}(G)$.
We close this section with a characterization of some finite subgroups in ${\mathcal W}(G)$.
\[finite-W(G)\] Let $G$ be a group and $H=\bigoplus_{i=1}^n H_i$ a finite subgroup such that $H\cap p^\omega G=0$ and each $H_i=\langle z_i\rangle$ is a cyclic group of exponent $e_i$. The following are equivalent:
1. $H\in{\mathcal W}(G)$;
2. 1. If $e_j\leq e_i$ then $U(z_i)\leq
U(z_j)\leq U(p^{e_i-e_j}z_i)$,
2. $H=\bigoplus_{i=1}^n H_i$ is a valuated direct sum of cyclic groups.
(1)$\Rightarrow$(2) In order to prove (a), let $i,j$ be two indices such that $e_j\leq e_i$. Then there are homomorphisms $f:H_i\to H_j$ with $f(z_i)=z_j$ and $g:H_j\to H_i$ with $f(z_j)=p^{e_i-e_j}z_i$. Since these homomorphisms can be extended to endomorphisms of $H$, they can be extended to endomorphisms of $G$. The inequalities $U(z_i)\leq U(z_j)\leq U(p^{e_i-e_j}z_i)$ follow from the fact that endomorphisms do not decrease heights.
The statement (b) is a consequence of Lemma \[valuated\].
(2)$\Rightarrow$(1) As in the proof of Theorem \[finite-Q(G)\], it is enough to prove that every homomorphism of $f:H_i\to H$ does not decrease the valuation.
Let $f:H_j\to H$ be a homomorphism defined for some $j\in\{1,\dots,n\}$. Since $U(mz)=U(z)$ for all integers $m$ with $(m,p)=1$, it is enough to prove that $U(p^kz_j)\leq
U(p^k(f(z_j)))$ for all $0\leq k<e_j$. Since for every element $x$ and for every positive integer $k$ the indicator $U(p^kx)$ can be obtained by deleting the first $k$ components of $U(x)$, it is enough to prove $U(z_j)\leq U(f(z_j))$.
Let $f(z_j)=\sum_{i=1}^{n}m_iz_i$. Note that if $e_j< e_i$ then $p^{e_i-e_j}$ divides $m_i$. Then $$\textstyle{ f(z_j)=(\sum_{e_j<e_i}n_ip^{e_i-e_j}z_i)+(\sum_{e_i\leq
e_j}m_iz_i),}$$ hence $$U(f(z_j))=\min \{U(n_ip^{e_i-e_j}z_i)\mid
e_j<e_i\}\cup\{U(m_i z_i)\mid e_i\leq e_j\}\geq U(z_j),$$ and the proof is complete.
In the following example we show that both pairs of conditions (a) and (b) in Theorem \[finite-Q(G)\] and Theorem \[finite-W(G)\] respectively are necessary.
\[example-nonQ-valuated\] Let $G=\langle a\rangle\oplus \langle b\rangle\oplus \langle
c\rangle$ be a group with $\exp(a)=1$ and $\exp(b)=\exp(c)=2$. Then $x=(a,pb,0)$ and $y=(a,0,pc)$ generate direct summands. We have $\langle x,y\rangle=\langle x\rangle\oplus \langle
y\rangle=\langle x\rangle\oplus \langle (0,-pb,pc)\rangle$. The direct sum $\langle x\rangle\oplus \langle y\rangle$ is not a valuated direct sum, while $\langle x\rangle\oplus
\langle (0,-pb,pc)\rangle$ is a valuated direct sum, but $\langle (0,-pb,pc)\rangle\notin {\mathcal Q}(G)$.
\[counterexample\] The result of Theorem \[finite-W(G)\] cannot be extended to infinite direct sums of cyclic groups. Pierce [@P63 Theorem 15.4] has constructed an example of a separable $p$–group $G$ with standard basic subgroup (i.e. $B=\bigoplus_{i=1}^\infty {\mathbb Z}(p^i))$ such that ${\operatorname{End}}(G)=J +E $ where $J$ is the rank 1 torsion–free complete $p$–adic module generated by the identity and $E$ is the ideal of small endomorphisms. On the other hand, ${\operatorname{End}}(B)$ has infinite torsion–free $p$–adic rank, so $B\not\in{\mathcal W}(G)$.
Finitely generated subgroups in ${\mathcal {P}}(G)$
===================================================
Jain and Singh proved in [@JS67] that if $R$ is a principal ideal domain, then all pseudo–injective modules are quasi–injective. In this section we prove a stronger version of this for $R={\mathbb Z}$: if $G$ is a group then all finitely generated subgroups in ${\mathcal {P}}(G)$ are in ${\mathcal Q}(G)$.
*In this section, $G$ is an arbitrary abelian group.*
In order to prove ${\mathcal Q}_f(G)={\mathcal {P}}_f(G)$ for all groups $G$ we start with the case of $p$-groups.
Let $G$ be a $p$-group and $H\in{\mathcal {P}}(G)$. If $K$ is a cyclic direct summand of $H$ then $K\in{\mathcal {P}}(G)$.
Let $L$ be a direct complement of $K$ in $H$, so $H=K\oplus L$, and let $\varphi:K\to G$ be a monomorphism.
If $\varphi(K)\cap L=0$, then the homomorphism $\psi:K\oplus L\to
G$, $\psi(x,y)=\varphi(x)+y$ is a monomorphism, hence it can be extended to an endomorphism $\overline{\varphi}\in{\operatorname{End}}(G)$. It is easy to see that $\overline{\varphi}$ also extends $\varphi$.
If $\varphi(K)\cap L\neq 0$, we first observe that the socle $\varphi(K)[p]$ is contained in $L$ since $\varphi(K)$ is a cyclic $p$-group (hence its subgroup lattice is a finite chain). Let $\varphi':K\to G$ be the homomorphism defined by $\varphi'(x)=\varphi(x)-x$. Suppose that $\varphi'$ is not a monomorphism. Then there exists a non-zero element $x\in K$ such that $\varphi'(x)=0$. Then $\varphi(x)=x\in K$, hence $\varphi(K)\cap K\neq 0$. It follows that $\varphi(K)[p]\subseteq
K$, and this contradicts $K\cap L=0$. Hence $\varphi'$ is a monomorphism. Suppose that $\varphi'(K)\cap L\neq 0$. Then there exists $x\in K$ such that $0\neq \varphi(x)-x\in L$. If $e$ is the exponent of $x$ then $p^{e-1}(\varphi(x)-x)\in L[p]$. But $p^{e-1}\varphi(x)\in\varphi(K)[p]\subseteq L[p]$, hence $p^{e-1}x\in L[p]$, a contradiction. Then $\varphi'(K)\cap L=0$ and we can apply what we proved so far to observe that there exists an endomorphism $\psi$ of $G$ which extends $\varphi$. Then for every $x\in K$ we have $\varphi(x)=\psi(x)+x$, hence $\psi+1_G$ extends $\phi$.
We need the following technical result:
\[lemma-dsvc\] Let $G$ be a $p$-group and $H=\bigoplus_{i=1}^n H_i$ a finite subgroup such that all $H_i=\langle h_i\rangle$ are cyclic groups such that
1. $\exp(h_1)\leq \exp(h_2)\leq\dots\leq\exp(h_n)$,
2. for all $m\in\{1,\dots,n\}$ and for all $x\in
\bigoplus_{i=1}^m H_i$ we have $U(h_m)\leq U(x)$, and
3. if $\exp(h_i)=\exp(h_j)$ then $U(h_i)=U(h_j)$.
Then the direct sum $\bigoplus_{i=1}^n H_i$ is a valuated direct sum of cyclic $p$-groups.
The proof is by induction. For $n=1$ the property is obvious. Suppose that (ii) is valid for all $m<n$. Then $\bigoplus_{i=1}^{n-1} H_i$ is a valuated direct sum of cyclic groups. Let $k$ be the minimal index such that $\exp(H_n)=\exp(H_k)$.
We observe that the sequence $U(h_i),\ i=1,\dots,n$ is a decreasing sequence such that $U(h_i)=U(h_j)$ if and only if $\exp(H_i)=\exp(H_j)$. Moreover, it follows by (b) that $U(h_n)\leq U(y)$ for all $y\in H$.
If $U$ is an Ulm sequence, we denote, as in [@HRW77 p.100], $$H(U)=\{x\in H:\, U(x)\geq U\},$$ $$H(U)^*=\{x\in H:\,
U(x)> U\}$$ and we consider the ${\mathbb Z}(p)$-vector space $$H_U=\frac{H(U)+pH}{H(U )^*+pH}.$$ Recall that a *$v$-basis* for $H$ is constructed in the following way: for every $U $ we fix a basis in $H_U $, and we choose one representative whose Ulm sequence is $U $ for each element in this basis; the union of all these representatives is a $v$–basis for $H$. It is proved in [@HRW77 Theorem 3] that $H$ is a valuated direct sum of cyclics if and only if the cardinal of a $v$–basis coincides to the rank of $H$. Moreover, in this hypothesis every $v$–basis is linearly independent and it generates $H$. Therefore, it is enough to prove that $\{h_i:\,
i=1,\dots,n\}$ is a $v$–basis for $H$.
Since $K=\bigoplus_{i=1}^{n-1} H_i$ is a direct sum of cyclic valuated groups, it follows that the set $\{h_i:\,
i=1,\dots,n-1\}$ is a $v$–basis for $K$.
Let $V $ be the Ulm sequence of $h_n$. If $U< V $ then $H(U)=H(U )^*=H$, so $H_U =0$. If $U $ and $V $ are not comparable then $H(U )=H(U )^*$ since $V $ is minimal as Ulm sequence of an element of $H$, so $H_U =0$.
It is easy to see that $H(V )=H$, and $H(V
)^*=(\bigoplus_{i<k}H_i)\oplus(\bigoplus_{i=k}^n pH_i)$, so $h_{k},\dots,h_{n}$ represent a basis in $H_V $.
If $V <U $ then $$\textstyle{H(U )+pH=(\bigoplus_{U(h_i)\geq
U }H_i)+pH=K(U )\oplus pH_n}$$ and $$\textstyle{H(U )^*+pH=(\bigoplus_{U(h_i)> U }H_i)+pH=K(U )^*\oplus
pH_n.}$$ Therefore every set in $K=\bigoplus_{i=1}^{n-1} H_i$ which represents a basis in $K_U $ is also a representative set for a basis in $H_U $.
It follows that $\{h_1,\dots,h_n\}$ is a $v$-basis, and an application of the proof of [@HRW77 Theorem 3] will complete the proof.
\[basic-in\] Let $G$ be a $p$-group and $x,y\in G$. If $U(x+y)=U(x)$ then $U(x)\leq U(y)$.
Suppose that $U(x)\nleqq U(y)$. Then there exists a positive integer $k$ such that $h(p^k y)<h(p^k x)$. It follows that $h(p^k(x+y))=h(p^k y)\neq h(p^k x)$, and this contradicts our hypothesis.
\[Pf=Qf\] Let $G$ be a group. Then ${\mathcal {P}}_f(G)={\mathcal Q}_f(G)$.
Since only the inclusion ${\mathcal {P}}_f(G)\subseteq {\mathcal Q}_f(G)$ requires a proof, we start with a finitely generated subgroup $H\in{\mathcal {P}}(G)$.
If $G$ is a $p$-group, we write $H=\bigoplus_{i=1}^n H_i$ such that all $H_i=\langle h_i\rangle$ are cyclic groups with $$\exp(h_1)\leq \exp(h_2)\leq\dots\leq\exp(h_n).$$ We will prove that this decomposition satisfies the conditions (ii) and (iii) from Lemma \[lemma-dsvc\].
Let $m\in\{1,\dots,n\}$ and $j<m$. Then $H=\bigoplus_{i=1}^n
H'_i$, where $H'_i=\langle h_i\rangle$ for all $i\neq m$ and $H'_i=\langle h_m+h_j\rangle$. Since $H\in{\mathcal {P}}(G)$, the isomorphism $\varphi:H\to H$ defined by $\varphi(h_i)=h_i$ for all $i\neq m$ and $\varphi(h_m)=h_m+h_j$ can be extended to an endomorphism of $G$. Then $U(h_m)\leq U(h_m+h_j)$. But $\varphi^{-1}$ also can be extended to an endomorphism of $G$, hence $U(h_m)\geq U(h_m+h_j)$. Therefore, $U(h_m)= U(h_m+h_j)$, and applying Lemma \[basic-in\] we obtain that the condition (ii) is satisfied.
In order to prove (iii), it is enough to observe that if $\exp(h_i)=\exp(h_j)$ with $i<j$ we can replace in the direct decomposition of $H$ as direct sum of cyclic groups either of the two direct summands $\langle h_i\rangle$ or $\langle h_j\rangle$ by $\langle h_i+h_j\rangle$. By what we just proved for (ii) we have $U(h_i)=U(h_j)$.
Therefore, we proved that ${\mathcal Q}_f(G)={\mathcal {P}}_f(G)$ for all $p$-groups. It is not hard to extend this property to all torsion groups and we now show that result can be extended to all abelian groups.
Let $G$ be a group and $H\in {\mathcal {P}}_f(G)$. Since $H$ is finitely generated, $H=F\oplus K$, with $F$ a free subgroup of finite rank and $K$ a finite subgroup. We claim that $F$ and $K$ are in ${\mathcal Q}(G)$ and every homomorphism $\varphi:F\to G$ can be extended to an endomorphism $\overline{\varphi}$ of $G$ such that $\overline{\varphi}(K)=0$.
Since $K\leq T(G)$, every monomorphism $\varphi:K\to G$ can be extended to a monomorphism $\varphi':H\to G$ such that $\varphi'(x)=x$ for all $x\in F$. Then there is an endomorphism $\overline{\varphi}$ of $G$ which extends $\varphi'$. Since $T(G)$ is fully invariant, $\overline{\varphi}$ induces an endomorphism of $G$ which extends $\varphi$. Therefore $K\in{\mathcal {P}}_f(T(G))={\mathcal Q}_f(T(G))$, and it follows that we can embed $K$ in a finite direct summand $L$ of $G$.
In order to prove $F\in{\mathcal Q}(G)$, let $\varphi:F\to G$ be a homomorphism. For every positive integer $i$ we consider the subgroup $$\textstyle{U_i=\{x\in F\mid \varphi(x)=ix\}\leq F,}$$ and we will prove by induction on $n$ that $$\textstyle{\sum_{i=1}^nU_i=\bigoplus_{i=1}^nU_i}$$ for all $n>0$. Since the case $n=1$ is obvious, suppose that $\sum_{i=1}^nU_i=\bigoplus_{i=1}^nU_i$. Let $x\in
(\sum_{i=1}^nU_i)\cap U_{n+1}$. Then $x=\sum_{i=1}^nx_i$ with $x_i\in U_i$, hence $$\textstyle{(n+1)\sum_{i=1}^nx_i=\varphi(x)=\sum_{i=1}^n\varphi(x_i)=\sum_{i=1}^nix_i.}$$ Then $\sum_{i=1}^n(n+1-i)x_i=0$, and by the induction hypothesis $(n+1-i)x_i=0$ for all $i=1,\dots,n$. Since $F$ is torsion free, it follows that $x=0$. Then $\sum_{i>0}U_i=\bigoplus_{i>0}U_i\leq
F$. But $F$ is of finite rank, hence we can find an integer $N>0$ such that $U_n=0$ for all $n\geq N$. Let $q>N$ be a prime such that $\gcd(q,|K|)=0$. Therefore the homomorphism $\psi:F\to G$, $\psi(x)=\varphi(x)-qx$ is a monomorphism. Then $\psi(F)$ is torsion–free, and as in the first part of the proof it can be extended to a monomorphism $\psi':H\to G$ such that $\psi'(x)=-qx$ for all $x\in K$. Since $H\in{\mathcal {P}}(G)$ there is an endomorphism $\overline{\psi}$ of $G$ which extends $\psi'$. Then $\overline{\varphi}=\overline{\psi}+q1_G$ is an endomorphism of $G$ which extends $\varphi$ and $\overline{\varphi}(K)=0$.
Now we will prove that every homomorphism $\varphi:K\to G$ can be extended to an endomorphism $\overline{\varphi}$ of $G$ such that $\overline{\varphi}(F)=0$. Let $\varphi:K\to G$ be a homomorphism. If $\varphi':G\to G$ extends $\varphi$ then the restriction $\varphi'|_{F}:F\to G$ can be extended to an endomorphism $\psi$ of $G$ such that $\psi(K)=0$. Then $\overline{\varphi}=\varphi'-\psi$ has the required properties.
In order to complete the proof, let $\varphi:\,H\to G$ be a homomorphism. Then it induces by restrictions two homomorphisms $\varphi_1:F\to G$ and $\varphi_2:K\to G$. But by what we just proved we can extend these homomorphisms to two endomorphisms $\overline{\varphi}_1$ and $\overline{\varphi}_2$ of $G$ such that $\overline{\varphi}_1$ extends $\varphi_1$ and $\overline{\varphi}_1(K)=0$, and $\overline{\varphi}_2$ extends ${\varphi}_2$ and $\overline{\varphi}_2(F)=0$. Then $\overline{\varphi}=\overline{\varphi}_1+\overline{\varphi}_2$ extends $\varphi$ and the proof is complete.
In the case $G$ is a $p$-group with $p\geq 3$ there is a simpler proof. In fact, in order to prove $H\in {\mathcal Q}(G)$ for all finite subgroups $H\in{\mathcal Q}(G)$ it is enough to prove that Lemma \[valuated\] is valid for $K\oplus L\in{\mathcal {P}}(G)$. For the case $p\geq 3$ it is easy to see that the homomorphism $\varphi:K\to
L\to G$, $\varphi(k,\ell)=2k+\ell$ is a monomorphism, hence it can be extended to an endomorphism $\overline{\varphi}$ of $G$. Then $\overline{\varphi}-1_G$ extends the canonical projection $K\oplus
L\to K$ and the proof presented for Lemma \[valuated\] also works for ${\mathcal {P}}(G)$.
A careful analysis of the previous proof reveals the fact that $F$ can be replaced by any finite rank torsion-free group. Therefore we obtain:
Let $G$ be a group and $H$ a subgroup of $G$ of finite rank. Then $H\in{\mathcal Q}(G)$ if and only if $H\in{\mathcal {P}}(G)$.
We have been unable to determine whether Theorem \[Pf=Qf\] can be extended to all subgroups in ${\mathcal {P}}(G)$, i.e. if there exists an abelian group $G$ such that ${\mathcal Q}(G)\neq {\mathcal {P}}(G)$.
[9]{}
D. M. Arnold, , Springer–Verlag CMS Books in Mathematics, 2000.
G. Birkhoff, *Subgroups of abelian groups*, Proc. London Math. Soc. **38** (1934), 385–401.
M. Dugas, *Localizations of torsion-free abelian groups*, J Algebra **278** (2004), 411–429.
M. Dugas and R. Göbel, *Applications of abelian Groups and Model Theory to Algebraic Structures*, in: , Ravello 1994, de Gruyter (1996), 41–62.
N. Er, S. Singh and A. K. Srivastava, *Rings and modules which are stable under automorphisms of their injective hulls*, J. Algebra **379** (2013), 223–229.
C. Faith, , Lecture Notes in Mathematics **49**, Springer–Verlag, 1967.
L. Fuchs, , vol.1, Academic Press, 1970.
L. Fuchs. , vol. 2, Academic Press, 1973.
R. Hunter, F. Richman and E. Walker, *Finite direct sums of cyclic valuated $p$-groups*, Pacific J. Math. **69** (1977), 97–104.
S.K. Jain and S. Singh, *On pseudo-injective modules and self-pseudo-injective rings*, J. Math. Sciences **2** (1967), 23–31.
M. Kilp, *Quasi-injective Abelian Groups*, Vestnik Moskov. Univ. Ser. I Mat. Meh. **22** (1967), 3–4.
A. P. Mišina, *On automorphisms and endomorphisms of Abelian Groups*, Vestnik Moskov. Univ. Ser. I Mat. Meh. **17** (1962), 39–43.
R. S. Pierce, *Homomorphisms of primary abelian groups* in , Scott, Foresman and Co., 1963, 215–310.
C. M. Ringel and M. Schmidmeier, *Submodule categories of wild representation type*, J. Pure and Applied Alg. **205** (2006), 412–422.
C. M. Ringel and M. Schmidmeier, *The Auslander-Reiten translation in submodule categories*, Trans. Amer. Math. Soc. **360** (2008), 691–716.
F. Richman and E. Walker. *Subgroups of $p^5$ bounded groups*, in **Abelian Groups and Modules**, Birkhäuser, Boston, 1999, 55–74.
R. Sanderson, *A characterization of quasi-injective Abelian Groups*, J. Elisha Mitchell Sci. Soc. **89** (1973), 143–146.
S. Singh, *On pseudo-injective modules*, Riv. Mat. Univ. Parma **9** (1968), 59–65.
M. L. Teply, *Pseudo-injective modules that are not quasi–injective*, Proc. Amer. Math. Soc. **49** (1975), 305–310.
[^1]: S. Breaz is supported by the CNCS-UEFISCDI grant PN-II-RU-TE-2011-3-0065
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addtoreset[equation]{}[section]{}
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[In this work we classify the subalgebras satisfied by non-geometric $Q$-fluxes in type IIB orientifolds on ${\mbox{\small$\T^6/(\Z_2 \times \Z_2)$}}$ with three moduli $(S,T,U)$. We find that there are five subalgebras compatible with the symmetries, each one leading to a characteristic flux-induced superpotential. Working in the 4-dimensional effective supergravity we obtain families of supersymmetric ${\rm AdS}_4$ vacua with all moduli stabilized at small string coupling $g_s$. Our results are mostly analytic thanks to a judicious parametrization of the non-geometric, RR and NSNS fluxes. We are also able to leave the flux-induced $C_4$ and $C_8$ RR tadpoles as free variables, thereby enabling us to study which values are allowed for each $Q$-subalgebra. Another novel outcome is the appearance of multiple vacua for special sets of fluxes. However, they generically have $g_s > 1$ unless the net number of O3/D3 or O7/D7 sources needed to cancel the tadpoles is large. We also discuss briefly the issues of axionic shift symmetries and cancellation of Freed-Witten anomalies. ]{}
Introduction {#sec:intro}
============
The study of flux compactifications in string theory has been pursued intensively in recent years [@generalreviews]. One important motivation is the possibility to stabilize the massless moduli at a minimum of the potential induced by the fluxes. The simplest scenarios for this mechanism are provided by type IIB and type IIA 1 orientifolds with $p$-form fluxes turned on [@generalreviews]. In IIA compactifications the mixture of NSNS and RR fluxes generates a superpotential that depends on all closed string moduli allowing to stabilize them without invoking non-perturbative effects [@Grimm; @Derendinger; @vz1; @DeWolfe; @cfi]. Moreover, in the IIA setup it is natural to add the so-called geometric $f$-fluxes that determine the isometry algebra of the internal space [@Derendinger; @vz1; @cfi]. The case of nilpotent algebras was studied in [@gmpt; @andriot; @caviezel] and an example with internal $\mathfrak{su(2)^2}$ was spelled out in [@af].
To recover T-duality between IIA and IIB compactifications, it is necessary to introduce new parameters referred to as non-geometric fluxes [@stw1; @stw2; @wecht]. The original observation is that performing a T-duality to NSNS $\bar H$-fluxes leads to geometric $f$-fluxes [@glmw; @kstt]. Further T-dualities give rise to generalized $Q$ and $R$-fluxes [@stw1]. The $Q$’s are called non-geometric because the emerging background after two T-dualities can be described locally but not globally. The third T-duality is formal, evidence for the $R$-fluxes comes rather from T-duality at the level of the effective superpotential [@stw1]. Moreover, the $Q$ and $R$-fluxes logically extend [@stw1; @dabholkar] the set of structure constants of the gauge algebra, generated by isometries and shifts of the $B$ field, that is known to contain the geometric and NSNS fluxes [@ss; @km].
In this article we consider type IIB orientifolds with O3/O7-planes in which only NSNS $\bar H$ and non-geometric $Q$-fluxes are invariant under the orientifold action. These fluxes together induce a superpotential that depends on all closed string moduli. One advantage of working with IIB is that the $Q$-fluxes by themselves appear as the structure constants of a subalgebra of the full gauge algebra. However, one must keep in mind that the $\bar H$ and $Q$ in IIB map into all kinds of fluxes in type IIA with O6-planes, and into non-geometric $R$ plus geometric $f$ in IIB with O9/O5-planes. Similar examples with generalized fluxes have been considered by several authors [@stw1; @acfi; @stw2; @vz2; @benmachiche; @tasinato; @ihl; @palti; @camara].
Our guiding principle is precisely the classification of the subalgebras satisfied by the non-geometric $Q$-fluxes. We will discuss a simplified scheme with additional symmetries in order to reduce the number of fluxes. Concretely, we study compactification on ${\mbox{\small$(\T^2 \times \T^2 \times \T^2)/(\Z_2 \times \Z_2)$}}$, and further impose invariance under exchange of the internal $\T^2$’s. In this way we obtain the same model with moduli $(S,T,U)$ proposed in [@stw1] and generalized in [@acfi]. We have classified the allowed subalgebras of the $Q$-fluxes of the $(S,T,U)$-model. There are five inequivalent classes, namely $\mathfrak{so(4)}$, $\mathfrak{so(3,1)}$, $\mathfrak{su(2)+u(1)^3}$, $\mathfrak{iso(3)}$ and the nilpotent algebra denoted $n(3.5)$ in [@gmpt]. The non-semisimple solutions are contractions of $\mathfrak{so(4)}$ consistent with the symmetries. A compelling byproduct is that each subalgebra yields a characteristic flux-induced superpotential. The corresponding 12-dimensional gauge algebras can be easily identified after a convenient change of basis.
We are mostly interested in discovering supersymmetric flux backgrounds with non-geometric fluxes switched on, and all moduli stabilized. To this end we work exclusively with the 4 effective action. We widen the search of vacua of [@stw2] in several respects. A key difference is that in most cases we can solve the F-flat conditions analytically and can therefore derive explicit expressions for the moduli vevs in terms of the fluxes. The computations are facilitated by using a transformed complex structure , invariant under the modular group $SL(2,\Z)_U$. The independent non-geometric fluxes are precisely parametrized by $\Gamma=\genfrac{(}{)}{0pt}{}{\a \, \b}{\g \, \d}$. The parametrization of NSNS and RR fluxes is also dictated by $\Gamma$. By exploiting the variable $\cz$ we can effectively factor out the vacuum degeneracy due to modular transformations.
There is a further vacuum degeneracy originating from special constant translations in the axions $\re S$ and $\re T$. We argue that vacua connected by this type of translations are identical because the full background including the RR fluxes is invariant under such axionic shifts.
In our analysis the values of the flux-induced $C_4$ and $C_8$ RR tadpoles are treated as variables. To cancel these tadpoles in general requires to add D-branes besides the orientifold planes. These D-branes are also constrained by cancellation of Freed-Witten anomalies [@cfi; @vz2]. In our concrete setup, D3-branes and unmagnetized D7-branes wrapping an internal $\T^4$ are free of anomalies and can be included. However, such D-branes do not give rise to charged chiral matter.
By treating the flux tadpoles as variables we can deduce in particular that the vacua found in [@stw2], having O3-planes and no O7/D7 sources, can only arise when the $Q$-subalgebra is the compact $\mathfrak{so(4)}$. For completeness we study the supersymmetric ${\rm AdS}_4$ minima due to the fluxes of all compatible $Q$-subalgebras, including the non-compact $\mathfrak{so(3,1)}$. In general, such vacua exist in all cases but unusual types of sources might be needed to cancel the tadpoles. Interestingly, in models based on semisimple subalgebras we find that there can exist more than one vacuum for some combinations of fluxes.
It is well known that supersymmetric or no-scale Minkowski vacua in IIB orientifolds with RR and NSNS fluxes require sources of negative RR charge such as O3-planes or wrapped D7-branes [@gkp]. However, working with the effective 4 formalism we find that O3-planes and/or D7-branes can be bypassed in fully stabilized supersymmetric ${\rm AdS}_4$ vacua, provided specific non-geometric fluxes are turned on. It is conceivable that such vacua only occur in the effective theory and will not survive after lifting to a full string background. Helpful hints in this direction can come from our results relating properties of the vacua with the gauge algebra. It might well be that only models built on certain algebras can be lifted to full backgrounds. The newly proposed formulation of non-geometric fluxes based on compactification on doubled twisted tori suggests that the gauge algebra has to be compact or admit a discrete cocompact subgroup [@reid; @prezas]. It is also feasible that the recent description of non-geometric fluxes in the context of generalized geometry [@minasian] could be applied to deduce the generalized flux configurations which allow supersymmetric vacua. A discussion of these issues is beyond our present scope.
We now outline the paper. In section \[sec:gen\] we review the properties of the fluxes and write down the flux-induced effective quantities needed to investigate the vacua. The classification of the $Q$-subalgebras is carried out in section \[sec:alg\], where we also obtain the parametrization of the non-geometric and NSNS fluxes that is crucial in the subsequent analysis. In section \[sec:newvars\] we introduce the transformed complex structure $\cz$ motivated by modular invariance. Using this variable then points to the efficient parametrization of the RR fluxes given in the appendix. In the end we are able to derive very compact expressions for the flux-induced superpotential and tadpoles according to the particular $Q$-subalgebra. In section \[sec:vac\] we solve the F-flat conditions and collect the results that distinguish the vacua with moduli stabilized. The salient features of these vacua are discussed in section \[sec:lands\]. Section \[sec:end\] is devoted to some final comments.
Generalities {#sec:gen}
============
In this section we outline our notation to describe the non-geometric fluxes introduced in [@stw1]. To be specific we will work in the context of toroidal orientifolds with O3/O7-planes. We will discuss the case of generic untwisted moduli, and also the simpler isotropic model considered in [@stw1].
Fluxes {#ssec:fluxes}
------
The starting point is a type IIB string compactification on a six-torus $\T^6$ whose basis of 1-forms is denoted $\eta^a$. Moreover, we assume the factorized geometry \^6=\^2 \^2 \^2 : (\^[1]{},\^[2]{}) ( \^[3]{},\^[4]{} ) ( \^[5]{},\^[6]{} ) . \[factorus\] As in [@stw1], we will use greek indices $\alpha,\beta,\gamma$ for horizontal $\,``-"$ $x$-like directions $(\eta^{1},\eta^{3},\eta^{5})$ and latin indices $i,j,k$ for vertical $\,``|"$ $y$-like directions $(\eta^{2},\eta^{4},\eta^{6})$ in the 2-tori.
The $\Z_2$ orientifold involution denoted $\sigma$ acts as : ( \^[1]{},\^[2]{},\^[3]{},\^[4]{},\^[5]{},\^[6]{} ) ( -\^[1]{},-\^[2]{},-\^[3]{},-\^[4]{},-\^[5]{},-\^[6]{} ) . \[osigma\] There are 64 O3-planes located at the fixed points of $\sigma$. We further impose a $\Z_2 \times \Z_2$ orbifold symmetry with generators acting as \[orbifold1\] \_1 & : & ( \^[1]{},\^[2]{},\^[3]{},\^[4]{},\^[5]{},\^[6]{} ) (\^[1]{},\^[2]{},-\^[3]{},-\^[4]{},-\^[5]{},-\^[6]{} ) ,\
\_2 & : & (\^[1]{},\^[2]{},\^[3]{},\^[4]{},\^[5]{},\^[6]{} ) (-\^[1]{},-\^[2]{},\^[3]{},\^[4]{},-\^[5]{},-\^[6]{} ) . Clearly, there is another order-two element $\theta_3 = \theta_1 \theta_2$. Under this $\Z_{2} \times \Z_{2}$ orbifold group, only 3-forms with one leg in each 2-torus survive. This also occurs in the compactification with an extra $\Z_{3}$ cyclic permutation of the three 2-tori that was studied in [@stw1; @stw2]. In that case there are only O3-planes and two geometric moduli, namely the overall Kähler and complex structure parameters. In contrast, in our setup, the full symmetry group $\Z_2^3$ includes additional orientifold actions $\sigma \theta_I$ that have fixed 4-tori and lead to , $I=1,2,3$. Another difference is that in principle we have one Kähler and one complex structure parameter for each 2-torus $\T_I^2$.
The Kähler form and the holomorphic 3-form that encode the geometric moduli of the internal space can be written in a basis of invariant forms that also enters in the description of background fluxes. Under the $\Z_2 \times \Z_2$ orbifold action the invariant 3-forms are just
[lclclcl]{} \_[0]{}=\^[135]{} & ; & \_[1]{}=\^[235]{} & ; & \_[2]{}=\^[451]{} & ; & \_[3]{}=\^[613]{} ,\
\^[0]{}=\^[246]{} & ; & \^[1]{}=\^[146]{} & ; & \^[2]{}=\^[362]{} & ; & \^[3]{}=\^[524]{} .
\[basisab\] where, e.g. $\eta^{135}= \eta^1 \wedge \eta^3 \wedge \eta^5$. Clearly, these forms are all odd under the orientifold involution $\sigma$. On the other hand, the invariant 2-forms and their dual 4-forms are
[lclcl]{} \_[1]{}=\^[12]{} & ; & \_[2]{}=\^[34]{} & ; & \_[3]{}=\^[56]{} ,\
\^[1]{}=\^[3456]{} & ; & \^[2]{}=\^[1256]{} & ; & \^[3]{}=\^[1234]{} .
\[inv2form\] These forms are even under $\sigma$. We choose the orientation and normalization \_[\_6]{} \^[123456]{}=\_6 . \[normal1\] The positive constant $\cv_6$ gives the volume of the internal space that we generically denote $\M_6$. Notice that the basis satisfies \_[\_6]{} \_[0]{} \^[0]{}= -\_6 , \_[\_6]{} \_[I]{} \^[J]{}= \_[\_6]{} \_[I]{} \^[J]{}= \_6 \_[I]{}\^[J]{} , I,J=1,2,3. \[normal2\] The $\Z_{2} \times \Z_{2}$ orbifold symmetry restricts the period matrix $\tau^{ij}$ to be diagonal. Then, up to normalization, the holomorphic 3-form is given by \[holoexpan\] = (\^1 + \_1 \^2) (\^3 + \_2 \^4) (\^5 + \_3 \^6) =\_[0]{} + \_[K]{} \_[K]{} + \^[K]{} + \^[0]{}\_1 \_[2]{} \_[3]{} , with the $H^{3}(\M_6,\Z)$ basis displayed in (\[basisab\]).
The next step is to switch on background fluxes for the NSNS and RR 3-forms. Since both $H_3$ and $F_3$ are odd under the orientifold involution, the allowed background fluxes can be expanded as |H\_3 & = & b\_[3]{} \_[0]{} + b\_2\^[(I)]{} \_[I]{} + b\_[1]{}\^[(I)]{} \^[I]{} + b\_[0]{} \^[0]{} , \[H3expan\]\
|F\_3 & = & a\_[3]{} \_[0]{} + a\_[2]{}\^[(I)]{} \_[I]{} + a\_[1]{}\^[(I)]{} \^[I]{} + a\_[0]{} \^[0]{} . \[F3expan\] All flux coefficients are integers because the integrals of $\bar H_3$ and $\bar F_3$ over 3-cycles are quantized. To avoid subtleties with exotic orientifold planes we take all fluxes to be even [@frey; @kst].
As argued originally in [@glmw; @kstt], applying one T-duality transformation to the NSNS fluxes can give rise to geometric fluxes $f^a_{bc}$ that correspond to structure constants of the isometry algebra of the internal space. Performing further T-dualities leads to generalized fluxes denoted $Q_c^{ab}$ and $R^{abc}$ [@stw1]. The $Q_c^{ab}$ are called non-geometric fluxes because the resulting metric after two T-dualities yields a background that is locally but not globally geometric [@stw2; @wecht]. Compactifications with $R^{abc}$ fluxes are not even locally geometric but these fluxes are necessary to maintain T-duality between type IIA and type IIB. The geometric and the R-fluxes must be even under the orientifold involution and are thus totally absent in type IIB with O3/O7-planes. On the other hand, the non-geometric fluxes must be odd and are fully permitted.
The main motivation of this work is to study supersymmetric vacua in toroidal type IIB orientifolds with NSNS, RR and non-geometric $Q$-fluxes turned on. In our construction, the $\Z_2 \times \Z_2$ symmetry only allows 24 components of the flux tensor $Q_c^{ab}$, namely those with one leg on each 2-torus. This set of non-geometric fluxes is displayed in table \[tableNonGeometric\]. All components of the tensor $Q$ are integers that we take to be even.
Type Components Fluxes
----------------------------------------------- --------------------------------------------- ------------------------------------------------------------------------------
$Q_{-}^{--} \equiv Q_{\alpha}^{\beta \gamma}$ $ Q_{1}^{35}\,,\,Q_{3}^{51}\,,\,Q_{5}^{13}$ $\tilde{c}_{1}^{\,(1)}\,,\,\tilde{c}_{1}^{\,(2)}\,,\,\tilde{c}_{1}^{\,(3)}$
$Q_{|}^{|-} \equiv Q_{k}^{i \beta} $ $ Q_{4}^{61}\,,\,Q_{6}^{23}\,,\,Q_{2}^{45}$ $ \hat{c}_{1}^{\,(1)}\,,\,\hat{c}_{1}^{\,(2)}\,,\,\hat{c}_{1}^{\,(3)}$
$Q_{|}^{-|} \equiv Q_{k}^{\alpha j}$ $ Q_{6}^{14}\,,\,Q_{2}^{36}\,,\,Q_{4}^{52}$ $ \check{c}_{1}^{\,(1)}\,,\,\check{c}_{1}^{\,(2)}\,,\,\check{c}_{1}^{\,(3)}$
$Q_{|}^{--} \equiv Q_{k}^{\alpha\beta}$ $Q_{2}^{35}\,,\,Q_{4}^{51}\,,\,Q_{6}^{13}$ $ c_{0}^{\,(1)}\,,\,c_{0}^{\,(2)}\,,\,c_{0}^{\,(3)}$
$Q_{-}^{||} \equiv Q_{\gamma}^{i j}$ $ Q_{1}^{46}\,,\,Q_{3}^{62}\,,\,Q_{5}^{24}$ $ c_{3}^{\,(1)}\,,\,c_{3}^{\,(2)}\,,\,c_{3}^{\,(3)}$
$Q_{-}^{|-} \equiv Q_{\gamma}^{i \beta}$ $Q_{5}^{23}\,,\,Q_{1}^{45}\,,\,Q_{3}^{61}$ $\check{c}_{2}^{\,(1)}\,,\,\check{c}_{2}^{\,(2)}\,,\,\check{c}_{2}^{\,(3)}$
$Q_{-}^{-|} \equiv Q_{\beta}^{\gamma i}$ $ Q_{3}^{52}\,,\,Q_{5}^{14}\,,\,Q_{1}^{36}$ $\hat{c}_{2}^{\,(1)}\,,\,\hat{c}_{2}^{\,(2)}\,,\,\hat{c}_{2}^{\,(3)}$
$Q_{|}^{||} \equiv Q_{k}^{i j}$ $Q_{2}^{46}\,,\,Q_{4}^{62}\,,\,Q_{6}^{24}$ $\tilde{c}_{2}^{\,(1)}\,,\,\tilde{c}_{2}^{\,(2)}\,,\,\tilde{c}_{2}^{\,(3)}$
: Non-geometric $Q$-fluxes.[]{data-label="tableNonGeometric"}
Effective action {#ssec:action}
----------------
The NSNS, RR and non-geometric fluxes induce a potential for the closed string moduli. We will focus on the untwisted moduli of the toroidal orientifold. To write explicitly the effective action, recall first that the axiodilaton and the complex structure moduli are given by S = C\_0 + i e\^[-]{} ; U\_I = \_I ; I=1,2,3 , \[sumoduli\] where $C_0$ is the RR 0-form, $\phi$ is the 10-dimensional dilaton and the $\tau_I$ are the components of the period matrix. The Kähler moduli $T_I$ are instead extracted from the expansion of the complexified Kähler 4-form $\cj$, i.e. $\cj=-\sum T_{I} \, \tilde{\omega}^{I}$. In turn, the real (axionic) part of $\cj$ arises from the RR 4-form $C_4$ whereas the imaginary part is $e^{-\phi} J\wedge J/2$, where $J$ is the fundamental Kähler form. In fact, $\im T_I$ is basically the area of the 4-cycle dual to the 4-form $\tilde\omega^I$.
We are interested in compactifications that preserve 1 supersymmetry in four dimensions. In this case we know that the scalar potential can be computed from the Kähler potential and the superpotential. The Kähler potential for the moduli is given by the usual expression K =-\_[K=1]{}\^[3]{}( -i(U\_[K]{}-|[U]{}\_[K]{})) - ( -i(S-|[S]{})) - \_[K=1]{}\^[3]{} ( -i(T\_[K]{}-|[T]{}\_[K]{})) , which is valid to first order in the string and sigma model perturbative expansions. The NSNS and RR fluxes induce a superpotential only for $S$ and the $U_I$. In absence of non-geometric fluxes Kähler moduli do not enter in the superpotential and non-perturbative effects such as gaugino condensation are required to get vacua with all moduli fixed. The $Q$-fluxes generate new couplings involving Kähler fields, thereby opening the possibility to stabilize all types of closed string moduli.
The general superpotential can be computed from [@acfi] \[WInt\] W=\_[\_[6]{}]{} (G\_[3]{} +Q ) , where $G_{3}= \bar F_{3}-\,S\,\bar H_{3}$, and $Q\cj$ is a 3-form with components defined by (Q)\_[abc]{}= Q\_[\[a]{}\^[mn]{} \_[bc\]mn]{} . \[qjcomp\] Being a 3-form, $Q\cj$ can be expanded in the basis (\[basisab\]). We obtain \[QJexpan\] Q=T\_[K]{} ( c\_[3]{}\^[(K)]{} \_[0]{} - \_[2]{}\^[(I K)]{} \_[I]{} - \_[1]{}\^[(I K)]{} \^[I]{} + c\_[0]{}\^[(K)]{} \^[0]{} ) , where $\cc_1$ and $\cc_2$ are the non-geometric flux matrices \_[1]{}=(
[lll]{} -\_[1]{}\^[(1)]{} & \_[1]{}\^[(3)]{} & \_[1]{}\^[(2)]{}\
\_[1]{}\^[(3)]{} & -\_[1]{}\^[(2)]{} & \_[1]{}\^[(1)]{}\
\_[1]{}\^[(2)]{} & \_[1]{}\^[(1)]{} & -\_[1]{}\^[(3)]{}\
) ,\_[2]{}=(
[lll]{} -\_[2]{}\^[(1)]{} & \_[2]{}\^[(3)]{} & \_[2]{}\^[(2)]{}\
\_[2]{}\^[(3)]{} & -\_[2]{}\^[(2)]{} & \_[2]{}\^[(1)]{}\
\_[2]{}\^[(2)]{} & \_[2]{}\^[(1)]{} & -\_[2]{}\^[(3)]{}\
) . \[c1c2mat\] The expansion for the 3-form $G_3$ that combines the NSNS and the RR fluxes can be read off from (\[H3expan\]) and (\[F3expan\]). Substituting the expansions of the holomorphic 3-form and the background fluxes in (\[WInt\]) shows that the superpotential takes the form W=P\_[1]{}(U) + P\_[2]{}(U)S + \_[K=1]{}\^[3]{} P\_[3]{}\^[(K)]{}(U)T\_[K]{} . \[fullW\] The $P$’s are cubic polynomials in the complex structure moduli given by P\_[1]{}(U) & = & a\_[0]{} -\_[K=1]{}\^[3]{} a\_[1]{}\^[(K)]{}U\_[K]{} + \_[K=1]{}\^[3]{} a\_[2]{}\^[(K)]{} - a\_[3]{} U\_[1]{}U\_[2]{}U\_[3]{} , \[p1gen\]\
P\_[2]{}(U) & = & -b\_[0]{} +\_[K=1]{}\^[3]{} b\_[1]{}\^[(K)]{}U\_[K]{} - \_[K=1]{}\^[3]{} b\_[2]{}\^[(K)]{} + b\_[3]{} U\_[1]{}U\_[2]{}U\_[3]{} , \[p2gen\]\
P\_[3]{}\^[(K)]{}(U) & = & c\_[0]{}\^[(K)]{} +\_[L=1]{}\^[3]{} \_[1]{}\^[(L K)]{}U\_[L]{} - \_[L=1]{}\^[3]{} \_[2]{}\^[(L K)]{} -c\_[3]{}\^[(K)]{} U\_[1]{}U\_[2]{}U\_[3]{} . \[p3gen\] The main feature of the flux superpotential is that it depends on all untwisted closed string moduli.
At this point we have a model with seven moduli whose potential depends on forty flux parameters. Finding vacua in this generic setup is rather cumbersome. For this reason we consider a simpler configuration in which the fluxes are isotropic. Concretely, we make the Ansatz \_[1]{}\^[(I)]{} \_[1]{} ; && \_[1]{}\^[(I)]{} \_[1]{} ; \_[1]{}\^[(I)]{} \_[1]{} ; \_[2]{}\^[(I)]{} \_[2]{} ; \_[2]{}\^[(I)]{} \_[2]{} ; \_[2]{}\^[(I)]{} \_[2]{} ,\
&& b\_[1]{}\^[(I)]{}b\_[1]{} ; b\_[2]{}\^[(I)]{} b\_[2]{} ; a\_[1]{}\^[(I)]{} a\_[1]{} ; a\_[2]{}\^[(I)]{} a\_[2]{} . \[isofluxes\] Isotropic fluxes are summarized in tables \[tableIsoNSRR\] and \[tableIsoNon-Geometric\].
$\bar{F}_{---}$ $\bar{F}_{|--}$ $\bar{F}_{-||}$ $\bar{F}_{|||}$ $\bar{H}_{---}$ $\bar{H}_{|--}$ $\bar{H}_{-||}$ $\bar{H}_{|||}$
----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- -----------------
$a_{3}$ $a_{2}$ $a_{1}$ $a_{0}$ $b_{3}$ $b_{2}$ $b_{1}$ $b_{0}$
: NS and RR isotropic fluxes. []{data-label="tableIsoNSRR"}
$Q_{-}^{--}$ $Q_{|}^{|-}$ $Q_{|}^{-|}$ $Q_{|}^{--}$ $Q_{-}^{||}$ $Q_{-}^{|-}$ $Q_{-}^{-|}$ $Q_{|}^{||}$
----------------- ---------------- ------------------ -------------- -------------- ----------------- --------------- -----------------
$\tilde{c}_{1}$ $ \hat{c}_{1}$ $ \check{c}_{1}$ $ c_{0}$ $c_{3}$ $\check{c}_{2}$ $\hat{c}_{2}$ $\tilde{c}_{2}$
: Non-geometric isotropic fluxes.[]{data-label="tableIsoNon-Geometric"}
The Ansatz of isotropic fluxes is compatible with vacua in which the geometric moduli are also isotropic, namely U\_[1]{}=U\_[2]{}=U\_[3]{}U ; T\_[1]{}=T\_[2]{}=T\_[3]{}T . \[isoUT\] This means, that there is only one overall complex structure modulus $U$ and one Kähler modulus $T$. The model also includes the axiodilaton. In this case, the Kähler potential and the superpotential reduce to K & = & -3( -i(U-|[U]{})) - ( -i(S-|[S]{})) - 3( -i(T-|[T]{}))\
W & = &P\_[1]{}(U) + P\_[2]{}(U)S + P\_[3]{}(U)T . \[kwiso\] The $P$’s are now cubic polynomials in the single complex structure moduli. They are given by P\_[1]{}(U) & = & a\_[0]{}-3a\_[1]{}U+3a\_[2]{}U\^[2]{}-a\_[3]{}U\^[3]{} , \[P1Iso\]\
P\_[2]{}(U) & = & -b\_[0]{}+3b\_[1]{}U-3b\_[2]{}U\^[2]{}+b\_[3]{}U\^[3]{} , \[P2Iso\]\
P\_[3]{}(U) & = & 3 ( c\_[0]{}+ (\_[1]{}+\_[1]{}-\_[1]{}) U - (\_[2]{}+\_[2]{}-\_[2]{})U\^[2]{} - c\_[3]{}U\^[3]{} ) . \[P3Iso\] This is the model considered in [@stw1; @stw2].
Bianchi identities and tadpoles {#ssec:bianchi}
-------------------------------
The NSNS and generalized fluxes that follow from the T-duality chain can be regarded as structure constants of an extended symmetry algebra of the compactification [@stw1; @dabholkar]. This algebra includes isometry generators $Z_{a}$ as well as gauge symmetry generators $X^{a}$, $a=1,\ldots, 6$, coming from the reduction of the $B$-field on $T^{6}$ with fluxes. We are interested in type IIB with O3/O7-planes where geometric and $R$-fluxes are forbidden. In this case the algebra is given by &=&Q\_[c]{}\^[ab]{}X\^[c]{} ,\
&=&Q\_[a]{}\^[bc]{}Z\_[c]{} , \[zxalgebra\]\
&=&|H\_[abc]{}X\^[c]{} . Notice that the $X^a$ span a 6-dimensional subalgebra in which the non-geometric $Q_c^{ab}$ are the structure constants.
Computing the Jacobi identities of the full 12-dimensional algebra we obtain the constraints \[BianchiGen\] |H\_[x\[bc]{}Q\^[ax]{}\_[d\]]{} =0 ; Q\_[x]{}\^[\[ab]{}Q\^[c\]x]{}\_[d]{}=0 . In the following we will refer to these identities in the shorthand notation $\bar H Q=0$ and $Q Q=0$. The constraints on the fluxes can also be interpreted in terms of a nilpotency condition $\cd^2=0$ on the operator $\cd=H\wedge + Q\cdot$ introduced in [@stw2].
The RR fluxes are also constrained by Bianchi identities of the type $\cd \bar F=\cs$, where $\cs$ is a generalized form due to sources that are assumed smeared instead of localized. These Bianchi identities can be understood as tadpole cancellation conditions on the RR 4-form $C_4$ and $C_8$ that couple to the sources. The sources are just the orientifold O3/O7-planes and D3/D7-branes that can be present. In the IIB orientifold that we are considering there is a flux-induced $C_4$ tadpole due to the coupling \_[\_[4]{} \_[6]{}]{} C\_[4]{} |H\_[3]{} |F\_[3]{} . \[c4tad\] There are further $C_4$ tadpoles due to O3-planes and to D3-branes that can also be added. The total orientifold charge is -32, equally distributed among 64 O3-planes located at the fixed points of the orientifold involution $\sigma$. Each D3-brane has charge $+1$ and if they are located in the bulk, as opposed to fixed points of $\Z_2^3$, images must be included. Adding the sources to the flux tadpole (\[c4tad\]) leads to the cancellation condition \[O3tad\] a\_[0]{}b\_[3]{} - a\_[1]{}\^[(K)]{}b\_[2]{}\^[(K)]{} + a\_[2]{}\^[(K)]{}b\_[1]{}\^[(K)]{} - a\_[3]{}b\_[0]{}=N\_[3]{} , where $N_3=32-N_{\rm D3}$ and $N_{\rm D3}$ is the total number of D3-branes.
The non-geometric and RR fluxes can also combine to produce a tadpole for the RR $C_8$ form. The contraction $Q \bar F_3$ is a 2-form and the flux-induced tadpole is due to the coupling \_[\_[4]{} \_[6]{}]{} C\_[8]{} (Q |F\_[3]{}) \[c8tad\] Expanding the 2-form $(Q \bar{F}_{3})$ in the basis of 2-forms $\omega_I$, $I=1,2,3$, yields coefficients ( Q |[F]{}\_[3]{})\_[I]{}=a\_[0]{}c\_[3]{}\^[(I)]{}+a\_[1]{}\^[(K)]{}\_[2]{}\^[(K I)]{} - a\_[2]{}\^[(K)]{}\_[1]{}\^[(K I)]{}-a\_[3]{}c\_[0]{}\^[(I)]{} ; I=1,2,3 . \[o3d3tad\] This means that there are induced tadpoles for $C_8$ components of type $C_8 \sim d{\rm vol}_4 \wedge \widetilde{\omega}^I$, where $d{\rm vol}_4$ is the space-time volume 4-form and $\widetilde{\omega}^I$ is the 4-form dual to $\omega_I$. On the other hand, there are also $C_8$ tadpoles due to that have a total charge $+32$ for each $I$. As discussed before, due to the orbifold group $\Z_2 \times \Z_2$, there are located at the 4 fixed tori of $\sigma \theta_I$, where $\theta_I$ are the three order-two elements of $\Z_2 \times \Z_2$. In the end we find the three tadpole cancellation conditions \[O7tad\] a\_[0]{}c\_[3]{}\^[(I)]{}+a\_[1]{}\^[(K)]{}\_[2]{}\^[(K I)]{} - a\_[2]{}\^[(K)]{}\_[1]{}\^[(K I)]{}-a\_[3]{}c\_[0]{}\^[(I)]{} =N\_[7\_I]{} ; I=1,2,3 , where $N_{7_I}=-32+N_{{\rm D}_{7_I}}$ and $N_{{\rm D}_{7_I}}$ is the number of that are generically allowed.
In this work we mostly consider isotropic fluxes so that we will again make the Ansatz (\[isofluxes\]). Jacobi identities as well as tadpoles cancellation conditions become simpler. Computing $QQ=0$ constraints from (\[BianchiGen\]) leave us with \[BianchiXhatcheck\] \_[2]{}\_[1]{} - \_[1]{}\_[2]{} + \_[1]{}\_[2]{} - c\_[0]{}c\_[3]{}=0 & ; & c\_[3]{}\_[1]{} - \_[2]{}\^[2]{} + \_[2]{}\_[2]{} - \_[1]{}c\_[3]{}=0 ,\
c\_[3]{}c\_[0]{} - \_[2]{}\_[1]{} + \_[2]{}\_[1]{} - \_[1]{}\_[2]{}=0 & ; & c\_[0]{}\_[2]{} - \_[1]{}\^[2]{} + \_[1]{}\_[1]{} - \_[2]{}c\_[0]{}=0 , plus one additional copy of each condition with $\check{c}_{i} \leftrightarrow \hat{c}_{i}$. An important result is that saturating[^1] this ideal with respect to the conditions $\check{c}_{i} \not= \hat{c}_{i}$ automatically implies that $\tilde{c}_{i}$ is complex. Therefore, it must be that \_[1]{}=\_[1]{} c\_[1]{} ; \_[2]{}=\_[2]{} c\_[2]{} . \[oneci\] The cubic polynomial that couples the complex structure and Kähler moduli, c.f. (\[P3Iso\]), then reduces to \[P3Iso2\] P\_[3]{}(U)=3 ( c\_[0]{}+ (2c\_[1]{}-\_[1]{}) U - (2c\_[2]{}-\_[2]{}) U\^[2]{} - c\_[3]{}U\^[3]{} ) . Recall that the non-geometric fluxes are integer parameters.
Upon using (\[oneci\]), the Jacobi constraints satisfied by the non-geometric fluxes become \[BianchiC\] c\_0 (c\_2-\_2)+ c\_1(c\_1-\_1) &=&0 ,\
c\_2(c\_2-\_2)+c\_3 (c\_1-\_1) &=&0 ,\
c\_0 c\_3-c\_1 c\_2 &=&0 . This system of equations is easy to solve explicitly. The solution variety has three disconnected pieces of different dimensions. The first piece has dimension four and it is characterized by fluxes
[lclcl]{} c\_[3]{}= \_[p]{}k\_2 & ; & c\_[2]{}= \_[p]{}k\_1 & ; & \_[1]{}= \_[q]{}k\_2 + k\_[1]{} ;\
c\_[1]{}= \_[q]{}k\_2 & ; & c\_[0]{}= \_[q]{}k\_1 & ; & \_[2]{}= \_[p]{}k\_1 - k\_[2]{} .
\[PieceA\] Here $\lambda=1$, $(k_1,k_2)$ are two integers not zero simultaneously, and $(\lambda_p, \lambda_q)$ are two rays given by \_[p]{}=1+ ; \_[q]{}=1+ , \[raylambda\] where $p, \, q, \in \Z$. By convention ${\rm GCD}(n,0)=|n|$. With coefficients given by the fluxes (\[PieceA\]) the polynomial $P_3(U)$ turns out to factorize as P\_[3]{}(U)=3 (k\_1 + k\_2 U) (\_q - U - \_[p]{}U\^[2]{}) . \[p3fact\] Notice that we have taken into account that the non-geometric fluxes are integers. The second piece of solutions is three dimensional, the set of fluxes can still be characterized by (\[PieceA\]) and $P_3(U)$ by (\[p3fact\]), but with $\lambda\equiv 0$ and $\lambda_p\equiv1 $. Finally, the third piece has only two dimensions with fluxes and $P_3(U)$ specified by setting $\lambda\equiv 0$, $\lambda_p\equiv 0$ and $\lambda_q\equiv 1$.
As a byproduct of the above analysis we have isolated the real root of $P_3(U)$ that always exist. In the next section we will explain how the nature of the remaining two roots is correlated with the type of algebra fulfilled by the $X^a$ generators. For example, we will see that in the third piece of solutions with $k_2=0$, the algebra is nilpotent.
Let us now consider the constraints $\bar H Q=0$ that mix non-geometric and NSNS fluxes. Inserting the isotropic fluxes in (\[BianchiGen\]), and using (\[oneci\]), we find \[BianchiB\] b\_2 c\_0-b\_0 c\_2+b\_1(c\_1- \_1) &=&0 ,\
b\_3 c\_0-b\_1 c\_2+b\_2 (c\_1-\_1)&=&0 ,\
b\_2 c\_1-b\_0 c\_3-b\_1(c\_2-\_2) &=&0 ,\
b\_3 c\_1-b\_1 c\_3-b\_2 (c\_2-\_2)&=&0 . These conditions restrict the NSNS fluxes $b_A$ that determine the coupling between the complex structure and the dilaton moduli through the polynomial $P_2(U)$ in (\[P2Iso\]). In the next section we will discuss solutions to the full set of constraints that will lead to specific forms for the polynomials $P_2(U)$ and $P_3(U)$.
The tadpole cancellation relations also become simpler in the isotropic case. In particular, the three constraints in (\[O7tad\]), depending on $I$, reduce to just one condition. Substituting the isotropic Ansatz and (\[oneci\]) we obtain \[O3tadIso\] a\_[0]{}b\_[3]{}-3a\_[1]{}b\_[2]{}+3a\_[2]{}b\_[1]{}-a\_[3]{}b\_[0]{}=N\_[3]{} , \[O7tadIso\] a\_[0]{}c\_[3]{}+a\_[1]{}(2c\_[2]{} -\_[2]{})-a\_[2]{}(2c\_[1]{} -\_[1]{})-a\_[3]{}c\_[0]{}=N\_[7]{} . These conditions constraint the RR fluxes. We consider the net O3/D3 and O7/D7 charges, $N_3$ and $N_7$, to be free parameters.
Algebras and fluxes {#sec:alg}
===================
In this section we discuss solutions to the Jacobi identities satisfied by the NSNS and the non-geometric $Q$ fluxes. The key idea is twofold. First, the generators $X^a$ in (\[zxalgebra\]) span a six-dimensional subalgebra whose structure constants are precisely the $Q^{ab}_c$. Second, when these fluxes are invariant under the $\Z_2^3$ symmetry described in section \[ssec:fluxes\], this subalgebra is rather constrained. We expect only a few subalgebras to be allowed and our strategy is to identify them. In this way we will manage to provide explicit parametrizations for non-geometric fluxes that satisfy the identity $QQ=0$. Once this is achieved, we will also be able to find the corresponding NSNS fluxes that fulfill $\bar H Q=0$.
We want to consider in detail the set of isotropic non-geometric fluxes given in table \[tableIsoNon-Geometric\] plus the conditions $\check{c}_{1}=\hat{c}_{1} \equiv c_{1}$, $\check{c}_{2}=\hat{c}_{2} \equiv c_{2}$. In this case the subalgebra simplifies to & = & \_[IJK]{} ( c\_1 X\^[2K-1]{} + c\_0 X\^[2K]{}) ,\
& = & \_[IJK]{} (c\_2 X\^[2K-1]{} + c\_1 X\^[2K]{}) \[subiso\] ,\
& = & \_[IJK]{} (c\_3 X\^[2K-1]{} + c\_2 X\^[2K]{}) , where $I,J,K=1,2,3$. The Jacobi identities of this algebra are given in (\[BianchiC\]). To reveal further properties, it is instructive to compute the Cartan-Killing metric, denoted $\cam$, with components \^[ab]{}= Q\_[c]{}\^[ad]{} Q\_[d]{}\^[bc]{} . \[ckmetric\] For the above algebra of isotropic fluxes we find that the six-dimensional matrix $\cam$ is block-diagonal, namely = (\_2, \_2, \_2) . \[mkcblock\] The $2\times 2$ matrix $\cx_2$ turns out to be \_2=-2(
[ll]{} c\_1\^2 + 2 c\_0c\_2 + c\_1\^2 & c\_1 c\_2 + c\_1 c\_2 + c\_0 c\_3 + c\_1 c\_2\
c\_1 c\_2 + c\_1 c\_2 + c\_0 c\_3 + c\_1 c\_2 & c\_2\^2 + 2 c\_1c\_3 + c\_2\^2\
) . \[mkc2\] Since $\cx_2$ is symmetric, we conclude that $\cam$ can have up to two distinct real eigenvalues, each with multiplicity three.
The full 12-dimensional algebra also enjoys distinctive features. In the isotropic case the remaining algebra commutators involving NSNS fluxes are given by & = & \_[IJK]{} (b\_3 X\^[2K-1]{} + b\_2 X\^[2K]{}) ,\
& = & \_[IJK]{} (b\_2 X\^[2K-1]{} + b\_1 X\^[2K]{}) , \[zzxiso\]\
& = & \_[IJK]{} (b\_1 X\^[2K-1]{} + b\_0 X\^[2K]{}) . The mixed piece of the algebra is determined by the non-geometric fluxes as & = & \_[IJK]{} (c\_1 Z\_[2K-1]{} + c\_2 Z\_[2K]{}) ,\
& = & \_[IJK]{} (c\_2 Z\_[2K-1]{} + c\_3 Z\_[2K]{}) ,\
& = & \_[IJK]{} (c\_0 Z\_[2K-1]{} + c\_1 Z\_[2K]{}) , \[zxziso\]\
& = & \_[IJK]{} (c\_1 Z\_[2K-1]{} + c\_2 Z\_[2K]{}) . Besides the Jacobi identities purely involving non-geometric fluxes, there are the additional mixed constraints (\[BianchiB\]).
Computing the full Cartan-Killing metric, denoted $\cam_{12}$, shows that there are no mixed $XZ$ terms. In fact, the matrix is again block-diagonal \_[12]{} = (\_2, \_2, \_2, \_2, \_2, \_2) , \[fullmkcblock\] with $\cx_2$ shown above. The new $2\times 2$ matrix $\cz_2$ is found to be \_2=-4(
[ll]{} b\_3c\_1 + 2 b\_2c\_2 + b\_1 c\_3 & b\_2(c\_1+c\_1) +b\_1(c\_2 + c\_2)\
b\_2(c\_1+c\_1) +b\_1(c\_2 + c\_2) & b\_0c\_2 + 2 b\_1c\_1 + b\_2 c\_0\
) . \[fullmkc2\] Here we have simplified using the Jacobi identities (\[BianchiB\]). We conclude that the allowed 12-dimensional algebras are such that the Cartan-Killing matrix can have up to four distinct eigenvalues, each with multiplicity three.
Let us now return to the subalgebra spanned by the $X$ generators and the task of solving the constraints (\[BianchiC\]) that arise from the Jacobi identities $QQ=0$. The idea is to fulfill these constraints by choosing the non-geometric fluxes to be the structure constants of six-dimensional Lie algebras whose Cartan-Killing matrix has the simple block-diagonal form (\[mkcblock\]). To proceed it is convenient to distinguish whether $\cam$ is non-degenerate or not, i.e. whether the algebra is semisimple or not. If , and $\cam$ is negative definite, the only possible algebra is the compact $\mathfrak{so(4)} \sim \mathfrak{su(2)^2}$. On the other hand, the only non-compact semisimple algebra with the required block structure is $\mathfrak{so(3,1)}$. When $\det \cam\! =\! 0$, the algebra is non-semisimple. In this class to begin we find two compatible algebras, namely the direct sum $\mathfrak{su(2) + u(1)^3}$ and the semi-direct sum $\mathfrak{su(2) \oplus u(1)^3}$ that is isomorphic to the Euclidean algebra $\mathfrak{iso(3)}$. The remaining possibility is that the non-semisimple algebra be completely solvable. One example is the nilpotent $\mathfrak{u(1)^6}$ that we disregard because the non-geometric fluxes vanish identically. A second non-trivial solvable algebra, that is actually nilpotent, will be discussed shortly.
After classifying the allowed 6-dimensional subalgebras the next step is to find the set of corresponding non-geometric fluxes. Except for the nilpotent example, all other cases have an $\mathfrak{su(2)}$ factor. This suggests to make a change of basis from $(X^{2I-1}, X^{2I})$, $I=1,2,3$, to new generators $(E^I, \widetilde E^I)$ such that basically one type, say $E^I$, spans $\mathfrak{su(2)}$. The $\Z_2^3$ symmetries of the fluxes require that we form combinations that transform in a definite way, For instance, $E^I$ can only be a combination of $X^{2I-1}$ and $X^{2I}$ with the same $I$. Furthermore, for isotropic fluxes it is natural to make the same transformation for each $I$. We will then make the $SL(2, \RR)$ transformation (
[c]{} E\^I\
\^I
) = (
[cc]{} -&\
-&
) (
[c]{} X\^[2I-1]{}\
X\^[2I]{}
) , \[chbasis\] for all $I=1,2,3$. Here $|\Gamma|=\alpha\delta - \beta\gamma$, and it must be that $|\Gamma|\not=0$. In the following we will refer to $(\a, \b, \g, \d)$ as the $\Gamma$ parameters.
Substituting in (\[subiso\]) it is straightforward to obtain the algebra satisfied by the new generators $E^I$ and $\wt E^J$. This algebra will depend on the non-geometric fluxes as well as on the parameters $(\a, \b, \g, \d)$. We can then prescribe the commutators to have the standard form for the allowed algebras found previously. For instance, in the direct product examples we impose $\big[E^I, \wt E^J\big]=0$.
In the following sections we will discuss each compatible 6-dimensional algebra in more detail. The goal is to parametrize the non-geometric fluxes in terms of $(\a, \b, \g, \d)$. By construction these fluxes will satisfy the Jacobi identities of the algebra. We will then solve the mixed constraints involving the NSNS fluxes. The main result will be an explicit factorization of the cubic polynomials $P_3(U)$ and $P_2(U)$ that dictate the couplings among the moduli.
Semisimple algebras
-------------------
The algebra is semisimple when the Cartan-Killing metric is non-degenerate. This means $\det \cam \not=0$ and hence $\det \cx_2 \not= 0$. Now, six-dimensional semisimple algebras are completely classified. If $\cam$ is negative definite the algebra is compact so that it must be $\mathfrak{so(4) \sim su(2) + su(2)}$. When $\cam$ has positive eigenvalues the algebra is non-compact and it could be $\mathfrak{so(3,1)}$ or $\mathfrak{so(2,2)}$ but the latter does not fit the required block-diagonal form (\[mkcblock\]).
### $\mathfrak{so(4) \sim su(2)^2}$ {#subsubso4}
The standard commutators of this algebra are =\_[IJK]{} E\^K ; =\_[IJK]{}E\^K ; =0 . \[su2su2\] After performing the change of basis in (\[subiso\]) we find that the non-geometric fluxes needed to describe this algebra can be parametrized as \[LimC\]
[lcl]{} c\_[0]{}= (+) & ; & c\_[3]{}=- (+) ,\
c\_[1]{} = (+) & ; & c\_[2]{}=- (+) ,\
\_[2]{}= \^[2]{} + \^[2]{}& ; & \_[1]{}=- ( \^[2]{} + \^[2]{}) ,
provided that $|\Gamma|=(\a\d-\b\g) \not= 0$. It is easy to show that these fluxes verify the Jacobi identities (\[BianchiC\]). What we have done is to trade the six non-geometric fluxes, constrained by two independent conditions, by the four independent parameters $(\alpha,\beta,\g, \d)$. These parameters are real but the resulting non-geometric fluxes in (\[LimC\]) must be integers.
For future purposes we need to determine the cubic polynomial $P_3(U)$ that corresponds to the parametrized non-geometric fluxes. Substituting in (\[P3Iso2\]) yields P\_3(U)=3(U + )(U + ) . \[p3so4\] This clearly shows that in this case $P_3$ has three real roots. Moreover, the roots are all different because $|\Gamma|\not=0$. We will prove that for other algebras $P_3$ has either complex roots or degenerate real roots. The remarkable conclusion is that $P_3$ has three different real roots if and only if the algebra of the non-geometric fluxes is the compact $\mathfrak{so(4) \sim su(2) + su(2)}$. Alternatively, we may start with the condition that the polynomial has three different real roots that we can choose to be at $0$, $-1$ and $\infty$ without loss of generality. These roots can then be moved to arbitrary real locations by a linear fractional transformation = . \[zdef\] with $(\a, \b, \g, \d) \in \RR$ and $|\Gamma|\not=0$. By comparing the roots of $P_3$ in terms of the fluxes with those in terms of the transformation parameters we rediscover the map (\[LimC\]) and the associated $\mathfrak{su(2)^2}$ algebra. In the next sections we will see that the variable $\cz$ introduced above plays a very important physical rôle.
We now turn to the Jacobi constraints (\[BianchiB\]) involving the NSNS fluxes. Inserting the non-geometric fluxes (\[LimC\]) we find that the $b_A$ can be completely fixed by the $\Gamma$ parameters plus two new real variables $(\eps_1, \eps_2)$ as follows b\_[0]{}&=&-(\_[1]{} \^[3]{} + \_[2]{} \^[3]{}) ,\
b\_[1]{}&=& \_[1]{} \^[2]{} + \_[2]{} \^[2]{} , \[bso4\]\
b\_[2]{}&=& -(\_[1]{} \^[2]{}+ \_[2]{}\^[2]{} ) ,\
b\_[3]{}&=&\_[1]{} \^[3]{} + \_[2]{} \^[3]{} . We also need to compute the polynomial $P_2(U)$ that depends on the NSNS fluxes. Substituting the above $b_A$ in (\[P2Iso\]) yields P\_2(U)=\_1 (U + )\^3 + \_2(U + )\^3 . \[p2so4\] It is easy to show that because $|\Gamma| \not= 0$, $P_2$ has complex roots whenever $\eps_1\eps_2\not=0$. Contrariwise, $P_2$ has a triple real root if either $\eps_1$ or $\eps_2$ vanishes.
We may expect that the full 12-dimensional algebra has special properties when $P_2$ has a triple root. Indeed, inserting the fluxes in (\[fullmkc2\]) yields $\det \cz_2 = 16 \eps_1\eps_2|\Gamma|^6$. Hence, the full Cartan-Killing matrix $\cam_{12}$ happens to be degenerate when $\eps_1\eps_2=0$. To learn more about the full algebra it is convenient to switch from the original $Z_a$ generators to a new basis $(D_I, \wt D_I)$ defined by (
[c]{} D\_I\
D\_I
) = (
[cc]{} &\
&
) (
[c]{} Z\_[2I-1]{}\
Z\_[2I]{}
) , \[chzbasis\] for $I=1,2,3$. It is straightforward to compute the piece of the full algebra generated by the $(D_I, \wt D_I)$. Substituting the parametrized fluxes in (\[zzxiso\]) and (\[zxziso\]) we obtain
[lcl]{} =-\_1 \_[IJK]{} E\^K & ; & = -\_2 \_[IJK]{}E\^K ,\
=\_[IJK]{} D\_K & ; & =\_[IJK]{} D\_K .
\[morealg\] All other commutators do vanish.
A quick inspection of the whole algebra encoded in (\[su2su2\]) and (\[morealg\]) shows that when either $\eps_1$, or $\eps_2$, is zero, the $D_I$, or the $\wt D_I$, generate a 3-dimensional invariant Abelian subalgebra. Moreover, when say $\eps_1=0$ and $\eps_2\not=0$, the $\cz_2$ block of the full Cartan-Killing metric has one zero and one non-zero eigenvalue which is negative for $\eps_2 < 0$ and positive for $\eps_2 > 0$. The upshot is that when $\eps_1\eps_2=0$, the 12-dimensional algebra is $\mathfrak{iso(3) + g}$, where $\mathfrak{g}$ is either $\mathfrak{so(4)}$ or $\mathfrak{so(3,1)}$. On the other hand, when $\eps_1 \eps_2 < 0$, the algebra is $\mathfrak{so(4) + so(3,1)}$, whereas for $\eps_1, \eps_2 < 0$ it is $\mathfrak{so(4)^2}$, and for $\eps_1, \eps_2 > 0$ it is $\mathfrak{so(3,1)^2}$.
The methods developed in this section will be applied shortly to other subalgebras. In summary, the non-geometric and NSNS fluxes can be parametrized using auxiliary variables $(\a,\b, \g, \d)$ and $(\eps_1, \eps_2)$ in such a way that the Jacobi identities are satisfied and flux-induced superpotential terms are explicitly factorized. The full 12-dimensional algebras can be simply characterized after the changes of basis (\[chbasis\]) and (\[chzbasis\]) are performed.
The auxiliary variables are constrained by the condition that the resulting fluxes be integers. This issue deserves further explanation. There are two cases depending on whether the polynomial $P_2(U)$ has complex roots or not. If it does not, we can take $\epsilon_1=0$ to be concrete. From the structure of the NSNS fluxes in (\[bso4\]) it is then obvious that, for $\a \not=0$, the quotient $\b/\a$ is a rational number. Going back to the non-geometric fluxes it can be shown that the ratios $\g/\a$ and $\d/\a$, as well as $\a^3$ and $\eps_2$ also belong to $\mathbb{Q}$. If $P_2(U)$ admits complex roots the generic result is that $\eps_2/\eps_1$, $\b/\a$, $\a^3$, etc., involve square roots of rationals. However, it happens that when at least one of the non-geometric parameters $(\a,\b, \g, \d)$ is zero then all well defined quotients are again rational numbers.
### $\mathfrak{so(3,1)}$
This is the Lorentz algebra. We can take $E^I$ to be the angular momentum, and $\wt E^J$ to be the boost generators. Thus, the algebra can be written as =\_[IJK]{} E\^K ; =-\_[IJK]{} E\^K ; =\_[IJK]{} E\^K . \[s031\] In this case the non-geometric fluxes that produce the algebra are found to be \[LimCSO31\]
[lcl]{} c\_[0]{}=-(\^2+\^2) & ; & c\_[3]{}= (\^2+\^2) ,\
c\_[1]{}= -(\^2+\^2) & ; & c\_[2]{}= (\^2+\^2) ,\
\_[2]{}= - (\^2-\^2)-2 & ; & \_[1]{}=(\^2-\^2) + 2 ,
as long as $|\Gamma| \not= 0$.
Substituting the resulting non-geometric fluxes in (\[P3Iso2\]) gives the $P_3(U)$ polynomial P\_3(U)=-3(U+) . \[p3so31\] Since $\Gamma\not=0$, $P_3$ always has complex roots. We will see that for non-semisimple algebras all roots of $P_3$ are real, as for the compact $\mathfrak{so(4)}$. Hence, the important observation now is that $P_3$ has complex roots if and only if the algebra of the non-geometric fluxes is the non-compact $\mathfrak{so(3,1)}$.
The Jacobi constraints (\[BianchiB\]) for the NSNS fluxes can again be solved in terms of the $\Gamma$ parameters plus two real constants that we again denote by $(\eps_1, \eps_2)$. Concretely, b\_[0]{}&=&-(\^2 - 3\^2) \_1 -(\^2-3 \^2) \_2 ,\
b\_[1]{}&=& (\^2 - 2 - \^2) \_1 +(\^2 - 2 - \^2) \_2 , \[bso31\]\
b\_[2]{}&=& (\^2 + 2 - \^2) \_1 +(\^2+2 - \^2 ) \_2 ,\
b\_[3]{}&=& (\^2 - 3\^2) \_1 +(\^2-3 \^2) \_2 . These fluxes give rise to P\_2(U)=(U+)\^3(\_1 \^3 - 3\_2 \^2 - 3 \_1 + \_2) , \[p2so31\] where $\cz=(\a U + \b)/(\g U + \d)$ as before. The discriminant of this cubic polynomial is always negative. Therefore, $P_2$ has three different real roots.
Non-semisimple algebras
-----------------------
In this case the algebra is the semidirect sum of a semisimple algebra and a solvable invariant subalgebra. Lack of simplicity is detected imposing $\det \cam=0$ which requires $\det \cx_2=0$, where $\cx_2$ is shown in (\[mkc2\]). Combining with the Jacobi identities (\[BianchiC\]) we deduce that up to isomorphisms there are only two solutions in which the solvable invariant subalgebra has dimension less than six. In practice this means that $\cx_2$ has only one zero eigenvalue. As expected from the underlying symmetries, this invariant subalgebra can only have dimension three and be $\mathfrak{u(1)^3}$. The semisimple piece can only be $\mathfrak{su(2)}$. The two solutions are the direct and semidirect sum discussed below.
The remaining possibility consistent with the symmetries is for the solvable invariant subalgebra to have dimension six. The criterion for solvability is that the derived algebra $\mathfrak{[g,g]}$ be orthogonal to the whole algebra $\mathfrak{g}$ with respect to the Cartan-Killing metric. In our case this means $Q^{ab}_c \cam^{dc}=0$, $\forall a,b,d$. The non-geometric fluxes further satisfy the Jacobi identities $Q_{x}^{[ab}\,Q^{c]x}_{d}=0$. On the other hand, the stronger condition for nilpotency is $\cam^{dc}=0$. For our algebra of isotropic fluxes given in (\[subiso\]), we find that all solvable flux configurations are necessarily nilpotent. The proof can be carried out using the algebraic package [*Singular*]{} to manipulate the various ideals. This result is consistent with the fact that in our model $\cam$ is block-diagonal so that when $\det \cam=0$, it has three or six null eigenvalues and in the latter situation $\cam$ is identically zero. One obvious nilpotent algebra is $\mathfrak{u(1)^6}$, but it is uninteresting because the associated fluxes vanish identically. There is a second solution described in more detail below.
The allowed non-semisimple subalgebras can all be obtained starting from $\mathfrak{su(2)^2}$ and performing contractions consistent with the underlying symmetries of the isotropic fluxes. For example, setting $E^{\prime\, I} = E^I$, $\wt E^{\prime\, I} = \lambda \wt E^I$ in (\[su2su2\]) and then letting $\lambda \to 0$ obviously gives the direct sum $\mathfrak{su(2)+ u(1)^3}$. More generically we can take $E^{\prime\, I} = \lambda^a(E^I+ \wt E^I)$, $\wt E^{\prime\, I} = \lambda^b(E^I- \wt E^I)$, with $a\ge 0$, $b\ge 0$. The limit $a=0$, $b >0$, $\lambda \to 0$ yields the Euclidean algebra $\mathfrak{iso(3)}$. Letting instead $2b=a >0$ and contracting gives the nilpotent algebra.
In the coming sections we present the explicit configurations of non-geometric fluxes associated to the non-semisimple subalgebras. The parametrization of NSNS fluxes is also computed. Evaluating the full 12-dimensional algebras in each case is straightforward.
### $\mathfrak{su(2)+ u(1)^3}$
Since the algebra is a direct sum and one factor is Abelian, the brackets take the simple form =\_[IJK]{} E\^K ; =0 ; =0 . \[su2d\] Requiring that upon the change of basis the algebra (\[subiso\]) is of this type returns the following non-geometric fluxes \[LimCFac\]
[lcl]{} c\_[0]{}= \^2 & ; & c\_[3]{}=- \^2,\
c\_[1]{} = & ; & c\_[2]{}= - ,\
\_[2]{}= \^[2]{} & ; & \_[1]{}= -\^[2]{} ,
assuming $|\Gamma| \not= 0$. These fluxes automatically satisfy the Jacobi identities (\[BianchiC\]). They also satisfy the additional condition $c_0 c_2 = c_1 \tilde c_1$ arising from $\det \cx_2=0$.
The non-geometric fluxes of the algebra $\mathfrak{su(2)+ u(1)^3}$ lead to the $P_3(U)$ polynomial P\_3(U)=3(U+)(U + )\^2 . \[p3su2d\] Evidently, $P_3$ has one single and one double real root.
The Jacobi identities $\bar H Q=0$ again fix the NSNS fluxes as in the previous cases. The solution in terms of the free parameters is given by b\_[0]{}&=&-(\_[1]{} \^[3]{} + \_[2]{} \^[3]{}) ,\
b\_[1]{}&=& \_[1]{} \^[2]{} + \_[2]{} \^[2]{} , \[bsu2d\]\
b\_[2]{}&=& -(\_[1]{} \^[2]{}+ \_[2]{}\^[2]{} ) ,\
b\_[3]{}&=&\_[1]{} \^[3]{} + \_[2]{} \^[3]{} . For the associated polynomial $P_2(U)$ we then find P\_2(U)=\_1 (U + )\^3 + \_2(U + )\^3 . \[p2su2d\] As in the compact case, this $P_2$ has complex roots whenever $\eps_1 \eps_2 \not= 0$.
### $\mathfrak{su(2)\oplus u(1)^3 \sim iso(3)}$
According to Levi’s theorem, in general this algebra can be characterized as =\_[IJK]{} (E\^K + E\^K ) ; =0 ; =\_[IJK]{} E\^K . \[su2u13sd\] The typical form of the Euclidean algebra in three dimensions is recognized after the isomorphism $(E^I-\wt E^I) \to \widehat E^I$. The non-geometric fluxes needed to reproduce the above commutators turn out to be \[LimCFacSemi\]
[lcl]{} c\_[0]{}=-\^2(-) & ; & c\_[3]{}=\^2(-) ,\
c\_[1]{}= -\^2(-) & ; & c\_[2]{}=\^2(-) ,\
\_[2]{}= \^2(+)- 2 & ; & \_[1]{}=-\^2 (+) + 2 ,
for $|\Gamma| \not= 0$. Besides the Jacobi identities these fluxes satisfy $4 c_0c_2=-(c_1-\tilde c_1)^2$, by virtue of $\det \cx_2=0$.
For the flux configuration of this algebra the $P_3(U)$ polynomial becomes P\_3(U)=3(U + )\^2 . \[p3su2sd\] As in the direct sum $\mathfrak{su(2) + u(1)^3}$, $P_3$ has one single and one double real root.
The NSNS fluxes can be determined from the Jacobi identities (\[BianchiB\]). Introducing again parameters $(\eps_1,\eps_2)$ leads to b\_[0]{}&=&-\^2 ( \_1+ \_2) ,\
b\_[1]{}&=& ( + 2 )\_1 + \^2 \_2 , \[bsu2sd\]\
b\_[2]{}&=& - ( +2 ) \_1 - \^2 \_2 ,\
b\_[3]{}&=& \^2 ( \_1+ \_2) , The companion polynomial $P_2(U)$ of NSNS fluxes is fixed as P\_2(U)=(U + )\^2 . \[p2su2sd\] Analogous to the non-compact case, this $P_2$ has only real roots, but one of them is degenerate.
### Nilpotent algebra
To search for flux configurations that generate a nilpotent algebra we impose that the Cartan-Killing metric vanishes. Now, in our model $\cam=0$ implies the much simpler conditions $\det \cx_2=0$ and $\Tr \cx_2=0$. Up to isomorphisms, we find only one non-trivial solution. This is the expected result based on the known classification of 6-dimensional nilpotent algebras[^2].
From the 34 isomorphism classes of nilpotent algebras, besides $\mathfrak{u(1)^6}$, only one is compatible with isotropic fluxes invariant under $\Z_2 \times \Z_2$. The algebra is 2-step nilpotent and its brackets can be written as = \_[IJK]{} E\^K ; =0 ; =0 . \[nilal\] Up to isomorphisms this is the algebra labelled $n(3.5)$ in Table 4 of [@gmpt].
The change of basis from the original $(X^{2I-1}, X^{2I})$ generators to the $(E^I, \wt E^I)$ is still given by (\[chbasis\]). Starting from the $X$ commutators in (\[subiso\]) we can then deduce fluxes such that the nilpotent algebra (\[nilal\]) is reproduced. In this way we obtain \[LimCNilp\]
[lcl]{} c\_[0]{}= \^3 & ; & c\_[3]{}=- \^3 ,\
c\_[1]{} = \^2 & ; & c\_[2]{}=- \^2 ,\
\_[2]{}= \^[2]{} & ; & \_[1]{}=- \^[2]{} .
Notice that these fluxes only depend on two independent parameters. This occurs because besides the Jacobi constraints there are two more conditions $\det \cx_2=0$ and $\Tr \cx_2=0$. The non-geometric fluxes of the nilpotent algebra generate the $P_3(U)$ polynomial P\_3(U)=3(U+)\^3 . \[p3nil\] Clearly, $P_3$ always has one triple real root.
In analogy with all previous examples, the $\bar H Q=0$ Jacobi identities determine the NSNS fluxes in terms of two additional parameters $(\eps_1, \eps_2)$. Inserting the non-geometric fluxes of the nilpotent algebra in (\[BianchiB\]) readily yields b\_[0]{}&=&-\^2 (\_2+\_1) ,\
b\_[1]{}&=& \^2 \_2 - (\^2 -2 \^2) \_1 , \[bnil\]\
b\_[2]{}&=& - \^2 \_2 + (2 \^2 - \^2 ) \_1 ,\
b\_[3]{}&=& \^2 ( \_2-\_1) . Substituting in (\[P2Iso\]) we easily obtain the corresponding polynomial P\_2(U)=(U + )\^2 . \[p2nil\] As in $\mathfrak{su(2) \oplus u(1)^3}$, this $P_2$ has one single and one double real root. Without loss of generality we can choose $\a=-\d$ and $\b=\g$ in order to write $P_2$ in terms of the variable $\cz=(\a U + \b)/(\g U + \d)$ as P\_2(U)=(U+)\^3(\_1 + \_2) . \[p2nils\] The advantage of this choice of parameters will become evident when we perform a transformation from $U$ to $\cz$ in the scalar potential.
New variables and RR fluxes {#sec:newvars}
===========================
In type IIB orientifolds, the superpotential depends on the complex structure parameter $U$ through the three cubic polynomials $P_1(U)$, $P_2(U)$ and $P_3(U)$ induced respectively by RR, NSNS and non-geometric $Q$-fluxes. Our results in last section show that the last two polynomials can be concisely written as P\_2(U)=(U + )\^3 \_2() ; P\_3(U)=(U + )\^3 \_3() , \[cp23def\] where $\cz=(\a U+ \b)/(\g U + \d)$. The real parameters $(\a, \b, \g, \d)$, with , encode the non-geometric fluxes. For the NSNS fluxes two additional real constants $(\eps_1, \eps_2)$ are needed. As summarized in table \[tablecps\], $\cp_2(\cz)$ and $\cp_3(\cz)$ take very specific forms according to the subalgebra of the $Q$-fluxes.
A very nice property of the variable $\cz$ is its invariance under the $SL(2,\Z)_U$ modular transformations U\^= ; k, , m , n ; kn-m=1 . \[umodt\] Since this is a symmetry of the compactification, the effective action must be invariant. The Kähler potential, $K=-3\log[-i(U-\bar U)] + \cdots$, clearly transforms as K\^= K + 3 | m U + n|\^2 . \[kmodt\] Therefore, the physically important quantity $e^K |W|^2$ is invariant as long as the superpotential satisfies W\^= . \[wmodt\] In order for $W$ to fulfill this property the fluxes must transform in definite patterns. In fact, it follows that (\[wmodt\]) holds separately for each of the flux induced polynomial $P_i(U)$.
We claim that the fluxes transform under $SL(2,\Z)_U$ precisely in such a manner that $\cz^\prime=\cz$. The proof begins by first finding how the $Q$-fluxes mix among themselves from the condition $P_3^\prime=P_3/(m U + n)^3$. For example, under $U^\prime=-1/U$, the non-geometric fluxes transform as c\_0\^=-c\_3 , c\_1\^=c\_2 , c\_2\^=-c\_1 , c\_3\^=c\_0 , c\_1\^[ ]{}= c\_2 , c\_2\^[ ]{}= -c\_1 . \[cduals\] Next we read off the corresponding transformation of the parameters $(\a, \b, \g, \d)$ that are better thought of as the elements of a matrix $\Gamma$. The result is \^= (
[ll]{} \^& \^\
\^& \^
) =(
[ll]{} &\
&
) (
[cc]{} n & -\
-m & k
) \[Gmodt\] It easily follows that $\cz^\prime=\cz$. Notice that $|\Gamma^\prime|=|\Gamma|$.
For the NSNS fluxes we can study the transformation of $P_2$ with coefficients given by the $b_A$. Alternatively, we may start from $P_2$ written as function of $\cz$ as in (\[cp23def\]). The conclusion is that the transformation of the $b_A$ is also determined by $\Gamma^\prime$ together with $(\eps_1^\prime,\eps_2^\prime)=(\eps_1, \eps_2)$. This is valid for all $Q$-subalgebras.
At this point it must be evident that we want to change variables from $U$ to $\cz$. It is also convenient to trade the axiodilaton $S$ and the Kähler modulus $T$ by new fields defined by = S + \_s ; =T + \_t , \[csctdef\] where the shifts $\xi_s$ and $\xi_t$ are some real parameters. The motivation is that such shifts in the axions $\re S$ and $\re T$ can be reabsorbed into RR fluxes as explained in the following.
Parametrization of RR fluxes {#ss:rr}
----------------------------
The systematic procedure is to express the RR fluxes $a_A$ in such a way that their contribution to the superpotential is of the form P\_1(U)=(U + )\^3 \_1() , \[cp1hatdef\] in complete analogy with (\[cp23def\]). To arrive at this factorization we must relate the four RR fluxes $a_A$ to the parameters $(\a, \b, \g, \d)$ that define $\cz=(\a U+ \b)/(\g U + \d)$, and to four additional independent variables. Obviously, $\widehat \cp_1(\cz)$ can be expanded in the monomials $(1,\cz,\cz^2,\cz^3)$. However, a more convenient basis contains the already known polynomials $\cp_3$ and $\cp_2$ that are generically linearly independent. We still need two independent polynomials and these are taken to be the duals $\widetilde \cp_3$ and $\widetilde \cp_2$. The dual $\widetilde \cp$ is such that $\cp \to \widetilde \cp/\cz^3$ when $\cz \to -1/\cz$. The last two subalgebras in table \[tablecps\] must be treated slightly different because linear independence of $\cp_3$ and $\cp_2$ fails for particular properties of the NSNS flux parameter $\eps_1$.
We concretely make the expansion \_1() = \_s \_2() + \_t \_3() + \_1() . \[cp1hatexp\] In the full superpotential the first two terms in $\widehat \cp_1$ will precisely offset the axionic shifts in the new variables $\cs$ and $\ct$. Let us now discuss the remaining piece $\cp_1(\cz)$ that also depends on the $Q$-subalgebra and is displayed in table \[tablecps\]. As explained before, for the first three subalgebras in the table we can further choose \_1() = \_7 \_3() - \_3 \_2() . \[cp1defa\] A motivation for this choice is that the RR tadpoles turn out to depend on the RR fluxes only through the coefficients $(\xi_3, \xi_7)$.
For the last two subalgebras in table \[tablecps\], $\cp_3$ and $\cp_2$ are not independent when $\eps_1$ takes a particular critical value. For $\mathfrak{su(2)\oplus u(1)^3}$ this happens when $\eps_1=-\eps_2$, whereas for the nilpotent algebra the critical value is $\eps_1=0$. To take into account these possibilities, compensating at the same time for the axionic shifts, we still make the decomposition (\[cp1hatexp\]) but with \_1() = 3\_1 + 3 \_2 \^2 + \_3 \^3 . \[cp1defb\] Away from the critical values of $\eps_1$ we can take $\lambda_1=0$ because $\xi_s$ and $\xi_t$ are independent parameters. At the critical value necessarily $\lambda_1 \not=0$ but in this case $\xi_s$ and $\xi_t$ enter in the RR fluxes in only one linearly independent combination. The RR tadpoles happen to depend just on the parameters $(\lambda_2,\lambda_3)$.
The next step is to compare the expansion of $P_1(U)$ in $U$ with its factorized form, c.f. (\[cp1hatdef\]) and (\[P1Iso\]). In this way we can obtain an explicit parametrization of the RR fluxes $a_A$ in terms of the variables that determine $\widehat \cp_1(\cz)$, namely $(\xi_s,\xi_t)$ together with $(\xi_3,\xi_7)$ or $(\lambda_1, \lambda_2, \lambda_3)$, depending on the $Q$-subalgebra. These results are collected in the appendix. We stress that the $\xi$’s and $\lambda$’s are real parameters but the emerging RR fluxes must be integers.
A vacuum solution in which the moduli $(\cz, \cs, \ct)$ are fixed generically requires specific values of the non-geometric, NSNS and RR fluxes. These fluxes also generate RR tadpoles that must be balanced by adding orientifold planes or D-branes. To determine the type of sources that must be included we need to evaluate the RR tadpole cancellation conditions using all parametrized fluxes. Substituting in (\[O3tadIso\]) and (\[O7tadIso\]) we arrive at the very compact expressions for the number of sources $N_3$ and $N_7$ gathered in table \[tabletads\]. As advertised before, the RR fluxes only enter either through the parameters $(\xi_3, \xi_7)$ or $(\lambda_2,\lambda_3)$. The non-geometric and NSNS fluxes only contribute through $|\Gamma|^3$ and $(\eps_1,\eps_2)$. We will see that there is also a clear correlation of the tadpoles with the vevs of the moduli.
Finally, let us remark that, just like $(\eps_1, \eps_2)$, the $\xi$ and $\lambda$ variables are all invariant under modular transformations of the complex structure $U$. Indeed, from the explicit parametrization of the RR fluxes $a_A$ we deduce that their correct behavior under $SL(2,\Z)_U$, analogous to (\[cduals\]), precisely follows from the transformation of $(\a, \b, \g, \d)$ in (\[Gmodt\]). This is of course consistent with the fact that the number of sources $N_3$ and $N_7$ in the tadpoles are physical quantities that must be modular invariant.
Moduli potential in the new variables
-------------------------------------
We have just seen how a systematic parametrization of the fluxes has guided us to new moduli fields denoted $(\cz, \cs, \ct)$. As we may expect, the effective action in the transformed variables also takes a form more suitable for finding vacua. The shifts in the axionic real parts of the axiodilaton and the Kähler field do not affect the Kähler potential $K$ whereas in the superpotential $W$ they can be reabsorbed in RR fluxes. On the other hand, the change from the complex structure $U$ to $\cz$ is the $SL(2,\RR)$ transformation $U=(\b-\d\cz)/(\g\cz-\a)$ whose effect on $K$ and $W$ is completely analogous to a modular transformation except for factors of $|\Gamma|=(\a\d-\b\g)$. Combining previous results we obtain $e^K|W|^2 \to e^\ck|\cw|^2$, where the transformed Kähler potential $\ck$ and superpotential $\cw$ are given by & = &-3 ( -i(-|)) - ( -i(-|)) - 3 (- i(-|) ) , \[KModular\]\
& = & ||\^[3/2]{} . \[WModular\] The flux-induced polynomials $\cp_i(\cz)$ are displayed in table \[tablecps\] for each $Q$-subalgebra. In the effective 4-dimensional action with 1 supergravity the functions $\ck$ and $\cw$ determine the scalar potential of the moduli according to V = e\^{ \_[=,,]{} \^[|]{} |D\_|\^2 - 3||\^2 } . \[VModular\] We are interested in supersymmetric minima for which $D_\Phi \cw = \partial_\Phi \cw + \cw \partial_\Phi \ck =0$, for all fields.
Supersymmetric vacua {#sec:vac}
====================
This section is devoted to searching for supersymmetric vacua of the moduli potential induced by RR, NSNS and non-geometric fluxes together. We will show that by using our new variables the problem simplifies substantially and analytic solutions are feasible.
Supersymmetric vacua are characterized by the vanishing of the F-terms. In our setup the conditions are D\_&=& + =0 ,\
D\_&=& + =0 , \[FFlat\]\
D\_&=& + =0 . The task is to determine whether there are solutions with moduli completely stabilized at vevs denoted \_0 = x\_0 + i y\_0 ; \_0 = s\_0 + i\_0 ; \_0 = t\_0 + i \_0 . \[vzst\] The vacua are either Minkowski or AdS because the potential (\[VModular\]) at the minimum is given by $V_0 = - 3 e^{\ck_0} |\cw_0|^2 \leq 0$.
Besides stabilization, there are further physical requirements. At the minimum the imaginary part of the axiodilaton, $\sigma_0$, must be positive for the reason it is the inverse of the string coupling constant $g_s$. It can be argued that the geometric moduli are subject to similar conditions. The main assumption is that they arise from the metric of the internal space, which is $\T^6$ in absence of fluxes. In particular, the Kähler modulus has $\im \ct = e^{-\phi} A$, where $A$ is the area of a 4-dimensional subtorus. Hence, it must be $\mu_0 > 0$. Notice also that the internal volume is measured by $V_{int}=(\mu_0/\sigma_0)^{3/2}$. For the transformed complex structure $\cz$ it happens that $\im \cz = |\Gamma| \im U/|\g U + \d|^2$. Therefore, necessarily $\im \cz_0 = y_0 \not= 0$ because for $\im U_0=0$ the internal space is degenerate. Without loss of generality we choose that $\im U_0$ is always positive.
Another physical issue is whether the moduli take values such that the effective supergravity action is a reliable approximation to string theory. Specifically, the string coupling $g_s = 1/\sigma_0$ is expected to be small to justify the exclusion of non-perturbative string effects. Conventionally, there is also a requirement of large internal volume to disregard corrections in $\alpha^\prime$. However, in presence of non-geometric fluxes the internal space might be a T-fold in which there can exist cycles with sizes related by T-duality [@hull; @dabholkar]. Thus, for large volume there could be tiny cycles whose associated winding modes would be light. To date these effects are not well understood. At any rate, in this work we limit ourselves to finding supersymmetric vacua of an effective field theory defined by a very precise Kähler potential and flux-induced superpotential. A more detailed discussion of the landscape of vacua is left for section \[sec:lands\]. We will see that the moduli can be fixed at small string coupling and cosmological constant.
In the following we will first consider supersymmetric Minkowski vacua that have at the minimum. In our approach it is straightforward to show that for isotropic fluxes such vacua are disallowed. We then turn our attention to the richer class of ${\rm AdS}_4$ vacua. Since superpotential terms adopt very specific forms depending on the particular subalgebra satisfied by the non-geometric fluxes, we will study the corresponding vacua case by case. We will mostly focus on the model associated to the non-geometric fluxes of the compact $\mathfrak{su(2)^2}$ but will also consider other allowed subalgebras to some extent.
Minkowski vacua
---------------
Minkowski solutions with zero cosmological constant require that the potential vanishes. Imposing supersymmetry further implies that the superpotential must be zero at the minimum $(\cz_0, \cs_0, \ct_0)$. A key property of the superpotential (\[WModular\]) is its linearity in $\cs$ and $\ct$. This implies in particular that the F-flat conditions $D_\cs\cw=0$ and $D_\ct\cw=0$, together with $\cw=0$, reduce just to \_3(\_0)=\_2(\_0)=\_1(\_0)=0 . \[minkcon\] The third condition $D_\cz\cw=0$ yields a linear relation between $\cs_0$ and $\ct_0$ so that not all moduli can be stabilized. The situation is actually worse because (\[minkcon\]) cannot be fulfilled appropriately. Indeed, for the specific polynomials for each subalgebra shown in table \[tablecps\], it is evident that $\cp_3$ and $\cp_2$ can only have a common real root $\cz_0$. But then $\im U_0=\im \cz_0=0$ and this is inconsistent with a well defined internal space.
It must be emphasized that we are assuming that non-geometric fluxes, and their induced $\cp_3$, are non-trivial. Our motivation is to fix the Kähler modulus without invoking non-perturbative effects. If only RR and NSNS fluxes are turned on there do exist physical supersymmetric Minkowski vacua in which only the axiodilaton and the complex structure are stabilized [@kst; @dgkt]. In such solutions the RR and NSNS fluxes must still satisfy a non-linear constraint [@dgkt; @gray].
No-go results for supersymmetric Minkowski vacua in presence of non-geometric fluxes have been obtained previously [@acfi; @tasinato; @gray] [^3]. In [@acfi] the existence was disproved supposing special solutions for the Jacobi identities (\[BianchiC\]). We are now extending the proof to all possible non-trivial [*isotropic*]{} non-geometric fluxes solving these constraints.
${\rm AdS}_4$ vacua {#sub:ads}
-------------------
We now want to solve the supersymmetry conditions when $\cw \not=0$. The three complex equations $D_\Phi\cw=0$, $\Phi=\cz,\cs,\ct$, in principle admit solutions with all moduli fixed at values $\cz_0 = x_0 + i y_0$, $\cs_0 = s_0 + i\sigma_0$, and $\ct_0 = t_0 + i \mu_0$. We will also impose the physical requirements $\sigma_0 > 0$, $\mu_0 > 0$ and $\im U_0 > 0$ which implies $|\Gamma| y_0 > 0$. In general existence of such solutions demands that the fluxes satisfy some specific properties.
In the ${\rm AdS}_4$ vacua, $\cp_2$ and $\cp_3$ are necessarily different from zero. Moreover, combining the equations $D_\cs\cw=0$ and $\D_\ct\cw=0$ shows that at the minimum $\im\left(\cp_3/\cp_2\right)=0$, or equivalently (\_3 \_2\^\* - \_3\^\* \_2)|\_[0]{}.=0 . \[parcond\] From this condition we can quickly extract useful information. For example, for the polynomials of the nilpotent subalgebra we find that $\eps_1=0$. Similarly, for the semidirect product $\mathfrak{su(2) \oplus u(1)^3}$, it follows that $\eps_1=-\eps_2$. Thus, in these two cases $\cp_2$ and $\cp_3$ are forced to be parallel and equation (\[parcond\]) is inconsequential for the moduli. Having one equation less means that all moduli cannot be fixed. In fact, what happens is that only a linear combination of the axions $s_0$ and $t_0$ is determined [@cfi].
Another instructive example is that of the $\mathfrak{su(2) + u(1)^3}$ subalgebra. With the polynomials provided in table \[tablecps\] the condition (\[parcond\]) implies \_2 - 2 \_1 x\_0(x\_0\^2 + y\_0\^2) = 0 , \[direx\] where we already used that $y_0\not=0$. Now we see that forcefully $\eps_1\not=0$ because otherwise $\eps_2$, and thus $\cp_2$ itself, would vanish. However, it could be $\eps_2=0$ and then $x_0=0$. If $\eps_2 \not=0$ we will just have one equation that gives $y_0$ in terms of $x_0$.
In other examples with $\cp_2$ and $\cp_3$ not parallel there are analogous results. It can happen that (\[parcond\]) already fixes $x_0$ or it gives $y_0$ as function of $x_0$. The remaining five equations can be used to obtain $\cs_0$ and $\ct_0$ in terms of $y_0$ or $x_0$, and to find a polynomial equation that determines $y_0$ or $x_0$. This procedure can be efficiently carried out using the algebraic package [*Singular*]{}. The results are described below in more detail.
The superpotential for each $Q$-subalgebra is constructed with the flux-induced polynomials listed in table \[tablecps\]. The numbers of sources needed to cancel tadpoles are given in table \[tabletads\]. Recall that O3-planes (D3-branes) make a positive (negative) contribution to $N_3$, whereas O7-planes (D7-branes) yield negative (positive) values of $N_7$.
Each supersymmetric vacua can be distinguished by the modular invariant values of the string coupling constant $g_s$ and the potential at the minimum $V_0$ that is equal to the cosmological constant up to normalization. In the models at hand these quantities are given by V\_0 =- ; g\_s = 1[\_0]{} . \[vacdata\] In all examples the vevs of the moduli $y_0$, $\sigma_0$, $\mu_0$, as well as the value $\cw_0$ of the superpotential at the minimum, can be completely determined and will be given explicitly. It is then straightforward to evaluate the characteristic data $(V_0, g_s)$.
### Nilpotent subalgebra {#sss:nilpotentres}
When $\eps_1=0$, the model based on the non-geometric fluxes of the nilpotent subalgebra is $U \leftrightarrow T$ dual to a IIA orientifold with only RR and NSNS fluxes already considered in the literature [@DeWolfe; @cfi]. Supersymmetry actually requires $\eps_1=0$. There are some salient features that are easily reproduced in our setup. For instance, a solution exists only if $\lambda_3 \not=0$ and $(\lambda_1\lambda_3 - \lambda_2^2) > 0$. The axions $s_0$ and $t_0$ can only be fixed in the linear combination 3t\_0 + \_2 s\_0 = (3\_1\_2 - 2 \_2\^2) . \[nilaxions\] The rest of the moduli are determined as x\_0=- ; y\_0\^2 = ; \_0 = - ; \_0 = \_2 \_0 . \[nilrest\] The cosmological constant can be computed using $\cw_0=2i\mu_0 |\Gamma|^{3/2}$.
From the results we deduce that $\eps_2 > 0$, and $\lambda_3 > 0$ for $y_0 < 0$. Then $\im U_0 > 0$ requires $|\Gamma| < 0$ as it happens for the nilpotent algebra. The tadpole conditions then verify $N_3 = - \lambda_3 \eps_2 |\Gamma|^3 > 0$ and $N_7 = \lambda_3 |\Gamma|^3 < 0$. The relevant conclusion is that the model necessarily requires O3-planes and O7-planes.
### Semidirect sum $\mathfrak{su(2) \oplus u(1)^3}$ {#sss:semidirectres}
The non-geometric fluxes of this subalgebra are $U \leftrightarrow T$ dual to NSNS plus [*geometric*]{} fluxes in a IIA orientifold. Models of this type have been studied previously [@Derendinger; @vz1; @cfi]. For completeness we will briefly summarize our results that totally agree with the general solution presented in [@cfi]. Existence of a supersymmetric minimum imposes the constraint $\eps_1=-\eps_2$. In this case it occurs again that the axions $s_0$ and $t_0$ can only be determined in a linear combination given by 3t\_0 + \_2 s\_0 = 3\_1 + 3\_2(9-7x\_0) + 3\_3 x\_0(9-8x\_0) . \[semiaxions\] The imaginary parts of the axiodilaton and the Kähler field are stabilized at values \_0 = \_2 \_0 ; \_2 \_0 = 6(\_2 + \_3 x\_0) y\_0 . \[semimusigma\] Notice that $\eps_2$ must be positive. It also follows that $\cw_0=2i\mu_0 (1-x_0 - iy_0)|\Gamma|^{3/2}$. The vevs of $x_0$ and $y_0$ depend on whether the RR flux parameter $\lambda_3$ is zero or not.
When $\lambda_3=0$ we obtain x\_0=1 ; 3\_2 y\_0\^2 = -(\_1+\_2) . \[semil3zero\] Notice that $\lambda_2 \not=0$ to guarantee $\sigma_0 \not=0$. In fact, choosing $y_0 > 0$ it must be $\lambda_2 > 0$. For the number of sources we find $N_3 = - \lambda_2 \eps_2 |\Gamma|^3 < 0$ and $N_7 = \lambda_2 |\Gamma|^3 > 0$. Therefore, D3 and D7-branes must be included.
When $\lambda_3\not=0$ we instead find \_3 y\_0\^2 = 15(x\_0-1)(\_2+\_3 x\_0) , \[semil3difzero\] whereas $x_0$ must be a root of the cubic equation 160(x\_0-1)\^3 + 294(1+ )(x\_0-1)\^2 + 135(1+ )\^2(x\_0-1) +(\_3 + 3\_2 + 3\_1)=0 . \[semix0\] The solution for $x_0$ must be real and such that $y_0 ^2 > 0$. For the tadpoles we now have $N_7=|\Gamma|^3(\lambda_2+\lambda_3)$ and $N_3=-\eps_2 N_7$. Thus, in general $N_3$ and $N_7$ have opposite signs. The remarkable feature is that now they can be zero simultaneously. This occurs when the RR parameters satisfy $\lambda_2=-\lambda_3$, in which case the cubic equation for $x_0$ can be solved exactly.
### Direct sum $\mathfrak{su(2)+ u(1)^3}$ {#sss:directres}
As explained before, necessarily $\eps_1 \not=0$. Let us consider $\eps_2=0$ which is the condition for $\cp_2$ to have only real roots. Now it happens that all moduli can be determined. The axions are fixed at $x_0=0$, $s_0=0$ and $t_0=0$, whereas the real parts have vevs y\_0\^2 = ; \_0 = - ; \_0 = 2 \_7 y\_0 . \[alldprod\] The cosmological constant is easily found substituting $\cw_0=-2\mu_0 y_0 |\Gamma|^{3/2}$. Clearly, the solution exists only if $\xi_3 \not=0$ and $\xi_7\not=0$. Moreover, $\eps_1\xi_3\xi_7 > 0$ and if we take $y_0 > 0$, $\xi_3 < 0$, $\xi_7 > 0$ and $\eps_1 < 0$. The numbers of sources satisfy $N_3 < 0$ and $N_7 > 0$, so that D3 and D7-branes are needed.
Taking $\eps_2\not=0$ we deduce that there are no solutions at all when $\xi_7=0$ and $\xi_3\not=0$. However, there are minima that require $\eps_1 < 0$ and $N_7 > 0$ when $\xi_3=0$.
### Non-compact $\mathfrak{so(3,1)}$ {#sss:noncompactres}
This is the only flux configuration for which $\cp_3(\cz)$ has complex roots. It also happens that $\cp_2(\cz)$ always has three different real roots. We will briefly discuss the vacua according to whether the NSNS flux parameter $\eps_2$ vanishes or not.
In this setup the axions are determined to be $x_0=0$, $s_0=0$ and $t_0=0$. For the imaginary parts of the Kähler modulus and the axiodilaton we obtain \_0 = ; \_1\_0 = . \[so31ip\] To evaluate the potential at the minimum we use $\cw_0=2\mu_0 y_0 (1-y_0^2)|\Gamma|^{3/2}$. Notice that $\xi_3$ and $\xi_7$ cannot be zero simultaneously and that $y_0^2=1$ is not allowed. Actually, the imaginary part of the transformed complex structure satisfies a third order polynomial equation in $y_0^2$ given by \_1\_3(5 y\_0\^6 + 13 y\_0\^4 + 15 y\_0\^2 -1) - \_7(y\_0\^2-1)(5 y\_0\^4 + 6y\_0\^2 -3) = 0 . \[sol31y0\] We are interested in real roots $y_0 \not=0$ and $y_0\not=\pm1$.
Although we have not made an exhaustive analysis, it is clear that the solutions of (\[sol31y0\]) depend on the range of the ratio $\xi_7/\eps_1\xi_3$. For instance, there are values for which there is no real root at all, as it occurs e.g. for $2\xi_7=-\eps_1\xi_3$. For other values there might be only one real positive solution for $y_0^2$. An special example happens when $\xi_3=0$ and the net O3/D3 charge $N_3$ is zero, while the net O7/D7 charge $N_7$ is negative as implied by the conditions $\mu_0 > 0$ and $|\Gamma| y_0 > 0$. Similarly, when $\xi_7=0$ , there is only one solution in which $N_7=0$ while $N_3 < 0$.
The third possibility is to have two allowed solutions. For instance, taking $\xi_7=2\eps_1 \xi_3$ gives roots $y_0^2=1/5$ and $y_0^2=1+2\sqrt 2$. However, in principle the corresponding vacua cannot be realized simultaneously because the net charges would have to jump. In fact, for $y_0^2 < 1$, it happens that $N_3 N_7 > 0$, whereas for $y_0^2 > 1$, it must be $N_3 N_7 < 0$. It can also arise that both solutions have $y_0^2 < 1$. For example, when $\xi_7=-30\eps_1 \xi_3$ each of the two vacua has $N_3 > 0$ and $N_7 < 0$. We will explore the phenomenon of multiple AdS vacua in more detail for the non-geometric fluxes of the $\mathfrak{su(2)^2}$ algebra.
We have only studied the special cases when one of the flux-tadpoles $N_3$ or $N_7$ is zero. We find that when $\eps_1=0$ the F-flat conditions can not be solved but for $\eps_1 > 0$ there are consistent solutions for a particular range of $|\eps_2/\eps_1|$. Vacua with $\xi_3 = 0$ exist provided that $\xi_7 < 0$. Vacua with no O7/D7 flux-tadpoles, i.e. with $\xi_7=0$, require $\xi_3 < 0$. One important conclusion is that for the fluxes of the non-compact $Q$-subalgebra solutions with $N_7=0$ must have $N_3 < 0$.
### Compact $\mathfrak{su(2)^2}$ {#sss:compactres}
This is the only situation in which the polynomial $P_3(U)$ induced by the non-geometric fluxes has three different real roots. The polynomial $P_2(U)$ generated by NSNS fluxes has complex roots whenever $\eps_1\eps_2\not=0$, and one triple real root otherwise. We will study the vacua in both cases in some detail.
The full model based on the non-geometric fluxes of $\mathfrak{su(2)^2}$ has an interesting residual symmetry that exchanges the NSNS auxiliary parameters. It can be shown that the effective action is invariant under $\eps_1 \leftrightarrow \eps_2$, $\xi_3 \to \xi_3$ and $\xi_7 \to \xi_7$, together with the field transformations 1/\^\* ; -\^\* ; -\^\* . \[extrasym\] This symmetry leaves one of the $\cp_3$ roots invariant while exchanging the other two.
### .1 $P_2(U)$ with triple real root {#su2zero .unnumbered}
Due to the symmetry (\[extrasym\]) it is enough to consider $\eps_1=0$ and $\eps_2 \not=0$. In this model the axions are stabilized at vevs x\_0=-12 ; \_2s\_0=3\_7 - ; t\_0=\_7 - . \[su2rst\] The imaginary parts of the Kähler modulus and the axiodilaton are fixed in terms of $y_0$ according to \_0 = - ; \_2\_0 = -y\_0 . \[su2ip\] At the minimum $\cw_0=2i \eps_2 \sigma_0 |\Gamma|^{3/2}$. Clearly $\xi_3$ and $\xi_7$ cannot vanish simultaneously so that the model always requires additional sources to cancel tadpoles. Observe that necessarily $\eps_2 < 0$.
The modulus $y_0$ is determined by the fourth order polynomial equation \_2\_3(4y\_0\^2-1)(4y\_0\^2+5) - 8\_7(4y\_0\^2-5)= 0 . \[sol2y0\] In the two special cases $\xi_7=0$ and $\xi_3=0$ an exact solution is easily found. When $\xi_3 \xi_7 \not=0$ there can be two AdS solutions. The corresponding vacua, which can be characterized by the net tadpoles $N_3$ and $N_7$, are described more extensively in the following.
When $\xi_7=0$ the vevs have the very simple expressions y\_0\^2 = 14 ; \_0 = ; \_0 = -2\_2 \_0 ; V\_0 = . \[n7zero\] Since both $\mu_0$ and $\sigma_0$ are positive, it must be $\eps_2 <0$, and taking $y_0 > 0$, $\xi_3 > 0$. Therefore, $N_3 > 0$ and O3-planes must be included.
This is the case $\xi_3=0$. The moduli and the cosmological constant are fixed at values y\_0\^2 = 54 ; \_2\_0 = -3\_7 y\_0 ; \_0 = -23 \_2 \_0 ; V\_0 = . \[n3zero\] Necessarily $\eps_2 < 0$, and choosing $y_0 > 0$, $\xi_7 > 0$. Hence, $N_7 > 0$ and D7-branes are required.
The solutions for $y_0$ depend on the ratio $\xi_7/\eps_2\xi_3$. A detailed analysis can be easily performed because the polynomial equation (\[sol2y0\]) is quadratic in $y_0^2$. We find that there are no real solutions in the interval ${\mbox{\small$1/8 < \xi_7/\eps_2\xi_3 < (7+2\sqrt{10})/4$}}$. On the other hand, when ${\mbox{\small$0 < \xi_7/\eps_2\xi_3 < 1/8$}}$, there is only one real positive solution for $y_0^2$ and it requires $N_3 > 0$ and $N_7 < 0$. For ${\mbox{\small$\xi_7/\eps_2\xi_3 \leq 0$}}$ there is only one acceptable root for $y_0^2$ and it leads to $N_3 > 0$ and $N_7 \geq 0$. A more interesting range of parameters is ${\mbox{\small$\xi_7/\eps_2\xi_3 > (7+2\sqrt{10})/4$}}$ because there are two allowed solutions for $y_0^2$ and for both it must be that $N_3 < 0$ and $N_7 > 0$. The upshot is that there can be metastable AdS vacua in the presence of D3 and D7-branes.
### .2 $P_2(U)$ with complex roots {#su2nonzero .unnumbered}
The F-flat conditions can be unfolded to obtain analytic expressions for the vevs of all moduli. However, for generic range of parameters, a higher order polynomial equation has to be solved to determine $y_0$ in the end. The main interesting feature is the appearance of multiple vacua even when $N_3 N_7 = 0$, i.e. when there are either no O7/D7 or no O3/D3 net charges present. We will first describe the overall picture and then present examples. For definiteness we always choose $y_0 > 0$ so that $|\Gamma| > 0$ is required to have $\im U_0 > 0$ for the complex structure.
To obtain and examine the results it is useful to make some redefinitions. The idea is to leave as few free parameters as possible in the F-flat equations. Since $\epsilon_1$ is different from zero we can work with the ratio = . \[newps\] By virtue of the residual symmetry (\[extrasym\]) there is an invariance under $\rho \to 1/\rho$. Therefore, we can restrict to the range $-1 \leq \rho \leq 1$, where the boundary corresponds to the fixed points of the inversion. Furthermore, as discussed at the end of section \[subsubso4\], the parameter $\rho$ is either a rational number or involves at most square roots of rationals.
When $\xi_3 \not=0$ it is also convenient to introduce new variables as =\_1 \_3 ; = \_3 ; \_7 = \_1 \_3 (\^2+1) . \[newus\] The definition of the parameter $\eta$ seems awkward but it simplifies the results. Notice that $\eta \to \eta\rho$ under (\[extrasym\]). In the new variables the superpotential becomes =||\^[3/2]{} \_1 \_3 \[3 (+1) + (\^3 + ) + (1-\^3) + 3(1+\^2) (1-) \] . \[w1\] Since the F-flat conditions are homogeneous in $\cw$ the resulting equations will only depend on the parameters $\rho$ and $\eta$. When $\xi_3=0$ we just make different field redefinitions, i.e. $\ct=\epsilon_1 \xi_7 \,\hat\ct$ and $\cs=\xi_7 \,\hat\cs$, so that the free parameters will be $\rho$ and $\xi_7/\eps_1$.
Manipulating the F-flat conditions enables us to find the vevs $\ct_0$ and $\cs_0$ as functions of $(x_0, y_0)$. The expressions are tractable but bulky so that we refrain from presenting them. The exception is the handy relation between the size and string coupling moduli \_0= , \[msvacgen\] which is valid when $x_0\not=\ds{{\mbox{\small$-\frac12$}}}$ and $y_0^2\not=\ds{{\mbox{\small$\frac34$}}}$. There is a solution with $x_0 =\ds{{\mbox{\small$-\frac12$}}}$ and $y_0^2 =\ds{{\mbox{\small$\frac34$}}}$ but it has $\mu_0=-\eps_1(1+\rho)\sigma_0$, $\mu_0=3\xi_7y_0$, and it requires $\eta=-(1+\rho)/(\rho^2-7\rho+1)$. There is another vacuum with $x_0=\ds{{\mbox{\small$-\frac12$}}}$ that occurs when $\rho \to \infty$ ($\eps_1=0$) and was discussed in section \[su2zero\]. The case $x_0^2=y_0^2$, which is better treated separately, requires $\xi_7\not=0$ unless $\rho=0$.
The residual unknowns $(x_0, y_0)$ are determined from the coupled system & & y\_0\^4+2x\_0(1+x\_0)y\_0\^2-(2x\_0+1) + x\_0\^3(x\_0+2) = 0 \[eqab\] ,\
& & y\_0\^6 ( 1+2x\_0-2 ) + ( 1+30 \^[3]{}-[x\_0]{}\^[2]{}+18\^[2]{}-6) y\_0\^4\
& & + x\_0 ( 54\^[4]{}+11 [x\_0]{}\^[3]{}+42 \^[3]{}+8 [x\_0]{}\^[2]{}+12 x\_0-4 x\_0-6 ) y\_0\^2 \[eqab2\]\
& & + ( 2+4x\_0+11 [x\_0]{}\^[3]{}+13 [x\_0]{}\^[4]{} ) ( 2+2\^[3]{}+[x\_0]{}\^[2]{}+x\_0 ) = 0 . The corresponding equations when $\xi_3=0$ can be obtained taking the limit $\eta \to \infty$. Eliminating $y_0$ for generic parameters gives a ninth-order polynomial equation for $x_0$.
For some range of parameters the above equations can admit several solutions for $\cz_0=x_0+iy_0$, which in turn yield consistent values for the remaining moduli. The existence of multiple vacua is most easily detected in the limiting cases in which one of the net tadpoles $N_7$ or $N_3$ vanishes, equivalently when $\xi_7=0$ ($\eta=0$) or $\xi_3=0$ ($\eta \to \infty$). In either limit the NSNS parameter $\rho$ can still be adjusted. We expect the results to be invariant under $\rho \to 1/\rho$ and this is indeed what happens.
We have mostly looked at models having no O7/D7 net charge, namely with $\eta=0$. It turns out that the solutions require $\xi_3 > 0$ so that $N_3 > 0$ and O3-planes must be present. Below we list the main results.
1\. For $\rho=1$ there are no minima with moduli stabilized.
2\. For $\rho=-1$ there is only one distinct vacuum with data \[solrmu\] Notice that necessarily $\xi_3 > 0$ and $\epsilon_1 < 0$. Actually, for $\rho=-1$, there is a second consistent solution but it is related to the above by the residual symmetry (\[extrasym\]).
3\. There can be only one solution when $\rho_c \leq \rho < 1$, where $\rho_c=-0.7267361874$. The critical value $\rho_c$ is such that the discriminant of the polynomial equation that determines $x_0$ is zero. Consistency requires $\eps_1 < 0$ and $\xi_3 > 0$ so that O3-planes are needed. For instance, when $\rho=0$ the solution is exact and has \[solrzero\] As expected, upon the transformation (\[extrasym\]) this vacuum coincides with that having $\xi_7=0$ and $\eps_1=0$, given in (\[n7zero\]). For other values of $\rho$ the solution is numerical. For example, taking $\rho={\mbox{\small$\frac12$}}$ leads to the vevs \[solrm12\]
4\. The important upshot is that in the interval $-1 < \rho < \rho_c$ there can be two distinct solutions for the same set of fluxes. An example with $\rho={\mbox{\small$-\frac45$}}$ is shown in table \[solezero\]. Notice that the last two solutions can exist for $\xi_3 > 0$ and $\epsilon_1 > 0$. The first solution can also occur but for $\xi_3 > 0$ and $\epsilon_1 < 0$.
$\cz_0$ $\cs_0/\xi_3$ $\ct_0/\xi_3 \epsilon_1$ $V_0\, \xi_3^2 \, \epsilon_1/|\Gamma|^3$
--------------------------------------- ---------------------------------- -------------------------------- ------------------------------------------
$\!\! -0.91105442 + 1.14050441 \, i$ $ -0.26002362 + 0.19059447 \, i$ $0.53128071 - 0.27572497\, i$ 3.353
$\!\! -0.43550654 + 0.73478523 \, i$ $ 0.28605555 + 0.55017649 \, i$ $0.60410811 + 0.12407321 \, i$ -2.168
$\!\! -0.40368586 + 0.57866160\, i$ $ 0.49215445 + 0.33255331 \, i$ $0.57101568 + 0.26593032\, i$ -1.880
: Degenerate vacua for $\xi_7=0$ and $\rho={\mbox{\small$-\frac45$}}$.[]{data-label="solezero"}
For models having no O3/D3 net charge a detailed analysis is clearly feasible but we have only sampled narrow ranges of the adjustable parameter $\rho$. Consistent solutions must have $\eps_1 < 0$ and $\xi_7 > 0$. Hence, $N_7 > 0$ and D7-branes must be included. There are values of $\rho$, e.g. $\rho=-1$, for which there are no vacua with stabilized moduli. For $\rho=1$ there is only one minimum which can be computed exactly. More interestingly, models of this type can also exhibit multiple vacua. In table \[soleinfty\] we show one example with $\rho={\mbox{\small$\frac34$}}$. Observe that both solutions exist for $ \eps_1 < 0$ and $\xi_7 >0$.
$\cz_0$ $\eps_1\cs_0/\xi_7$ $\ct_0/\xi_7$ $V_0\, \xi_7^2 /\epsilon_1|\Gamma|^3$
---------------------------------- ----------------------------------- ---------------------------------- ---------------------------------------
$-0.88312113 + 0.74580943 \, i$ $ -6.1818994 - 1.6867660 \, i$ $-4.20643209 + 3.92605399 \, i$ 0.026
$0.20646056 + 0.89488895 \, i$ $ 0.03039439 - 2.49813344 \, i$ $-0.06455485 + 1.18981502 \, i$ 0.084
: Vacua for $\xi_3=0$ and $\rho={\mbox{\small$\frac34$}}$.[]{data-label="soleinfty"}
Aspects of the non-geometric landscape {#sec:lands}
======================================
In this section we discuss the main aspects of the ${\rm AdS}_4$ vacua in our models that are standard examples of type IIB toroidal orientifolds with O3/O7-planes. Besides the axiodilaton $S$, after an isotropic Ansatz the massless scalars reduce to the overall complex structure $U$ and the size modulus $T$. Fluxes of the RR and NSNS 3-forms generate a potential that gives masses only to $S$ and $U$. The new ingredient here are non-geometric $Q$-fluxes, that are required to restore T-duality between type IIA and type IIB, and that induce a superpotential for the Kähler field $T$. The various fluxes must satisfy certain constraints arising from Jacobi or Bianchi identities. The problem is then to minimize the scalar potential while solving the constraints. The question is whether there are solutions with all moduli stabilized. We have seen that the answer is affirmative and now we intend to analyze it in more detail.
It is instructive to begin by recounting the findings of the previous sections. The initial step is to classify the subalgebras whose structure constants are the $Q$’s. With the isotropic Ansatz there are only five classes. For each type, the non-geometric fluxes can be written in terms of four auxiliary parameters $\genfrac{(}{)}{0pt}{}{\a \, \b}{\g \, \d}= \Gamma$, in such a way that the Jacobi identities are automatically satisfied. Other fluxes can also be parametrized using $\Gamma$ plus additional variables: $(\eps_1, \eps_2)$ for NSNS, and $(\xi_3, \xi_7, \xi_s, \xi_t)$ or $(\lambda_1, \lambda_2, \lambda_3, \xi_s, \xi_t)$ for RR. The significance of $\Gamma$ is that it defines a transformed complex structure $\cz=(\a U + \b)/(\g U + \d)$ that is invariant under the modular group $SL(2,\Z)_U$. The effective action can be expressed in terms of $\cz$ according to the $Q$-subalgebra. Once the subalgebra is chosen the vacua will depend only on the variables $\Gamma$, $(\eps_1, \eps_2)$, and $(\xi_3, \xi_7)$ or $(\lambda_1, \lambda_2, \lambda_3)$, that in turn determine the values of the cosmological constant and the string coupling $(V_0, g_s)$, as well as the net tadpoles $(N_3, N_7)$. In many examples, the vevs of the moduli can be determined in closed form.
Our approach to analyze the vacua in presence of non-geometric fluxes has the great advantage that the degeneracy due to modular transformations of the complex structure is already taken into account. Inequivalent vacua are just labelled by the vevs $(\cz_0, S_0, T_0)$ that are modular invariant. In practice this means that we can study families of modular invariant vacua by choosing a particular structure for $\Gamma$. In section \[sub:fam\] we will give concrete examples.
There is an additional vacuum degeneracy because the characteristic data $(V_0, g_s)$ happen to be independent of the parameters $(\xi_s, \xi_t)$. The explanation is that they correspond to shifts of the axions $\re S$ and $\re T$ which can be reabsorbed in the RR fluxes. The flux-induced RR tadpoles $(N_3, N_7)$ are blind to $(\xi_s, \xi_t)$ as well. Apparently, generic shifts in $\re S$ and $\re T$ are not symmetries of the compactification, so that two vacua differing only in the RR flux parameters $(\xi_s, \xi_t)$ would be truly distinct. We argue below that the vacua are equivalent because the full background is symmetric under $S \to S - \xi_s$, and $T \to T - \xi_t$.
In absence of non-geometric fluxes the 3-form RR field strength that appears in the 10-dimensional action is given by $F_3=dC_2- H_3 \wedge C_0 + \bar F_3$, where $H_3 = dB_2 + \bar H_3$. The natural generalization to include non-geometric fluxes is F\_3=dC\_2- H\_3 C\_0 + QC\_4 + |F\_3 , \[f3q\] where $QC_4$ is a 3-form that we can extract from (\[QJexpan\]) because $\re \cj = C_4$. In fact, $C_4=-\re T \sum_I \tilde \omega^I$, where $\tilde \omega ^I$ are the basis 4-forms. Recall also that $C_0 = \re S$. Notice then that $F_3$ involves the axions in question. The relevant result is that $F_3$ is invariant[^4] under the shifts $S \to S - \xi_s$, and $T \to T - \xi_t$. To show this we first compute the variation of $\bar F_3$ using the universal terms (\[uniRR\]) in the parametrization of the RR fluxes and then substitute in (\[f3q\]). In the effective 4 action the result is simply that the superpotential is invariant under these axionic shifts and the corresponding transformation of the RR fluxes. In turn this follows from (\[P3Iso\]) after substituting (\[uniRR\]).
Overview {#ss:view}
--------
We now describe in order some prominent features of the ${\rm AdS}_4$ vacua with non-geometric $Q$-fluxes switched on.
1\. The explicit results of section \[sub:ads\] indicate that in all models the vevs $\sigma_0 = \im S_0$ and $\mu_0 = \im T_0$ are correlated. This generic property follows from the F-flat conditions simply because the superpotential is linear in the axiodilaton and the Kähler modulus. Recall that the vevs in question determine physically important quantities, namely the string coupling $g_s=1/\sigma_0$, and the overall internal volume $V_{int}=(\mu_0/\sigma_0)^{3/2}$. To trust the perturbative string approximation $g_s$ must be small and we will shortly explain, as already shown in [@stw2], that generically there are regions in flux space in which both $g_s$ and the cosmological constant are small, while $V_{int}$ is large. We stress again the caveat that even at large overall volume there could still exist light winding string states when non-geometric fluxes are in play. These effects are certainly important in trying to lift the solutions to full string vacua. In this paper we only claim to have found vacua of the effective field theory with a precise set of massless fields and interactions due to generalized fluxes.
2\. Another common feature of all models is the relation between moduli vevs and net RR charges. In type IIB toroidal orientifolds it is known that in Minkowski supersymmetric vacua the contribution of RR and NSNS fluxes to the $C_4$ tadpole is positive ($N_3 >0$) and this occurs if and only if $\im S_0 > 0$ [@kst]. The interpretation is that to cancel the tadpole due to $\bar F_3$ and $\bar H_3$ it is mandatory to include O3-planes, whereas D3-branes can be added only as long as $N_3$ stays positive. This is also true for no-scale Minkowski vacua in which supersymmetry is broken by the F-term of the Kähler field. Turning on non-geometric fluxes enables to stabilize all moduli at a supersymmetric ${\rm AdS}_4$ minimum. At the same time, the $Q$-fluxes induce a $C_8$ tadpole of magnitude $N_7$ that can be cancelled by adding O7-planes and/or D7-branes. We find in general that the vevs $\im S_0$ and $\im T_0$, that must be positive, are correlated to the tadpoles $(N_3, N_7)$. According to the $Q$-subalgebra there are several possibilities for the type of sources that have to be included. For example, the models considered in [@stw2], having $N_3 > 0$ and $N_7 =0$, proceed only with the fluxes of the compact $\mathfrak{su(2)^2}$.
For the $Q$-fluxes of the nilalgebra, and the semidirect sum $\mathfrak{su(2) \oplus u(1)^3}$, there is a relation $N_3 = -\eps_2 N_7$, with $\eps_2 > 0$. Only in the latter case it is allowed to have $N_3=N_7=0$, and the sources can be avoided altogether. For the fluxes of $\mathfrak{su(2) + u(1)^3}$ it turns out that orientifold planes are unnecessary to cancel tadpoles, but both D3 and D7-branes must be added ($N_3 < 0$, $N_7 > 0$).
The fluxes of the semisimple subalgebras are more flexible. In particular, it can happen that one flux-tadpole vanishes while the other must have a definite sign. Moreover, the sign is opposite for the compact and non-compact cases. For instance, when $N_7=0$, $N_3 > 0$ and O3-planes are obligatory for the $\mathfrak{su(2)^2}$ fluxes, while for $\mathfrak{so(3,1)}$ $N_3 < 0$ and D3-branes are required.
The magnitudes of the vevs are also proportional to the net tadpoles. This then implies that the string coupling typically decreases when $N_3$ and/or $N_7$ increase. However, the number of D-branes cannot be increased arbitrarily without taking into account their backreaction.
3\. Consistency of the vacua can in fact be related to the full 12-dimensional algebra in which the $\bar H$ and $Q$-fluxes are the structure constants. The reason is that the conditions $\im S_0 > 0$ and $\im T_0 > 0$ also impose restrictions on the signs of the NSNS parameters $(\eps_1, \eps_2)$. For instance, in section \[sss:compactres\] we have seen that for $Q$-fluxes of the compact $\mathfrak{so(4) \sim su(2)^2}$, the solutions with $\eps_1=0$ require $\eps_2 < 0$. This in turn implies, as explained in section \[subsubso4\], that the full gauge algebra is $\mathfrak{so(4) + iso(3)}$. Another simple example is the model based on the $\mathfrak{su(2)+ u(1)^3}$ $Q$-subalgebra. The vacua of \[sss:directres\] with $\eps_2=0$ require $\eps_1 < 0$ and it can then be shown that the full gauge algebra is $\mathfrak{so(4) + u(1)^6}$. A more detailed study of the 12-dimensional algebras is left for future work [@guarino].
4\. We defer to section \[sub:fam\] a more thorough discussion of the landscape of values attained by the string coupling $g_s$ and the cosmological constant $V_0$, for the fluxes of the compact $\mathfrak{su(2)^2}$ $Q$-subalgebra. The situation for $\mathfrak{so(3,1)}$ is similar and can be analyzed using the results of section \[sss:noncompactres\]. The model based on the direct product $\mathfrak{su(2)+u(1)^3}$ is different because both $N_3$ and $N_7$ must be non-zero, but it can still be shown that there exist vacua with small $g_s$ and $V_0$. The models built using the nilpotent and semidirect $Q$-subalgebras have been studied in their T-dual IIA formulation in refs. [@DeWolfe; @cfi], where it was found that there are infinite families of vacua within the perturbative region.
5\. A peculiar result is the appearance of multiple vacua for certain combination of fluxes. These events occur only in models based on the semisimple $Q$-subalgebras. They can have $N_3 N_7=0$ or $N_3 N_7 \not=0$, but in the former case both NSNS parameters $(\eps_1, \eps_2)$ must be non-zero. Reaching small string coupling and cosmological constant typically requires that $N_3$ and/or $N_7$ be sufficiently large.
6\. To cancel RR tadpoles it might be necessary to add stacks of D3 and/or D7-branes. These additional D-branes could also generate a charged chiral spectrum but more generally a different sector of D-branes will serve this purpose. In any case, the D-branes that can be included are constrained by cancellation of Freed-Witten anomalies [@cfi; @vz2]. In absence of non-geometric fluxes the condition amounts to the vanishing of $\bar H_3$ when integrated over any internal 3-cycle wrapped by the D-branes. For unmagnetized D7-branes in $\T^6/\Z_2 \times \Z_2$, with $\bar H_3$ given in (\[H3expan\]), it is easy to see that the condition is met, whereas for D3-branes it is trivial. When $Q$-fluxes are switched on the modified condition [@vz2] is still satisfied basically because the 3-form $Q\cj$, defined in (\[QJexpan\]), can be expanded in the same basis as $\bar H_3$.
D3-branes and unmagnetized D7-branes in $\T^6/\Z_2 \times \Z_2$ do not give rise to charged chiral matter. Therefore the models will not have $U(1)$ chiral anomalies. This is consistent with the fact that the axions $\re S$ and $\re T$ are generically stabilized by the fluxes and having acquired a mass they could not participate in the Green-Schwarz mechanism to cancel the chiral anomalies[^5].
To construct a more phenomenologically viable scenario one could introduce magnetized D9-branes as in the $\T^6/\Z_2 \times \Z_2$ type IIB orientifolds with NSNS and RR fluxes that were considered some time ago [@magnetized]. Now, care has to be taken because magnetized D9-branes suffer from Freed-Witten anomalies. They are actually forbidden in absence of non-geometric fluxes when $\bar H_3 \not=0$.
The effect of the $Q$-fluxes can be studied as explained in [@vz2]. Cancellation of Freed-Witten anomalies translates into invariance of the superpotential under shifts and , where the real charges $(q_s,q_t)$ depend on the $U(1)$ gauged by the D-brane. Applying this prescription we conclude that in our setup with isotropic fluxes magnetized D9-branes could be introduced only in models based on the nilpotent and semidirect sum $\mathfrak{su(2) \oplus u(1)^3}$ . The reason is that only in these cases the flux-induced polynomials $P_2(U)$ and $P_3(U)$ can be chosen parallel and then $W$ can remain invariant under the axionic shifts. Equivalently, only in these cases the axions $\re S$ and $\re T$ are not fully determined and the residual massless linear combination can give mass to an anomalous $U(1)$. For other the polynomials $P_2(U)$ and $P_3(U)$ are linearly independent and both axions are completely stabilized.
It would be interesting to study the consistency conditions on magnetized D9-branes in models with non-isotropic fluxes. In principle there could exist configuration of fluxes such that the general superpotential (\[fullW\]-\[p3gen\]) is invariant under axionic shifts of $S$ and the Kähler moduli $T_I$.
Families of modular invariant vacua {#sub:fam}
-----------------------------------
To generate specific families of vacua we first choose the $Q$-subalgebra and then select the parameters in $\Gamma$. In general $\Gamma$ can be chosen so that the non-geometric fluxes are even integers. The NSNS fluxes turn out to be even integers by picking $(\eps_1, \eps_2)$ appropriately. One can also start from given non-geometric and NSNS even integer fluxes and deduce the corresponding $\Gamma$ and $(\eps_1, \eps_2)$. Similar remarks apply to the RR fluxes. We will illustrate the procedure for the compact $\mathfrak{su(2)^2}$.
If one of the parameters vanishes, say $\g=0$, it can be shown from (\[LimC\]) that the ratios $\d/\a$ and $\b/\a$ are rational numbers (recall that $|\Gamma|\not=0$ so that $\a, \d \not=0$). It then follows that by a modular transformation, c.f. (\[Gmodt\]), we can go to a canonical gauge in which also $\b=0$.
The canonical diagonal gauge $\g=\b=0$ is completely generic when $\eps_2=0$ ($\eps_1 \not=0$). In this case we find that $\b/\a$ and $\g/\d$ are rational because they are given respectively by quotients of NSNS and non-geometric fluxes. Therefore, $\b$ and $\g$ can be gauged away by modular transformations. If instead $\eps_1=0$, but $\eps_2\not=0$, we can take $\a=\d=0$.
When $\eps_1 \eps_2 \not=0$ we can still use the canonical gauge but it will not give the most general results that are obtained simply by considering $\a, \b, \g, \d \not= 0$.
### Canonical families for $\mathfrak{su(2)^2}$ fluxes {#ss:canonical}
For each subalgebra we can obtain families of vacua starting from the canonical gauge defined by $\g=\b=0$. In the $\mathfrak{su(2)^2}$ case only the non-geometric fluxes $\tilde c_1$ and $\tilde c_2$ are different from zero and can be written as c\_1 = -2m ; c\_2 = 2n ; m, n . \[nongeocan\] From (\[LimC\]) we easily find $\a/\d=n/m$, $\d^3=2m^2/n$, so that $|\Gamma|^3=4 nm$. The non-zero NSNS and RR fluxes are easily found to be b\_0 & = & - \_2 ; b\_3 = \_1 ; a\_0 = (\_1 \_3 + \_2 \_s) , \[abcan\]\
a\_1 & = & -2m(\_t + \_7) ; a\_2 = 2n(\_t - \_7) ; a\_3 = -(\_1\_s - \_2 \_3) . Since the $b$’s and $a$’s are (even) integers, it is obvious that $(\eps_1, \eps_2)$ and $(\xi_3, \xi_7, \xi_s, \xi_t)$ are all rational numbers.
The moduli vevs depend on $(\xi_3, \xi_7)$ and $(\eps_1, \eps_2)$. For concreteness, and to compare with the results of [@stw2], we focus on the case $\xi_7=0$. Other cases can be studied using the results of section \[sss:compactres\]. When $\xi_7=0$ the RR fluxes $a_1$ and $a_2$ are spurious, they can be eliminated by setting $\xi_t=0$, i.e. by a shift in $\re T$.
To continue we have to distinguish whether one of the NSNS parameters $\eps_1$ or $\eps_2$ is zero. Recall that in this case the flux induced polynomial $P_2$ does not have complex roots.
Let us consider $\eps_2=0$. Then, also $a_3$, or $\xi_s$, is irrelevant and can be set to zero by a shift in $\re S$. The important physical parameters are $\eps_1$ and $\xi_3$, they can be deduced from $b_3$ and $a_0$. Notice also that at this point $N_3=a_0 b_3$. Using (\[solrzero\]) we obtain the values of the cosmological constant and the string coupling V\_0 = ; g\_s = . \[candata\] Consistency requires $\eps_1 < 0$ and $\xi_3 > 0$, or equivalently $V_0 < 0 $ and $g_s > 0$. For the purpose of counting distinct vacua we can safely assume $b_3 > 0$ and then $m, n < 0$.
As noticed in [@stw2], the important outcome is that $g_s$ and $V_0$ can be made arbitrarily small by keeping $b_3$ and $m$ fixed while letting $n \to \infty$.
In our approach it is also easy to see that $(V_0, g_s)$ always take values of the form (\[candata\]) whenever $P_2$ has only real roots. This follows because all vacua are related by modular transformations plus axionic shifts. However, if as in [@stw2] we want to count the vacua with fluxes bounded by an upper limit $L$, it does not suffice to just consider the canonical gauge. The reason is that by performing modular transformations and axionic shifts we can reach larger effective values of $b_3$ that seem to violate the tadpole condition. Rather than an elaborate argument we will just provide a simple example. We can go to a non-canonical gauge with $\g=0$ but $\b\not=0$ and also take $\xi_t=0$ but $\xi_s \not=0$. With these choices it is straightforward to show that $N_3=a_0 b_3 - a_3 b_0$, which would allow to take e.g. $b_3=N_3$ that is forbidden when $b_0=0$ ($\b=0$), or $a_3=0$ ($\xi_s=0$), because $a_0$ must be even. To do detailed vacua statistics it is necessary to use generic gauge and axionic shifts.
As in section \[su2nonzero\] we set $\eps_2=\rho \eps_1$. In the canonical gauge the parameter $\rho$ is a rational number that we assume to be given. We choose to vary the NSNS flux $b_3$ that determines \_1= ; b\_0 = - , \[bzero\] where $m, n$ are the integers coming from the non-geometric fluxes. The vacuum data have been found to be V\_0 = ; g\_s = , \[candatagen\] where we used $|\Gamma|^3=4nm$. The numerical factors $F_V$ and $F_g$ depend on $\rho$. For instance, for $\rho=0$, $F_V=6$ and $F_g={\mbox{\small$1/8$}}$. Other examples are given in section \[su2nonzero\]. We remark that for $\rho$ in a particular range there can be multiple vacua, meaning that for some $\rho$ the above numerical factors might take different values (e.g. table \[solezero\]).
It is most convenient to extract $\xi_3$ from the tadpole relation $N_3=4mn\eps_1^2(1+\rho^2)\xi_3$, which in terms of the integer fluxes reads $N_3=a_0 b_3 - a_3 b_0$. Combining all the information we readily find V\_0 = ; g\_s = . \[candata2\] Unlike the case when $\rho=0$, in general we cannot keep $m$ and $b_3$ fixed while letting $n \to \infty$. The reason is that the NSNS flux $b_0$ in (\[bzero\]) must be an integer.
The main conclusion is that it is not always possible to obtain small string coupling and cosmological constant. In fact, when $\rho\not=0$, there are no vacua with $g_s < 1$ unless the tadpole $N_3$ is sufficiently big. To prove this, notice first that the string coupling can be rewritten as $g_s={\mbox{\small$- b_3 b_0 (1+\rho^2)/(F_s \rho N_3)$}}$. The most favorable situation occurs when $\rho=-1$ for which $F_s=0.238$. The smallest allowed NSNS fluxes are $b_0=b_3=2$ (compatible with $\rho=-1$). Hence, the minimum value of the coupling is $g_s^{min}={\mbox{\small$8/(F_s N_3)$}}$ and $g_s^{min} < 1$ would require $N_3 > 33$. The situation is worse for values of $\rho$ such that multiple vacua can appear. The problem is that since such $\rho$’s are rational, $b_3$ must be largish for $b_0$ to be integer. Going to a more general gauge does not change the conclusion.
We have just provided a quantitative, almost analytic, explanation of why there are no perturbative vacua when the flux polynomial $P_2$ has complex roots and $N_3$ is not large enough. This observation was first made in [@stw2] based on a purely numerical analysis.
Final remarks {#sec:end}
=============
In this paper we have investigated supersymmetric flux vacua in a type IIB orientifold with RR, NSNS and non-geometric $Q$-fluxes turned on. We enlarged the related analysis of [@stw2] by considering the most general fluxes solving the Jacobi identities, and by including variable numbers of O3/D3 and O7/D7 sources to cancel the flux-induced RR tadpoles.
Our approach is based on the classification of the subalgebras satisfied by the non-geometric fluxes. A convenient parametrization of the $Q$-fluxes leads to an auxiliary complex structure that turns out to be invariant under modular transformations. Writing the superpotential in terms of this invariant field simplifies solving the F-flat conditions and enables us to obtain analytic expressions for the moduli vevs. We have found families of supersymmetric ${\rm AdS}_4$ vacua in all models defined by the inequivalent $Q$-subalgebras. General properties of the solutions were discussed in section \[sec:lands\]. The vacua typically exist in all cases, provided that arbitrary values of the flux-induced RR tadpoles are allowed.
In type IIB orientifolds with only RR and NSNS fluxes there is a non-trivial induced tadpole that must be cancelled by O3-planes or wrapped D7-branes. But including non-geometric fluxes can require other types of sources. For instance, similar to well understood ${\rm AdS}_4$ models in type IIA, the induced flux-tadpoles might vanish implying that sources can be avoided. There are also examples in which sources of positive RR charge are sufficient to cancel the tadpoles. As one might expect, these latter exotic vacua occur in models built using $Q$-fluxes satisfying the non-compact $\mathfrak{so(3,1)}$ subalgebra. Such solutions might be ruled out once a deeper understanding of non-geometric fluxes has been developed.
We discussed a simplified set of fluxes but our methods could be used to study other configurations. The starting point would be the classification of the $Q$-subalgebras consistent with the underlying symmetries.
Although our main goal was to explore supersymmetric vacua with moduli stabilized, our results could have further applications. We have succeeded in connecting properties of the vacua to the underlying gauge algebra and this can help towards extending the description of non-geometric fluxes beyond the effective action limit. At present one of the most challenging problems in need of new insights is precisely to formulate string theory on general backgrounds at the microscopic level.
[**Acknowledgments**]{}
We are grateful to P. Cámara, B. de Carlos, L. Ibáñez, R. Minasian, G. Tasinato, S. Theisen and G. Weatherill for useful comments. A.F. thanks the Max-Planck-Institut für Gravitationsphysik, as well as the Instituto de Física Teórica UAM/CSIC, for hospitality and support at several stages of this paper, and CDCH-UCV for a research grant No. PI-03-007127-2008. A.G. acknowledges the financial support of a FPI (MEC) grant reference BES-2005-8412. This work has been partially supported by CICYT, Spain, under contract FPA 2007-60252, the Comunidad de Madrid through Proyecto HEPHACOS S-0505/ESP-0346, and by the European Union through the Marie Curie Research and Training Networks [*Quest for Unification*]{} (MRTN-CT-2004-503369) and [*UniverseNet*]{} (MRTN-CT-2006-035863).
Appendix: Parametrized RR fluxes {#appA .unnumbered}
================================
In this appendix we give the explicit expressions for the original RR fluxes $a_A$ in terms of the axionic shifts $(\xi_s, \xi_t)$ and the tadpole parameters $(\xi_3, \xi_7)$ or $(\lambda_2, \lambda_3)$, depending on the $Q$-subalgebra. For the semidirect sum $\mathfrak{su(2)\oplus u(1)^3}$ and the nilpotent algebra there is another auxiliary variable $\lambda_1$ as explained in \[ss:rr\]. In all cases there is a non-singular rotation matrix from the $a_A$’s to the new variables.
In principle the $\xi$’s and $\lambda$’s are just real constants but the resulting $a_A$ fluxes must be integers. The exact nature of these parameters can be elucidated starting with the non-geometric fluxes of each subalgebra. For example, following the discussion at the end of section \[subsubso4\], for $\mathfrak{su(2)^2}$ when $\eps_1\eps_2=0$ it transpires that $(\xi_3, \xi_7, \xi_s, \xi_t) \in \mathbb{Q}$.
There is a universal structure in the RR fluxes that is worth noticing. For all $Q$-subalgebras the dependence on the axionic shift parameters $(\xi_s, \xi_t)$ is of the form a\_0 &=& - b\_0 \_s + 3 c\_0 \_t +\
a\_1 &=& - b\_1 \_s - (2c\_1 - c\_1) \_t + \[uniRR\]\
a\_2 &=& - b\_2 \_s - (2c\_2 - c\_2) \_t +\
a\_3 &=& - b\_3 \_s + 3 c\_3 \_t + where $\cdots$ stands for extra terms depending on the tadpole parameters.
. a\_[0]{} &=& \^3(\_[1]{} \_3 + \_[2]{}\_s) + \^3(\_1 \_s - \_2 \_3) + 3 \^2(\_t - \_7) + 3 \^2(\_t+\_7)\
a\_[1]{} &=& -\^2(\_[1]{} \_3 + \_[2]{} \_s) - \^2 (\_1 \_s - \_2 \_3) - (+2)(\_t - \_7) - (+2)(\_t+\_7)\
a\_[2]{} &=& \^2(\_[1]{} \_3 + \_[2]{} \_s) + \^2 (\_1 \_s - \_2 \_3) + (+ 2)(\_t - \_7) + (+2)(\_t+\_7)\
a\_[3]{} &=& -\^3(\_[1]{} \_3 + \_[2]{}\_s) - \^3(\_1 \_s - \_2 \_3) - 3 \^2(\_t - \_7) - 3 \^2(\_t+\_7)
. a\_[0]{} &=& (\^2-3\^2)(\_[1]{} \_3 + \_[2]{}\_s) + (\^2-3\^2)(\_1 \_s - \_2 \_3) - 3(\^2 + \^2)(\_t - \_7)\
a\_[1]{} &=& (\^2+ 2- \^2)(\_[1]{} \_3 + \_[2]{}\_s) + (\^2 + 2- \^2)(\_[1]{} \_s - \_[2]{}\_3)\
&& + (\^2 + \^2)(\_t - \_7) + 2 (+ )(\_t - \_7)\
a\_[2]{} &=& (\^2- 2- \^2)(\_[1]{} \_3 + \_[2]{}\_s) + (\^2 - 2- \^2)(\_[1]{} \_s - \_[2]{}\_3)\
&& - 2 (+ )(\_t - \_7) - (\^2 + \^2)(\_t - \_7)\
a\_[3]{} &=& - (\^2-3\^2)(\_[1]{} \_3 + \_[2]{}\_s) - (\^2-3\^2)(\_1 \_s - \_2 \_3) + 3(\^2 + \^2)(\_t - \_7)
. a\_[0]{} &=& \^3(\_[1]{} \_3 + \_[2]{}\_s) + \^3(\_1 \_s - \_2 \_3) + 3 \^2 \_t - 3\^2 \_7\
a\_[1]{} &=& -\^2(\_[1]{} \_3 + \_[2]{} \_s) - \^2 (\_1 \_s - \_2 \_3) - (+2)\_t + (+2)\_7\
a\_[2]{} &=& \^2(\_[1]{} \_3 + \_[2]{} \_s) + \^2 (\_1 \_s - \_2 \_3) + (+2)\_t - (+ 2)\_7\
a\_[3]{} &=& -\^3(\_[1]{} \_3 + \_[2]{}\_s) - \^3(\_1 \_s - \_2 \_3) - 3\^2\_t + 3 \^2\_7
. a\_[0]{} &=& \^3(\_2 \_s + 3\_t) + \^2(\_1 \_s - 3\_t + 3\_1) + 3 \^2 \_2 + \^3 \_3\
a\_[1]{} &=& -\^2(\_2 \_s + 3\_t) - (+ 2)(\_1 \_s - 3\_t + 3\_1) - (+2)\_2 - \^2\_3\
a\_[2]{} &=& \^2(\_2 \_s + 3\_t) + (+ 2)(\_1 \_s - 3\_t + 3\_1) + (+2)\_2 + \^2\_3\
a\_[3]{} &=& -\^3(\_2 \_s + 3\_t) - \^2(\_1 \_s - 3\_t + 3\_1) - 3 \^2 \_2 - \^3 \_3
. a\_[0]{} &=& \^3(\_2 \_s + 3\_t) + \^2(\_1 \_s + 3\_1) + 3 \^2 \_2 + \^3 \_3\
a\_[1]{} &=& -\^2(\_2 \_s + 3\_t) + (\^2-2\^2)(\_1 \_s + 3\_1) - (\^2-2\^2)\_2 + \^2\_3\
a\_[2]{} &=& \^2(\_2 \_s + 3\_t) + (\^2 - 2\^2)(\_1 \_s + 3\_1) + (\^2 -2\^2)\_2 + \^2\_3\
a\_[3]{} &=& -\^3(\_2 \_s + 3\_t) + \^2(\_1 \_s + 3\_1) - 3 \^2 \_2 + \^3 \_3
[98]{}
For reviews and references, see:\
M. Graña, [*Flux compactifications in string theory: A comprehensive review*]{}, Phys. Rept. 423 (2006) 91 \[arXiv:hep-th/0509003\];\
M. R. Douglas and S. Kachru, [*Flux compactification*]{}, Rev. Mod. Phys. [**79**]{} (2007) 733 \[arXiv:hep-th/0610102\];\
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, [*Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes*]{}, Phys. Rept. [**445**]{} (2007) 1 \[arXiv:hep-th/0610327\]. T. W. Grimm and J. Louis, [*The effective action of type IIA Calabi-Yau orientifolds*]{}, Nucl. Phys. B [**718**]{} (2005) 153 \[arXiv:hep-th/0412277\]. J. P. Derendinger, C. Kounnas, P. M. Petropoulos and F. Zwirner, [*Superpotentials in IIA compactifications with general fluxes*]{}, Nucl. Phys. B [**715**]{} (2005) 211 \[arXiv:hep-th/0411276\];\
[*Fluxes and gaugings: N = 1 effective superpotentials*]{}, Fortsch. Phys. [**53**]{} (2005) 926 \[arXiv:hep-th/0503229\]. G. Villadoro and F. Zwirner, [*N = 1 effective potential from dual type-IIA D6/O6 orientifolds with general fluxes*]{}, JHEP [**0506**]{} (2005) 047 \[arXiv:hep-th/0503169\]. O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, [*Type IIA moduli stabilization*]{}, JHEP [**0507**]{} (2005) 066 \[arXiv:hep-th/0505160\]. P. G. Cámara, A. Font and L. E. Ibáñez, [*Fluxes, moduli fixing and MSSM-like vacua in a simple IIA orientifold*]{}, JHEP [**0509**]{} (2005) 013 \[arXiv:hep-th/0506066\]. M. Graña, R. Minasian, M. Petrini and A. Tomasiello, [*A scan for new N=1 vacua on twisted tori*]{}, JHEP [**0705**]{}, 031 (2007) \[arXiv:hep-th/0609124\]. D. Andriot, [*New supersymmetric flux vacua with intermediate SU(2) structure*]{}, arXiv: 0804.1769 \[hep-th\]. C. Caviezel, P. Koerber, S. Körs, D. Lüst, D. Tsimpis and M. Zagermann, [*The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets*]{}, arXiv:0806.3458 \[hep-th\]. G. Aldazabal and A. Font, [*A second look at N=1 supersymmetric ${\rm AdS}_4$ vacua of type IIA supergravity*]{}, JHEP [**0802**]{} (2008) 086 \[arXiv:0712.1021 \[hep-th\]\]. J. Shelton, W. Taylor and B. Wecht, [*Nongeometric flux compactifications*]{}, JHEP [**0510**]{} (2005) 085 \[arXiv:hep-th/0508133\]. J. Shelton, W. Taylor and B. Wecht, [*Generalized flux vacua*]{}, JHEP [**0702**]{} (2007) 095 \[arXiv:hep-th/0607015\]. For a review and references, see:\
B. Wecht, [*Lectures on Nongeometric Flux Compactifications*]{}, Class. Quant. Grav. [**24**]{} (2007) S773 \[arXiv:0708.3984 \[hep-th\]\]. S. Gurrieri, J. Louis, A. Micu and D. Waldram, [*Mirror symmetry in generalized Calabi-Yau compactifications*]{}, Nucl. Phys. B [**654**]{} (2003) 61 \[arXiv:hep-th/0211102\]. S. Kachru, M. B. Schulz, P. K. Tripathy and S. P. Trivedi, [*New supersymmetric string compactifications*]{}, JHEP [**0303**]{} (2003) 061 \[arXiv:hep-th/0211182\]. A. Dabholkar and C. Hull, [*Generalised T-duality and non-geometric backgrounds*]{}, JHEP [**0605**]{} (2006) 009 \[arXiv:hep-th/0512005\]. J. Scherk and J. H. Schwarz, [*Spontaneous Breaking Of Supersymmetry Through Dimensional Reduction*]{}, Phys. Lett. B [**82**]{} (1979) 60; [*How To Get Masses From Extra Dimensions*]{}, Nucl. Phys. B [**153**]{} (1979) 61. N. Kaloper and R. C. Myers, [*The O(dd) story of massive supergravity*]{}, JHEP [**9905**]{} (1999) 010 \[arXiv:hep-th/9901045\]. G. Aldazabal, P. G. Cámara, A. Font and L. E. Ibáñez, [*More dual fluxes and moduli fixing*]{}, JHEP [**0605**]{} (2006) 070 \[arXiv:hep-th/0602089\]. G. Villadoro and F. Zwirner, [*D terms from D-branes, gauge invariance and moduli stabilization in flux compactifications*]{}, JHEP [**0603**]{} (2006) 087 \[arXiv:hep-th/0602120\]. I. Benmachiche and T. W. Grimm, [*Generalized N = 1 orientifold compactifications and the Hitchin functionals*]{}, Nucl. Phys. B [**748**]{} (2006) 200 \[arXiv:hep-th/0602241\]. A. Micu, E. Palti and G. Tasinato, [*Towards Minkowski vacua in type II string compactifications*]{}, JHEP [**0703**]{} (2007) 104 \[arXiv:hep-th/0701173\]. M. Ihl, D. Robbins and T. Wrase, [*Toroidal Orientifolds in IIA with General NS-NS Fluxes*]{}, JHEP [**0708**]{} (2007) 043 \[arXiv:0705.3410 \[hep-th\]\]. E. Palti, [*Low Energy Supersymmetry from Non-Geometry*]{}, JHEP [**0710**]{} (2007) 011 \[arXiv:0707.1595 \[hep-th\]\]. P. G. Cámara, E. Dudas, T. Maillard and G. Pradisi, [*String instantons, fluxes and moduli stabilization*]{}, Nucl. Phys. B [**795**]{} (2008) 453 \[arXiv:0710.3080 \[hep-th\]\]. S. B. Giddings, S. Kachru and J. Polchinski, [*Hierarchies from fluxes in string compactifications*]{}, Phys. Rev. D [**66**]{} (2002) 106006 \[arXiv:hep-th/0105097\]. C. M. Hull and R. A. Reid-Edwards, [*Gauge Symmetry, T-Duality and Doubled Geometry*]{}, JHEP [**0808**]{} (2008) 043 \[arXiv:0711.4818 \[hep-th\]\]. G. Dall’Agata, N. Prezas, H. Samtleben and M. Trigiante, [*Gauged Supergravities from Twisted Doubled Tori and Non-Geometric String Backgrounds*]{}, Nucl. Phys. B [**799**]{} (2008) 80 \[arXiv:0712.1026 \[hep-th\]\]. M. Graña, R. Minasian, M. Petrini and D. Waldram, [*T-duality, Generalized Geometry and Non-Geometric Backgrounds*]{}, arXiv:0807.4527 \[hep-th\]. A. R. Frey and J. Polchinski, [*N = 3 warped compactifications*]{}, Phys. Rev. D [**65**]{} (2002) 126009 \[arXiv:hep-th/0201029\]. S. Kachru, M. B. Schulz and S. Trivedi, [*Moduli stabilization from fluxes in a simple IIB orientifold*]{}, JHEP [**0310**]{} (2003) 007 \[arXiv:hep-th/0201028\]. G.-M. Greuel, G. Pfister and H. Schönemann, [Singular]{} 3.0 - [*A Computer Algebra System for Polynomial Computations*]{}. In M. Kerber and M. Kohlhase: Symbolic Computation and Automated Reasoning, The Calculemus-2000 Symposium. (2001), 227-233. [http://www.singular.uni-kl.de/]{}
C. M. Hull, [*A geometry for non-geometric string backgrounds*]{}, JHEP [**0510**]{} (2005) 065 \[arXiv:hep-th/0406102\]. O. DeWolfe, A. Giryavets, S. Kachru and W. Taylor, [*Enumerating flux vacua with enhanced symmetries*]{}, JHEP [**0502**]{} (2005) 037 \[arXiv:hep-th/0411061\]. J. Gray, Y. H. He, A. Ilderton and A. Lukas, [*STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology*]{}, arXiv:0801.1508 \[hep-th\]. R. Blumenhagen, D. Lüst and T. R. Taylor, [*Moduli stabilization in chiral type IIB orientifold models with fluxes*]{}, Nucl. Phys. B [**663**]{} (2003) 319 \[arXiv:hep-th/0303016\];\
J. F. G. Cascales and A. M. Uranga, [*Chiral 4d N = 1 string vacua with D-branes and NSNS and RR fluxes*]{}, JHEP [**0305**]{} (2003) 011 \[arXiv:hep-th/0303024\];\
D. Lüst, S. Reffert and S. Stieberger, [*Flux-induced soft supersymmetry breaking in chiral type IIb orientifolds with D3/D7-branes*]{}, Nucl. Phys. B [**706**]{} (2005) 3 \[arXiv:hep-th/0406092\];\
F. Marchesano and G. Shiu, [*MSSM vacua from flux compactifications*]{}, Phys. Rev. D [**71**]{} (2005) 011701 \[arXiv:hep-th/0408059\]; [*Building MSSM flux vacua*]{}, JHEP [**0411**]{} (2004) 041 \[arXiv:hep-th/0409132\];\
M. Cvetic and T. Liu, [*Three-family supersymmetric standard models, flux compactification and moduli stabilization*]{}, Phys. Lett. B [**610**]{} (2005) 122 \[arXiv:hep-th/0409032\]. A. Guarino et al., work in progress.
[^1]: This can be done using a computational algebra program as Singular [@singular] and solving over the real field. In [@stw1], an analogous result is obtained manipulating this set of polynomial constraints by hand.
[^2]: A table and references to the original literature are given in [@gmpt].
[^3]: In [@tasinato] it is further shown that Minkowski vacua with all moduli stabilized can exist in more general setups having more complex structure than Kähler moduli (in IIB language).
[^4]: We thank P. Cámara for giving us this hint.
[^5]: We thank L. Ibáñez for discussions on this point.
|
---
abstract: 'Let $p$ be an odd prime, ${\mathbb{F}}$ a finite field of characteristic $p$ and let $\bar{\rho}:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2({\mathbb{F}})$ be a continuous Galois representation. Denote by $\bar{\chi}$ the mod $p$ cyclotomic character. We assume that the local representation $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$ is flat and irreducible and $\text{det}\bar{\rho}=\bar{\chi}$. The celebrated theorem of Khare and Wintenberger asserts that if $\bar{\rho}$ satisfies some natural conditions, there exists a normalized Hecke-eigenform $f=\sum_{n\geq 1} a_n q^n$ and a prime $\mathfrak{p}|p$ in its field of Fourier coefficients such that the associated $\mathfrak{p}$-adic representation ${\rho}_{f,\mathfrak{p}}$ lifts $\bar{\rho}$. In this manuscript we prove a refined lifting theorem. Let $\iota_{\mathfrak{p}}$ be the inclusion of the field of Fourier coefficients ${\mathbb{Q}}(f)$ of $f$ into ${\mathbb{Q}}(f)_{\mathfrak{p}}$ and let $v_p$ be a $p$-adic valuation on $\bar{{\mathbb{Q}}}_p$ normalized by $v_p(p)=1$. We show that for any choice of $\lambda\in \mathbb{Z}_{\geq 1}$, there exists a cuspidal Hecke eigenform $f$ of weight $2$ over $\Gamma_1(N)$ for $(N,p)=1$ and a prime $\mathfrak{p}|p$ such that $v_p(\iota_{\mathfrak{p}}(a_p))=\lambda$ and the residual representation $\bar{\rho}$ lifts to $\rho_{f,\mathfrak{p}}$. Thus all slopes ${\mathbb{Z}}_{\geq 1}\subset {\mathbb{Q}}_{>0}$ are realized as we vary over all Galois representations associated to cuspidal Hecke eigenforms which lift $\bar{\rho}$.'
address: |
Cornell University\
Department of Mathematics\
Malott Hall, Ithaca, NY 14853-4201 USA
author:
- Anwesh Ray
title: A Refined Lifting Theorem for Supersingular Galois Representations
---
Introduction
============
Statement of the Main Result
----------------------------
Let $p$ be an odd rational prime and ${\mathbb{F}}$ a finite field of characteristic $p$. Let $v_p$ be a $p$-adic valuation on the algebraic closure $\bar{{\mathbb{Q}}}_p$ normalized by $v_p(p)=1$. Let $f$ be a normalized Hecke eigenform and let ${\mathbb{Q}}(f)$ denote the number field generated by the Fourier coefficients of $f$ and $\mathcal{O}_f\subset {\mathbb{Q}}(f)$ its ring of integers. Choose a prime $\mathfrak{p}|p$ of ${\mathbb{Q}}(f)$ and denote by $\iota_{\mathfrak{p}}:{\mathbb{Q}}(f)\rightarrow {\mathbb{Q}}(f)_{\mathfrak{p}}$ the corresponding inclusion. Letting $\mathcal{O}_{f,\mathfrak{p}}$ denote the $\mathfrak{p}$-adic completion of the ring of integers $\mathcal{O}_f$, the $\mathfrak{p}$-adic Galois representation attached to $f$ is denoted by $$\rho_{f, \mathfrak{p}}:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2(\mathcal{O}_{f,\mathfrak{p}}).$$
Let $\bar{\rho}:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2({\mathbb{F}})$ be an odd absolutely irreducible Galois representation which is unramified at all but finitely many primes. The theorem of Khare and Wintenberger (previously referred to as Serre’s conjecture) asserts that $\bar{\rho}$ lifts to a modular $\mathfrak{p}$-adic Galois representation $\rho_{f, \mathfrak{p}}$ for a normalized Hecke eigenform $f$ and a prime $\mathfrak{p}|p$ of ${\mathbb{Q}}(f)$. We examine the case when $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$ is flat and irreducible. In this case, an eigenform $f$ for which $\rho_{f, \mathfrak{p}}$ lifts $\bar{\rho}$ is supersingular at $p$, i.e. $v_p(\iota_{\mathfrak{p}}(a_p))\in {\mathbb{Q}}_{>0}$.
Let $\chi$ denote the $p$-adic cyclotomic character and $\bar{\chi}$ denote its reduction and let $\omega_2$ be a choice of fundamental character of level $2$.
\[main\] Let $\bar{\rho}:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2({\mathbb{F}})$ be a Galois representation for which
1. $\bar{\rho}$ is unramified outside a set of finitely many primes $S$,
2. the local representation $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$ is flat and irreducible,
3. $\det \bar{\rho}=\bar{\chi}$,
and let $\lambda\in {\mathbb{Z}}_{\geq 1}$.
There exists a eigenform $f=\sum_{n\geq 1} a_n q^n\in S_2(\Gamma_1(N))$ such that $(N,p)=1$ and for a certain $\mathfrak{p}|p$ in $\mathcal{O}_f$
1. the residue field $k(\mathfrak{p})\subseteq {\mathbb{F}}$ and the ramification index $e(\mathfrak{p}|p)=1$ and consequently, $$\mathcal{O}_{f,\mathfrak{p}}\subseteq W({\mathbb{F}}).$$
2. With respect to the embedding $\iota_{\mathfrak{p}}:{\mathbb{Q}}(f)\hookrightarrow {\mathbb{Q}}(f)_{\mathfrak{p}}$, $$v_p(\iota_{\mathfrak{p}}(a_p))=\lambda.$$
3. There is a finite set of primes $X$ disjoint from $S$ such that the Galois representation $\rho_{f,\mathfrak{p}}$ is unramified outside $S\cup X$.
4. The characteristic zero representation $\rho_{f,\mathfrak{p}}$ lifts $\bar{\rho}$
$$\begin{tikzpicture}[node distance = 2.0cm, auto]
\node (GSX) {$\text{G}_{{\mathbb{Q}},S\cup X}$};
\node (GS) [right of=GSX] {$\text{G}_{{\mathbb{Q}},S}$};
\node (GL2) [right of=GS]{$\text{GL}_{2}({\mathbb{F}}).$};
\node (GL2W) [above of= GL2]{$\text{GL}_2(W({\mathbb{F}}))$};
\draw[->] (GSX) to node {} (GS);
\draw[->] (GS) to node {$\bar{\rho}$} (GL2);
\draw[->] (GL2W) to node {} (GL2);
\draw[dashed,->] (GSX) to node {$\rho_{f, \mathfrak{p}}$} (GL2W);
\end{tikzpicture}$$
We are led to define the notion of the *slope* of any flat deformation of $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$. This notion for local Galois representations is an adaptation of the global notion of the slope of the $p$-th Fourier coefficient of a normalized Hecke eigencuspform giving rise to a deformation of $\bar{\rho}$. We show that for any fixed $\lambda\in {\mathbb{Z}}_{\geq 1}$, the slope-$\lambda$ flat deformations of $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$ are parametrized by a closed subscheme of the deformation space of all flat deformations of $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$. Using the method of [@KLR], we may lift $\bar{\rho}$ a number of steps so that when restricted to $\text{G}_{{\mathbb{Q}}_p}$ it has slope-$\lambda$. The slope-$\lambda$ deformation condition has the property that at this stage it is preserved by action of the entire tangent space of the functor of flat-deformations. Consequently, at this stage the global deformation problem becomes *balanced* and it becomes possible to lift it to characteristic zero by applying the classical method due to Ramakrishna which involves annihiliating the associated dual Selmer group of the deformation problem. The global deformation comes from a normalized cuspidal Hecke eigenform $f$. The field of Fourier coefficients of $f$ and the set of primes of ramification can be expected to be really large for any choice of $\lambda$.
There has been some interest in understanding the variation in the slope of the Fourier coefficient $a_p(f)$ for cuspidal Hecke eigenforms lifting a fixed residual Galois representation from the point of view of the $p$-adic Langlands program. The reader may contrast the global lifting theorem proved in this manuscript with the local results of Berger-Li-Zhu (see [@berger2004construction]), Buzzard-Gee ( see [@BuzzardGee] and [@BuzzardGee2]) and Ganguli-Ghate (see [@GanguliGhate]) who classify the local representation $\bar{\rho}_{f\restriction G_{{\mathbb{Q}}_p}}$ as $f$ ranges over all cuspidal Hecke eigenforms $f$ of weight $k\geq 2$ such that the slope of the Fourier coefficient $a_p(f)$ lies in various sub-intervals of $(0,\infty)$.
A Sketch of the Approach via Deformation Theory
-----------------------------------------------
We now provide a slightly more detailed sketch of the strategy involved. We lift our residual Galois representation to $\rho:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2(W({\mathbb{F}}))$ such that
1. $\rho$ is unramified at all but finitely many primes,
2. $\text{det}\rho=\chi$,
3. $\rho_{\restriction \text{G}_{{\mathbb{Q}}_p}}$ is flat and irreducible,
4. $\rho_{\restriction \text{G}_{{\mathbb{Q}}_p}}$ satisfies a certain deformation condition which will have the effect of prescribing the valuation of the $p$-th Fourier coefficient of a normalized cuspidal eigenform which whose associated Galois representation may satisfy this condition.
Fix $\lambda\in {\mathbb{Z}}_{\geq 1}$, at the prime $p$, we use a variation of the Ramakrishna’s flat deformation functor from [@Ram1]. If $f=\sum_{n\geq 1} a_n q^n$ is a normalized eigenform and $\mathfrak{p}|p$ is a prime of ${\mathbb{Q}}(f)$ such that $\bar{\rho}_{f, \mathfrak{p}}=\bar{\rho}$ and $\rho_{f, \mathfrak{p}}$ satisfies this local deformation condition at $p$, then $v_p(\iota_{\mathfrak{p}}(a_p))=\lambda$. Let $\mathcal{C}$ be the category of Noetherian coefficient rings over $W({\mathbb{F}})$, for $R\in\mathcal{C}$, its maximal ideal is denoted by $\mathfrak{m}_R$. Let $\mathcal{F}$ denote Ramakrishna’s flat deformation functor defined so that $\mathcal{F}(R)$ consists the deformations $\varrho$ of $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$ to $R$ $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}_p}$};
\node (A) at (3,0){$\text{GL}_2({\mathbb{F}})$};
\node (B) at (3,2){$\text{GL}_2(R)$};
\draw[->] (G) to node [swap]{$\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\varrho$} (B);
\end{tikzpicture}$$ such that $\varrho$ is flat and $\det \varrho=\chi$. It is a consequence of a theorem of Ramakrishna that $\mathcal{F}$ is represented by $\mathcal{R}=W({\mathbb{F}})[|X|]$ (see [@FontaineMazur Theorem B2]).
The functor of points of $\hat{\mathbb{A}}^1=W({\mathbb{F}})[|Y|]$ when evaluated at $R\in \mathcal{C}$ with maximal ideal $\mathfrak{m}_R$ is $$\hat{\mathbb{A}}^1(R)={{\rm{Hom}}}_{\mathcal{C}}(W({\mathbb{F}})[|Y|], R)\simeq \mathfrak{m}_R.$$
The strategy employed in this manuscript involves a number of steps. There is a subfunctor ${\mathcal{F}^{\lambda}}$ of $\mathcal{F}\times_{\text{Spec}W({\mathbb{F}})} \hat{\mathbb{A}}^1$ for which ${\mathcal{F}^{\lambda}}(R)$ consists of pairs $(\rho,U)\in \mathcal{F}(R)\times \mathfrak{m}_R$ with a prescribed slope relation. Let $\pi_1$ and $\pi_2$ be the first and second projections of $\text{Spec}\mathcal{R}\times_{\text{Spec}W({\mathbb{F}})} \hat{\mathbb{A}}^1$, we show that ${\mathcal{F}^{\lambda}}$ is representable by a closed subscheme of $X_{{\mathcal{F}^{\lambda}}}$ and that $\pi_2$ induces an isomorphism $\pi_2:X_{{\mathcal{F}^{\lambda}}}\xrightarrow{\sim} \hat{\mathbb{A}}^1$. At a prime $v\in S\backslash\{p\}$ the local deformation condition $\mathcal{C}_v$ is the condition described in Proposition 1 of [@RaviFM]. The projection $\pi_1$ induces a natural transformation $\pi_1:{\mathcal{F}^{\lambda}}\rightarrow \mathcal{F}$. Let $\mathcal{C}_p^{\lambda}$ be the local condition at $p$ prescribed by $$\mathcal{C}_p^{\lambda}(R):=\text{image}\{\pi_1(R):{\mathcal{F}^{\lambda}}(R)\rightarrow \mathcal{F}(R)\}.$$ We show that $\mathcal{C}_p^{\lambda}$ defines a liftable subfunctor of $\mathcal{F}$. Moreover, the slope relation implies that if $f=\sum_{n\geq 1} a_n q^n$ is a normalized eigenform and $\mathfrak{p}|p$ is a prime of ${\mathbb{Q}}(f)$ such that $\bar{\rho}_{f, \mathfrak{p}}=\bar{\rho}$ and satisfies $\mathcal{F}^{\lambda}$ then $v_p(\iota_{\mathfrak{p}}(a_p))=\lambda$. For a finite length ring $R\in \mathcal{C}$ the precise definition of the slope of a deformation of $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$ involves the analysis of the Fontaine-Laffaille module associated to it.
We apply the method in [@KLR] to lift $\bar{\rho}$ up to $$\rho_{\lambda+2}:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2(W({\mathbb{F}})/p^{\lambda+2})$$ after which the Ramakrishna method is employed to produce a finite set of auxiliary primes and a characteristic zero weight $2$ deformation $$\rho:\text{G}_{{\mathbb{Q}},S\cup X}\rightarrow \text{GL}_2(W({\mathbb{F}}))$$ subject to the local conditions $\mathcal{C}_v$. Since $\rho_{\restriction \text{G}_{{\mathbb{Q}}_p}}\in \mathcal{C}_p^{\lambda}$ this deformation is in particular flat and supersingular.
The Fontaine-Mazur conjecture for absolutely irreducible $\text{G}_{{\mathbb{Q}}}$-representations is fully established in [@KisinFM] for all but one exception in the ordinary case. In particular, rank 2 super singular representations which satisfy the conditions of the Fontaine-Mazur conjecture are modular and consequently $\rho$ coincides with a family of representations with slope $\lambda$ and varying weight whose points of classical weight coincide with representations arising from cusp forms of slope $\lambda$.
The methods in this manuscript do not as of yet carry over to non-integral $\lambda\in {\mathbb{Q}}_{>0}$. The Ramakrishna method as originally conceived applies to produce deformations of $\bar{\rho}$ to $\rho:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2(\mathcal{O})$ where $\mathcal{O}$ is the valuation ring of an unramified extension of ${\mathbb{Q}}_p$.
Acknowledgements
----------------
I am very grateful to my advisor Ravi Ramakrishna for introducing me to his Galois-theoretic lifting method and for some helpful suggestions. I would also like to thank Brian Hwang, Aftab Pande and Stefan Patrikis for some fruitful conversations.
Notation
========
In this section we summarize some basic notation in this manuscript.
- The prime $p$ is an odd prime and ${\mathbb{F}}$ is a finite field of characteristic $p$.
- The $p$-adic valuation $v_p:\bar{{\mathbb{Q}}}_p\rightarrow {\mathbb{Q}}\cup\{\infty\}$ is normalized by $v_p(p)=1$.
- The completion of $\bar{{\mathbb{Q}}}_p$ w.r.t the valuation $v_p$ is denoted by ${\mathbb{C}}_p$. We simply denote the extension of the valuation to ${\mathbb{C}}_p$ by $v_p$ which takes values in $\mathbb{R}\cup \{\infty\}$. The valuation ring is denoted by $\mathcal{O}_{{\mathbb{C}}_p}$.
- The ring of Witt vectors with residue field ${\mathbb{F}}$ is denoted by $W({\mathbb{F}})$ and $K:=W({\mathbb{F}})[1/p]$. The field $K$ is the unramified extension of ${\mathbb{Q}}_p$ with residue field ${\mathbb{F}}$.
- Let $f$ be a normalized cuspidal Hecke eigenform, the field of Fourier coefficients of $f$ is the extension of ${\mathbb{Q}}$ generated by the Fourier coefficients of the $q$-expansion of $f$ and is denoted by ${\mathbb{Q}}(f)$. This is a number field. The ring of integers of ${\mathbb{Q}}(f)$ is denoted by $\mathcal{O}_f$ and its completion at a finite prime $v$ of ${\mathbb{Q}}(f)$ is denoted by $\mathcal{O}_{f,v}$.
- The residual representation $\bar{\rho}$ satisfies the hypotheses of Theorem $\ref{main}$. The adjoint representation $\text{Ad}^0\bar{\rho}$ is the ${\mathbb{F}}[\text{G}_{{\mathbb{Q}}}]$ module of trace zero matrices $$\text{Ad}^0\bar{\rho}=\left\{ { \left( {\begin{array}{cc}
a & b \\
c & -a \\
\end{array} } \right)}\mid a,b,c\in {\mathbb{F}}\right\}$$where $g\in \text{G}_{{\mathbb{Q}}}$ acts through conjugation via $\bar{\rho}$ $$g\cdot X=\bar{\rho}(g)X\bar{\rho}(g)^{-1}.$$
- Associated to a normalized Hecke-eigenform $f$ and a prime $\mathfrak{p}|p$ of the field of Fourier coefficients ${\mathbb{Q}}(f)$ is the associated $\mathfrak{p}$-adic Galois representation $$\rho_{f, \mathfrak{p}}:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2(\mathcal{O}_{f,\mathfrak{p}}).$$
- Let $K=W({\mathbb{F}})[1/p]$ and let $\sigma\in \text{Gal}(K/{\mathbb{Q}})$ denote the Frobenius automorphism of $K$.
- ${\text{Rep}_{K}^f(\text{G}_{{\mathbb{Q}}_p})}$ is the category of finite dimensional continuous $G_{{\mathbb{Q}}_p}$ representations over $K$.
- ${\text{Rep}^f(\text{G}_{{\mathbb{Q}}_p})}$ is the category of finite length $W({\mathbb{F}})[G_{{\mathbb{Q}}_p}]$-modules.
- Let $\mathcal{C}$ be the category of finitely generated noetherian local $W({\mathbb{F}})$ algebras $R$ with maximal ideal $\mathfrak{m}$ and a prescribed mod $\mathfrak{m}$ isomorphism $R/\mathfrak{m}\xrightarrow{\sim} {\mathbb{F}}$. We shall refer to $\mathcal{C}$ as the category of *coefficient rings* over $W({\mathbb{F}})$. A map $f:(R_1, \mathfrak{m}_1)\rightarrow (R_2, \mathfrak{m}_2)$ in $\mathcal{C}$ is a map of local rings compatible with reduction isomorphisms. Let ${\mathcal{C}^f}$ be the full subcategory of finite length algebras.
- Let $Z$ be a finite set of primes containing $S$ and $M$ an ${\mathbb{F}}[\text{G}_{{\mathbb{Q}},Z}]$-module, for $i=0,1,2$, the $\Sha^i$-group is the kernel of the restriction map $$\Sha_Z^i(M):=\text{ker}\{H^1(G_{{\mathbb{Q}},Z},M)\rightarrow \bigoplus_{l\in Z} H^1(G_{{\mathbb{Q}}_l},M)\}.$$
- The functor of crystalline weight $2$ deformations of $\bar{\rho}$ is a functor $$\mathcal{F}:\mathcal{C}\rightarrow \text{Sets}$$on $\mathcal{C}$ by $\mathcal{F}(R)$ consists of all deformations $\rho:\text{G}_{{\mathbb{Q}}_p}\rightarrow \text{GL}_2(R)$ of $\bar{\rho}$ that are crystalline and with determinant $\det\rho=\chi$.
Recollections on the Fontaine-Laffaille Functor
===============================================
In this section, we review some properties of the Fontaine-Laffaille functor and make preparations to describe a fixed slope subfunctor of the functor of flat deformations of $\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$. The standard references for this section are [@FontaineLaffaille] and [@patrikisphd].
Let $K=W({\mathbb{F}})[1\slash p]$ and $\sigma\in \operatorname{Gal}(K/{\mathbb{Q}}_p)$ the Frobenius element and let ${\mathbb{C}}_p$ denote the completion of $\bar{{\mathbb{Q}}}_p$ w.r.t the valuation $v_p$. We choose a distinguished element $$\varepsilon=(\varepsilon_i)\in {\mathcal{O}^{\flat}_{\mathbb{C}_p}}:=\varprojlim_{x\mapsto x^p}\mathcal{O}_{{\mathbb{C}}_p}/p$$ for which $\varepsilon_0\neq 1$ is a $p$-th root of unity. For $x\in {\mathcal{O}^{\flat}_{\mathbb{C}_p}}$ let $[x]$ denote the Teichmüller lift of $x$ in $W({\mathcal{O}^{\flat}_{\mathbb{C}_p}})$. The continuous Galois-equivariant ring homomorphism $\theta: W({\mathcal{O}^{\flat}_{\mathbb{C}_p}})\rightarrow {\mathcal{O}^{\flat}_{\mathbb{C}_p}}$, defined by $\theta\left(\sum_i [x^{(i)}] p^i\right):=\sum_i x^{(i)}_0 p^i$ is open and surjective, the kernel of which is a principal ideal generated by a special element $\xi\in W({\mathcal{O}^{\flat}_{\mathbb{C}_p}})$ with some key properties stated in [@ConradBrinon Proposition 4.4.3]. Let $\text{A}_{cris}$ denote the divided power envelope of $W({\mathcal{O}^{\flat}_{\mathbb{C}_p}})$ with respect to the ideal $\ker \theta$, in other words, $\text{A}_{cris}=W({\mathcal{O}^{\flat}_{\mathbb{C}_p}})[\xi^m/m!]_{m\geq 1}$. The $p$-adic completion ${\text{B}_{cris}}^+$ is a local ring with residue field ${\mathbb{C}}_p$ and its valuation and the crystalline period ring ${\text{B}_{cris}}$ is then defined to be the ring obtained on inverting the period $t:=\log[\varepsilon]\in \text{A}_{cris}$. The period ring ${\text{B}_{cris}}:={\text{B}_{cris}}^+[1/t]$ has an induced Galois stable filtration of ${\mathbb{Q}}_p$ vector spaces given by $F^i{\text{B}_{cris}}:=t^i \text{A}_{cris}$.
A filtered $\varphi$-module $M$ is a $K$ vector space equipped with a semilinear bijective map $\varphi$ with respect to $\sigma$ and a decreasing filtration $\{F^i M\}$. For this filtration, $F^i M=M$ for $i\ll 0$ and $F^i M=0$ for $i\gg 0$.
Fontaine’s crystalline period functor ${\text{D}_{cris}}$ from ${\text{Rep}_{K}^f(\text{G}_{{\mathbb{Q}}_p})}$ to the category of finitely filtered ${\varphi}$-modules is defined as follows, for $V\in {\text{Rep}_{K}^f(\text{G}_{{\mathbb{Q}}_p})}$ $${\text{D}_{cris}}(V):=\left(V\otimes_{{\mathbb{Q}}_p} {\text{B}_{cris}}\right)^{G_{{\mathbb{Q}}_p}}$$ with filtration $$F^i {\text{D}_{cris}}(V):=\left(V\otimes_{{\mathbb{Q}}_p} F^i{\text{B}_{cris}}\right)^{G_{{\mathbb{Q}}_p}}.$$ The representation $V$ is *crystalline* if $$\dim_K V=\dim_K \text{D}_{\text{cris}}(V).$$A filtered ${\varphi}$-module $V$ is weakly-admissible if and only if there exists a $W({\mathbb{F}})$-lattice $M\subset V$ with induced filtration $F^i M=M\cap F^i V$ for which
1. ${\varphi}(F^iM)\subseteq p^i M$
2. $\sum_{i} p^{-i} \varphi (F^i M)=M$.
Theorem A of [@ColmezFontaine] asserts that to a crystalline representation $\rho$, the ${\varphi}$-module ${\text{D}_{cris}}(V_{\rho})$ is weakly-admissible and all weakly-admissible $\varphi$-modules arise this way.
A filtered Dieudonné $W({\mathbb{F}})$-module also known as a Fontaine-Laffaille Module it is furnished with a decreasing, exhaustive, separated filtration of $W({\mathbb{F}})$-submodules $\{F^i M\}$ and for each integer $i$ a $\sigma$-semilinear map $$\varphi^i=\varphi_M^i:F^i M\rightarrow M$$ such that
1. the following compatibility relation is satisfied $\varphi^{i+1}=p \varphi^i$,
2. $\sum_i \varphi^i(F^i M)=M$.
Denote by ${\varphi}={\varphi}^0:M\rightarrow M$.
A map $f:(M,{\varphi}^i_M)\rightarrow (N, {\varphi}^i_N)$ of Fontaine-Laffaille modules is a $W({\mathbb{F}})$-module map such that $f(F^i M)\subseteq F^i N$ and ${\varphi}^i_N\circ f_{\restriction F^i M}=f\circ {\varphi}^i_M$.
Let $\text{MF}_{tor}^{f}$ denote the category of finite length Fontaine-Laffaille modules $M$ with morphisms satisfying the conditions alluded to above. For $a<b$ let $\text{MF}_{tor}^{f,[a,b]}$ let the full subcategory of $\text{MF}_{tor}^{f}$ whose underlying modules $M$ satisfy $F^a M=M$ and $F^b M=0$.
Let ${\text{Rep}^f(\text{G}_{{\mathbb{Q}}_p})}$ is the category of continuous $W({\mathbb{F}})[G_{{\mathbb{Q}}_p}]$ modules that are finite-length ${\mathbb{Z}}_p$-modules. Fontaine and Laffaille in [@FontaineLaffaille] showed that $\text{MF}_{tor}^{f}$ and $\text{MF}_{tor}^{f,[a,b]}$ are abelian categories and defined a contravariant functor $$\text{U}_S:\text{MF}_{tor}^{f,[0,p]}\rightarrow {\text{Rep}^f(\text{G}_{{\mathbb{Q}}_p})}.$$
\[dualFM\] Let $M\in \text{MF}_{tor}^{[a,b]}$, its dual $M^*\in \text{MF}_{tor}^{[1-b,1-a]}$ is the module $$M^*={{\rm{Hom}}}_{W({\mathbb{F}})}(M, K/W({\mathbb{F}}))$$ with filtration $$F^i(M^*)={{\rm{Hom}}}_{W({\mathbb{F}})}(M/F^{1-i}M, K/W({\mathbb{F}})).$$ We proceed to describe the maps $\varphi^i_{M^*}:F^i(M^*)\rightarrow M^*$. Let $f\in F^i(M^*)$ and $m\in M$. Since $M=\sum_{j} \varphi^j_M(F^j M)$, the element $m$ may be represented as a sum $m=\sum_j \varphi^j_M(m_j)$. To prescribe $\varphi^i_{M^*}(f)(m)$ it suffices to define $\varphi^i_{M^*}(f)(\varphi_M^j(m_j))$. These are taken as follows $$\varphi^i_{M^*}(f)(\varphi^j_M(m_j))=\begin{cases}0 \text{ for }j>-i;\\
f(p^{-i-j}m_j) \text{ for }j\leq -i.\\
\end{cases}$$
The reader may check that the maps $\varphi_{M^*}^i$ are well defined. We now describe Tate-twists.
For an object $M\in \text{MF}_{tor}^{f,[a,b]}$ and $m\in {\mathbb{Z}}$, the $m$-fold Tate twist of $M$ is the module $M(m)\in \text{MF}_{tor}^{f,[a+m,b+m]}$ whose underlying module is $M$ and $F^i(M(m))=F^{i-m}(M)$ and $\varphi^i_M=\varphi^{i-m}_{M(m)}$.
We proceed to describe the covariant adaptations of the Fontaine-Laffaille functor which satisfy a suitable compatibility relation with the tensor product. This makes these functors more suitable to work with.
We let $$\text{T}:\text{MF}_{tor}^{[2-p,1]}\rightarrow {\text{Rep}^f(\text{G}_{{\mathbb{Q}}_p})}$$ defined by $\text{T}(M)=\text{U}_S(M^*)$. We define $$\text{T}_1:\text{MF}_{tor}^{[3-p,2]}\rightarrow {\text{Rep}^f(\text{G}_{{\mathbb{Q}}_p})}$$ by $\text{T}_1(M):=\text{U}_S(M^*(1))$.
We now collect a few facts about the functors $\text{T}$ and $\text{T}_1$.
The functor $\text{T}$ is full and faithful and as a consequence, so is $\text{T}_1$.
For each object $M$ of ${\text{MF}_{tor}^{[2-p,1]}}$, $$\text{length}_{W({\mathbb{F}})}M= \text{length}_{W({\mathbb{F}})}\text{T}(M)=\text{length}_{W({\mathbb{F}})}\text{T}_1(M).$$
Let $\mathcal{C}$ be the category of finitely generated noetherian local $W({\mathbb{F}})$-algebras $R$ with maximal ideal $\mathfrak{m}$ and a prescribed mod $\mathfrak{m}$ isomorphism $R/\mathfrak{m}\xrightarrow{\sim} {\mathbb{F}}$ and refer to $\mathcal{C}$ as the category of *coefficient rings* over $W({\mathbb{F}})$. A map $f:(R_1, \mathfrak{m}_1)\rightarrow (R_2, \mathfrak{m}_2)$ in $\mathcal{C}$ is a map of local rings compatible with reduction isomorphisms. Let ${\mathcal{C}^f}$ be the full subcategory of finite-length algebras.
For $R\in \mathcal{C}$, a deformation $\rho:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2(R)$ of $\bar{\rho}$ is said to be *crystalline* if for every finite-length quotient $R'$ of $R$ in $\mathcal{C}$ the pushforward $\rho\otimes_R R'$ lies in the essential image of $T$.
The residual representation $\bar{\rho}:\text{G}_{{\mathbb{Q}}_p}\rightarrow \text{Aut}_{{\mathbb{F}}_p}(V_{\bar{\rho}})$ is flat, and consequently, there exists $M_0\in \text{MF}_{tor}^{[0,2]}$ such that $\text{T}_1(M_0)\simeq V_{\bar{\rho}}$. This is shown for instance in [@Ram1].
Define the functor $$\mathcal{F}:\mathcal{C}\rightarrow \text{Sets}$$on $\mathcal{C}$ by $\mathcal{F}(R)$ consists of all deformations $\rho:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2(R)$ of $\bar{\rho}$ that are flat and with determinant $\det\rho=\chi$.
The restriction of this functor to $\mathcal{C}^f$ has the following description: an object of $\mathcal{F}(R)$ for $R\in {\mathcal{C}^f}$ can be identified with $\text{T}_1(M)$ for $M\in \text{MF}_{tor}^f$. In fact, $M\in \text{MF}_{tor}^{f,[0,2]}$ and as $\text{T}_1$ is full and faithful, the $R$ action on the free $R$-module $\text{T}_1(M)$ carries over to a faithful $R$ action on $M$. In greater detail, if $r\in R$, the endomorphism induced by multiplication by $r$ on $\text{T}_1(M)$ is an $R[\text{G}_{{\mathbb{Q}}_p}]$-module endomorphism. Also since $\text{T}_1$ is full and faithful, this multiplication by $r$ endomorphism concides with an endomorphism $$\text{T}_1^{-1}(r):M\rightarrow M$$ in ${\text{MF}_{tor}^{[2-p,1]}}$, i.e.
- $\text{T}_1^{-1}(r):M\rightarrow M$ is a $W({\mathbb{F}})$-module endomorphism
- $\text{T}_1^{-1}(r)(M^i)\subseteq M^i$
- ${\varphi}^i\circ \text{T}_1^{-1}(r)_{\restriction M^i}=\text{T}_1^{-1}(r)\circ {\varphi}^i.$
Denote ${\varphi}^0$ by ${\varphi}$ and $W({\mathbb{F}})$-algebra of ${\varphi}$ equivariant $W({\mathbb{F}})$-module homomorphisms of $M$ by $\text{End}_{{\varphi}}(M)$. As a consequence of the faithfulness of $T_1$, we obtain a ring homomorphism $$\text{T}_1^{-1}: R\rightarrow \text{End}_{{\varphi}}(M)$$ taking $r\in R$ to the endomorphism $\text{T}_1^{-1}(r)$. We summarize these observations by saying that $M$ is an $R$-module in the category $\text{MF}_{tor}^f$. In particular, it is an $R[{\varphi}]$-module and that the $R$ action on $M$ is uniquely determined by that on $\text{T}_1(M)$ and preserves the filtration on $M$. In fact the module $M$ can be canonically identified with a free $R$-module of rank $2$. The following result follows from Lemma 4.2 in [@patrikisphd] where the analogous result for the functor $\text{T}$ is stated. That the same implication is true for $\text{T}_1$ is because $\text{T}_1(M)=\text{T}(M(-1))$.
\[freemodprop\] Let $R\in \mathcal{C}^f$ and $M\in \text{MF}_{tor}^{[0,2]}$ with $\text{T}_1(M)\in \mathcal{F}(R)$. With respect to the $R$-module structure on $M$ induced from the $R$-module structure on the rank $2$ free $R$-module $\text{T}_1(M)$, $M$ is also a free $R$-module of rank $2$ and ${\varphi}:M\rightarrow M$ is an $R$ linear endomorphism.
Let $R\in \mathcal{C}_f$ and $M\in \text{MF}_{tor}^{[0,2]}$ with $\text{T}_1(M)\in \mathcal{F}(R)$. Note that by Proposition $\ref{freemodprop}$, the $R$-module $M$ is free of rank $2$. The $R$-linear operator ${\varphi}:={\varphi}^0$ thus may be described by a $2\times 2$ matrix with entries in $R$. The trace of ${\varphi}$ is the sum of the diagonal elements of this matrix. We denote the trace by ${\text{tr}\varphi}_{\restriction M}$, or sometimes to ease notation when working with Galois representations we use ${\text{tr}\varphi}_{\restriction \text{T}_1(M)}$ interchangeably.
Let $R\in {\mathcal{C}^f}$, we use $R$ to refer to an object of $\text{MF}_{tor}^f$ which is concentrated in degree zero and $\varphi^0$ is the identity. Then $\text{T}(R)=R$ with trivial Galois action. Let $\alpha\in \text{Aut}_{W({\mathbb{F}})}(R)$, we denote by $R(\alpha)$ the object concentrated at $0$ and with ${\varphi}^0=\alpha$. Then $\text{T}(R(\alpha))$ is the $R$-module of rank $1$ with unramified $\text{G}_{{\mathbb{Q}}_p}$-action by $\alpha^{-1}$ (see Lemma 4.6 [@patrikisphd]).
Let $R\in \mathcal{C}^f$ and $A,B\in \text{MF}_{tor}^f$ be $R$-modules such that the $R$-module action is compatible with the $W({\mathbb{F}})$-action, the filtrations and $\varphi^i$-maps. The tensor product $A\otimes_R B$ has a natural filtration $F^m(A\otimes_R B)=\bigoplus_{i+j=m} F^i(A)\otimes_R F^j (B)$ and $\varphi^m_{A\otimes_R B}=\bigoplus_{i+j=m} \varphi^i_A \otimes_R \varphi^j_B$.
\[Tensor\] Let $A\in \text{MF}_{tor}^{f,[0,2]}$ and $B\in \text{MF}_{tor}^{f,[0,1]}$ and $R\in \mathcal{C}^f$ such that $T_1(A)$ and $T(B)$ have the structure of $R[\text{G}_{{\mathbb{Q}}_p}]$-modules compatible with their $W({\mathbb{F}})[\text{G}_{{\mathbb{Q}}_p}]$-module structure. Then, there is an isomorphism of $R[\text{G}_{{\mathbb{Q}}_p}]$-modules $\text{T}_1(A)\otimes_{R} \text{T}(B)\xrightarrow{\sim} \text{T}_1(A\otimes_R B)$.
This statement follows from Lemma 4.3 of [@patrikisphd].
Let $g:R_1\rightarrow R_2$ be a map of coefficient rings in $\mathcal{C}^f$ and $\text{T}_1(M_1)\in \mathcal{F}(R_1)$. The push-forward of $\text{T}_1(M_1)$ to $\mathcal{F}(R_2)$ is $g_* \text{T}_1(M_1):=\text{T}_1(M_1)\otimes_{R_1} R_2$. It follows from Lemma $\ref{Tensor}$ that there is an $R_2[\text{G}_{{\mathbb{Q}}_p}]$-module isomorphism $\text{T}_1(M_1)\otimes_{R_1} R_2\simeq \text{T}_1(M_1\otimes_{R_1} R_2)$ and as a result, we may identify $g_*\text{T}_1(M_1)$ with $\text{T}_1(M_2)$ where $M_2:=M_1\otimes_{R_1} R_2$. The following relation comes as a consequence of this identification.
With respect to notation introduced above, for $\varrho\in \mathcal{F}(R_1)$, the following relation holds $$\label{relation}{\text{tr}\varphi}_{\restriction g_*\varrho}=g\left({\text{tr}\varphi}_{\restriction \varrho}\right).$$
The Fixed-Slope Functor
=======================
We recall that $K=W({\mathbb{F}})[1/p]$ with Frobenius $\sigma$ and $N>0$ coprime to $p$. Let $f\in S_2(\Gamma_1(N))$ be a normalized eigenform with nebentype character $\psi$. Let $\mathfrak{p}|p$ in the number field generated by the Fourier coefficients ${\mathbb{Q}}(f)$ of $f$. The $\mathfrak{p}$-adic Galois representation $\rho_{f,\mathfrak{p}}:\text{G}_{{\mathbb{Q}}}\rightarrow \text{GL}_2(\mathcal{O}_{f,\mathfrak{p}})$ where $\mathcal{O}_{f,\mathfrak{p}}$ is the valuation ring of ${\mathbb{Q}}(f)_{\mathfrak{p}}$ with residue field $k(\mathfrak{p})$. Assume that the following conditions are satisfied
1. $\rho_{f,\mathfrak{p}}$ is supersingular, i.e $f$ is supersingular at $\mathfrak{p}$,
2. ${\mathbb{Q}}(f)_{\mathfrak{p}}$ embeds in $K$, or in other words, the residue field $k(\mathfrak{p})\subseteq {\mathbb{F}}$ and the ramification index of $e(\mathfrak{p}|p)=1$,
3. $\rho_{f,\mathfrak{p}}$ lifts $\bar{\rho}$.
In addition to these conditions, since $M$ is coprime to $p$ it follows that $\rho_{f,\mathfrak{p}}$ satisifes the flat deformation condition. We have not made any assumptions on the slope of $\iota_{\mathfrak{p}}(a_p(f))$ except that it is positive. Let $V_f$ denote the $2$-dimensional $\mathfrak{p}$-adic $G_{{\mathbb{Q}}_p}$-representation induced by $f$. This representation is flat and in particular, crystalline and hence, $$\dim_{K} {\text{D}_{cris}}(V_f)=\dim_{K} V_f=2.$$ The following theorem is alluded to in [@BuzzardGee] and follows from the main theorem of [@saito].
\[Saito\] The characteristic polynomial of ${\varphi}$ on the $2$ dimensional $K$ vector space ${\text{D}_{cris}}(V_f)$ is $$ch(X)=X^2-\iota_{\mathfrak{p}}(a_p) X+\psi(p)p^{k-1}.$$
In order to describe a *fixed slope* deformation condition we proceed to describe the notion of the *slope* of any flat deformation of $\bar{\rho}_{\restriction\text{G}_{{\mathbb{Q}}_p}}$ to any Artinian coefficient ring $R$ $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}_p}$};
\node (A) at (3,0){$\text{GL}_2({\mathbb{F}})$.};
\node (B) at (3,2){$\text{GL}_2(R)$};
\draw[->] (G) to node [swap]{$\bar{\rho}_{\restriction \text{G}_{{\mathbb{Q}}_p}}$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\varrho$} (B);
\end{tikzpicture}$$ The following result gives us control on the $p$-adic valuation of $\iota_{\mathfrak{p}}(a_p(f))$ via torsion Fontaine-Laffaille theory.
\[comparisonprop\] Let $\rho:G_{{\mathbb{Q}}}\rightarrow \text{GL}_2(W({\mathbb{F}})/p^m)$ be a flat deformation of $\bar{\rho}$ such that there exists a cuspidal Hecke eigenform $f\in S_2(\Gamma_1(M))$ of level $M$ coprime to $p$ such that $\rho_{f,\mathfrak{p}}$ lifts $\rho$ $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}_p}$};
\node (A) at (3,0){$\text{GL}_2(W({\mathbb{F}})/p^m)$.};
\node (B) at (3,2){$\text{GL}_2(W({\mathbb{F}}))$};
\draw[->] (G) to node [swap]{$\rho$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\rho_{f,\mathfrak{p}}$} (B);
\end{tikzpicture}$$ Then ${\text{tr}\varphi}_{\restriction \rho}=\iota_{\mathfrak{p}}(a_p(f))\mod{p^m}$.
Let $V_f=V_{f,\mathfrak{p}}$ be the underlying vector space on which $\text{G}_{{\mathbb{Q}}_p}$ acts via $\rho_{f,\mathfrak{p}}$ and let $L\simeq W({\mathbb{F}})\oplus W({\mathbb{F}})$ be the Galois stable submodule of $V_f$. The module $\mathcal{M}:=\varprojlim_n \text{T}_1^{-1}(L/p^n L)$ is equipped with a $\varphi$-action which is by definition the inverse limit of the $\varphi^0$-actions on $\text{T}_1^{-1}(L/p^n L)$ for $n\geq 1$. It follows from Proposition $\ref{freemodprop}$ that $\mathcal{M}$ is a free module of rank $2$ over $W({\mathbb{F}})$. Following [@hattori2018integral] we denote by $\hat{\text{T}}^*_{cris}(\mathcal{M}):={{\rm{Hom}}}_{MF}(\mathcal{M},\text{A}_{cris})$ with its induced $\text{G}_{{\mathbb{Q}}_p}$-action. If we show that there is a $\text{G}_{{\mathbb{Q}}_p}$-equivariant isomorphism $$\label{technicaliso}L\otimes_{W({\mathbb{F}})}K\simeq {{\rm{Hom}}}_{W({\mathbb{F}})}(\hat{\text{T}}^*_{cris}(\mathcal{M}),W({\mathbb{F}})(1))\otimes_{W({\mathbb{F}})}K$$ it follows from Theorem 2.11 of [@hattori2018integral] that there is a $\varphi$-equivariant isomorphism $\text{D}_{cris}(V_f)\simeq \mathcal{M}\otimes_{W({\mathbb{F}})} K$. In greater detail, letting $V:=\hat{\text{T}}_{cris}^*(\mathcal{M})\otimes_{W({\mathbb{F}})} K$, Theorem 2.11 of [@hattori2018integral] states that $\text{D}_{cris}(V^*)\simeq \mathcal{M}\otimes_{W({\mathbb{F}})} K$ as $\varphi$-modules. On the other hand, it would follow from $\ref{technicaliso}$ that $$\begin{split}
V^*=&{{\rm{Hom}}}_{K}(V, K(1))\\
\simeq &{{\rm{Hom}}}_{W({\mathbb{F}})}(\hat{\text{T}}_{cris}^*(\mathcal{M}),W({\mathbb{F}})(1))\otimes_{W({\mathbb{F}})}K\\
\simeq& L\otimes_{W({\mathbb{F}})}K\simeq V_f.
\end{split}$$ The statement of the proposition follows from this and Theorem $\ref{Saito}$, and all that remains to be proved is the isomorphism $\ref{technicaliso}$.
Following [@hattori2018integral] $\text{U}_S\simeq \text{T}_{cris}^*$ where $\text{T}_{cris}^*$ is prescribed by $$\text{T}_{cris}^*(M)={{\rm{Hom}}}_{MF}(M,\varinjlim_{n}\text{A}_{cris}/p^n),$$ and thus, $$\text{T}_1(M)={{\rm{Hom}}}_{W({\mathbb{F}})}(\text{T}_{cris}^*(M),K/W({\mathbb{F}})(1)).$$We now proceed to show $\ref{technicaliso}$. We observe that $$\begin{split}
&{{\rm{Hom}}}_{W({\mathbb{F}})}(\hat{\text{T}}^*_{cris}(\mathcal{M}),W({\mathbb{F}})(1))\\
=&{{\rm{Hom}}}_{W({\mathbb{F}})}({{\rm{Hom}}}_{MF}(\mathcal{M},\text{A}_{cris}),W({\mathbb{F}})(1))\\
\simeq &{{\rm{Hom}}}_{W({\mathbb{F}})}({{\rm{Hom}}}_{MF}(\mathcal{M},\text{A}_{cris})\otimes_{W({\mathbb{F}})}K/W({\mathbb{F}}),K/W({\mathbb{F}})(1))\\
\simeq &{{\rm{Hom}}}_{W({\mathbb{F}})}(\varinjlim_{n}{{\rm{Hom}}}_{MF}(\mathcal{M},\text{A}_{cris})/p^n,K/W({\mathbb{F}})(1)).\\
\end{split}$$ On the other hand, $$\begin{split}
L\simeq & \varprojlim_m\text{T}_1(\text{T}_1^{-1}(L_m))\\
\simeq &\varprojlim_m{{\rm{Hom}}}_{W({\mathbb{F}})}({{\rm{Hom}}}_{MF}(\text{T}_1^{-1}(L_m),\varinjlim_{n}\text{A}_{cris}/p^n),K/W({\mathbb{F}})(1))\\
\simeq & {{\rm{Hom}}}_{W({\mathbb{F}})}(\varinjlim_m{{\rm{Hom}}}_{MF}(\text{T}_1^{-1}(L_m),\varinjlim_{n}\text{A}_{cris}/p^n),K/W({\mathbb{F}})(1))\\
\simeq & {{\rm{Hom}}}_{W({\mathbb{F}})}(\varinjlim_m{{\rm{Hom}}}_{MF}(\mathcal{M}/p^m,\varinjlim_{n}\text{A}_{cris}/p^n),K/W({\mathbb{F}})(1))\\
\simeq & {{\rm{Hom}}}_{W({\mathbb{F}})}(\varinjlim_m{{\rm{Hom}}}_{MF}(\mathcal{M},\text{A}_{cris}/p^m),K/W({\mathbb{F}})(1)).
\end{split}$$ The $\text{G}_{{\mathbb{Q}}_p}$-equivariant inclusion $$\varinjlim_{n}{{\rm{Hom}}}_{MF}(\mathcal{M},\text{A}_{cris})/p^n
\hookrightarrow \varinjlim_n{{\rm{Hom}}}_{MF}(\mathcal{M},\text{A}_{cris}/p^n)$$ induces a $\text{G}_{{\mathbb{Q}}_p}$-equivariant surjection $$L\otimes_{W({\mathbb{F}})}K\twoheadrightarrow {{\rm{Hom}}}_{W({\mathbb{F}})}(\hat{\text{T}}^*_{cris}(\mathcal{M}),W({\mathbb{F}})(1))\otimes_{W({\mathbb{F}})}K\simeq V^*.$$ On the other hand, we observe that $$\dim_K L\otimes_{W({\mathbb{F}})}K=\text{rank}_{W({\mathbb{F}})} L=2$$ and $$\dim_K V^*\geq \dim_K \text{D}_{cris}(V^*)=\dim_K \mathcal{M}\otimes_{W({\mathbb{F}})}K=2,$$ and thus this surjection must be an isomorphism.
For $R\in \mathcal{C}_f$ with maximal ideal $\mathfrak{m}_R$ and $\lambda\in {\mathbb{Z}}_{\geq 1}$ we define $${\mathcal{F}^{\lambda}}(R)\subseteq (\mathcal{F}\times \hat{\mathbb{A}}^1)(R)= \mathcal{F}(R)\times \mathfrak{m}_R$$ the set of pairs $(\rho,U)$ such that $\rho$ lifts $\bar{\rho}_{|G_{{\mathbb{Q}}_p}}$ $${\text{tr}\varphi}_{\restriction \rho}=p^{\lambda}(1+U).$$That $\mathcal{F}^{\lambda}$ is a functor follows from equation $\ref{relation}$.
We recall that it is a consequence of a theorem of Ramakrishna that $\mathcal{F}$ is pro-represented by $\mathcal{R}=W({\mathbb{F}})[|X|]$ (see [@FontaineMazur Theorem B2]). The functor $\mathcal{F}\times_{\text{Spec}W({\mathbb{F}})} \hat{\mathbb{A}}^1$ is represented by the fibered product $$\text{Spec}\mathcal{R}\times_{\text{Spec}W({\mathbb{F}})}\hat{\mathbb{A}}^1\simeq \text{Spec}W({\mathbb{F}})[|X,Y|].$$ Let $\mathcal{I}$ be an ideal in $\mathcal{R}=W({\mathbb{F}})[|X|]$ such that $\mathcal{R}\slash \mathcal{I}$ is Artinian. Let $\rho_{\mathcal{R}}$ be the universal representation representing $\mathcal{F}$ and let $\rho_{\mathcal{R}\slash \mathcal{I}}$ be the associated mod $\mathcal{I}$. Let $\mathcal{I}'$ be an ideal which contains $\mathcal{I}$, then by equation $\ref{relation}$, $${\text{tr}\varphi}_{|\rho_{\mathcal{R}\slash \mathcal{I}'}}={\text{tr}\varphi}_{|\rho_{\mathcal{R}\slash \mathcal{I}}}\mod \mathcal{I'},$$ let $\Phi(X)\in \mathcal{R}=W({\mathbb{F}})[|X|]$ be the limit over ideals $\mathcal{I}$ for which $\mathcal{R}\slash \mathcal{I}$ is Artinian $$\Phi(X):=\varprojlim_{\mathcal{I}} {\text{tr}\varphi}_{|\rho_{\mathcal{R}\slash \mathcal{I}}}.$$
The functor $\mathcal{F}^{\lambda}$ is represented by $$X_{\mathcal{F}^{\lambda}}=\text{Spec}\tilde{\mathcal{R}}^{\lambda}=\text{Spec}\left(\frac{W({\mathbb{F}})[|X,Y|]}{(\Phi(X)-p^{\lambda}(1+Y))}\right).$$ Let $\mathfrak{m}_{\mathcal{R}}$ be the maximal ideal of $\mathcal{R}=W({\mathbb{F}})[|X|]$, we observe that $${\text{tr}\varphi}_{|\bar{\rho}}=\Phi(X)\:\text{mod}\: \mathfrak{m}_{\mathcal{R}}.$$ It follows from the theorem of Khare and Wintenberger that $\bar{\rho}=\bar{\rho}_{g,\mathfrak{q}}$ where $g$ is a cuspidal Hecke eigenform and $\mathfrak{q}|p$ is a prime in the field of Fourier coefficients ${\mathbb{Q}}(g)$. Since $\bar{\rho}_{\restriction G_{{\mathbb{Q}}_p}}$ is irreducible, $g$ is supersingular at $\mathfrak{q}$, i.e. $p|\iota_{\mathfrak{q}}(a_p(g))$. It follows from Proposition $\ref{comparisonprop}$ the trace of the residual representation ${\text{tr}\varphi}_{|\bar{\rho}}=0$ and as a consequence $\Phi(X)\in \mathfrak{m}_{\mathcal{R}}$. The projection to the second factor is a natural transformation $\pi_2:\mathcal{F}\times_{\text{Spec} W({\mathbb{F}})}\hat{\mathbb{A}}^1\rightarrow \hat{\mathbb{A}}^1$ and restricts to a natural transformation $\pi_2: \mathcal{F}^{\lambda}\rightarrow \hat{\mathbb{A}}^1$ which induces the map of $W({\mathbb{F}})$-algebras $$\pi_2^*:W({\mathbb{F}})[|Y|]\rightarrow \tilde{\mathcal{R}}^{\lambda}=\frac{W({\mathbb{F}})[|X,Y|]}{(\Phi(X)-p^{\lambda}(1+Y))}$$ for which $\pi_2^*(Y)=Y\:\text{mod}(\Phi(X)-p^{\lambda}(1+Y))$.
\[pi2\] The map $\pi_2^*$ is an isomorphism of $W({\mathbb{F}})$-algebras and thus $\mathcal{F}^{\lambda}$ is representable by a power series ring in one variable over $W({\mathbb{F}})$.
It follows from Proposition $\ref{comparisonprop}$ the trace of the residual representation ${\text{tr}\varphi}_{|\bar{\rho}}=0$ and as a consequence $\Phi(X)\in \mathfrak{m}_{\mathcal{R}}$. The map $\pi_2^*$ is an isomorphism provided the coefficient of $X$ in the power series expansion of $\Phi(X)$ is a unit. In greater detail, suppose that the coefficient of $X$ in the power series expansion of $\Phi(X)$ is a unit, then, $\pi_2^*$ is injective since $\Phi(X)\neq 0$. On the other hand since the coefficient of $X$ is a unit, we may write $X$ as a power series with coefficients in $Y$. In other words, $X$ will lie in the image of $\pi_2^*$ and therefore the $\pi_2^*$ will be surjective as well.
We proceed to prove that the coefficient of $X$ is a unit. Let $M_0\in \text{MF}_{tor}^{f,[0,2]}$ be such that $\text{T}_1(M_0)\simeq \bar{\rho}_{|G_{{\mathbb{Q}}_p}}$. Choose a basis $\{e,k\}$ of $M_0$ such that $k$ spans $F^1 M_0$. The Fontaine-Laffaille module $M_0$ is characterized by a single matrix $\left( {\begin{array}{c|c}
a & c \\
b & d \\
\end{array} } \right)\in \text{M}_2({\mathbb{F}})$ where $$\varphi^0(e)=ae+bk\text{ and }\varphi^1(k)=ce+dk.$$The relation $\varphi^0_{\restriction F^1 M_0}=p\varphi^1=0$ implies that $\varphi^0(f)=0$. The trace $a={\text{tr}\varphi}_{|\bar{\rho}_{|G_{{\mathbb{Q}}_p}}}=0$ since $\bar{\rho}_{|G_{{\mathbb{Q}}_p}}$ is supersingular. Let $\tilde{M}_0$ be a Fontaine-Laffaille module of dimension $4$ over ${\mathbb{F}}$ provided with the structure of a free ${\mathbb{F}}[\epsilon]$-module which we proceed to describe. We let $\tilde{M}_0={\mathbb{F}}[\epsilon] \cdot\tilde{e}\oplus {\mathbb{F}}[\epsilon] \cdot\tilde{k}$ with filtration and $\varphi^i$ defined as follows:
- $F^0\tilde{M}_0:=\tilde{M}_0$, $F^1\tilde{M}_0:={\mathbb{F}}[\epsilon]\cdot \tilde{k}$ and $F^2\tilde{M}_0:=0$,
- the maps $\varphi^j$ for $j=0,1$ are ${\mathbb{F}}[\epsilon]$-module maps, which similar to $M_0$ is characterized by a matrix $\left( {\begin{array}{c|c}
\tilde{a} & \tilde{c} \\
\tilde{b} & \tilde{d} \\
\end{array} } \right)\in \text{M}_2({\mathbb{F}}[\epsilon])$ where $$\varphi^0(\tilde{e})=\tilde{a}\cdot \tilde{e}+\tilde{b}\cdot \tilde{k} \text{ and }\varphi^1(\tilde{k})=\tilde{c}\cdot \tilde{e}+\tilde{d}\cdot \tilde{k}.$$
We further assume that $\tilde{r}$ lifts $r$ for $r\in \{a,b,c,d\}$, then the map $\mathcal{Q}:\tilde{M}_0\rightarrow M_0$ mapping $\tilde{e}\mapsto e$ and $\tilde{f}\mapsto f$ is a map of Fontaine-Laffaille modules since $\varphi^j\circ\mathcal{Q}=\mathcal{Q}\circ \varphi^j$ for $j=0,1$.
It follows from previous discussions that $\text{T}_1(\tilde{M}_0)$ is a rank $2$ Galois representation $\varrho$ which lifts $\bar{\rho}_{|G_{{\mathbb{Q}}_p}}$ $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}_p}$};
\node (A) at (3,0){$\text{GL}_2({\mathbb{F}})$.};
\node (B) at (3,2){$\text{GL}_2({\mathbb{F}}[\epsilon])$};
\draw[->] (G) to node [swap]{$\bar{\rho}_{\restriction \text{G}_{p}}$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\varrho$} (B);
\end{tikzpicture}$$ We observe that ${\text{tr}\varphi}_{|\varrho}=\tilde{a}\in {\mathbb{F}}[\epsilon]$ lifts $a=0$. The element $\tilde{a}$ is thus in fact invariant in the isomorphism class of $\tilde{M}_0$. We simply specify $\tilde{M}_0$ by choosing the matrix $\left( {\begin{array}{c|c}
\tilde{a} & \tilde{c} \\
\tilde{b} & \tilde{d} \\
\end{array} } \right)$ such that $\tilde{a}\neq 0$, clearly such a choice can be made. The Galois representation $\varrho$ coincides with a $W({\mathbb{F}})$-algebra homomorphism $H_{\varrho}:\mathcal{R}\rightarrow {\mathbb{F}}[\epsilon]$ such that $$H_{\varrho}(\Phi(X))=\Phi(H_{\varrho}(X))=\tilde{a}\neq 0.$$This implies that the coefficient of $X$ in the expansion of $\Phi(X)$ is a unit from which the statement of the Proposition follows.
The Local Deformation Condition at $p$
======================================
Throughout this section we fix a choice of $\lambda\in {\mathbb{Z}}_{\geq 1}$. Let $\mathcal{C}_p^{\lambda}$ be the sub-functor of $\mathcal{F}$ prescribed by $$\label{DefintionCp}\mathcal{C}_p^{\lambda}:=\text{image}\{\mathcal{F}^{\lambda}\rightarrow \mathcal{F}\},$$ and thus $$\mathcal{C}_p^{\lambda}(R)=\{\rho\in \mathcal{F}(R)\mid \text{ there exists } U\in \mathfrak{m}_R\text{ such that } {\text{tr}\varphi}_{|\rho}=p^{\lambda}(1+U)\}.$$ We shall prescribe the deformation condition $\mathcal{C}_p^{\lambda}$ in applying the Ramakrishna method of lifting the global representation $\bar{\rho}$. We observe that if $f=\sum_{n\geq 1} a_n q^n$ is an eigenform with field of Fourier coefficients $K_f$ and $\mathfrak{p}|p$ is a prime of $K_f$ such that the ramification index $e(\mathfrak{p}|p)=1$, then if the associated representation $\varrho=\rho_{f,\mathfrak{p}}$ is a characteristic zero deformation of $\bar{\rho}$ $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}}$};
\node (A) at (3,0){$\text{GL}_2({\mathbb{F}})$.};
\node (B) at (3,2){$\text{GL}_2(W({\mathbb{F}}))$};
\draw[->] (G) to node [swap]{$\bar{\rho}$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\varrho$} (B);
\end{tikzpicture}$$ such that ${\varrho}_{\restriction G_{{\mathbb{Q}}_p}}\in \mathcal{C}_p^{\lambda}$, then it follows that $\iota_{\mathfrak{p}}(a_p)={\text{tr}\varphi}_{|\varrho}=p^{\lambda}(1+U)$ for some $U\in {\mathbb{Z}}_p$ for which $p|U$ and consequently, the slope of $a_p$ is $\lambda$.
We shall show that $\bar{\rho}$ lifts to a *geometric* representation $\rho$, it follows from the results of Kisin [@KisinFM] that such a representation $\rho$ is modular, i.e. there exists an eigenform $f$ and a prime $\mathfrak{p}|p$ in $K_f$ for which $e(\mathfrak{p}|p)=1$ such that $\rho=\rho_{f, \mathfrak{p}}$. We proceed to provide a brief outline of the lifting strategy which is an adaptation of that in [@RaviFM].
There are several steps in the lifting construction:
1. \[Step 1\] we show that the local deformation condition $\mathcal{C}_p^{\lambda}$ is liftable and that there is a one dimensional space $\mathcal{N}_p^{\lambda}\subset H^1(G_p,\text{Ad}^0\bar{\rho})$ such that for any $N>\lambda$, deformation $\varrho$ of $\bar{\rho}_{|G_{{\mathbb{Q}}_p}}$, $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}_p}$};
\node (A) at (3,0){$\text{GL}_2({\mathbb{F}})$.};
\node (B) at (3,2){$\text{GL}_2(W({\mathbb{F}})/p^N)$};
\draw[->] (G) to node [swap]{$\bar{\rho}_{|G_{{\mathbb{Q}}_p}}$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\varrho$} (B);
\end{tikzpicture}$$and $X\in \mathcal{N}_p^{\lambda}$, the *twist* $$(\text{Id}+p^{N-1}X)\varrho\in \mathcal{C}_p^{\lambda}(W({\mathbb{F}})/p^N).$$
2. \[Step 2\] At primes $l\in S\slash\{p\}$ we specify local liftable deformation conditions (see [@RaviFM]) $\mathcal{C}_l$ with *tangent spaces* $\mathcal{N}_l\subseteq H^1(G_{{\mathbb{Q}}_l}, Ad^0\bar{\rho})$ for which $$\dim \mathcal{N}_l=\dim H^0(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho}).$$ At a prime $l\notin S$ which is a *nice* prime, we may choose the local deformation condition for $\bar{\rho}_{|G_{{\mathbb{Q}}_l}}$ denoted by $\mathcal{C}_l$ and a tangent space $\mathcal{N}_l$ specified in [@RaviFM], we recall that $$\dim \mathcal{N}_l=\dim H^0(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho}).$$
3. \[Step 3\] We show that for a certain finite set of primes $X$ containing $S$, such that $X\slash S$ consists of *nice* primes defined in loc. cit., one may exhibit a lift of $\bar{\rho}$ to a mod $p^{\lambda+2}$ representation $\rho_{\lambda+2}$ $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}},X}$};
\node (A) at (3,0){$\text{GL}_2({\mathbb{F}})$.};
\node (B) at (3,2){$\text{GL}_2(W({\mathbb{F}})/p^{\lambda+2})$};
\draw[->] (G) to node [swap]{$\bar{\rho}$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\rho_{\lambda+2}$} (B);
\end{tikzpicture}$$ such that for $l\in X$, ${\rho_{\lambda+2}}_{\restriction G_{{\mathbb{Q}}_l}}\in \mathcal{C}_l$.
4. \[Step 4\] We lift $\rho_{\lambda+2}$ to a geometric representation one step at a time. Suppose that $\rho_m$ is a lift of $\rho_{\lambda+2}$ for $m\geq \lambda+2$ which
- is unramified at all primes $l\notin X$,
- for $l\in X$, the local representation ${\rho_{m}}_{|G_{{\mathbb{Q}}_l}}\in \mathcal{C}_l$,
we lift $\rho_{m}$ to $\rho_{m+1}$ such that $\rho_{m+1}$ satisifes the same conditions $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}}$};
\node (A) at (4,0){$\text{GL}_2(W({\mathbb{F}})/p^{\lambda+2})$.};
\node (B) at (4,1.5){$\text{GL}_2(W({\mathbb{F}})/p^{m})$};
\node(C) at (4,3){$\text{GL}_2(W({\mathbb{F}})/p^{m+1})$};
\draw[->] (G) to node [swap]{$\rho_{\lambda+2}$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node [swap]{$\rho_{m}$} (B);
\draw[->] (C) to node{} (B);
\draw[->] (G) to node{$\rho_{m+1}$} (C);
\end{tikzpicture}$$ This can be done for any $m$ for an enlarged choice of primes $X$, the argument goes back to Ramakrishna. The key step involves showing that a set of *nice* primes $X\slash S$ may be chosen so that the associated dual Selmer group $$H^1_{\mathcal{N}^{\perp}}(G_{{\mathbb{Q}},X}, \text{Ad}^0\bar{\rho}^*):=\text{ker}\{H^1(G_{{\mathbb{Q}},X}, \text{Ad}^0\bar{\rho}^*)\xrightarrow{\text{res}}\bigoplus_l H^1(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho}^*)\slash \mathcal{N}_l^{\perp}\}$$ is zero. As a result of the local deformation conditions being large enough, i.e. the dimensions of $\mathcal{N}_l$ for $l\in X$ being as specified, it follows that we are in a *balanced* setting, i.e. the dimensions of the Selmer and dual Selmer groups match $$\dim H^1_{\mathcal{N}}(G_{{\mathbb{Q}},X}, \text{Ad}^0\bar{\rho})=\dim H^1_{\mathcal{N}^{\perp}}(G_{{\mathbb{Q}},X}, \text{Ad}^0\bar{\rho}^*).$$ We show that the Selmer group can be made to vanish on choosing the appropriate auxiliary primes $X\slash S$.
To complete the proof of the main theorem only steps $\ref{Step 1}$ and $\ref{Step 3}$ require some argument, we refer to [@RaviFM] for a complete description of the steps $\ref{Step 2}$ and $\ref{Step 4}$. In this section we lay out some of the properties which $\mathcal{C}_p^{\lambda}$ satisfies and fully establish step $\ref{Step 1}$. In the next section we shall do so for step $\ref{Step 3}$.
\[defcondition\]We say that a sub-functor $\mathcal{F}'$ of $\mathcal{F}$ is a *deformation condition* if the following conditions 1 to 3 are satisfied, and that $\mathcal{F}'$ is a *liftable deformation condition* if condition 4 is also satisfied:
1. \[dc1\] $\mathcal{F}'({\mathbb{F}})=\bar{\rho}_{\restriction G_{{\mathbb{Q}}_p}}.$
2. \[dc2\] Let $R_1$ and $R_2$ be in $\mathcal{C}$, $\rho_1\in \mathcal{F}'(R_1)$ and $\rho_2\in \mathcal{F}'(R_2)$. Let $I_1$ be an ideal in $R_1$ and $I_2$ an ideal in $R_2$ such that there is an isomorphism $\alpha:R_1/I_1\xrightarrow{\sim} R_2/I_2$ satisfying $$\alpha(\rho_1 \;\text{mod}\;{I_1})=\rho_2 \;\text{mod}\;{I_2}.$$ Let $R_3$ be the fibred product $$R_3=\lbrace(r_1,r_2)\mid \alpha(r_1\;\text{mod}\; I_1)=r_2\; \text{mod} \;I_2\rbrace$$ and $\rho_1\oplus \rho_2$ the induced $R_3$-representation, then $\rho_1\oplus \rho_2\in \mathcal{F}'(R_3)$.
3. \[dc3\] Let $R\in \mathcal{C}$ with maximal ideal $\mathfrak{m}_R$. If $\rho\in \mathcal{F}(R)$ and $\rho\in \mathcal{F}'(R/\mathfrak{m}_R^n)$ for all $n>0$ it follows that $\rho\in \mathcal{F}'(R)$. In other words, the functor $\mathcal{F}'$ is continuous.
4. \[dc4\] Let $R\in \mathcal{C}$ and $I$ an ideal such that $I.\mathfrak{m}_R=0$. Then for $\rho\in \mathcal{F}'(R/I)$ there exists $\rho'\in \mathcal{F}'(R)$ such that $\rho=\rho'\;\text{mod} \;I$.
The subfunctor $\mathcal{C}_p^{\lambda}$ of $\mathcal{F}$ is a liftable deformation condition.
We proceed to check the conditions of Definition $\ref{defcondition}$.
Condition $\ref{dc1}$ requires that $\mathcal{C}_p^{\lambda}({\mathbb{F}})=\bar{\rho}_{|G_{{\mathbb{Q}}_p}}$, is shown by only observing that $\mathcal{C}_p^{\lambda}({\mathbb{F}})$ is nonempty. The condition requiring ${\text{tr}\varphi}_{|\bar{\rho}}=p^{\lambda}(1+U)$ is clearly satisfied since ${\text{tr}\varphi}_{|\bar{\rho}}=0$ and $\lambda\geq 1$ and hence $p^{\lambda}(1+U)=0$ as well.
We next check condition $\ref{dc2}$. Let $R_j$ and $I_j$ for $j=1,2$ and $\alpha:R_1/I_2\xrightarrow{\sim} R_2/I_2$ be as in condition $\ref{dc2}$. Suppose that $\rho_j\in \mathcal{C}_p^{\lambda}(R_j)$ for $j=1,2$ such that $$\alpha(\rho_1 \;\text{mod}\;{I_1})=\rho_2 \;\text{mod}\;{I_2}.$$ By assumption there exist $U_j\in \mathfrak{m}_{R_j}$ for $j=1,2$ such that $${\text{tr}\varphi}_{|\rho_j}=p^{\lambda}(1+U_j).$$ Let $\bar{U_j}$ denote the reduction of $U_j$ modulo $\text{Ann}_{R_j}(p^{\lambda})$ in $R_j/\text{Ann}_{R_j}(p^{\lambda})$. Let $R_3$ be the fibred product $$R_3=\lbrace(r_1,r_2)\mid \alpha(r_1\;\text{mod}\; I_1)=r_2 \;\text{mod}\; I_2\rbrace$$ and $\rho_3$ the induced $R_3$-representation. We have that ${\text{tr}\varphi}_{|\rho_3}=p^{\lambda}(1+U_1, 1+U_2)\in R_3$. Since $p^{\lambda}(1+U_1, 1+U_2)\in R_3$ see that $(\bar{U_1},\bar{U_2})\in R_3/\text{Ann}_{R_3}(p^{\lambda})$ where we identify $$\begin{split}R_3/\text{Ann}_{R_3}(p^{\lambda})=&\{(\bar{r_1}, \bar{r_2})\in R_1/\text{Ann}_{R_1}(p^{\lambda})\times R_2/\text{Ann}_{R_2}(p^{\lambda})\\
&\mid \alpha(\bar{r_1}\:\text{mod}(pR_1+I_1))=\bar{r_2}\:\text{mod}(pR_2+I_2)\}.
\end{split}$$ We choose a lift $(\bar{U_1},\bar{U_2})\in R_3/p^{\lambda} R_3$ to $V=(V_1,V_2)\in R_3$. We observe that for $j=1,2$ since $V_j-U_j\in \text{Ann}_{R_j}(p^{\lambda})$, $$p^{\lambda}(1+V_j)=p^{\lambda}(1+U_j)$$ and we conclude that $${\text{tr}\varphi}_{|\rho_3}=p^{\lambda}(1+V)$$ and consequently $\rho_3\in \mathcal{C}_p^{\lambda}(R_3)$.
Condition $\ref{dc3}$ is an immediate consequence of $\mathcal{F}$ being a deformation condition. By Proposition $\ref{pi2}$ it follows that the functor $\mathcal{F}^{\lambda}\simeq \hat{\mathbb{A}}^1$ and thus it follows that condition $\ref{dc4}$ is satisfied.
\[localatp\] Let $\lambda\in{\mathbb{Z}}_{\geq 1}$ and $\mathcal{N}_p^{\lambda}\subseteq H^1(G_{{\mathbb{Q}}_p}, \text{Ad}^0\bar{\rho})$ be the one-dimensional tangent space to the flat deformation functor (which is independent of $\lambda$). The space $\mathcal{N}_p^{\lambda}$ stabilizes mod $p^m$ lifts of $\bar{\rho}_{\restriction G_{{\mathbb{Q}}_p}}$ in $\mathcal{C}_p^{\lambda}$ for $m> \lambda+1$. In other words, for $m> \lambda+1$, $X\in \mathcal{N}_p^{\lambda}$ and $\varrho_m\in \mathcal{C}_p^{\lambda}(W({\mathbb{F}})/p^m)$, the twist $$(\text{Id}+p^{m-1}X)\varrho_m\in \mathcal{C}_p^{\lambda}(W({\mathbb{F}})/p^m).$$
Recall that $\mathcal{N}_p^{\lambda}$ is the subspace of $H^1(G_{{\mathbb{Q}}_p}, \text{Ad}^0\bar{\rho})$ that constitutes the tangent space to the functor of flat deformations (cf. [@RaviFM]) and that $\mathcal{N}_p^{\lambda}$ is one-dimensional and is independent of $\lambda$. The $p$ deformations $\varrho^{(i)}=(\text{Id}+p^{m-1}X_i)\varrho_m$ as $X_i$ ranges over $\mathcal{N}_p^{\lambda}$ as $i=1,\dots, p$ are flat deformations of $\varrho_{m-1}=\varrho_m\mod{p^{m-1}}$. For ease of notation, we denote by $\varrho':=\varrho_m\:\text{mod}\:p^{m-\lambda-1}$ as is depicted below $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}_p}$};
\node (A) at (3,0){$\text{GL}_2(W({\mathbb{F}})/p^{m-\lambda-1})$.};
\node (B) at (3,2){$\text{GL}_2(W({\mathbb{F}})/p^{m})$};
\draw[->] (G) to node [swap]{$\varrho'$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\varrho_m$} (B);
\end{tikzpicture}$$ Since $\varrho_m\in \mathcal{C}_p^{\lambda}$ there exists $U\in W({\mathbb{F}})/p^m$ such that ${\text{tr}\varphi}_{|\varrho_m}=p^{\lambda}(1+U)$. Setting $U'=U\mod{p^{m-\lambda-1}}$ we observe that $(\varrho_m,U)$ is a lift of $(\varrho', U')$.
We choose $U_1,\dots, U_p\in W({\mathbb{F}})/p^m$ such that $U_j\equiv U\mod{p^{m-\lambda-1}}$ and such that they are all distinct modulo $p^{m-\lambda}$. Since $U_j\equiv U\mod{p^{m-\lambda-1}}$ it follows that $$p^{\lambda}(1+U_j)\equiv p^{\lambda}(1+U)\mod{p^{m-1}}.$$ On the other hand, since $U_j$ are distinct modulo $p^{m-\lambda}$ it follows that $p^{\lambda}(1+U_j)$ are distinct mod $p^{m}$. Let $\bar{U_j}=U_j\mod{p^{m-1}}$, the pair $(\varrho_{m-1}, \bar{U_j})\in \mathcal{F}^{\lambda}(W({\mathbb{F}})/p^{m-1})$ for $j=1,\dots,p$. The functor $\mathcal{F}^{\lambda}\simeq \hat{\mathbb{A}}^1$ and thus in particular, there exists a lift $(\tilde{\varrho}^{(j)}, Q_j)\in \mathcal{F}^{\lambda}(W({\mathbb{F}})/p^m)$ of $(\varrho_{m-1}, \bar{U_j})$.
We observe that $\tilde{\varrho}^{(j)}\in \mathcal{C}_p^{\lambda}(W({\mathbb{F}})/p^m)$ since $(\tilde{\varrho}^{(j)}, Q_j)\in \mathcal{F}^{\lambda}(W({\mathbb{F}})/p^m)$ and the set of deformations $\tilde{\varrho}^{(j)}$ are deformations of $\varrho_{m-1}$ for $j=1,\dots, p$. We show that $\tilde{\varrho}^{(j)}$ are distinct deformations. The result will follow from this since it shall follow that $$\{\varrho^{(j)}\mid j=1,\dots, p\}=\{\tilde{\varrho}^{(j)}\mid j=1,\dots, p\}\subseteq \mathcal{C}_p^{\lambda}(W({\mathbb{F}})/p^m)$$ and therefore, $\varrho^{(j)}\in \mathcal{C}_p^{\lambda}(W({\mathbb{F}})/p^m)$. For $X\in \mathcal{N}_p^{\lambda}=\{X_1,\dots, X_p\}$ we have that $X=X_k$ for some choice of $k$ and the twist $(\text{Id}+p^{m-1}X)\varrho_m=\varrho^{(k)}\in \mathcal{C}_p^{\lambda}(W({\mathbb{F}})/p^m)$.
Therefore, to conclude the proof, we proceed to show that for $i\neq j$, the deformations of $\varrho_{m-1}$ are not strictly equivalent deformations, $\tilde{\varrho}^{(i)}\not\simeq\tilde{\varrho}^{(j)}$. Since $\lambda\geq 1$ and $Q_j\equiv U_j\mod{p^{m-1}}$ (they are both lifts of $\bar{U_j}$) it follows that $${\text{tr}\varphi}_{|\tilde{\varrho}^{(j)}}=p^{\lambda}(1+Q_j)=p^{\lambda}(1+U_j)$$ and likewise, $${\text{tr}\varphi}_{|\tilde{\varrho}^{(i)}}=p^{\lambda}(1+Q_i)=p^{\lambda}(1+U_i).$$ Since $U_i\not\equiv U_j\mod{p^{m-\lambda}}$ it follows that $${\text{tr}\varphi}_{|\tilde{\varrho}^{(i)}}=p^{\lambda}(1+U_i)\neq p^{\lambda}(1+U_j)={\text{tr}\varphi}_{|\tilde{\varrho}^{(j)}}$$ and therefore, $\tilde{\varrho}^{(i)}\not\simeq\tilde{\varrho}^{(j)}$. The proof is now complete.
Lifting to mod $p^{\lambda+2}$
==============================
In this section we show that $\bar{\rho}$ may be lifted a mod $p^{\lambda+2}$ representation $\rho_{\lambda+2}$ $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}}$};
\node (A) at (3,0){$\text{GL}_2({\mathbb{F}})$.};
\node (B) at (3,2){$\text{GL}_2(W({\mathbb{F}})/p^{\lambda+2})$};
\draw[->] (G) to node [swap]{$\bar{\rho}$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\rho_{\lambda+2}$} (B);
\end{tikzpicture}$$
A prime $l$ is said to be a nice prime if
1. $l\not\equiv \pm 1\mod{p}$,
2. $\bar{\rho}$ is unramified at $l$,
3. $\bar{\rho}(\sigma_l)$ has eigenvalues whose ratio equals $l$.
At a nice prime $l$, there is a smooth deformation condition $\mathcal{C}_l$ for which we refer to as Ramakrishna deformations. They were introduced in [@RaviFM]. A deformation of $\bar{\rho}_{|G_{{\mathbb{Q}}_l}}$ is said to be nice if it lies in $\mathcal{C}_l$. The tangent space of $\mathcal{C}_l$ at a nice prime $l$ is denoted by $\mathcal{N}_l$.
We enlarge the set of primes $S$ so that the $\Sha$-groups $\Sha^1_S(\text{Ad}^0\bar{\rho})$ and $\Sha^2_S(\text{Ad}^0\bar{\rho})$ are both trivial. For further details, the reader may refer to loc. cit. At every prime $l\in S$ we prescribe a smooth local deformation condition $\mathcal{C}_l$ with tangent space $\mathcal{N}_l$ such that $\dim \mathcal{N}_l=\dim H^0(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho})$ as in loc. cit. At the prime $p$ we let $\mathcal{C}_p^{\lambda}$ be the deformation condition $\ref{DefintionCp}$ and $\mathcal{N}_p^{\lambda}\subseteq H^1(G_{{\mathbb{Q}}_p},\text{Ad}^0\bar{\rho})$ the one-dimensional space of Proposition $\ref{localatp}$.
Suppose that $m \leq \lambda$ and $\rho_m$ is a mod $p^m$ lift of $\bar{\rho}$ which is unramified outside a finite set. We seek to replace $\rho_m$ by a twist $\rho_m'=(\text{Id}+p^{m-1} Z)\rho_m$ by a global cohomology class $Z$ which does lift to some $\rho_{m+1}$ as depicted $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}}}$};
\node (A) at (3,0){$\text{GL}_2(W({\mathbb{F}})/p^m)$.};
\node (B) at (3,2){$\text{GL}_2(W({\mathbb{F}})/p^{m+1})$};
\draw[->] (G) to node [swap]{$\rho_m'$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\rho_{m+1}$} (B);
\end{tikzpicture}$$ Furthermore we seek to do so so that $\rho_{m+1}$ like $\rho_m$ is ramified outside finitely many primes.
Assume that $m\geq 2$. Suppose that $X_m$ is a finite set of primes containing the primes above $p$, the set of primes $S$ and the set of primes at which $\rho_m$ is ramified. Let $Z_m$ be a finite set of nice primes disjoint from $X_m$, chosen so that a twist of $\rho_m$ lifts to $\rho_{m+1}$ such that $\rho_{m+1}$ is unramified outside $X_{m+1}:=Z_m\cup X_m$. For ease of notation we replace $\rho_m$ by this twist, $$\begin{tikzpicture}[node distance = 2.5 cm, auto]
\node at (0,0) (G) {$\text{G}_{{\mathbb{Q}},X_{m+1}}$};
\node (A) at (3,0){$\text{GL}_2(W({\mathbb{F}})/p^m)$.};
\node (B) at (3,2){$\text{GL}_2(W({\mathbb{F}})/p^{m+1})$};
\draw[->] (G) to node [swap]{$\rho_m$} (A);
\draw[->] (B) to node{} (A);
\draw[->] (G) to node {$\rho_{m+1}$} (B);
\end{tikzpicture}$$ We may arrive at a collection of nice primes $Z_m$ by applying the method of Khare, Larsen and Ramakrishna from [@KLR].
At each prime $l\in X_n$ there is a choice of a local cohomology class $z_l\in H^1(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho})$ such that the twist $(\text{Id}+p^{m-1} z_l)\rho_{m}\in \mathcal{C}_l$. Thus we arrive at a tuple $$(z_l)_{l\in X_m}\in \bigoplus_{l\in X_m}H^1(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho}) .$$ If $(z_l)_{l}$ is the image of a global cohomology class $Z\in H^1(G_{{\mathbb{Q}},X_m}, \text{Ad}^0\bar{\rho})$ then we may simply replace $\rho_m$ by its twist by $Z$, $$(\text{Id}+p^{m-1}Z)\rho_m.$$ Since the local representations $$(\text{Id}+p^{m-1}Z){\rho_m}_{\restriction G_{{\mathbb{Q}}_l}}\in \mathcal{C}_l$$ and $\mathcal{C}_l$ is a liftable deformation condition, the obstruction to lifting $\rho_m$ lies in $\Sha^2_{X_m}(\text{Ad}^0\bar{\rho})$ which we have arranged to be trivial. Therefore, if such a global cohomology class exists, a twist of $\rho_m$ by this global cohomology class lifts to $\rho_{m+1}$.
We make preparations to complete the argument in the case when there does not exist such a global cohomology class $Z$.
There exists a finite choice of primes $X_{m+1}$ containing $X_m$ such that on replacing $\rho_m$ by its twist w.r.t a cohomology class $f\in H^1(G_{{\mathbb{Q}},X_{m+1}},\text{Ad}^0\bar{\rho})$, the twist $(\text{Id}+p^{m-1}\rho_m)$ lifts to $\rho_{m+1}$ which is unramified at all primes outside $X_{m+1}$. As a consequence, $\bar{\rho}$ lifts to $\rho_{\lambda+2}$ which is unramified outside a finite set of primes $X=X_{\lambda+2}$.
Let $Q$ be a finite set of primes as in Lemma 8 of [@KLR] for which the restriction map $$H^1(G_{{\mathbb{Q}},X_m\cup Q}, \text{Ad}^0\bar{\rho})\rightarrow \left(\oplus_{l\in X_m} H^1(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho})\right)\oplus\left(\oplus_{q\in Q} H^1_{nr}(G_{{\mathbb{Q}}_q}, \text{Ad}^0\bar{\rho})\right)$$ is an isomorphism. Therefore, there exists a cohomology class $h\in H^1(G_{{\mathbb{Q}},X_{m}\cup Q},\text{Ad}^0\bar{\rho})$ such that $h_{\restriction G_{{\mathbb{Q}}_l}}=z_l$ and such that $(\text{Id}+p^{m-1} h)\rho_m$ is nice at all primes $q\in Q$. By Corollory 11 of [@KLR] there exists a finite set of $\rho_m$-nice primes $V$ disjoint from $X_m\cup Q$ such that there exists a cohomology class $g$ such that
- $(\text{Id}+p^{m-1} (g+h))\rho_m$ is unramified outside $X_m\cup Q\cup V$
- is unobstructed at all primes in $X_m\cup Q\cup V$.
We set $X_{m+1}=X_m\cup Q\cup V$, since $\Sha^2_{X_{m+1}}(\text{Ad}^0\bar{\rho})=0$, and there are no local obstructions to lifting $\rho_m$ the statement of the Proposition follows.
(of Theorem \[main\])
Let $X$ be a finite set of primes for which $\bar{\rho}$ lifts to $\rho_{\lambda+2}$ such that the ${\rho_{\lambda+2}}_{\restriction G_{{\mathbb{Q}}_l}}\in \mathcal{C}_l$ for $l\in X$. The Selmer condition $\{\mathcal{N}_l\}_{l\in X}$ is balanced, i.e. $$\dim H^1_{\mathcal{N}}(G_{{\mathbb{Q}},X},\text{Ad}^0\bar{\rho})=\dim H^1_{\mathcal{N}^{\perp}}(G_{{\mathbb{Q}},X},\text{Ad}^0\bar{\rho}^*).$$ It is shown for instance in [@RaviFM] that in this situation, on enlarging $X$ by a finite set of nice primes, the Selmer group and hence the dual Selmer group $$H^1_{\mathcal{N}^{\perp}}(G_{{\mathbb{Q}},X},\text{Ad}^0\bar{\rho}^*)=0.$$ Consequently, from the Poitou Tate long exact sequence, we arrive at an isomorphism $$\label{isoSDS}H^1(G_{{\mathbb{Q}},S\cup X}, \text{Ad}^0\bar{\rho})\xrightarrow{\sim}\bigoplus_{l\in X}H^1(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho})/\mathcal{N}_l.$$ Let $\rho_m$ be a lift of $\rho_{\lambda+2}$ for $m\geq \lambda+2$. We lift $\rho_m$ to $\rho_{m+1}$ by twisting $\rho_m$ by a global cohomology class $f\in H^1(G_{{\mathbb{Q}},X}, \text{Ad}^0\bar{\rho})$. In greater detail, for $l\in X$, there exists a class $z_l+\mathcal{N}_l\in H^1(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho})/\mathcal{N}_l $ such that for any representative $z_l\in H^1(G_{{\mathbb{Q}}_l}, \text{Ad}^0\bar{\rho})$ of this class, the twist $(\text{Id}+p^{m-1}z_l)({\rho_{m}}_{|G_{{\mathbb{Q}}_l}})\in \mathcal{C}_l$. By the isomorphism $\ref{isoSDS}$ there is a global cohomology class $Z\in H^1(G_{{\mathbb{Q}},X}, \text{Ad}^0\bar{\rho})$ such that for all $l\in X$, $Z_{\restriction G_{{\mathbb{Q}}_l}}=z_l$. We replace $\rho_m$ by its twist by $Z$, we observe that $$\left((\text{Id}+p^{m-1}Z)\rho_{m}\right)_{\restriction G_{{\mathbb{Q}}_l}}\in \mathcal{C}_l$$ for all $l\in X$ and hence is unobstructed at all primes $l\in X$. Since $\Sha^2_{X}(\text{Ad}^0\bar{\rho})=0$ it follows that $\rho_m$ does lift to $\rho_{m+1}$ where $\rho_{m+1}$ is unramified outside $X$. Once again replacing $\rho_{m+1}$ by its twist by a suitable global cohomology class, we may assume further that ${\rho_{m+1}}_{\restriction G_{{\mathbb{Q}}_l}}\in \mathcal{C}_l$ at all primes $l\in X$. The main theorem therefore follows as $\rho_{\lambda+2}$ therefore lifts to a characterisitic zero representation $\rho$ for which $\rho_{\restriction G_{{\mathbb{Q}}_p}}\in \mathcal{C}_p^{\lambda}$ and thus in particular, ${\text{tr}\varphi}_{|\rho}\in W({\mathbb{F}})$ such that $v_p({\text{tr}\varphi}_{|\rho})=\lambda$.
The representation $\rho$ satisfies the conditions of Fontaine and Mazur and thus by [@KisinFM] there is a supersingular normalized eigenform $f\in S_2(\Gamma_1(N))$ and a prime $\mathfrak{p}|p$ of the field of Fourier coefficients ${\mathbb{Q}}(f)$ of $f$ for which $e(\mathfrak{p}|p)=1$, such that the associated Galois representation $\rho_{f,\mathfrak{p}}\simeq \rho$. In particular, $\iota_{\mathfrak{p}}(a_p)={\text{tr}\varphi}_{|\rho}$ and as a consequence, $v_p(\iota_{\mathfrak{p}}(a_p))=\lambda$. Since $\rho_{f,\mathfrak{p}}$ is flat when restricted to $G_{{\mathbb{Q}}_p}$ it follows that $(N,p)=1$. Thus the proof of the theorem is complete.
[1]{} Berger, Laurent, Hanfeng Li, and Hui June Zhu. “Construction of some families of 2-dimensional crystalline representations.” Mathematische Annalen 329.2 (2004): 365-377.
Brinon, O., and B. Conrad. “p-adic Hodge theory, notes from the 2009 Clay Mathematics Institute summer school.”
Buzzard, Kevin, and Toby Gee. “Explicit reduction modulo p of certain two-dimensional crystalline representations.” International Mathematics Research Notices 2009.12 (2009): 2303-2317.
Buzzard, Kevin, and Toby Gee. “Explicit reduction modulo p of certain 2-dimensional crystalline representations, II.” Bulletin of the London Mathematical Society 45.4 (2013): 779-788.
Colmez, Pierre, and Jean-Marc Fontaine. “Construction des représentations p-adiques semi-stables.” Inventiones mathematicae 140.1 (2000): 1-43.
Fontaine, Jean-Marc, and Guy Laffaille. “Construction de représentations $ p $-adiques.” Annales scientifiques de l’Ecole Normale Supérieure. Vol. 15. No. 4. 1982.
Fontaine, Jean-Marc, and Barry Mazur. “Geometric Galois representations.” Elliptic curves, modular forms and Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I (1995): 41-78.
Ganguli, Abhik, and Eknath Ghate. “Reductions of Galois representations via the mod p Local Langlands Correspondence.” Journal of Number Theory 147 (2015): 250-286.
Hamblen, Spencer, and Ravi Ramakrishna. “Deformations of certain reducible Galois representations, II.” American journal of mathematics 130.4 (2008): 913-944.
Hattori, Shin. “Integral p-adic Hodge theory and ramification of crystalline representations.” preprint (2018).
Khare, Chandrashekhar, Michael Larsen, and Ravi Ramakrishna. “Constructing semisimple p-adic Galois representations with prescribed properties.” American Journal of Mathematics 127.4 (2005): 709-734.
Khare, Chandrashekhar, and Ravi Ramakrishna. “Lifting torsion Galois representations.” Forum of Mathematics, Sigma. Vol. 3. Cambridge University Press, 2015.
Kisin, Mark. “The Fontaine-Mazur conjecture for $\text{GL}_2$” Journal of the American Mathematical Society 22.3 (2009): 641-690.
Patrikis, Stefan. Lifting symplectic galois representations. Diss. Harvard College, 2006.
Ramakrishna, Ravi. “On a variation of Mazur’s deformation functor.” Compositio Mathematica 87.3 (1993): 269-286.
Ramakrishna, Ravi. “Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur.” Annals of mathematics 156.1 (2002): 115-154.
Saito, Takeshi. “Modular forms and p-adic Hodge theory.” Inventiones mathematicae 129.3 (1997): 607-620.
|
---
abstract: 'We present a new approach for learning compact and intuitive distributed representations with binary encoding. Rather than summing up expert votes as in products of experts, we employ for each variable the opinion of the most reliable expert. Data points are hence explained through a partitioning of the variables into expert supports. The partitions are dynamically adapted based on which experts are active. During the learning phase we adopt a smoothed version of this model that uses separate mixtures for each data dimension. In our experiments we achieve accurate reconstructions of high-dimensional data points with at most a dozen experts.'
author:
- |
Marc Goessling\
Department of Statistics\
University of Chicago\
Chicago, IL 60637, USA\
`goessling@galton.uchicago.edu` Yali Amit\
Departments of Statistics and Computer Science\
University of Chicago\
Chicago, IL 60637, USA\
`amit@galton.uchicago.edu`
bibliography:
- 'references.bib'
title: Dynamic Partition Models
---
Introduction
============
We consider the task of learning a compact binary representation [e.g. @goessling2015compact]. That means we are seeking a parsimonious set of experts, which can explain a given collection of multivariate data points. In contrast to most existing approaches the emphasis here is on finding experts that are individually meaningful and that have disjoint responsibilities. Ideally, each expert explains only one factor of variation in the data and for each factor of variation there is exactly one expert that focuses on it.
Formally, the experts $\mathbb{P}_k$, $k=1,\ldots,K$, are probability distributions that depend on binary latent variables $\bm{h}(k)$. The latent state $\bm{h}$ specifies which experts are active and has to be inferred for each $D$-dimensional data point $\bm{x}$. The active experts then define a probability distribution $\mathbb{P}$. The goal of representation learning is to train experts such that the conditional likelihood $\mathbb{P}(\bm{x} \,|\, \bm{h})$ of the data given the latent activations is maximized.
We start by describing a simple model family, which forms the basis of our work. A partition model [@hartigan1990partition] makes use of a manually specified partitioning of the $D$ variables into subsets $$\{1,\ldots,D\} = \bigcup_{\ell=1}^L S_\ell.$$ For each subset of variables $\bm{x}(S_\ell) = (\bm{x}(d))_{d \in S_\ell}$ there exists a separate model $\mathbb{P}_\ell$. It is then typically assumed that variables in different subsets are conditionally independent, i.e., $$\mathbb{P}(\bm{x} \,|\, \bm{h}) = \prod_{\ell=1}^L \mathbb{P}_\ell(\bm{x}(S_\ell) \,|\, \bm{h}(\ell)).
\label{eq:partition}$$ The model is completed by specifying a prior distribution $\mathbb{P}(\bm{h})$ for the latent state $\bm{h}$. One advantage of partition models is that estimating $\mathbb{P}_\ell$ from observations is straightforward, while learning expert models in general requires computationally involved procedures [@bengio2013representation]. However, in order to be able to define a satisfactory partitioning of the variables some prior knowledge about the dependence structure is needed. For image data a common choice is to use a regular grid that divides the image into patches [e.g. @pal2002learning]. In general, a good partitioning is characterized by providing weakly dependent subsets of variables so that the conditional independence assumption (\[eq:partition\]) is reasonable and the distribution of the latent variables is easy to model. Unfortunately, often there simply is no single fixed partitioning that works well for the whole dataset because the set of variables, which are affected by different factors of variation, might overlap. This restricts the scenarios in which partition models are useful.
In this paper we extend partition models to allow for dynamically adapting partitionings. In Section \[sec:model\] we introduce the model and present an appropriate learning procedure. Related work is discussed in Section \[sec:related\_work\]. Special emphasis is given to the comparison with products of experts [@hinton2002training]. Experiments on binary and real-valued data are performed in Section \[sec:experiments\]. While it is important to explain high-dimensional data points through multiple experts, our work shows that it is possible to assign the responsibility for individual variables to a single expert (rather than having all active experts speak for every variable).
Dynamic partition models {#sec:model}
========================
Our main proposal is to define for each expert $\mathbb{P}_k$ its level of expertise $\bm{e_k} \in \mathbb{R}_+^D$ for all variables. We can then dynamically partition the variables based on the active experts. Specifically, for each variable we employ the most reliable (active) expert $$\mathbb{P}(\bm{x} \,|\,\bm{h}) = \prod_{d=1}^D \mathbb{P}_{k^\star(d)}(\bm{x}(d)), \qquad k^\star(d) = \operatorname*{argmax}_{k:\bm{h}(k)=1} \bm{e_k}(d).
\label{eq:dynamic_partition}$$ That means, each variable $\bm{x}(d)$ is explained by only a single expert $k^\star(d)$. The partitioning into expert supports $S_k(\bm{h}) = \{d \in \{1,\ldots,D\} : k^\star(d)=k \}$ is determined dynamically based on the latent configuration $\bm{h}$. We hence call our model a dynamic partition model.
Inference {#sec:inference}
---------
In the inference step we try to find for each data point $\bm{x_n}$ the subset of experts $\{k : \bm{h_n}(k)=1\}$ that maximizes $P(\bm{x_n} \,|\, \bm{h_n})$. To do this, we suggest to sequentially activate the expert that most improves the likelihood, until the likelihood cannot be improved anymore. This approach is called likelihood matching pursuit [@goessling2015compact]. The greedy search works well for our model because we are working with a small set of experts and each expert focuses on a rather different structure in the data. Consequently, the posterior distribution on the latent variables given $\bm{x_n}$ is often highly peaked at a state $\bm{h_n}$ (note that for high-dimensional data the effect of the prior $\mathbb{P}(\bm{h})$ is typically negligible).
Learning
--------
In contrast to traditional approaches, which combine multiple experts for individual variables, training the experts in a dynamic partition model is trivial. Indeed, the maximum-likelihood estimates are simply the empirical averages over all observations for which the expert was responsible. For example, the expert means can be estimated from training data $\bm{x_n}$, $n=1,\ldots,N$, as $$\bm{{\overset{\circ}{\mu}}_k}(d) = \frac{\sum\limits_{n=1}^N \mathbbm{1}\{k_n^\star(d){=}k\}\bm{x_n}(d)}{\sum\limits_{n=1}^N \mathbbm{1}\{k_n^\star(d){=}k\}}.
\label{eq:hard_update}$$ Here, $k^\star_n(d)$ denotes the expert with the highest level of expertise $\bm{e_k}(d)$ among all experts $k$ with $\bm{h_n}(k)=1$.
### Expertise-weighted composition {#sec:expertise_weighting}
In order to compute the estimator in (\[eq:hard\_update\]) the levels of expertise $\bm{e_k}$ have to be known. Since in this paper we are trying to train the experts as well as the associated levels of expertise we consider a smoothing of the maximum-expertise composition (\[eq:dynamic\_partition\]) to motivate our learning procedure. Rather than using the expert with the highest level of expertise, we form a mixture of the active experts, where the mixture weight is proportional to the level of expertise. Thus, the smoothed composition rule is $$\widetilde{\mathbb{P}}(\bm{x} \,|\,\bm{h}) = \prod_{d=1}^D \sum_{k=1}^K \bm{r_k}(d) \mathbb{P}_k(\bm{x}(d)), \qquad \bm{r_k}(d) =
\begin{cases}
\frac{\bm{e_k}(d)}{\sum_{k':\bm{h}(k')=1}\bm{e_{k'}}(d)} & \textrm{if } \bm{h}(k) = 1\\
0 & \textrm{if } \bm{h}(k) = 0
\end{cases}.
\label{eq:expertise_weighting}$$ In contrast to classical mixture models [e.g. @mclachlan2004finite] we use different mixture weights for each dimension $d \in \{1,\ldots,D\}$. The mixture weight $\bm{r_k}(d)$ is the degree of responsibility of $k$-th expert for the $d$-th dimension and depends on the latent state $\bm{h}$. An expert with a medium level of expertise assumes full responsibility if no other reliable expert is present and takes on a low degree of responsibility if experts with a higher level of expertise are present.
According to the total variance formula $$\mathbb{V}[\widetilde{\mathbb{P}}] = \mathbb{E}_{\bm{r_k}}[\mathbb{V}[\mathbb{P}_k]] + \mathbb{V}_{\bm{r_k}}[\mathbb{E}[\mathbb{P}_k]]$$ the variance of a mixture is always larger than the smallest variance of its components. In other words, the precision of the smoothed model is maximized when all the mixture weight (individually for each dimension) is concentrated on the most precise expert. We can thus learn a dynamic partition model in an EM manner [@dempster1977maximum] by interleaving inference steps with updates of the experts and levels of expertise in the smoothed model.
### Expert update
The sequential inference procedure (from Section \[sec:inference\]) provides for each data point $\bm{x_n}$ the latent representation $\bm{h_n}$. We denote the corresponding expert responsibilities (using the current estimates for the level of expertise) by $\bm{r_{nk}}$. The smooth analog to the hard update equation (\[eq:hard\_update\]) is a responsibility-weighted average of the training samples $$\bm{\mu_k}(d) = \frac{\sum\limits_{n=1}^N \bm{r_{nk}}(d)\bm{x_n}(d) + \epsilon\bm{\mu_0}}{\sum\limits_{n=1}^N \bm{r_{nk}}(d) + \epsilon}.
\label{eq:mean_update}$$ For stability we added a term that shrinks the updated templates towards some target $\bm{\mu_0}$ if the total responsibility of the expert is small. In our experiments we set $\bm{\mu_0}$ to the average of all training examples. The update rule implies that the experts have local supports, in the sense that they are uninformative about variables for which they are not responsible.
For binary data the mean templates $\bm{\mu_k}$ are all we need. Continuous data $\bm{x} \in \mathbb{R}^D$ is modeled through Gaussians and hence we also have to specify the variance $\bm{v_k}$ of the experts. We again use a responsibility-weighted average $$\bm{v_k}(d) = \frac{\sum\limits_{n=1}^N \bm{r_{nk}}(d)(\bm{x_n}(d)-\bm{\mu_k}(d))^2 + \epsilon\bm{v_0}}{\sum\limits_{n=1}^N \bm{r_{nk}}(d) + \epsilon},
\label{eq:variance_update}$$ where $\bm{v_0}$ is the empirical variance of all training samples.
### Expertise update
We now turn to the updates of the levels of expertise. The log-likelihood of the smoothed model (\[eq:expertise\_weighting\]) as a function of $\bm{e}_k$ is rather complex. Using gradient descent is thus problematic because the derivatives with respect to $\bm{e_k}$ can have very different scales, which makes it difficult to choose an appropriate learning rate and hence the convergence could be slow. However, exact optimization is not necessary because in the end only the order of the levels of expertise matters. Consequently, we propose to adjust $\bm{e_k}(d)$ only based on the sign of the gradient. We simply multiply or divide the current value by a constant $C$. If the gradient is very close to $0$ we leave $\bm{e_k}(d)$ unchanged. For all our experiments we used $C=2$. Larger values can speed up the convergence but sometimes lead to a worse solution. Using an exponential decay is common practice when learning levels of expertise [e.g. @herbster1998tracking]. In the learning procedure we perform the expertise update first. We then recompute the responsibilities using these new levels of expertise and update the experts. Our algorithm typically converges after about 10 iterations.
Related work {#sec:related_work}
============
@herbster1998tracking proposed an algorithm for tracking the best expert in a sequential prediction task. In their work it is assumed that a linear ordering of the variables is known such that the expert with the highest level of expertise is constant on certain segments. In contrast to that, our approach can be applied to an arbitrary permutation of the variables. Moreover, they consider a single sequence of variables with a fixed partitioning into experts supports. In our setup the partitioning changes dynamically depending on the observed sample. However, the greatest difference to our work is that @herbster1998tracking do not learn the individual experts but only focus on training the levels of expertise.
@lucke2008maximal studied a composition rule that also partitions the variables into expert supports. In their model the composed template is simply the maximum of the experts templates $\bm{\mu_k}$. This rule is only useful in special cases. A generalization, in which the composition depends on the maximum and the minimum of the expert templates $\bm{\mu_k}(d)$, was considered by [@goessling2015compact]. While the motivation for that rule was similar, the maximum-expertise rule in this paper is more principled and can be applied to continuous data.
In the work by @amit2007pop a simple average (i.e., an equal mixture) of the individual templates was used. With such a composition rule, all experts are equally responsible for each of the variables and hence specialization on local structures is not possible. To circumvent this problem, in their work $\bm{e_k}(d)$ was manually set to 1 for some subset of the dimensions (depending on a latent shift variable) and to 0 elsewhere.
A popular model family with latent binary representation are products of experts [@hinton2002training]. In such a model the individual distributions $\mathbb{P}_k$ are multiplied together and renormalized. Computation of the normalizing constant is in general intractable though. A special case, in which an explicit normalization is possible, are restricted Boltzmann machines [@hinton2002training]. In these models the experts are product Bernoulli distributions with templates $\bm{\mu_k} \in [0,1]^D$. The composed distribution is then also a product Bernoulli distribution with composed template $$\bm{\mu}_\textrm{RBM}(d) = \sigma\left(\sum\nolimits_{k:\bm{h}(k)=1} \bm{w_k}(d)\right),$$ where the weights $\bm{w_k}(d) = \log(\bm{\mu_k}(d)/(1-\bm{\mu_k}(d)) \in \mathbb{R}$ are the log-odds of the experts and $\sigma(t) = (1+\exp(-t))^{-1}$ is the logistic function. This sum-of-log-odds composition rule arises naturally from generalized linear models for binary data because the log-odds are the canonical parameter of the Bernoulli family. In a product of experts, the variance of the composition is usually smaller than the smallest variance of the experts. As a consequence, products of experts tend to employ many experts for each dimension (for more details on this issue see @goessling2015compact). Even with an L1-penalty on the votes $\bm{w_k}(d)$ the responsibility for individual variables $\bm{x}(d)$ is typically still shared among many experts. The reason for this is that under the constraint $\sum_k \bm{w_k}(d) = \bm{w}(d)$ the quantity $\sum_k |\bm{w_k}(d)|$ is minimized whenever $\bm{w_k}(d)$ has the same sign for all $k$. The usual inference procedure for products of experts independently activates experts based on their inner product with the data point. In particular, not just the most probable expert configuration is determined but the whole posterior distribution on latent states given the data is explored through Monte Carlo methods. For learning in products of experts, simple update rules like (\[eq:mean\_update\]) and (\[eq:variance\_update\]) cannot be used because for each expert the effects of all other experts have to be factored out. Dynamic partition models essentially decompose the expert votes $\bm{w_k}$ into expert opinions $\bm{\mu_k}$ and levels of expertise $\bm{e_k}$. Apart from the computational advantages for learning, this introduces an additional degree of flexibility because the expert supports are adjusted depending on which other experts are present (cf. Figure \[fig:mnist\_support\]). Moreover, the decomposition into opinions and levels of expertise avoids ambiguities. For example, a vote $\bm{w_k}(d) \approx 0$ could mean that $\bm{\mu_k}(d) \approx 1/2$ or that $\bm{e_k}(d) \approx 0$.
Another common model for representation learning are autoencoders [@vincent2008extracting], which can be considered as mean-field approximations of restricted Boltzmann machines that use latent variables $\bm{h}(k)$ with values in $[0,1]$. To obtain a sparse representation a penalty on the number of active experts can be added [@ng2011sparse]. Such approaches are also known as sparse dictionaries [e.g., @elad2010sparse] and are based on opinion pools of the form $\sum_k \bm{h}(k) \bm{w_k}(d)$. The strength of the sparsity penalty is an additional tuning parameter which has to be tuned. In dynamic partition models sparse activations are inherent. In the next section, we experimentally compare products of experts, autoencoders and sparse dictionaries to our proposed model.
Experiments {#sec:experiments}
===========
{width="\textwidth"}
{height=".215\textwidth"} {height=".215\textwidth"} {height=".215\textwidth"}
Synthetic data
--------------
We consider a synthetic example and try to learn the underlying factors of variation. The dataset consists of the 32-element subset $\{(0,1),(1,0)\}^5 \subset \{0,1\}^{10}$. Note that there are 5 factors of variation corresponding to the state of the pairs $(\bm{x}(2\ell{-}1),\bm{x}(2\ell))$ for $\ell=1,\ldots,5$ with the two factor levels $(0,1)$ and $(1,0)$. Indeed, the distribution can be easily expressed through a partition model with partitioning $$\{1,2\} \cup \{3,4\} \cup \{5,6\} \cup \{7,8\} \cup \{9,10\}$$ and corresponding models $$\mathbb{P}_\ell(\bm{x}(2\ell{-}1),\bm{x}(2\ell)) = \tfrac{1}{2}\cdot \mathbbm{1}\{\bm{x}(2\ell{-}1){=}0,\,\bm{x}(2\ell){=}1\} + \tfrac{1}{2}\cdot \mathbbm{1}\{\bm{x}(2\ell{-}1){=}1,\,\bm{x}(2\ell){=}0\}.$$ We show that our dynamic partition model is able to learn these factors of variation without requiring a manual specification of the partitioning. Here, the total number of experts we need to accurately reconstruct all data points happens to be equal to the number of dimensions. However, in other cases the number of required experts could be smaller or larger than $D$. We ran our learning algorithm for 15 iterations starting from a random initialization of the experts. The resulting templates after 3, 5 and 15 iterations are shown in Figure \[fig:synthetic\_EM\]. We see that each of the final experts specializes in exactly two dimensions $d$ and $d+1$. Its opinion for these variables are close to 0 and 1, respectively, while the opinions for the remaining variables are about 1/2. Every data point can now be (almost) perfectly reconstructed by using exactly 5 of these experts.
For comparison we trained various other models with 10 experts, which use a sum-of-log-odds composition. We first tried an autoencoder [@vincent2008extracting], which in principle could adopt the identity map because it uses (in contrast to our model) a bias term for the observable and latent variables. However, the gradient descent learning algorithm with tuned step size yielded a different representation (Figure \[fig:synthetic\_others\], 1st panel). While the reconstruction errors are rather low, they are clearly nonzero and the factors of variations have not been disentangled. Next, we considered a dictionary with a sparse representation [e.g., @elad2010sparse]. The sparsity penalty was adjusted so that the average number of active dictionary elements was around 5. The learning algorithm again yielded highly dependent experts (Figure \[fig:synthetic\_others\], 2nd panel). Finally, we trained a restricted Boltzmann machine through batch persistent contrastive divergence [@tieleman2008training] using a tuned learning rate. Note that a restricted Boltzmann machine in principle only requires 5 experts to model the data appropriately because it uses bias terms. However, we again learned 10 experts (Figure \[fig:synthetic\_others\], 3rd panel). While the results look better than for the previous two models they are still far from optimal. In earlier work [@goessling2015compact] we performed a quantitative comparison for a similar dataset, which showed that the reconstruction performance of models with sum-of-log-odds composition is indeed suboptimal.
{height=".475\textwidth"} {height=".475\textwidth"}
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MNIST digits
------------
We now consider the MNIST digits dataset [@lecun1998gradient], which consists of 60,000 training samples and 10,000 test samples of dimension $28 \times 28 = 784$. We ran our learning algorithm for 10 iterations and trained 100 experts (Figure \[fig:mnist\_experts\]). We see that some experts specialize on local structures while others focus on more global ones. In Figure \[fig:mnist\_greedy\] we visualize the inference procedure for some test samples using these 100 learned experts. On average 12 experts were activated for each data point. For easier visualization we show at most 10 iterations of the likelihood matching pursuit algorithm. The reconstructions are overall accurate and peculiarities of the samples are smoothed out. In Figure \[fig:mnist\_support\] we illustrate how the expert supports change based on the latent representation. Depending on which other experts are present the supports can vary quite a bit.
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Weizmann horses
---------------
The following experiment shows that our model is able to cope with very high-dimensional data. The Weizmann horse dataset [@borenstein2008combined] consists of 328 binary images of size $200 \times 240$. We used the first 300 images to train 20 experts (Figure \[fig:weizmann\_experts\]) and used the remaining 28 images for testing. Some of the experts are responsible for the background and the central region of the horse while other experts focus on local structures like head posture, legs and tail. In Figure \[fig:horse\_decompositions\] we illustrate the partitioning of the test examples into expert opinions. For simplicity we used exactly 4 experts to reconstruct each sample. Not all characteristics of the samples are perfectly reconstructed but the general pose is correctly recovered. The same dataset was used to evaluate the shape Boltzmann machine [@eslami2014shape], where 2,000 experts were learned. For those experiments the images were downsampled to $32 \times 32$ pixels. This is a factor 50 smaller than the full resolution of 48,000 dimensions that we use.
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Caltech motorcycles
-------------------
We also experimented with real-valued data using the Caltech-101 motorcycle dataset [@fei2007learning], which consists of 798 images of size $100 \times 180$. The first 750 images were used for training and the remaining 48 images for testing. We trained 50 experts by running our learning procedure for 10 iterations. In Figure \[fig:motorbike\_reconstructions\] we visualize the reconstructed test examples. The reconstructions are a bit blurry since we use a fairly sparse binary representation. Indeed, for each data point on average only 7 experts were employed. Note that the shapes of the motorcycles are reconstructed quite accurately.
Discussion
==========
In order to improve the reconstructions for continuous image data we could use real-valued latent variables in addition to binary ones (as in @hinton1998hierarchical). This would allow us to model intensities and contrasts more accurately. The inference procedure would have to be adapted accordingly such that continuous activations can be returned.
Our work focused on product distributions. In order to apply the proposed approach to models with dependence structure one can make use of an autoregressive decomposition [e.g., @goessling2016mixtures]. If the joint distribution is written as a product of conditional distributions then we can employ the same composition rule as before. Indeed, we can model composed the conditionals as $$\mathbb{P}(\bm{x}(d) \,|\, \bm{x}(1{:}d{-}1), \bm{h}) = \mathbb{P}_{k^\star(d)}(\bm{x}(d) \,|\, \bm{x}(1{:}d{-}1)),$$ where $\mathbb{P}_k$ are autoregressive expert models and $k^\star(d)$ is the active expert with the highest level of expertise for dimension $d$.
Derivatives
===========
We provide here the derivatives of the log-likelihood in the expertise-weighted compositional model (\[eq:expertise\_weighting\]) with respect to the expert parameters.
Bernoulli model
---------------
The Bernoulli log-likelihood is $$f(\mu) = x \log \mu + (1-x) \log(1-\mu),$$ where the composition rule for the probability is $$\mu = \sum_k r_k \mu_k, \quad r_k = \frac{e_k}{\sum_{k'}e_{k'}}.$$
### Derivatives with respect to the composed probability
The first and second derivative of the log-likelihood with respect to the composed probability are $$\frac{df}{d\mu} = \frac{x}{\mu} - \frac{1-x}{1-\mu} = \frac{x-\mu}{\mu(1-\mu)},$$ $$\frac{d^2f}{d\mu^2} = -\frac{x}{\mu^2} - \frac{1-x}{(1-\mu)^2} = -\frac{(x-\mu)^2}{\mu^2(1-\mu)^2}.$$
### Derivatives with respect to the expert probabilities {#sec:derivative_expert_probas}
The first and second derivative of the composed probability with respect to the expert probabilities are $$\frac{d\mu}{d\mu_k} = r_k, \quad \frac{d^2\mu}{d\mu_k^2} = 0.$$ Consequently, the derivatives of the log-likelihood with respect to the expert probabilities are $$\frac{df}{d\mu_k} = \frac{df}{d\mu} \cdot \frac{d\mu}{d\mu_k} = r_k \frac{x-\mu}{\mu(1-\mu)},$$ $$\frac{d^2f}{d\mu_k^2} = \frac{d^2f}{d\mu^2} \cdot \left(\frac{d\mu}{d\mu_k}\right)^2 + \frac{df}{d\mu} \cdot \frac{d^2\mu}{d\mu_k^2} = -r_k^2\frac{(x-\mu)^2}{\mu^2(1-\mu)^2}.$$ We see that $d^2f/d\mu_k^2 < 0$ for $\mu \in (0,1)$, i.e., the log-likelihood is a strictly concave function of $\mu_k$.
### Derivative with respect to the levels of expertise
The derivative of the composed probability with respect to the levels of expertise is $$\frac{d\mu}{de_k} = \frac{\mu_k E - \sum e_{k'}\mu_{k'}}{E^2} = \frac{\mu_k-\mu}{E},$$ where $E = \sum_{k'} e_{k'}$. The derivative of the log-likelihood with respect to the levels of expertise can be computed as $$\frac{df}{de_k} = \frac{df}{d\mu} \cdot \frac{d\mu}{de_k}.$$
Gaussian model
--------------
The Gaussian log-likelihood is $$f(\mu,v) = -\frac{(x-\mu)^2}{2v} -\frac{1}{2}\log(v) -\frac{1}{2}\log(2\pi),$$ where the composition rules for the mean and variance are $$\mu = \sum_k r_k \mu_k, \quad v = \sum_k r_k (v_k+\mu_k^2) -\mu^2, \quad r_k = \frac{e_k}{\sum_{k'}e_{k'}}.$$
### Derivative with respect to the composed mean and variance
The derivative of the log-likelihood with respect to the composed mean and variance are $$\frac{df}{d\mu} = \frac{x-\mu}{v}, \quad \frac{df}{dv} = \frac{(x-\mu)^2}{2v^2} - \frac{1}{2v} = \frac{(x-\mu)^2-v}{2v^2}.$$
### Derivative with respect to the levels of expertise
The derivative of the composed mean and variance with respect to the levels of expertise are $$\frac{d\mu}{de_k} = \frac{\mu_k E - \sum e_{k'}\mu_{k'}}{E^2} = \frac{\mu_k-\mu}{E},$$ $$\frac{dv}{de_k} = \frac{q_k E - \sum e_{k'}q_{k'}}{E^2} - 2\mu\frac{d\mu}{de_k} = \frac{q_k-q}{E} - 2\mu\frac{\mu_k-\mu}{E} = \frac{v_k-v+(\mu_k-\mu)^2}{E},$$ where $E = \sum_{k'} e_{k'}$ and $q_k = v_k + \mu_k^2$, $q = v+\mu^2$. The derivative of the log-likelihood with respect to the levels of expertise can be computed as $$\frac{df}{de_k} = \frac{df}{d\mu} \cdot \frac{d\mu}{de_k} + \frac{df}{dv} \cdot \frac{dv}{de_k}.$$
Numerical optimization
======================
For binary data, the log-likelihood of the smoothed model is a concave function of $\bm{\mu_k}(d)$, see Section \[sec:derivative\_expert\_probas\]. We could therefore in principal perform an optimization for the experts opinions using Newton’s method. There are a few complications though. One problem is that the second derivative is proportional to the squared responsibility and hence close to 0 if the level of expertise is small. Consequently, template updates in regions with low expertise would be unstable. To deal with that we could add a penalty on the squared log-odds for example. Another problem is that the Newton steps may lead to probability estimates outside of $[0,1]$. This can be dealt with by pulling the estimates back into the unit interval. Note that working on the log-odds scale is not possible because the log-likelihood of our model is not concave in the expert log-odds. Because of these complications we use the simple, fast and robust heuristic (\[eq:mean\_update\]) instead of Netwon’s method.
|
---
abstract: 'In many applications of vehicle routing, a set of time windows are feasible for each visit, giving rise to the Vehicle Routing Problem with Multiple Time Windows (VRPMTW). We argue that such disjunctions are problematic for many solution methods, and exemplify this using a state of the art Adaptive Large Neighbourhood Search heuristic. VRPMTW comes in two variants depending on whether the time used en route must be minimised. A more compact and corrected mathematical formulation for both variants of the problem is presented, and new best solutions for all but six benchmark instances of VRPMTW without time minimisation is found. A new solution representation for VRPMTW with time minimisation is presented, its importance is demonstrated and it is used to find new best solutions for all but one benchmark instance of VRPMTW with time minimisation.'
address: 'DTU Management Engineering, Technical University of Denmark'
author:
- Rune Larsen
- Dario Pacino
bibliography:
- 'references.bib'
title: Fast delta evaluation for the Vehicle Routing Problem with Multiple Time Windows
---
Vehicle routing,multiple time windows,adaptive large neighbourhood search,dynamic programming
Introduction
============
The literature on is very extensive, both concerning applied methods and on variations of the problem [@Braekers2015TheReview]. A specific variation that has recently caught the attention of the research community is the . As mentioned in [@Belhaiza2014AWindows], applications of this problem can be found in delivery operations of furniture and electronic retailers, where a choice of delivery periods might be offered. In countries such as Denmark, deliveries to e.g. large grocery distribution centres require the booking of available time slots on an on-line system. Missing a booked window might result in extreme delays for the entire route. A solution to the , where the available time slots are modelled as multiple time windows, will, not only, optimise the route plan but also advice on which time slot to book. As similar case-study can also be found in Portuguese companies [@Amorim2014AStudy].
The is not a new problem in the literature, though the list of works is very small. To the best of the authors’ knowledge, the problem was introduced by [@Favaretto2007AntVisits]. The authors provided a novel set of benchmark instances and proposed an ant colony optimisation approach for the single and multiple visits variant of the problem. The current state-of-the-art approach for the is the hybrid variable neighbourhood tabu search presented by [@Belhaiza2014AWindows]. They generalised a set of well-known neighbourhood operators for the VRPMTW: 2-exchange, 2-opt, relocate, 3-node swap, and a 3-exchange. The operators are applied within a tabu search framework where shaking phases and restarts are applied to increase diversification. To assist in time minimisation, they present a recursive approach to calculate the optimal allocation of waiting times along the route, and thus implicitly selecting the time windows used. Subsequently, the same approach is used within a game theory framework for multiple-criterion optimisation by the same authors [@Belhaiza2016AProblem]. The use of multiple time windows has also been studied in other related problems, from which inspiration and knowledge can be drawn. Applied to the , [@Pesant1999OnProblem] show the flexibility of constraint programming when modelling e.g. multiple time windows. [@Baltz2014ExactSelection] show how simple feasibility checks during insertion operators can handle multiple time windows, while [@PaulsenHeuristicWindows] use dynamic programming to evaluate the optimal set of time windows to use when time minimising an entire route in the context of genetic algorithms. Multiple time windows have also been studied in the team orienting problem [@Souffriau2010TheWindows; @Lin2015AWindows; @Tricoire2010HeuristicsWindows], where they have mostly been handled by feasibility checks and not as part of the optimisation. Recently multiple time windows have also been included into a mathematical formulation of the long-haul transportation problem [@Neves-Moreira2016ALocations].
In this paper, we present a more compact and a corrected mathematical formulation for the problem, we describe how to adapt the framework [@ALNS2006] for the with a novel solution representation allowing delta evaluation to get the cost of insertions when time minimising for the *optimal* allocation of time windows. As opposed to most works on wherefrom we draw inspiration, our algorithm can be applied to both the distance and time minimisation variant of the problem. Computational results show that the adapted can find better solutions than the current state-of-the-art heuristic while retaining the modelling flexibility of the framework. We also measure the importance of each component of the solution method.
The remainder of the paper is organised as follows: First, we introduce a compact formulation for the problem (Section \[sec:problem\]). Section \[sec:ALNS\] describes the used in this paper, focusing on the fast evaluation of the insertions. Section \[sec:config\] documents the setup of the framework to solve the , and quantifies the importance of each component. We then demonstrate the effectiveness of the proposed heuristic on the instances from the literature in Section \[sec:results\], and finally, we conclude on our findings in Section \[sec:conclusions\].
Problem definition and formulation {#sec:problem}
==================================
We consider two variants of the VRPMTW. In both versions, we associate a cost with the travel distance and the number of vehicles used. In the second version, we are also minimising the time used by the vehicles.
The formulation is a simplified version of the model from [@Belhaiza2014AWindows]. We use fewer variables, but model the same problem. Eliminated variables include the flow carried along each arc $y^k_{ij}$, $r^k$ indicating if a vehicle is used, $w^k_i$ representing the waiting time at each customer, $q_d^k$ representing the amount loaded at depot, $d_k$ representing the duration of a vehicles route, and finally $b_i^k$ representing the departure time each vehicle at each customer. Furthermore, $v^p_i$ and $z_i^k$ have been merged into a new $z^k_{ij}$, indicating if a visit is served by a vehicle in a given time window. The number of constraint types are reduced from 20 in [@Belhaiza2014AWindows] to 13, easing readability further. Lastly, we adjusted the objective to reflect the one used in the evaluation of the instances in both papers. Namely the service time $s_{i}$ has been added when minimising time, which is currently missing in [@Belhaiza2014AWindows].
Assuming that an unlimited set of vehicles $K$ is available, the problem is formulated on a graph $G=(V,A)$, in which the set of vertices are partitioned into $V=N\cup \{d_{start},d_{end}\}$ with $N$ being a set of vertices representing the visits, $d_{start}$ being the start depot, and $d_{end}$ being the end depot for the vehicles. In the mathematical model we assume an heterogeneous fleet to match [@Belhaiza2014AWindows], where each vehicle $k\in K$ has a capacity $Q_k$ and a maximum route duration $T^k$. For the remainder of the paper we operate with a homogeneous fleet, as that is the case for the test instance. Each visit $i\in N$ has a cargo demand $q_i$ and a service time $s_i$. A vehicle is allowed to service a visit only within a fixed set of time-windows. Let $P_i$ be the set of time-windows for visit $i$. Each time-window $p\in P_i$ is then defined within the interval $[l^p_i,u^p_i]$. The travel time between two visits $i,j\in V$ is denoted $t_{ij}$, to which we associate a cost $c_{ij}^k$ if travelled by vehicle $k$. An additional cost $c_k$ is also paid for using vehicle $k$. An overview of the notation can be found in Table \[tab:notation\].
Parameter Description
--------------------------- ----------------------------------------------------------------------------------
$N$ The set of vertices representing the visits
$V$ The set of vertices including the depots
$P_i$ The set of time-windows for visit $i\in V$ and the time horizon for the depots
$t_{ij}\in \mathbb{R}$ The driving time between visit $i,j\in V$
$s_i\in \mathbb{R}$ The service time at visit or depot $i\in V$
$c^k\in \mathbb{R}$ The cost of using vehicle $k\in K$
$c^k_{ij}\in \mathbb{R}$ The cost of using vehicle $k\in K$ between visits $i,j \in V$
$q_i\in \mathbb{R}$ Demand of visit $i\in N$
$Q_k\in \mathbb{R}$ Capacity of vehicle $k\in K$
$l^p_i\in \mathbb{R}$ Lower bound of time window $p\in P_i$ of visit $i\in N$
$u^p_i\in \mathbb{R}$ Upper bound of time window $p\in P_i$ of visit $i\in N$
$T^k\in \mathbb{R}$ Maximum route duration for vehicle $k\in K$
$B\in \mathbb{B}$ 1 if and only if time usage is penalised
$M\in \mathbb{R}$ A large constant
Variable Description
$x^k_{ij} \in \mathbb{B}$ 1 if and only if vehicle $k\in K$ travels between visits $i,j \in V$
$w^k_i\in \mathbb{R}$ Waiting time for vehicle $k\in K$ at visit $i\in N$
$z^k_{ip}\in \mathbb{B}$ 1 if and only if vehicle $k\in K$ uses time window $p\in P_i$ for visit $i\in N$
$a^k_i\in \mathbb{R}$ Time at which service starts at visit $i\in N$
: Mathematical notation\[tab:notation\]
The following formulation is based on four group of decision variables: $x^k_{ij}\in \mathbb{B}$ indicating if vehicle $k\in K$ services visit $j\in V$ after visit $i\in V$. The variable $z^k_{ip}\in \mathbb{B}$ is $1$ if and only if vehicle $k$ services visit $i$ during time-window $p\in P_i$. The start of service and waiting times of the vehicles are decided by the variables $a^k_i\in \mathbb{R}$ and $w^k_i\in \mathbb{R}$ respectively.
$$\begin{aligned}
&\text{\bf minimise}\nonumber\\
&\sum_{k\in K}{ \sum_{i,j\in V}{ c^k_{ij}x^k_{ij} } }+B\sum_{i\in N}{\left(s_{i}+\sum_{k\in K}{w_i^k}\right)}+
\sum_{k\in K}{ c^k \left(\sum_{j\in N}{ x^k_{d_{start}j}} \right)}\label{eq:obj}\\
&\text{\bf subject to}\nonumber\\
& \sum_{k\in K}{\sum_{j \in V}{x^k_{ij}} } = 1 \quad \forall i\in N \label{eq:coverage}\\
& \sum_{j \in V} x_{ji}^k = \sum_{j \in V} x_{ij}^k \quad \forall i\in N, k\in K \label{eq:flow} \\
& \sum_{j \in V}{x^k_{d_{start}j}} \leq 1 \quad \forall k\in K \label{eq:leave}\\
& \sum_{j \in V}{x^k_{d_{end}j} } + \sum_{i \in V}{x^k_{i d_{start}} } = 0 \quad \forall k\in K \label{eq:reenter}\\
& \sum_{i \in V} \sum_{j\in N} q_j x_{ij}^k \leq Q_k \quad \forall k\in K \label{eq:capacity}\\
& \sum_{j \in V} x_{ij}^k = \sum_{p \in P_i} z_{ip}^k \quad \forall i\in V, k\in K \label{eq:tw_choose}\\
& a^k_j \geq a^k_i + t_{ij} + s_i - M(1-x^k_{ij}) \quad \forall k\in K, i,j\in V \label{eq:service_time}\\
& a^k_i \geq l^p_i - M(1-z^k_{i,p}) \quad \forall k\in K, i\in V, p\in P_i \label{eq:tw1}\\
& a^k_i \leq u^p_i + M(1-z^k_{i,p}) \quad \forall k\in K, i\in V, p\in P_i \label{eq:tw2}\\
& w^k_j \geq a^k_j - a^k_i - t_{ij} - s_i - M(1-x^k_{ij}) \quad \forall k\in K, i\in V ,j\in N \label{eq:waiting}\\
& T^k \geq a^k_{d_{end}} - a^k_{d_{start}} \quad \forall k\in K \label{eq:duration}\\
& w^k_i, a^k_i \geq 0 \quad \forall i \in N, k \in K \label{eq:domainR}\\
& x^k_{ij}, z^k_{i,p} \in \{0,1\} \quad \forall i,j \in N, k \in K, p\in P_i \label{eq:domainB}\end{aligned}$$
The objective function minimises three terms: the total cost of the route, the total service and waiting time and the cost of vehicles used. The inclusion of the service and waiting time into the objective is dictated by the parameter $B$, which is $1$ if we want to allow this minimisation and $0$ otherwise. Equations are the coverage constraints, while are the flow balance constraints. We ensure that each vehicle leaves its start depot at most once and that it cannot return to it or leave the end depot . The capacity of a vehicle is restricted by . The modelling of the multiple time-windows begins with Constraints . This constraint ensures that if a vehicle $k$ services visit $i$, it must choose one of the available time-windows. The selection will then be indicated by the variable $z^k_{ip}$. The time when the service of visit $i$ is started ($a^k_i$) is defined by . The service start time is then constrained to be within the selected time-window with Constraints and . The waiting time is calculated by and the route duration is restricted by . Finally, and define the domain of the variables.
The model is descriptive only as attempts to obtain bounds or solutions were unsuccessful.
An for the {#sec:ALNS}
===========
Typically ALNS uses a solution representation consisting of a list of visits, (earliest/latest) arrival times at each visit, and (forward/backward) slack describing how much forward/backward in time each visit can be moved. The information added to the list allows fast evaluation of insertion costs/feasibility, and can trivially be extended to by adding information about the time window chosen for each visit upon insertion.
We propose augmenting the list of visits with sets of labels for each visit, allowing efficient evaluation of the cheapest insertion for *any* allocation of time windows. Crucially, infeasible insertions can be detected in constant time.
We test our solution representation on a version of the framework from [@ALNS2006] adapted to the . is an extension of the Large Neighbourhood Search from [@Shaw1998], where, using a destroy operator, a portion of the current solution is destructed. A repair heuristic is then applied to re-build the solution. extends this search procedure by allowing multiple destroy and repair heuristics. At each iteration, it selects a destroy and a repair method using an adaptive engine. It applies the destroy method to remove visits, and applies the repair method to reinsert them. If the cost of the solution decreases, it is accepted. If it is not, the acceptance criterion from simulated annealing is applied, which results in a decreasing acceptance of worsening solutions as the search progresses, and makes acceptance increasingly unlikely as the cost increase grows. Afterwards, the temperature is adjusted using a geometric cooling schedule, and the likelihood of re-selecting the destroy and repair methods is updated based on the outcome of the destroy-repair operation. The procedure terminates once it reaches a time limit. We first present an overview of the algorithm, followed by a description of its components.
Step 1 - Initial Solution)
: [A greedy insertion heuristic is used to find a first feasible solution to the problem. We randomly iterate over visits. Once we find a route where the visit can be inserted we apply a best insertion. Should no route be feasible a new one is opened. The algorithm continues until all visits are assigned.]{}
Step 2 - Online temperature tuning
: [Since the cooling schedule of the simulated annealing acceptance criterion can have a large impact on the search, we implemented a form of online instance specific automated tuning. To do so, we execute the destroy and repair part of our for a limited number of iterations, allowing only improving solutions. During the search, the collected statistics are used to define an instance specific cooling schedule. More details about this step can be found in Section \[sec:onlineTuning\].]{}
Step 3 - Route minimisation
: [In order to minimise the number of routes used, we artificially increase the cost of each route, forbid adding routes and add a cost for having unassigned visits. Starting from the initial greedy solution, we unassign all the visits of the smallest route. We then look for a new solution, using the destroy and repair part of the , where the number of allowed routes is reduced by one. This process iterates until a time limit is reached. The best feasible solution that minimises the number of routes is then returned. A detailed description of the route minimisation step can be found in Section \[sec:routeMinimisation\].]{}
Step 4 - Solution optimisation
: [In the final step, the solution from Step 3 is used as starting point. Here, the route costs are returned to their original values and unassigned visits are disallowed. The destroy and repair part of the algorithm is run for the reminder of the time limit. ]{}
Step 2-4 all utilise the same destroy and repair routines, differing only in the objective function, strategy for saving the best found solution and a strategy for when to remove entire routes routes (during route minimisation).
Destroy methods
---------------
We apply seven different destroy methods. For all of them, the number of visits to remove from the solution is chosen randomly from the interval of $[10,40]$. Each destroy method is then tasked with removing the visits from the solution. ALNS is a well known method for solving s, therefore a number of efficient destroy and repair methods have already been studied [@ALNS2006; @ALNS2]. We apply them to .
Random
: from [@ALNS2006] selects the visits to remove randomly.
Cluster $y$
: for $y\in\{1,2,4\}$ are three destroy methods that select visits to remove randomly, but also remove $y$ successors to each visit. The method is based on ideas from the VeRoLog conference, but no publication of the method could be found.
Geometric
: adapted from [@ALNS2], selects a random seed visit. It then repeatedly selects a removed visit $v_{select}$, sorts the visits remaining in the solution in ascending order by distance to $v_{select}$, and selects the visit that lies on the $r^4*n_{rem}$’th position of the ordered list, where $r$ is a random number in the interval $[0,1]$ and $n_{rem}$ is the number of visits remaining in the solution.
Time
: adapted from [@ALNS2], selects visits like the geometric destroy, but sorts remaining visits in ascending order according to the difference between each visits start time and the start time of the randomly selected visit scheduled for removal $v_{select}$ in the current solution.
Solution history
: adapted from [@ALNS2], is based on an assumption that if edges are not typically found in good solutions, a better alternative is more likely to exist, and removing them from a solution is more likely to yield improvements. It keeps track of the cost of the best solution observed, that contains any edge $(i,j)$. It scores each visit using sum of the score of the two edges connected to it in the solution. All visits are sorted in descending order according to their score, and the visits are removed choosing the $r^4 \cdot n_{rem}$’th where $r$ is a random number in the interval $[0,1]$.
Concretely for a visit $i$ in a solution, connected with edges $(pre(i),i)$ and $(i,suc(i))$ in the incumbent solution: If the best solution observed using the edge $(pre(i),i)$ cost 5000 and the best solution using edge $(i,suc(i))$ cost 4500, the score of the visit is 9500.
Repair methods
--------------
After each destroy method has been employed, one repair method from [@ALNS2006; @ALNS2] is chosen and applied to all unassigned visits.
Regret-2
: was presented in [@ALNS2006] who adapted methods from [@regret]. We evaluate the cost of inserting each unassigned visit in each route, and rank the visits in decreasing order of the cost of their second best insertion. The visit with the highest cost of insertion into its second best route, is inserted into its best route.
Regret-2 randomised
: is a randomised version of Regret-2. It ranks all unassigned visits in increasing order of the cost of their insertion into the second best route, but each cost is increased with a percentage chosen uniformly randomly in \[0%,50%\].
### Caching insertion costs
Each time an insertion has been made, all unassigned visits have been evaluated for insertion in all routes. After the insertion, all previously calculated insertion costs are valid except those pertaining to the route wherein the visit was inserted. We employ this standard caching approach, where the insertion cost of each visit into each route is cached lazily. The cache is invalidated for all insertions pertaining to a route, each time a visit is inserted into the route.
Delta-evaluation and feasibility
--------------------------------
ALNS, as well as most common heuristics, needs an efficient calculation of the cost of inserting a visit into a route in a given position. This is typically achieved using $\Delta$-evaluation (also known as incremental updates) [@Hoos:2004:SLS:983505 pp48], where only a (small) part of the full objective needs to be recalculated. For a VRPTW this can be accomplished in constant time per position evaluated by caching enough information to establish an earliest possible start time, and a latest possible start time for each visit. The cost of maintaining these are typically $O(n)$ per update to the route, where $n$ is the size of the route. These methods can be generalised for both variants of the VRPMTW. Note that we include the start and the end depot in the routes and associate time windows to them.
### Without time minimisation {#sec:delta_no_time}
In the literature, the VRPTW and VRPMTW variants without time minimisation are the most common. The relative ease in evaluating feasibility and cost of insertions has been observed by others, and makes this version of the problem a good starting point to understand the complexities of VRPMTW. We easily adapt forward slack by [@Savelsbergh] to work on without time minimisation.
Let $pre(i)$ and $suc(i)$ define the predecessor and the successor respectively of the visit $i$ in a given solution.
Let $es_i$ and $ls_i$ for all $i \in V$ be the earliest and latest start time of visit $i$ respectively. We have that
$$\label{eq:es}
es_i = \min_{p\in P_i | u^p_i \geq es_{pre(i)}+t_{pre(i)i}} max(es_{pre(i)}+t_{pre(i)i},l^p_i)$$
with a base case of $es_i = \min_{p\in P_i} l^p_i$ if $i$ has no predecessors. $$\label{eq:ls}
ls_i = \max_{p\in P_i | l^p_i \leq ls_{suc(i)}-t_{isuc(i)}} min(ls_{suc(i)}-t_{isuc(i)},u^p_i)$$ with a base case of $ls_i = \max_{p\in P_i} u^p_i$ if $i$ has no successors.
Equation and can be evaluated in $O(|P_i|)$, making the whole update procedure take $O(n|P_{max}|)$ where $P_{max}= \max_{i\in V} |P_i|$.
Insertion of $j$ between $i$ and $suc(i)$ in the incumbent solution using time window $p\in P_j$ in a route is feasible if and only if: $$\label{eq:insfeas}
es_i+t_{ij} \leq u^p_j \land ls_{suc(i)}-t_{jsuc(i)}\geq l^p_j$$
and the cost of the insertion is given by:
$$\label{eq:cost}
cost=c_{ij}+c_{jsuc(i)}-c_{isuc(i)}$$
Note that Equation is verifiable in $O(|P_j|)$, and Equation is constant time. For the problems treated in this paper, the number of time-windows is limited, and further optimisations are not considered.
### With time minimisation
When time minimisation is included in the objective, the calculations from Section \[sec:delta\_no\_time\] are insufficient to establish the cost of an insertion. Two distinct methods can be applied when treating this problem:
1. When inserting a visit, keep track of and fix the time-window used for insertion.
2. Fix only the ordering of the visits, and evaluate insertions for all possible allocations of time-windows to visits.
The former suffers from routes being very hard to create good initial solutions for, as decisions regarding a visit expected position in the route is taken on insertion, including for the first inserted visit. Furthermore, difficulties with undoing the sub-optimal routes were observed during local search or destroy repair iterations, due to larger parts of the solution having to be changed in order to effect the change.
The second method suffers from the increased complexity in evaluating feasibility and cost of insertions. These challenges however, are not insurmountable.
When evaluating the insertion of a visit on a given position in a route, feasibility can be evaluated as in the case without time minimisation. To evaluate the resulting minimum route duration, we apply a labelling algorithm which is a version of dynamic programming. It works by keeping track of any promising (non dominated) ways the vehicle could arrive at the current visit. This includes when the vehicle should start, implicitly what time windows it should use on the way, and how much later the vehicle could start and still hit the time windows. Similar labels are kept describing any possible path from each visit, along the rest of the route and back to the depot: For each visit we maintain two sets of labels $L_i^f$ and $L_i^b$, namely the labels describing forwards and backwards paths from each visit $i\in V$. $L_i^b$ consists of labels containing the earliest start time $es_i$, the cost-less backwards slack $bs_i$, and resulting route start time $st_i$. $L_i^f$ consists of labels containing the latest start time $ls_i$, the cost-less forward slack $fs_i$, and the resulting route end time $et_i$.
Given two consecutive visits in a route $i$ and $j$, any backward labels $(es_i,bs_i,st_i)\in L_i^f$ and any forward label $(ls_j,fs_j,et_j)\in L_j^b$. The earliest start time for the visit we wish to insert ($o$) using the $p$’th time window $[l^p_o,u^p_o]$ is $es_o = max(es_i+t_{io},l^p_o)$ and the latest start time is $ls_o = min(ls_j-t_{oj}-s_o,u^p_o)$. The insertion is feasible if and only if $ls_o>es_o$. The resulting duration of the route is $et_j-st_i-min(ls_o-es_o, bs_i+fs_j)$. The term $min(ls_o-es_o, bs_i+fs_j)$ is due to compacting of the route by time shifting the forward and backward paths.
\(A) at (7,0); (B) at (0,5);
(0,0) – (A) node\[anchor=north\] [$space$]{}; (0,0) – (B) node\[anchor=south\] [$time$]{};
in [0,1,2,3,4,5,6]{} (cm,1pt) – (cm,-1pt) node\[anchor=north\] [$\x$]{};
in [0,1,2,3,4,5,6,7,8,9]{} (1pt, 0.5 cm) – (-1pt,0.5 cm) node\[anchor=east\] [$\y$]{};
(0,0) grid (7,5);
(1,0.5) coordinate (a\_1) – (1,1) coordinate (b\_1); (2,1) coordinate (a\_2) – (2,1.5) coordinate (b\_2); (3,1.75) coordinate (a\_3) – (3,2.25) coordinate (b\_3);
(5,2.5) coordinate (a\_5) – (5,3) coordinate (b\_5); (6,3.75) coordinate (a\_6) – (6,4.25) coordinate (b\_6);
(0,0.5) coordinate (a\_1) – (1,1) coordinate (a\_2); (1,1) coordinate (a\_1) – (2,1.5) coordinate (a\_2); (2,1.5) coordinate (a\_1) – (3,2) coordinate (a\_2); (3,2) coordinate (a\_1) – (4,2.5) coordinate (a\_2);
(0,0.25) coordinate (a\_1) – (1,0.75) coordinate (a\_2); (1,0.75) coordinate (a\_1) – (2,1.25) coordinate (a\_2); (2,1.25) coordinate (a\_1) – (3,1.75) coordinate (a\_2);
(3,1.75) coordinate (a\_1) – (4,2.25) coordinate (a\_2); (4,2.5) coordinate (a\_1) – (5,3) coordinate (a\_2);
(5,3) coordinate (a\_1) – (6,3.5) coordinate (a\_2); (6,3.5) coordinate (a\_1) – (6,3.75) coordinate (a\_2); (6,3.75) coordinate (a\_1) – (7,4.25) coordinate (a\_2);
at (0,0.25) ; at (7,4.25) ;
at (3,1.75) ; at (5,3) ;
at (4,2.25) ; at (4,2.5) ;
(1.5,1.25) coordinate (a\_6) – (1.5,1) coordinate (b\_6) node\[anchor=north\] [$bs_i$]{};
An example of inserting visit $o$ between visits $i$ and $j$ using the labels is illustrated in figure \[fig:twPaths\]. The backwards path (black, dashed) from space index 0 to space index 3 indicates the driving time to get to the visit at space index 3 as early as possible, without starting the route earlier than necessary ($st_i$). The path can be shifted forward in time (up to the green and dashed line) by $bs_i$ with a resulting change in start time up to $st_i+bs_i$. Further delays are possible, but the resulting start time will be capped at $st_i+bs_i$. The forward path from space index 5 to 7 (black, dashed) is the latest possible path to arrive at the earliest possible end time $et_j$. If this is shifted earlier, the resulting end time will remain unchanged, as the path is already constrained by the time window at space index 6, an thus contains no slack ($fs_j=0$).
### Label generation and elimination
Let the start depots backwards label set $L^b_{d_{start}}$ consist of the label $(0,\infty,0)$ and the end depots forward label set $L^f_{d_{end}}$ consist of the label $(\infty,\infty,\infty)$. With these base cases we define the remaining sets recursively.
Backward labels:
: Let $j=pre(i)$ and $(es_j,bs_j,st_j) \in L^b_{j}$ be a backward label of $j$. Using the $p$’th time window $[l^p_i,u^p_i]$ a new label $(es_i, bs_i, st_i)$ can be generated as $es_i = max(es_j+t_{ji}+s_{j}, l_i^p)$, $bs_i=max(0,bs_j-max(0,l^p_i-es_j-t_{ji}-s_j))$ and $st_i=st_j+bs_j-bs_i$.
Forward labels:
: Let $j=suc(i)$ and $(ls_j,fs_j,et_j) \in L^f_{j}$ be a forward label of $j$. Using the $p$’th time window $[l^p_i,u^p_i]$ a new label $(ls_i, fs_i, et_i)$ can be generated as $ls_i = min(ls_j-t_{ij}-s_{i}, u_i^p)$, $fs_i=max(0,fs_j-max(0,ls_j+t_{ij}+s_i-u^p_i)$ and $et_i=et_j-max(0,ls_j+t_{ij}+s_i+u^p_i)$.
Given backward labels $(es_i,bs_i,st_i)$ and $(es_j,bs_j,st_j)$, we say that the former dominates the latter if and only if $st_i+bs_i \geq st_j+bs_j \wedge es_i \leq es_j$.
Given forward labels $(ls_i,fs_i,et_i)$ and $(ls_j,fs_j,et_j)$, we say that the former dominates the latter if and only if $et_i-fs_i \leq et_j-fs_j \wedge ls_i \geq ls_j$.
$bestCost=\infty$
The cheapest insertion of visit $o$ between visits $i$ and $j$ can the be calculated using Algorithm \[alg:checking\]. It iterates through the backward labels for the previous visit $i$ (line \[alg:iterBL\]) and the forward labels for the next visit $j$ (line \[alg:iterFL\]), checking for compatibility ($es_i+t_{io}>ls_j-t_{oj}$) (line \[alg:compat\]). If the labels are compatible in the sense that driving time seems feasible without time windows, they are expanded for each matching time window (line \[alg:forTW\]) to visit $o$ as if it was already inserted (line \[alg:exp1\]-\[alg:exp2\]). The new forward and backward labels for $o$ are (if $es_o \leq ls_o$) then used to compute the route duration of the resulting route, and the delta cost of the insertion (line \[alg:newCalc1\]-\[alg:newCalc2\]). By carefully considering the conditions in line \[alg:compat\], \[alg:cond\] and \[alg:cond2\], iterations through either of the sets of labels can often be elided. This is crucial to obtain the performance observed in the paper.
Adaptive engine
---------------
ALNS uses an adaptive engine from [@ALNS2006] to select between destroy methods $D$ and repair methods $R$. The probability of selecting a specific destroy method $de$ is $\frac{weight_{de}}{\sum_{des\in D} weight_{des}}$, and the probability of selecting a repair method $re$ is $\frac{weight_{re}}{\sum_{res\in R} weight_{res}}$.
Each time a repair or destroy method is chosen, its weight is updated using a score and a decay factor $decay$. The score $sc$ depends on the result of the destruction and reconstruction, and is set to 1, $score_a$, $score_i$ or $score_b$ depending on if the new solution is rejected, accepted, improving or a new best respectively. The scores and the decay factors are parameters to be set during tuning. The new weight is calculated as $weight_{new}=weight_{old}\cdot decay + sc \cdot (1-decay)$.
The weight selection ensures that no destroy and repair method are ever entirely excluded as its weight can never be reduced below 1.
Online temperature tuning {#sec:onlineTuning}
-------------------------
We use an automatic tuning to control the simulated annealing part of the ALNS. The online temperature is parametrised by a number of iterations $n_{init}$ to run only accepting improving solutions, as well as $\Delta cost_{init}$ and $\Delta cost_{end}$ the desired worsening of the solution to be accepted with a 50% probability during the start and the end of the search respectively. The initial iterations establishes an approximation for the $cost$ of the solutions to be used in setting the temperature.
Recall that in simulated annealing the acceptance criteria for solutions worse than the current is controlled by $r<e^{\frac{\Delta cost}{cost}\cdot \tau}$, where $r\in[0,1]$ is a random number from a uniform distribution, and $\tau$ is the current temperature.
After the first $n_{iterTT}$ iterations, the start and the end temperature is determined. $$\label{eq:starttemp}
\tau_{start}=\frac{log(0.5) \cdot cost_{init}}{\Delta cost_{init}}, \tau_{end}=\frac{log(0.5) \cdot cost_{end}}{\Delta cost_{end}}$$ They are computed using with the given target $\Delta cost_{init}$ and $\Delta cost_{end}$. A geometric cooling scheme is used to adjust the temperature $\tau$ during the search.
Route minimisation {#sec:routeMinimisation}
------------------
The strategy for minimising the number of routes, is based on the concept presented in [@ALNS2006]. An initial solution is created using the greedy construction heuristics in Step 1. After the temperature tuning (Step 2), the remaining optimisation is performed using the destroy and repair part of the described in Section \[sec:ALNS\], with the following modifications.
The optimisation is divided into two phases: A route minimisation phase (Step 3) and an optimisation phase (Step 4). During the route minimisation phase, unassigned visits are penalised with a cost of $p_u=10000$ and routes are penalised with a cost of $p_r=1000000$. Thus any partial solution with fewer routes will be better cost wise if less than $\frac{p_r}{p_u}$ visits are unassigned. is prevented from storing a partial solution as the best solution $s_{best}$, unless all visits have been inserted into the solution. Each time a solution with all visits assigned is obtained, the route with the fewest visits is destroyed.
The shift from the route minimisation phase (Step 3) to the optimisation phase (Step 4) is triggered when the number of visits being unassigned is too high for the stage of the optimisation as depicted in Figure \[fig:minim\]. This corresponds to one of the following conditions are fulfilled: 1) Ten percent of the allotted time has passed, and no visits are unassigned in the current solution. 2) Ten percent of the time has passed, and more than 5 visits remain unassigned. (See parameter tuning in Section \[sec:params\]) 3) Ten percent of the time has passed, more than $percent/10-2$ visits remain unassigned, where $percent$ indicates the percentage of computation time is left. In the two latter cases, failed to make the partial solution feasible with the given number of routes, and then algorithm reverts to the best found solution $s_{best}$ that is guaranteed to be feasible.
\(A) at (5,0); (B) at (0,3);
(0,0) – (A) node\[anchor=west\] [Percent done]{}; (0,0) – (B) node\[anchor=south\] [Unassigned accepted]{};
in [0,10,20,30,40,50,60,70,80,90,100]{} (0.05 cm,1pt) – (0.05 cm,-1pt) node\[anchor=north\] [$\x$]{};
in [0,1,2,3,4,5]{} (1pt, 0.5 cm) – (-1pt,0.5 cm) node\[anchor=east\] [$\y$]{};
(0,0) grid (5,3); (0.5,2.5) coordinate (a\_1) – (0.5,3) coordinate (a\_2);
(0.5,2.5) coordinate (a\_1) – (2,2.5) coordinate (a\_2); (2,2.5) coordinate (a\_1) – (2,2) coordinate (a\_2);
(2,2) coordinate (a\_1) – (2.5,2) coordinate (a\_2); (2.5,2) coordinate (a\_1) – (2.5,1.5) coordinate (a\_2);
(2.5,1.5) coordinate (a\_1) – (3,1.5) coordinate (a\_2); (3,1.5) coordinate (a\_1) – (3,1) coordinate (a\_2);
(3,1) coordinate (a\_1) – (3.5,1) coordinate (a\_2); (3.5,1) coordinate (a\_1) – (3.5,0.5) coordinate (a\_2);
(3.5,0.5) coordinate (a\_1) – (4,0.5) coordinate (a\_2); (4,0.5) coordinate (a\_1) – (4,0) coordinate (a\_2);
Configuring the algorithm {#sec:config}
=========================
The just described, is composed of a set of components, each of which may have parameters to set, or may be omitted entirely. In this section we provide our rationalisation for the chosen parameters, and justify each components inclusion if it can be omitted.
Optimally each possible combination of parameters should be explored, along with each possible alternative implementation. Due to the sheer impossibility of this, a reduction in the search space is necessary. Where possible, we adopt values and methods known to work on VRPTW, and analyse deviations to increase the odds of at least a local optimum in the configuration space.
Initially we fixed the interval, from which the number of visits to release is chosen. Intervals start and end values based on ($[10,40]$ from [@ALNS2]) were investigated in increments of 5, but little change was observed around the chosen values, with a sharp degeneration in solution quality if the values increased by more than 15.
Next the strategy transitioning from route minimisation (Step 3) to solution optimisation (Step 4). Initial trials indicated a threshold of 5 unassigned customers after 10% of the search time had been expended, could be resolved in some cases. It was further observed that most subsequent improvements (Step 4) occur within the first 20% of the optimisation time, and thus this amount should be reserved for the solution optimisation (Step 4). A linear decline from 40% was chosen provisionally. Subsequent attempts to tune these parameters were unsuccessful due to the interdependence of the variables, and the big cost difference between solutions using varying number of vehicles being triggered both randomly and in a vast minority of instances.
Parameter tuning {#sec:params}
----------------
The algorithm is associated with a set of parameters to be tuned. To accomplish the tuning, we modify the instances by randomly reassigning the sets of time windows to the visits, thus creating a training set.
We have limited ourselves to setting two sets of parameters for all instances. One for the problem with time minimisation, and a set of parameters for the problem without time minimisation.
selects its repair and destroy methods using scores for previous performance. Recall that we apply the traditional selection procedures from [@ALNS2006], and score $score_{b}$, $score_{i}$ and $score_{a}$ points for a new best, an improvement to the current solution and an accepted solution respectively. The decay of the score $decay\in [0,1]$ determines the memory of the adaptive engine. Eg. $0.9$ meaning 10% of the score is based on the methods last score, and 90% on the previous score.
Parameters are tuned using the [*irace*]{} package from [@LopDubPerStuBir2016irace]. Irace works by sampling the domain of each parameter and running the resulting algorithm configurations against each other. Each tuning is given a budget of 15000 algorithm runs, and it will statistically eliminate configurations as they prove inferior. Amongst the survivors the best performing configurations are presented.
### Tuning without time minimisation
An initial run of the tuning was done to establish a baseline for each parameter, and test any assumptions made regarding minimum and maximum values of each parameter. These domains are specified in first line of table \[tab:initNoMin\].
$n_{iterTT}$ $\Delta cost_{init}$ $\Delta cost_{end}$ $score_{a}$ $score_{i}$ $score_{b}$ $decay$
-------------- ---------------------- --------------------- ------------- ------------- ------------- --------------
\[0,1000\] \[0,100\] \[1,20\] \[1,20\] \[1,20\] \[0.7,0.99\]
5641 213.93 4.38 4 10 15 0.77
5470 15.50 2.75 4 10 14 0.74
4801 201.14 1.16 4 10 15 0.75
4405 812.83 1.07 7 16 19 0.84
5312 46.97 4.57 4 10 14 0.74
: Initial tuning for the VRPMTW without time minimisation.[]{data-label="tab:initNoMin"}
The initial run resulted in the 5 configuration settings in table \[tab:initNoMin\]. All values are well within the specified domains rather and none of them are on an extreme value. In order to further converge the parameter settings, already established values $n_{iterTT}$ and $score_{a}$ are fixed. This also serves to remove equivalence causing symmetries from the set of algorithm configurations. All remaining parameter domains are further tightened to the values in Table \[tab:endNoMin\].
$n_{iterTT}$ $\Delta cost_{init}$ $\Delta cost_{end}$ $score_{a}$ $score_{i}$ $score_{b}$ $decay$
-------------- ---------------------- --------------------- ------------- ------------- ------------- --------------
5000 \[10,800\] \[0,5\] 4 \[10,20\] \[10,20\] \[0.7,0.99\]
5000 298.49 2.24 4 15 15 0.73
5000 223.80 1.78 4 16 20 0.73
5000 393.68 2.73 4 14 14 0.75
5000 332.21 2.58 4 15 18 0.76
5000 300.00 2.33 4 15 17 0.75
: Final tuning for the VRPMTW without time minimisation. The final line is the chosen configuration.[]{data-label="tab:endNoMin"}
Based on the parameter setting from in the top part of Table \[tab:endNoMin\] a the parameter tuning recommended the parameters in the final row of the table.
Comparing these values to other ALNS parameter settings, they are mainly characterised by an extreme emphasis on diversification. Both in terms of acceptance of worsening solutions by having a high $\Delta cost_{init}$, but also in a diverse selection of destroy/repair methods in having a low $decay$ factor, keeping more emphasis on recent results than historical.
### Tuning with time minimisation
The initial tuning/check was given the domains from Table \[tab:initMin\], and the following configurations were obtained:
$n_{iterTT}$ $\Delta cost_{init}$ $\Delta cost_{end}$ $score_{a}$ $score_{i}$ $score_{b}$ $decay$
-------------- ---------------------- --------------------- ------------- ------------- ------------- --------------
\[0,1000\] \[0,100\] \[1,20\] \[1,20\] \[1,20\] \[0.7,0.99\]
1028.33 19.46 4.91 2 11 14 0.93
1594.53 28.27 3.60 3 9 12 0.90
3942.04 518.73 2.14 7 11 14 0.90
570.89 20.63 6.02 2 11 14 0.92
: Initial tuning with time minimisation[]{data-label="tab:initMin"}
From Table \[tab:initMin\] a set of parameter domains for the final tuning was established and presented in the first row of Table \[tab:finalTuneMin\].
$n_{iterTT}$ $\Delta cost_{init}$ $\Delta cost_{end}$ $score_{a}$ $score_{i}$ $score_{b}$ $decay$
-------------- ---------------------- --------------------- ------------- ------------- ------------- --------------
\[10,600\] \[0,10\] \[1,7\] \[1,20\] \[10,20\] \[0.8,0.99\]
1999 44.65 4.95 2 9 17 0.83
1590 14.32 6.19 2 6 16 0.84
1219 11.16 5.31 3 11 12 0.84
1530 35.16 6.14 2 6 17 0.83
1600 26.00 5.50 2 8 16 0.83
: Final tuning with time minimisation[]{data-label="tab:finalTuneMin"}
The four candidate configurations presented in Table \[tab:finalTuneMin\] were reduced to the final algorithm configuration in the final row. These values continue the trend of diversification observed for the VRPMTW without time minimisation, but is adjusted for the increased time usage per iteration by having fewer pre-iterations ($n_{iterTT}$), and much lower start temperature.
### Importance of tuning
Note that some parameters such as $decay$ and $n_{iterTT}$ converge faster during the tuning, and thus can be assumed to have more impact on the final solution quality. Other parameters such as $\Delta cost_{init}$ seems to vary more. This lack of homogeneity between the configurations can be due to (a combination of) either a lack of tests to create statistical significance or lack of meaningful impact on the solution quality. In many applications tuning have shown to provide a great advantage, but our original test runs using the default configurations from Table \[tab:defaultConf\] gave solutions with only marginally worse average solutions, and one more best-known solutions. We thus conclude that further tuning is unwarranted.
$n_{iterTT}$ $\Delta cost_{init}$ $\Delta cost_{end}$ $score_{a}$ $score_{i}$ $score_{b}$ $decay$
-------------- ---------------------- --------------------- ------------- ------------- ------------- ---------
1000 10.00 3.0 2 4 10 0.9
: Original parameter values[]{data-label="tab:defaultConf"}
The value of the solver components
----------------------------------
To evaluate the importance of the solver components, we have omitted any component not strictly required. For implicit time windows (e.g. evaluating the new cost for the optimal allocation of time windows in the route) we implemented a modified solution representation that always inserts using the cheapest time window, and only changes time window upon reinsertion. When omitting online temperature tuning, we ran the algorithm with online temperature tuning initially, and fixed the average start and end temperatures selected. Each destroy and repair method can be omitted without complications.
[0.4]{} {width="\textwidth"}
[0.4]{} {width="\textwidth"}
Figure \[fig:omit\] indicates the increase in the average solution cost when running the time minimising algorithm for 60 seconds while omitting each of the tested components, while Figure \[fig:omitBest\] shows the increase in the best solution cost for the same settings. Since in the instances most costs components such as service times, vehicle costs and minimum distances are of greater magnitude than travelled distances and waiting times, small changes are more significant than they appear.
The solution history (Sol. Hist.) destroy is the most significant component when looking at average costs in Figure \[fig:omit\], which may be due to expanded options for choosing time windows in order to place visits close, and thus a greater chance that close visits can be place close together. Online temperature tuning is artificially important as instances can be divided into cheap instances with costs less than 5000 and instances with costs in excess of 10000, each requiring different temperatures. Third most important is our implicit time windows. The sharp increase in iterations resulting from deactivating them is far from able to compensate for the loss of flexibility.
Figure \[fig:omitBest\] indicates an increased importance of the implicit time window selection when looking for the best solutions rather than averages costs.
Computational results {#sec:results}
=====================
We test our adaptation of the on the instances proposed by [@Belhaiza2014AWindows]. The instances are generated based on the traditional Solomon instances [@solomon] with a set of newly generated time windows. As a result the instances have 100 customers, and vehicle capacity of 200 (cm1xx, rcm1xx, rm1xx), 700 (cm2xx) or 1000 (rcm2xx,rm2xx) All tests are performed on a Intel Core i7-4790K CPU @ 4.00GHz using a single thread. The results generated within a time limit of 60 seconds, which is slightly less than the average time used in [@Belhaiza2014AWindows](they do not operate with a fixed time limit).
For the instances, the cost is equivalent to the time for traversing the edge between the visits $c^k_{ij}=t_{ij}$, and the cost of using a vehicle $c^k$ is equal to its capacity $Q_k$. We further restricted the problem to assume $a^k_{d^k_{end}}\leq D^k$, meaning the route duration constraint is further restricted to be a deadline for returning to the depot. Any solution to this restricted problem is also a solution to the original as $a^k_{d^k_{end}}\leq D^k \Rightarrow a^k_{d^k_{end}}-a^k_{d^k_{start}}\leq D^k$ if and only if $a^k_{d^k_{start}}>0$.
[c c | r r | r r | r r ]{} & & &\
Instance & m & $B=0$ & & $B=0$ & 60 sec. & $B=0$ & 600 sec.\
& & Best R. & Avg. & Best R. & Avg. & Best R. & Avg.\
rm101 & 10 & 2977.2 & 3005.0 & **[2968.8]{} & **[2972.3]{} & **2968.8** & **2969.0**\
rm102 & 9 & 2759.4 & 2759.4 & **[2705.9]{} & **[2721.4]{} & **2705.9** & **2716.7**\
rm103 & 9 & 2692.5 & 2710.5 & **[2680.4]{} & **[2686.2]{} & **2680.4** & **2681.6**\
rm104 & 9 & 2696.6 & 2719.2 & **[2690.7]{} & **[2691.8]{} & **2690.7** & **2691.5**\
rm105 & 9 & 2688.8 & 2711.0 & **[2683.7]{} & **[2685.7]{} & **2683.7** & **2683.7**\
rm106 & 9 & **[2691.9]{} & **[2817.2]{} & 2700.9 & 2704.4 & 2700.9 & **2703.9**\
rm107 & 9 & 2690.8 & 2714.7 & **[2685.1]{} & **[2685.8]{} & **2685.1** & **2685.1**\
rm108 & 9 & 2729.1 & 2729.1 & **[2716.0]{} & **[2723.9]{} & **2716.0** & **2717.4**\
********************************
rm201 & **[2]{} & 3711.4 & 3720.5 & **[2750.0]{} & **[2769.0]{} & **2745.6** & **2756.0**\
rm202 & 2 & 2698.1 & 2717.3 & **[2681.3]{} & **[2685.0]{} & **2681.3** & **2685.0**\
rm203 & 2 & 2686.1 & 2702.0 & ***[2678.8]{}*** & **[2682.2]{} & 2679.0 & **2679.7**\
rm204 & 2 & 2680.5 & 2691.2 & **[2676.1]{} & **[2677.3]{} & **2672.8** & **2673.1**\
rm205 & 2 & **[2671.0]{}&2688.2&**[2671.0]{}& **[2671.6]{} & **2671.0** & **2671.1**\
rm206 & 2 & 2686.3 & 2704.9 & **[2679.0]{} & **[2681.9]{} & **2679.0** & **2679.0**\
rm207 & 2 & 2678.2 & 2696.2 & **[2674.7]{} & **[2679.7]{} & **2674.7** & **2678.2**\
rm208 & 2 & **[2673.9]{} & 2690.7 & 2675.7 & **[2677.7]{} & **2673.9** & **2676.2**\
**********************************
cm101 & 10 & **[3089.2]{} & **[3102.4]{} & 3151.4 & 3212.1 & **3073.0** & 3126.8\
cm102 & 12 & **[3426.9]{} & **[3426.9]{} & 3444.6 & 3488.9 & **3392.5** & 3463.0\
cm103 & **[11]{}& 3532.7 & 3572.7 & **[3465.3]{}& **[3526.2]{} & **3434.6** & **3477.1**\
cm104 & **[13]{}& 4051.3 & 4058.0 & **[3892.9]{}& **[3922.3]{} & **3859.6** & **3907.1**\
cm105 & **[10]{}& **[3060.6]{} & **[3077.3]{} & 3062.3 & 3093.9 & **3037.9** & **3051.7**\
cm106 & 10 & 2992.4 & 3020.2 & **[2982.2]{} & **[2990.3]{} & **2980.2** & **2983.3**\
cm107 & **[10]{}& 3256.5 & 3292.3 & **[3077.9]{}& **[3079.4]{} & **3077.9** & **3077.9**\
cm108 & 10 & 2968.7 & 2973.1 & **[2965.4]{} &**[2967.8]{} & **2965.4** & **2966.5**\
****************************************
cm201 & 5 & 4436.6 & 4452.5 & **[4367.6]{} & **[4400.1]{} & **4361.9** & **4374.4**\
cm202 & 6 & 4998.8 & 5024.9 & **[4983.2]{} & **[4990.0]{} & **4982.4** & **4987.4**\
cm203 & 5 & **[4445.8]{} & 4484.6& 4453.7 & **[4473.4]{} & **4440.3** & **4454.4**\
cm204 & 5 & 4335.2 & 4372.4 & **[4319.6]{} & **[4322.3]{} & **4317.5** & **4317.6**\
cm205 & 4 & 3863.5 & 3883.2 & **[3821.0]{} & **[3852.2]{} & **3816.8** & **3833.9**\
cm206 & 4 & 3722.0 & 3743.2 & **[3708.7]{} & **[3728.8]{} & **3699.0** & **3712.0**\
cm207 & 4 & 3968.4 & 3977.8 & **[3932.0]{} & **[3968.7]{} & **3917.4** & **3934.4**\
cm208 & 4 & 3771.1 & 3793.2 & **[3714.6]{} & **[3726.6]{} & **3714.6** & **3719.5**\
********************************
rcm101 & 10 & **[3062.0]{}&**[3062.0]{}&**[3062.0]{}& 3062.9 & **3062.0** & 3062.2\
rcm102 & 10 & 3132.2 & 3142.3 & **[3110.8]{} & **[3122.4]{} & **3110.8** & **3113.8**\
rcm103 & 10 & 3152.9 & 3163.8 & **[3121.3]{} & **[3127.0]{} & **3121.3** & **3121.9**\
rcm104 & 10 & 3119.6 & 3134.6 & **[3111.0]{} & **[3114.8]{} & **3111.0** & **3111.9**\
rcm105 & 10 & 3187.9 & 3210.7 & **[3166.2]{} & **[3173.0]{} & **3165.3** & **3169.5**\
rcm106 & 10 & 3218.9 & 3218.9 & **[3165.2]{} & **[3175.1]{} & **3158.2** & **3164.6**\
rcm107 & 11 & 3488.9 & 3514.0 & **[3487.7]{} & **[3494.4]{} & **3487.7** & **3490.2**\
rcm108 & 11 & 3592.7 & 3592.7 & **[3531.1]{} & **[3534.6]{} & **3531.1** & **3531.1**\
**********************************
rcm201 & 2 & 2804.0 & 2827.8 & **[2738.8]{} & **[2758.8]{} & **2715.4** & **2748.1**\
rcm202 & 2 & 2836.9 & 2846.8 & **[2734.0]{} & **[2757.7]{} & **2716.8** & **2733.8**\
rcm203 & 2 & 2721.9 & **[2725.4]{} & **[2705.0]{} & 2739.9 & **2704.8** & **2714.2**\
rcm204 & 2 & 2726.5 & 2743.1 & **[2698.6]{} & **[2712.7]{} & **2692.6** & **2699.3**\
rcm205 & 2 & 2754.5 & 2775.7 & **[2729.5]{} & **[2748.3]{} & **2718.8** & **2725.1**\
rcm206 & 2 & 2812.7 & 2830.6 & **[2732.3]{} & **[2766.6]{} & **2721.9** & **2743.1**\
rcm207 & **[2]{}& 3749.8 & 3786.8 & **[2861.0]{} & ****[3049.9]{}**** & **2861.0** & 3216.9\
rcm208 & 2 & 2791.4 & 2817.2 & **[2726.8]{} & **[2738.3]{} & **2722.7** & **2732.5**\
********************************
[c c | r r | r r | r r]{} & & &\
Instance & m & $B=1$ & & $B=1$ & 60 sec. & $B=1$ & 600 sec.\
& & Best R. & Avg. & Best R. & Avg. & Best R. & Avg.\
rm101 & 10 & 4041.9 & 4072.9 & **[4014.2]{} & **[4035.3]{} & **4014.0** & **4022.3**\
rm102 & 9 & 3765.1 & **[3765.1]{} & **[3732.0]{} & 3771.1 & **3729.9** & **3736.7**\
rm103 & 9 & 3708.5 & 3736.5 & ****[3699.7]{}**** & **[3705.2]{} & 3700.6 & **3701.5**\
rm104 & 9 & 3718.0 & 3722.9 & ****[3700.6]{}**** & **[3702.0]{} & **3700.6** & **3701.9**\
rm105 & 9 & 3688.8 & 3716.9 & **[3686.6]{} & **[3688.6]{} & **3686.6** & **3687.0**\
rm106 & 9 & ****[3692.9]{}**** & 3743.7 & 3708.0 & **[3713.3]{} & 3708.0 & **3711.4**\
rm107 & 9 & 3701.4 & 3714.1 & ****[3689.9]{}**** & **[3692.0]{} & **3689.9** & **3689.9**\
rm108 & 9 & 3729.1 & 3729.1 & ****[3719.9]{}**** & **[3725.6]{} & **3719.9** & **3720.5**\
**********************
rm201 & **[2]{} & 4808.2 & 4847.4 & **[3885.6]{} & **[4089.0]{} & **3815.0** & **3861.7**\
rm202 & 2 & **[3739.0]{} & 3775.7 & 3740.7 & **[3764.0]{} & **3725.9** & **3742.2**\
rm203 & 2 & 3710.3 & 3728.3 & **[3696.9]{} & **[3721.5]{} & **3696.1** & **3705.2**\
rm204 & 2 & **[3691.9]{} & 3708.8 & 3693.4 & **[3699.6]{} & **3681.5** & **3687.8**\
rm205 & 2 & 3689.9 & 3707.7 & **[3679.8]{} & **[3691.7]{} & **3678.7** & **3681.7**\
rm206 & 2 & 3703.4 & 3720.3 & **[3695.3]{} & **[3713.4]{} & **3690.2** & **3697.6**\
rm207 & 2 & 3701.7 & 3719.4 & **[3691.8]{} & **[3705.5]{} & **3688.3** & **3692.5**\
rm208 & 2 & 3682.8 & 3699.6 & **[3678.1]{} & **[3683.8]{} & **3676.2** & **3682.6**\
**********************************
cm101 & 10 & 12320.0 & **[12344.4]{} & **[12270.4]{}& 12366.8 & **12253.2** & **12295.5**\
cm102 & 12 & **[12492.1]{}&****[12492.1]{}****& 12554.8 & 12591.5 & **12467.5** & 12545.4\
cm103 & **[11]{}& 12641.2 & 12687.7 & **[12568.5]{}& **[12655.1]{} & **12506.0** & **12562.6**\
cm104 & **[13]{}& 13087.8 & 13117.9 & **[12913.0]{}& **[12969.1]{} & **12911.9** & **12917.6**\
cm105 & **[10]{}& **[12083.4]{}&****[12144.4]{}****& 12101.4 & 12144.5 & **12038.1** & **12064.6**\
cm106 & 10 & **[12073.9]{}& **[12133.9]{} & 12096.9 & 12146.4 & **12043.6** & **12073.8**\
cm107 & **[10]{}& 12324.2 & 12364.1 & ****[12106.2]{}**** & **[12147.9]{} & **12106.2** & **12106.5**\
cm108 & 10 & 11990.4 & 12012.6 & ****[11984.3]{}****& **[11986.1]{} & **11984.3** & **11984.3**\
********************************
cm201 & 5 & **[13520.1]{} & **[13591.7]{}& 13561.8 & 13654.0 & **13472.0** & **13551.2**\
cm202 & 6 & **[14027.3]{} & 14060.7 & 14045.7 & **[14060.5]{} & **14021.3** & **14049.3**\
cm203 & 5 & 13497.2 & **[13512.8]{}& **[13489.8]{} & 13566.1 & **13465.5** & **13504.9**\
cm204 & 5 & 13359.8 & 13413.7 & **[13346.8]{} & **[13375.5]{} & **13323.9** & **13342.3**\
cm205 & 4 & **[12884.1]{} & **[12963.1]{}& 12931.6 & 13018.7 & **12852.1** & **12914.9**\
cm206 & 4 & 12767.7 & 12811.2 & **[12753.4]{} & **[12809.1]{} & **12729.8** & **12774.3**\
cm207 & 4 & 13009.7 & **[13017.6]{} & **[12994.1]{} & 13060.7 & **12937.0** & **13007.2**\
cm208 & 4 & 12788.1 & 12805.2 & **[12737.7]{} & **[12764.8]{} & **12729.8** & **12741.2**\
********************************
rcm101 & 10 & 4098.9 & 4129.7 & ***[4079.6]{}*** & **[4082.6]{} & **4079.6** & **4081.0**\
rcm102 & 10 & 4222.6 & 4228.4 & **[4183.5]{} & **[4188.6]{} & **4178.4** & **4180.8**\
rcm103 & 10 & 4174.3 & 4185.4 & **[4144.6]{} & **[4149.5]{} & **4144.6** & **4146.5**\
rcm104 & 10 & 4156.3 & 4170.7 & **[4141.5]{} & **[4147.6]{} & **4137.0** & **4141.0**\
rcm105 & 10 & 4216.7 & 4227.0 & ****[4188.4]{}**** & **[4201.9]{} & **4188.4** & **4195.0**\
rcm106 & 10 & 4219.9 & 4236.3 & ****[4175.4]{}**** & **[4192.4]{} & **4175.4** & **4179.4**\
rcm107 & 11 & 4542.4 & 4560.8 & ****[4516.5]{}**** & **[4520.5]{} & **4516.5** & **4517.0**\
rcm108 & 11 & 4614.5 & 4614.5 & ****[4565.2]{}**** & **[4581.3]{} & **4565.2** & **4569.2**\
**********************
rcm201 & 2 & **[3783.6]{} &****[3824.5]{}****& 3816.0 & 4222.4 & **3733.8** & 3920.4\
rcm202 & 2 & 3847.1 & **[3847.1]{} & **[3792.6]{} & 4010.1 & **3756.1** & **3812.8**\
rcm203 & 2 &**[3721.9]{}&****[3725.4]{}****& 3746.7 & 3810.4 & **3716.5** & 3779.7\
rcm204 & 2 & 3726.5 & 3743.1 & **[3700.8]{} & **[3721.0]{} & **3699.8** & **3708.1**\
rcm205 & 2 & 3754.5 & 3775.7 & **[3732.0]{} & ****[3753.0]{}**** & **3731.8** & 3756.6\
rcm206 & 2 & 3812.7 & 3830.6 & **[3741.5]{} & **[3777.4]{} & 3744.7 & **3777.2**\
rcm207 & **[2]{}& 4764.2 & 4792.2 & **[3859.7]{} & **[4581.2]{} & **3859.1** & **4226.5**\
rcm208 & 2 & 3791.4 & 3817.2 & ****[3731.7]{}**** & ****[3746.2]{}**** & **3731.7** & 3747.5\
************************
The results in Table \[tab:resultsNTM\] indicate that our generally outperforms current state-of-the-art for VRPMTW without time minimisation, and provides new currently best known solutions for 40 of the 48 benchmark instances within 60 seconds. If the time limit is extended to 600 seconds, we find best known solutions for an additional 5 instances, totalling 45 of the 48 provided. Most of the solutions from the sets *rm1xx* and *rcm1xx* are identical to those obtained within 60 seconds, however, for the remaining sets the solutions are typically improved.
Table \[tab:resultsTM\] shows that generally outperforms current state-of-the-art also for the VRPMTW with time minimisation. Improved upper bounds are found in 36 of 48 cases for the average objective, and new currently best known solutions are found for 37 of the 48 instances. The gap between the average and the best found solutions, indicate a variance in the performance of the algorithm and a potential for further improvement. To further investigate this, the heuristic was also run with a 10 minute time limit using the parameters configuration from Table \[tab:defaultConf\]. As the solutions generally improve with the additional time, we can conclude that our hypothesis that further improvements are possible is correct. This resulted in new best known solutions for 47 of 48 instances.
In both problems, for six of the instances, ALNS also provides a solution with fewer routes than the previously best known, and five of these are new best known solutions.
On the challenges inherent to the problem
-----------------------------------------
The choice of a time window $p\in P_i$ for visit $i\in V$ is a disjunction. Models that accommodate such constraints often use a big-M approach as in Constraint . This modelling technique is typically expected to make mathematical models harder to solve, in part, due to a degeneration of the lower bound. This is also supported by the current model being incapable of providing good bounds for any relevant instance size.
Heuristics effectiveness depends on their ability to navigate in the search space. Denote by $sol_{opt}$ the optimal solution and assume we are given an objective function $cost(sol)$ and a neighbourhood $neighbour(sol_i)$ where $sol_j\in neighbour(sol_i)$ if and only if $sol_j$ is reachable from solution $sol_i$ in the local search algorithm. If there exists a sequence (search path) $sol_0, sol_1 \dots sol_{n}$ such that $sol_{n}=sol_{opt}$ and $sol_i \in neighbourhood(sol_{i-1})$ $\forall i\in 1\dots n$ for any initial solution $sol_0$, we call the solution space connected under the neighbourhood operator.
As the number of vehicles is not limited, it is clear that the search space is connected under neighbourhood operators such as move, where a single visit is moved at a time. Each visit can be moved to its own vehicle where after the optimal solution can be assembled visit by visit. The disjunctions will thus not impose a disconnected search space in traditional local search algorithms.
Any local search progresses through a search path, but as the potential number of search paths is prohibitive, the search is heuristically guided through a partial exploration. If a search path $sol_0, sol_1 \dots sol_{n}$ has the property $cost(sol_0) > cost(sol_1)> \dots >cost(sol_{n})$ in a minimisation problem, the search will not encounter a local minimum along its search path. If no such search path exists, any heuristic looking for the optimum solution must accept worsening solutions during the search as part of its diversification strategy. The difficulty of finding good solutions is typically considered to be positively correlated with number of consecutive non-improving neighbours that must be visited during a search.
The VRPMTW exhibits a potential for an increased number of non-improving moves between good solutions. Given a partial solution where a subset of visits are placed consecutively due to geographical closeness, a traditional VRPTW instance would contain time windows causing visits to be placed into the only feasible part of the route, eg. early, middle or late in the route. In VRPMTW they may not be scheduled to be visited during the optimal time window, and changing this time window directly may not be feasible with respect to the rest of the route. Moving the visits one by one is likely to increase the cost of the solution significantly before they potentially are rescheduled during an alternative time window.
The problem of swapping groups of customers between time windows is further compounded by the fact that scheduling a visit and choosing its time window immediately, already imposes precedence and compatibility constraints with unscheduled and thus unknown visits, as their time windows may not be compatible with promising orderings of the visits anymore. Our attempt to remedy these issues proved important for both the average cost of solutions (Figure\[fig:omit\]) and in the search for new best solutions (Figire \[fig:omitBest\]).
Conclusions {#sec:conclusions}
===========
We have formulated a more compact mathematical model for both variants of the , though bounds and solutions remain unobtainable the model is more readable, and an error regarding service times has been corrected.
We have presented an adaptation of to two variants of the VRPMTW. We have shown that the problem without time minimisation effectively can be solved using traditional heuristics based on insertion with an overhead scaling linearly in the number of time windows. We have furthermore found new best solutions for all but 6 instances.
We presented a labelling algorithm for evaluating delta costs of insertions for VRPMTW with time minimisation. Furthermore we demonstrated that using the delta evaluation is competitive with existing methods from the literature, and for all categories of instances outperform them. Lastly we found new best known solutions to all but 2 instances.
We have observed that an increased focus on diversification over intensification is needed when tuning the algorithm. Furthermore we noticed a large variance in the quality of results, making us suspect further work is warranted. This suspicion was confirmed when running the algorithm for 10 minutes, confirming that better solutions are obtainable.
Acknowledgements {#acknowledgements .unnumbered}
================
This project was partially funded by the Danish Transport Authority under the grant TS20707-00132.
References {#references .unnumbered}
==========
|
---
abstract: 'We obtain necessary and sufficient conditions with sharp constants on the distribution $\sigma$ for the existence of a globally finite energy solution to the quasilinear equation with a gradient source term of natural growth of the form $-\Delta_p u = |\nabla u|^p + \sigma$ in a bounded open set ${\Omega}\subset {\mathbb{R}}^n$. Here $\Delta_p$, $p>1$, is the standard $p$-Laplacian operator defined by $\Delta_p u={\rm div}\, (|\nabla u|^{p-2}\nabla u)$. The class of solutions that we are interested in consists of functions $u\in W^{1,p}_0({\Omega})$ such that $e^{{\mu} u}\in W^{1,p}_0({\Omega})$ for some ${\mu}>0$ and the inequality $$\int_{{\Omega}} |\varphi|^p |\nabla u|^p dx \leq A \int_{\Omega}|\nabla \varphi|^p dx$$ holds for all $\varphi\in C_c^\infty(\Omega)$ with some constant $A>0$. This is a natural class of solutions at least when the distribution $\sigma$ is nonnegative. The study of $-\Delta_p u = |\nabla u|^p + \sigma$ is applied to show the existence of globally finite energy solutions to the quasilinear equation of Schrödinger type $-\Delta_p v = \sigma\, v^{p-1}$, $v\geq 0$ in ${\Omega}$, and $v=1$ on $\partial{\Omega}$, via the exponential transformation $u\mapsto v=e^{\frac{u}{p-1}}$.'
address:
- '${}^1$ Department of Mathematical Sciences, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul 08826, South Korea.'
- '${}^2$ Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803, USA.'
author:
- 'Karthik Adimurthi$^{1}$'
- 'Nguyen Cong Phuc$^{2}$'
title: 'Nonlinear equations with gradient natural growth and distributional data, with applications to a Schrödinger type equation'
---
[^1] [^2]
Introduction
============
The main goal of this paper is to address the solvability of quasilinear elliptic equations with gradient nonlinearity of natural growth of the form $$\label{basic_pde}
\left\{ \begin{array}{ll}
-\Delta_p u = |\nabla u|^p + \sigma & {\quad \textrm{in}\quad } \Omega, \\
u = 0 & {\quad \textrm{on}\quad } \partial \Omega,
\end{array}
\right.$$ [in]{} a bounded open set $\Omega \subset {\mathbb{R}}^n$. Here $\Delta_p u:= \operatorname{div}(|\nabla u|^{p-2} \nabla u)$, $p>1$, is the $p$-Laplacian and the datum $\sigma$ is a distribution in ${\Omega}$. More generally, we also consider the equation $$\label{basic_pde2}
\left\{ \begin{array}{ll}
-\operatorname{div}\aa(x, u, \nabla u) = \mathcal{B}(x, u, \nabla u) + \sigma & {\quad \textrm{in}\quad } \Omega, \\
u = 0 & {\quad \textrm{on}\quad } \partial \Omega,
\end{array}
\right.$$ where the principal operator $\operatorname{div}\aa(x, u, \nabla u)$ is a Leray-Lions operator defined on $W_0^{1,p}({\Omega})$ and $|\mathcal{B}(x, u, \nabla u)| \lesssim |\nabla u|^p$.
The precise assumptions on the nonlinearities $\aa$, $B$ and the the precise definition of solutions to will be given in Section \[GenStru\]. Here we emphasize that in this paper we are interested only in [*finite energy solutions $u$*]{} with zero boundary condition in the sense that $u\in{W_0^{1,p}({\Omega})}$. The energy space ${W_0^{1,p}({\Omega})}$ is defined as the completion of $C_c^\infty({\Omega})$ under the semi-norm $\norm{\nabla (\cdot)}_{L^{p}({\Omega})}$. As an application of the study of , we also obtain existence of finite energy solution to the quasilinear Schrödinger type equation $$\label{basic_pde-schr}
-\Delta_p v = (p-1)^{1-p}\, \sigma\, v^{p-1} \text{ in } \Omega, \qquad v \geq 0 \text{ in } \Omega, \qquad v = 1 \text{ on } \partial \Omega.$$
Equation is a prototype for quasilinear equations with natural growth in the gradient that has attracted a lot of attention in the past years. It can be viewed as a quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi equation, also known as the Kardar-Parisi-Zhang equation, which appears in the physical theory of surface growth [@KPZ; @KS].
As far as existence is concerned, the nonlinearity $|\nabla u|^p$ in is considered “to have the bad sign” and by now it is well-known that in order for to have a solution the datum $\sigma$ must be both [*small and regular enough*]{}. In particular, if $\sigma$ is a nonnegative distribution in ${\Omega}$ (i.e., a nonnegative locally finite measure ), then a necessary condition for the first equation in to have a $W^{1,p}_{\rm loc}({\Omega})$ solution is that (see [@HMV; @JMV1; @JMV2]) $$\label{possig}
\int_{{\Omega}} |\varphi|^p d\sigma \leq \lambda \int_{\Omega}|\nabla \varphi|^p dx \quad \text{for all } \varphi\in C_c^\infty(\Omega),$$ with $\lambda=(p-1)^{p-1}$. Moreover, when $\sigma\geq 0$ the nonlinear term itself also obeys a similar Poincaré-Sobolev inequality $$\label{nablau-cond}
\int_{{\Omega}} |\varphi|^p |\nabla u|^pdx \leq A \int_{\Omega}|\nabla \varphi|^p dx \quad \text{for all } \varphi\in C_c^\infty(\Omega),$$ with $A=p^p$.
Thus a natural space of solutions associated to is the space $\mathcal{S}$ of functions $u\in W^{1,p}_0({\Omega})$ such that holds for some $A>0$. The main question we wish to address here is to find an optimal (largest) space $\mathcal{D}$ of ‘data’ so that whenever $\sigma\in \mathcal{D}$ with sufficiently small norm $\norm{\sigma}_{\mathcal{D}}$ then admits a solution in $\mathcal{S}$. In the case $\sigma\geq 0$ we can completely characterize the existence of finite energy solutions to in the following theorem. We remark again that in this case all $W^{1,p}_0({\Omega})$ solutions automatically belong to ${\mathcal{S}}$ and holds with $A=p^p$.
\[posmeas\] Let $\sigma$ be a nonnegative locally finite measure in ${\Omega}$. If has a solution in $u\in {W_0^{1,p}({\Omega})}$ then $\sigma\in (W^{1,p}_0({\Omega}))^*$ and holds with $\lambda=(p-1)^{p-1}$. Conversely, if $\sigma\geq 0$, $\sigma\in (W^{1,p}_0({\Omega}))^*$, and holds with $0<\lambda< (p-1)^{p-1}$ then has a *nonnegative* solution in $W^{1,p}_0({\Omega})$ such that $e^{\frac{\delta u}{p-1}}-1\in W^{1,p}_0({\Omega})$ for all $\delta\in [0, \delta_0)$ where $\delta_0=(p-1) \lambda^{\frac{-1}{p-1}}$.
In the linear case, $p=2$, these necessary and sufficient conditions have been observed in [@FV]. See also [@ADP] [(for $p=2$)]{} and [@HBV] for [certain]{} related results that were obtained by different [methods]{}. We remark that, under a mild restriction on the domain, by Hardy’s inequality (see [@Anc; @Lew]), Theorem \[posmeas\] covers the case of unbounded measure such as $\sigma =\varepsilon\, {\rm dist}(x, \partial{\Omega})^{-1}$ for some $\varepsilon>0$. It is also worth mentioning that in the case $p=2$ and $\sigma$ is a nonnegative locally finite measure, other sharp existence results for obtained in [@HMV] for $\Omega={\mathbb{R}}^n$ and recently in [@FV] for bounded domains $\Omega$ with $C^2$ boundary under a very weak notion of solution and boundary conditions.
The first part of Theorem \[posmeas\] follows from the known necessary condition , Hölder’s inequality, and the assumption that $\nabla u\in L^p(\Omega)$, since we have $$\sigma=-|\nabla u|^p - {\rm div}\, (|\nabla u|^{p-2}\nabla u)\leq - {\rm div}\, (|\nabla u|^{p-2}\nabla u).$$
On the other hand, the second part is a consequence of Theorem \[weakzero\] below that treats even sign changing distribution datum $\sigma$. This in fact is the main result that will be obtained in this paper.
\[weakzero\] [(i)]{} Suppose that has a solution in $u\in W^{1,p}_0({\Omega})$ such that holds for some $A>0$ then it necessarily holds that $\sigma= {\rm div}\, (F) - |F|^{\frac{p}{p-1}}$ for a vector field $F\in L^{\frac{p}{p-1}}({\Omega},{\mathbb{R}}^n)$ such that $$\label{Fpower-cond}
\int_{{\Omega}} |F|^{\frac{p}{p-1}} |\varphi|^p dx \leq A \int_{\Omega}|\nabla \varphi|^p dx \quad \text{for all } \varphi\in C_c^\infty({\Omega}).$$ In particular, both $\sigma$ and $|F|^{\frac{p}{p-1}}$ belong to the dual space $(W^{1,p}_0({\Omega}))^*$.
[(ii)]{} Conversely, suppose that $\sigma={\rm div}\, F + f$ where $F\in L^{\frac{p}{p-1}}({\Omega},{\mathbb{R}}^n)$ and $f$ is a locally finite signed measure in ${\Omega}$ with $|f|\in (W^{1,p}_0({\Omega}))^*$ such that $$\label{datasmallness}
p \int_{{\Omega}} |F| |\varphi|^{p-1} |\nabla \varphi|dx +\int_{{\Omega}} |\varphi|^p d|f| \leq \lambda \int_{\Omega}|\nabla \varphi|^p dx \qquad \forall \varphi\in C_c^\infty({\Omega}),$$ for some $\lambda\in (0, (p-1)^{p-1})$. Then equation has a (possibly sign changing) solution $u\in W^{1,p}_{0}({\Omega})$ such that $e^{\frac{\delta u}{p-1}}-1\in W^{1,p}_0({\Omega})$ for all $\delta\in [0, \delta_0)$ where $\delta_0=(p-1) \lambda^{\frac{-1}{p-1}}$. This solution satisfies the Poincaré-Sobolev inequality for some $A=A(p)>0$. [Moreover, if $\lambda\in (0, (p-1)^{{\rm min}\{1, p-1\}})$, then both $e^{\frac{u}{p-1}}-1$ and $e^{u}-1$ belong to $W^{1,p}_0({\Omega})$.]{}
Several remarks regarding Theorem \[weakzero\] are now in order.
By approximation and Fatou’s lemma, inequalities and actually hold for all $\varphi\in W^{1,p}_0({\Omega})$. The integral $\int_{{\Omega}} |\varphi|^p d|f|$ makes sense even for $\varphi\in W^{1,p}_0({\Omega})$ since $|f|$ is continuous with respect to the capacity ${\rm cap}_p(\cdot,{\Omega})$ and $\varphi$ has a ${\rm cap}_p$-quasicontinuous representative, whose values are defined ${\rm cap}_p$-quasieverywhere in ${\Omega}$. Here ${\rm cap}_p(\cdot,{\Omega})$ is the $p$-capacity associated to ${\Omega}$ defined for each compact set $K\subset{\Omega}$ by $${\rm cap}_{p}(K, {\Omega}):=\inf\left\{\int_{{\Omega}} |\nabla \phi|^p dx:\phi\in C_c^\infty({\Omega}) {\rm ~and~} \phi\geq \chi_K \right\}.$$
By Hölder’s inequality we see that if $F$ satisfies for some $A>0$ then $$p \int_{{\Omega}} |F| |\varphi|^{p-1} |\nabla \varphi|dx \leq p A^{\frac{p-1}{p}} \int_{\Omega}|\nabla \varphi|^p dx \qquad \forall \varphi\in C_c^\infty({\Omega}).$$ Thus by Theorem \[weakzero\](ii) if $F\in L^{\frac{p}{p-1}}({\Omega},{\mathbb{R}}^n)$ satisfies for some $0<A< (p-1)^p p^{-\frac{p}{p-1}}$ then the equation $-\Delta_p u =|\nabla u|^p +{\rm div}\, F$ .
Let $\mu$ be a nonnegative locally finite measure in ${\Omega}$. It is well-known that the inequality $$\int_{{\Omega}} |\varphi|^p d\mu \leq A_1 \int_{\Omega}|\nabla \varphi|^p dx \qquad \forall \varphi\in C_c^\infty({\Omega})$$ is equivalent to the condition $$\label{Cap-A2}
\mu(K)\leq A_2 \, {\rm cap}_{p}(K, {\Omega})$$ for all compact sets $K\subset{\Omega}$ (see [@Maz Chapter 2]).
Thus in , condition can be replaced by with $\mu=|F|^{\frac{p}{p-1}}+|f|$ for a sufficiently small constant $A_2>0$.
, if $f$ is a locally finite signed measure in ${\Omega}$ with $|f|\in (W^{1,p}_0({\Omega}))^*$ such that holds with $d\mu=d|f|$, the we have a decomposition $$f={\rm div}\, F -g,$$ where and $g\in L^1({\Omega}), \, g\geq0,$ such that $L^1$ function $\mu:=(|F|^{\frac{p}{p-1}}+g)$ also satisfies . See [@BGO; @FS] for a similar decomposition of measures that are continuous w.r.t the $p$-capacity.
Let $L^{s,\infty}({\Omega})$, $s\geq 1$, denote the weak $L^s$ space on ${\Omega}$ with quasinorm $$\norm{g}_{L^{s,\infty}({\Omega})}:= \sup_{t>0}t |\{ x\in{\Omega}: |g(x)|>t\}|^{1/s}.$$ it is known that $$\int_{\Omega}|\varphi|^p g dx \leq S_{n,p} \textcolor{black}{\norm{g}_{L^{\frac{n}{p},\infty}({\Omega})}} \int_{\Omega}|\nabla \varphi|^p dx \qquad \forall \varphi\in C_c^\infty({\Omega}),$$ where the constant $S_{n,p}$ is given by $$S_{n,p}= \left[ \frac{p}{\sqrt{\pi}(n-p)} \right]^p \Gamma(1+n/2)^{p/n}.$$
This shows that in Theorem \[weakzero\](ii), condition can be replaced by with a sufficiently small norm. Existence results under this weak norm condition have been obtained in [@FM3]. See also the earlier works [@FM1; @FM2] where the strong norm condition involving was used instead. More general existence results in which $|F|^{\frac{p}{p-1}}+|f|$ is assumed to be small in the norm of certain Morrey spaces can be found in the recent paper [@MP]. Those Morrey space conditions are also stronger than condition as they fall into the realm of Fefferman-Phong type conditions (see, e.g., [@DPT; @ChWW; @Fef; @P; @SW]).
We now discuss the Schrödinger type equation with distributional potential . This equation is interesting in its own right and has a strong connection to equation as being observed and exploited, e.g., in [@ADP; @HBV; @JMV1; @JMV2].
Formally, by making the change of unknowns $v=e^{\frac{u}{p-1}}$, equation is transformed into the Schrödinger type equation . Indeed, it is possible to show rigorously that Theorem \[weakzero\] implies the existence of finite energy solutions to :
\[Schro-type\] Suppose that $\sigma={\rm div}\, F + f$ where and $f$ is a locally finite signed measure in ${\Omega}$ with $|f|\in (W^{1,p}_0({\Omega}))^*$ such that $$p \int_{{\Omega}} |F| |\varphi|^{p-1} |\nabla \varphi|dx +\int_{{\Omega}} |\varphi|^p d|f| \leq \lambda \int_{\Omega}|\nabla \varphi|^p dx
\qquad \forall \varphi\in C_c^\infty({\Omega}),$$ for some $\lambda\in (0, (p-1)^{\min\{1, p-1\}})$. [Then equation has a nonnegative solution $v$ such that both belong to . Moreover, $v$ satisfies the following Poincaré-Sobolev inequality $$\label{WNforv}
\int_{{\Omega}} \left|\frac{\nabla v}{v}\right|^p |\varphi|^p dx \leq A \int_{\Omega}|\nabla \varphi|^p dx \qquad \forall \varphi\in C_c^\infty({\Omega}),$$ with a constant $A=A(p)>0$.]{}
If the factor $(p-1)^{1-p}$ on the right-hand side of is dropped then the smallness condition on $\lambda$ becomes $\lambda\in (0, p^{\#})$, where $p^{\#}=(p-1)^{2-p}$ if $p> 2$ and $p^{\#}=1$ if $p\leq 2$ as in [@JMV2]. The sharpness of $p^{\#}$ (and of $(p-1)^{\min\{1, p-1\}}$ for ) was also justified in [@JMV2].
[One could also treat the Schrödinger type equation in a more general fashion, where the standard $p$-Laplacian is replaced by a quasilinear elliptic operator with merely measurable ‘coefficients’. See Remark \[mea-coeff\] below and see also [@JMV2].]{}
We mention that the existence of finite energy solutions to in the case $\sigma \geq 0$ was obtained in [@HBV] by [a method that does not seem to work for sign changing $\sigma$]{} (see also [@ADP] for $p=2$). On the other hand, the work [@JMV2] (see also [@JMV1]) obtains a locally finite energy solution $v\in W^{1,p}_{\rm loc}({\Omega})$ to the first two equations in only under the mild restriction $$-\Lambda \int_{\Omega}|\nabla \varphi|^p dx \leq \langle \sigma, |\varphi|^p \rangle \leq \lambda \int_{\Omega}|\nabla \varphi|^p dx \quad \text{for all } \varphi\in C_c^\infty({\Omega})$$ for some $\lambda\in (0, (p-1)^{\min\{1, p-1\}})$ and $\Lambda\in (0,+\infty)$. Moreover, $v$ also satisfies for some $A>0$. Then, also under the restriction $\lambda\in (0, (p-1)^{\min\{1, p-1\}})$, by the logarithmic transformation $u=(p-1)\log(v)$ it was obtained in [@JMV2], a solution $u\in W^{1,p}_{\rm loc}({\Omega})$ to the first equation in (but without any boundary condition) that also satisfies for some $A>0$.
*In this paper, we follow an opposite route, i.e., we first treat equation directly and then deduce existence for the Schrödinger type equation from it.* This way, we are able to treat equation in its most general form, i.e., the nonlinear equation with general structure . Moreover, for equation we obtain larger upper bound for $\lambda$ in the existence condition $( {\rm i.e.,~} (p-1)^{p-1}$ versus $(p-1)^{\min\{ 1, p-1\}})$. Our approach to is a refinement of the approach of V. Ferone and F. Murat in [@FM2; @FM3]. The main difficulties to overcome here are the generality nature of $\sigma$ and the sharpness of the smallness constants. In particular, in this scenario one does not gain any higher integrability on the nonlinear term $\mathcal{B}(x,u, \nabla u)$, which makes it impossible to follow a compactness argument as in [@MP]. Moreover, in order for us to apply the existence results of to we need to find a solution $u$ of with the additional property that both $e^{\frac{u}{p-1}}-1$ and $e^{u}-1$ belong to $W^{1,p}_0({\Omega})$ as stated in Theorem \[weakzero\].
Equations with general nonlinear structure {#GenStru}
==========================================
As we have mentioned, existence results in the spirit of Theorem \[weakzero\](ii) also hold for equations with a more general nonlinear structure . For that we need the following assumptions on the nonlinearities $\mathcal{A}$ and $\mathcal{B}$:
[**Assumption on $\mathcal{A}$.**]{} The nonlinearity $\aa : {\Omega}\times {\mathbb{R}}\times {\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$ is a Carathédory function, i.e., $\aa(x, s, \xi)$ is measurable in $x$ for every $(s,\xi)$ and continuous in $(s,\xi)$ for a.e. $x\in{\Omega}$. For some $p>1$, it holds that $$\begin{gathered}
\label{monotone-strict} \langle \aa(x, s, \xi) - \aa(x, s, \eta), \xi - \eta \rangle >0,\\
\label{coercivity} \langle \aa(x, s, \xi), \xi\rangle \geq \alpha_0 |\xi|^p,\\
\label{growth-p} |\aa(x, s, \xi)| \leq a_0|\xi|^{p-1} + a_1 |s|^{p-1}\end{gathered}$$ for every $(\xi, \eta)\in {\mathbb{R}}^n \times {\mathbb{R}}^n$, $\xi\not=\eta$, and a.e. $x \in{\Omega}$. Here $\alpha_0>0$, and $a_0, a_1\geq 0$.
[**Assumption on $\mathcal{B}$.**]{} The nonlinearity $\mathcal{B} : {\Omega}\times {\mathbb{R}}\times {\mathbb{R}}^n \rightarrow {\mathbb{R}}$ is a Carathédory function which satisfies, for a.e. $x\in{\Omega}$, every $s\in{\mathbb{R}}$, and every $\xi\in{\mathbb{R}}^n$, $$\label{Bcond}
|\mathcal{B}(x,s,\xi)|\leq b_0 |\xi|^p +b_1 |s|^m, \quad \mathcal{B}(x,s,\xi){\rm sign}(s) \leq \alpha_0 \gamma_0 |\xi|^p,$$ where $m>0$, and $b_0, b_1$, $\gamma_0\geq 0$. Here $\alpha_0$ is as given in .
By a solution of we mean the following.
Under -, a function $u \in W^{1,p}_{0}({\Omega})$ is a solution of if $\mathcal{B}(x, u, \nabla u)\in L^1_{\rm loc}({\Omega})$ and $${\int_{{\Omega}}}\aa(x, u, \nabla u) \cdot \nabla \varphi \ dx = {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) \varphi \ dx + \langle\sigma, \varphi\rangle$$ for all test functions $\varphi \in C_c^{\infty} (\Omega)$.
We remark that in the case $\mathcal{B}(x, u, \nabla u)\in L^1({\Omega})$ and $\sigma\in (W^{1,p}_{0}({\Omega}))^{*}$, we can take any function $\varphi\in W^{1,p}_{0}({\Omega})\cap L^\infty({\Omega})$ as a test function in the above definition. This follows from a result of Brézis and Browder [@BB] as we have $\mathcal{B}(x, u, \nabla u)\in(W^{1,p}_{0}({\Omega}))^{*} \cap L^1({\Omega})$. It can also be seen by approximating $\varphi$ in $W^{1,p}_0({\Omega})$ by a sequence $\varphi_j\in C_c^\infty({\Omega})$ such that $|\varphi_j| \leq |\varphi|\leq M$ a.e. (using Theorem 9.3.1 in [@AH] and suitable convolutions).
We mention that in the special case $|\mathcal{B}(x, u, \nabla u)| \in (W^{1,p}_{0}({\Omega}))^{*} \cap L^1({\Omega})$ can even drop the condition $\varphi\in L^\infty({\Omega})$. In fact, we have the following more general result.
\[dis-mea\] Suppose that $f$ is a locally finite signed measure in ${\Omega}$ with $|f|\in (W^{1,p}_0({\Omega}))^*$. Then for any $\varphi\in W^{1,p}_0({\Omega})$ we have $$\langle f, \varphi\rangle= \int_{{\Omega}} \widetilde{\varphi}\, d f,$$ where $\widetilde{\varphi}$ is any ${\rm cap}_p$-quasicontinuous representative of $\varphi$.
In the case $f$ is nonnegative, the proof of Lemma \[dis-mea\] can be found in [@Mik Lemma 2.5]. The general case also follows from since $f=f^{+} + f^{-}$ and both $f^{+}$ and $f^{-}$ belong to $(W^{1,p}_0({\Omega}))^*$. In what follows, when dealing with pointwise behavior of functions in $W^{1,p}_0({\Omega})$ we will implicitly use their ${\rm cap}_p$-quasicontinuous representatives. Lemma \[dis-mea\] will be e.g., in below.
Under the above assumptions on $\mathcal{A}$ and $\mathcal{B}$, we obtain the following existence result.
\[MainExistence\] Let $\sigma={\rm div}\, F + f$ where and $f$ is a locally finite signed measure in ${\Omega}$ with $|f|\in (W^{1,p}_0({\Omega}))^*$ such that $$\label{smalllambda}
p \int_{{\Omega}} |F| |\varphi|^{p-1} |\nabla \varphi|dx +\int_{{\Omega}} |\varphi|^p d|f| \leq \lambda \int_{\Omega}|\nabla \varphi|^p dx$$ holds for all $\varphi\in C_c^\infty({\Omega})$, with $\lambda\in (0, \gamma_0^{1-p}\alpha_0(p-1)^{p-1}).$ Then there exists a solution $u\in W_0^{1,p}({\Omega})$ to the equation $$\label{basic_pde3}
-\operatorname{div}\aa(x, u, \nabla u) = \mathcal{B}(x, u, \nabla u) + \sigma {\quad \textrm{in}\quad } \Omega,$$ such that $e^{\frac{\delta|u|}{p-1}}-1\in W^{1,p}_0({\Omega})$ for all ${\delta}\in [\gamma_0, {\delta}_0)$, with
[ Moreover, for any $\delta_1 > \gamma_0$ such that holds with $$\label{lambdacond2}
\lambda<\left(\frac{p-1}{\delta_1}\right)^p \alpha_0 \left(\frac{\delta_1}{p-1} +\delta_1-\gamma_0\right),$$ we have $e^{\frac{\delta_1|u|}{p-1}}-1 \in {{W_0^{1,p}({\Omega})}}$.]{}
It is easy to check that, for $\delta_1>\gamma_0$ one has $$\left(\frac{p-1}{\delta_1}\right)^p \alpha_0 \left(\frac{\delta_1}{p-1} +\delta_1- \gamma_0\right) < \gamma_0^{1-p}\alpha_0(p-1)^{p-1}.$$ Moreover, $p>2$ and $\alpha_0=\gamma_0=1$ holds with $\lambda<p-1 \in (0, (p-1)^{p-1})$ we see that holds with $1\leq \delta< (p-1)(p-1)^{\frac{-1}{p-1}}$, but it does not allow us to take On the other hand, for $\lambda<p-1$ inequality holds with ${\delta}_1=p-1$ and thus $e^{|u|}-1 \in {{W_0^{1,p}({\Omega})}}$.
Due to the general structures of $\mathcal{A}$ and $\mathcal{B}$, here we do not claim that the solution $u$ obtained in Theorem \[MainExistence\] satisfies the Poincaré-Sobolev inequality .
The paper is organized as In Section \[ActualExistence\], we provide the proof of Theorem \[MainExistence\]. This proof is based on the existence of solutions to an [approximate]{} equation along with certain uniform bounds given in Section \[ApproxExist\]. These important uniform bounds are in turn deduced from the a priori estimate of Section \[AprioriEst\], though not directly. Finally, the proof of Theorems \[weakzero\] and \[Schro-type\] will be given in Section \[main-proofs\].
An a priori estimate {#AprioriEst}
====================
In this section, we obtain certain exponential type a priori bounds for solutions of $$\label{basic-ep}
-\operatorname{div}\aa(x, u, \nabla u) +\varepsilon\, |u|^{p-2} u= \mathcal{B}(x, u, \nabla u) + \sigma {\quad \textrm{in}\quad } \Omega,$$ where $\varepsilon\geq0$. The case $\varepsilon>0$ will be needed in the next section to absorb certain unfavorable terms in the approximating process; see below. Earlier, this idea was implemented by V. Ferone and F. Murat in [@FM3].
\[regularity\_Murat-Ferone\] Let $\sigma={\rm div}\, F + f$ where and $f$ is a locally finite signed measure in ${\Omega}$ with $|f|\in (W^{1,p}_0({\Omega}))^*$ such that holds for all $\varphi\in C_c^\infty({\Omega})$, with $\lambda\in (0, \gamma_0^{1-p}\alpha_0(p-1)^{p-1}).$ Then for any $\varepsilon \geq 0$ and any $W_0^{1,p}({\Omega})$ solution $u$ to equation such that $e^{\frac{\delta|u|}{p-1}}-1\in W^{1,p}_0({\Omega}),$ we have $$\label{apri1}
\|u\|_{{W_0^{1,p}({\Omega})}} + \|e^{\frac{\delta|u|}{p-1}}-1\|_{{W_0^{1,p}({\Omega})}} \leq M_{{\delta}}.$$ provided ${\delta}\in [\gamma_0, {\delta}_0)$ where $\delta_0=(p-1) (\alpha_{0}/\lambda)^{\frac{1}{p-1}}$. Here $M_\delta$ is independent of $u$ and $\varepsilon$.
[Moreover, for any $\delta_1 > \gamma_0$ such that $e^{\frac{\delta_1|u|}{p-1}}-1\in W^{1,p}_0({\Omega})$, and holds with $\lambda$ satisfying , we have]{} $$\label{apri2}
\|e^{\frac{\delta_1|u|}{p-1}}-1\|_{{W_0^{1,p}({\Omega})}} \leq M_{{\delta}_{1}} + C_{{\delta}_{1}} \norm{\nabla u}_{L^p({\Omega})}.$$ The constants $M_{{\delta}_{1}}$ and $C_{{\delta}_{1}}$ are independent of $u$ and $\varepsilon$.
Let $u\in {W_0^{1,p}({\Omega})}$ be a solution of and define $$w={\rm sign}(u)[e^{\mu|u|}-1]/\mu, \qquad {\rm with~} \mu=\delta/(p-1),$$ where ${\rm sign}(u)=0$ if $u=0$, ${\rm sign}(u)=1$ if $u>0$, and ${\rm sign}(u)=-1$ if $u<0$. Then from the assumption $e^{\mu|u|}-1\in {W_0^{1,p}({\Omega})}$, we see that $w\in {W_0^{1,p}({\Omega})}$ with $$\label{deriofw}
\nabla w=e^{\mu|u|} \nabla u.$$
Indeed, for $\varepsilon>0$ define $f_\varepsilon(x)= \frac{x}{\sqrt{x^2 +\varepsilon^2}}$, $x\in{\mathbb{R}}$, [and denote by $T_s$, $s>0$, the two-sided truncation operator at level $s$, i.e., that $T_s(u)\in W^{1,p}_0({\Omega})$ for any $s>0$ and $$\begin{aligned}
\nabla\left[ f_\varepsilon(T_s(u)) (e^{\mu|T_s(u)|}-1)/\mu\right]&=&\frac{\nabla T_s(u) {\varepsilon}^2}
{(T_s(u)^2 + {\varepsilon}^2)^{3/2}} (e^{\mu|T_s(u)|}-1)/\mu \\
&& +\, f_{\varepsilon}(T_s(u)) \nabla (e^{\mu|T_s(u)|}-1)/\mu\end{aligned}$$ in the weak sense. Note that $$\varepsilon^2 (e^{\mu|T_s(u)|}-1)/\mu\leq \frac{e^{\mu s}}{\mu s} \, \varepsilon^2 |T_s(u)|\leq \frac{e^{\mu s}}{\mu s} \, (|T_s(u)|^2 + {\varepsilon}^2)^{3/2},$$ $$f_\varepsilon(T_s(u))\rightarrow {\rm sign}(T_s(u))={\rm sign}(u) \text{ as } \varepsilon\rightarrow 0^{+},$$ and thus by Dominated Convergence Theorem we find $$\nabla\left[ {\rm sign}(u) (e^{\mu|T_s(u)|}-1)/\mu\right]={\rm sign}(u) \nabla (e^{\mu|T_s(u)|}-1)/\mu=e^{\mu|T_s(u)|}\nabla T_s(u).$$ using the assumption $e^{\mu|u|}|\nabla u|\in L^p({\Omega})$ and letting $s\rightarrow \infty$, we obtain .]{}
For each $s>0$, we will use the following test function for : $$v_s=e^{{\delta}|u_s|} w_s,$$ where $u_s= T_s(u)$ and $w_s={\rm sign}(u) [e^{\mu |u_s|} -1]/\mu$ with $\mu=\delta/(p-1)$.
From the definition of $w_s$ we have $|w_s| \leq |w|$ and $\nabla w_s =e^{\mu |u_s|}\nabla u_s$. Thus both $w_s$ and $v_s$ belong to ${W_0^{1,p}({\Omega})}\cap L^\infty({\Omega})$ and moreover, $$\nabla v_s = \Big[e^{{\delta}|u|} \nabla w + {\delta}|w| e^{{\delta}|u|} \nabla u \Big]\chi_{\{|u|\leq s\}}.$$
Using $v_s$ as a test function in , we get $$\begin{aligned}
\lefteqn{{\int_{{\Omega}}}\aa(x, u, \nabla u) \cdot \nabla w e^{{\delta}|u|} \chi_{\{|u|\leq s\}} \ dx + \varepsilon {\int_{{\Omega}}}|u|^{p-2} u e^{\delta |u_s|} w_s \ dx}\\
&=&- {\int_{{\Omega}}}{\delta}|w| e^{{\delta}|u|} \aa(x, u, \nabla u)\cdot \nabla u \chi_{\{|u|\leq s\}} \ dx +\\
&& + \ {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) e^{{\delta}|u_s|} w_s \ dx + {\int_{{\Omega}}}F \cdot \nabla v_s \ dx + {\int_{{\Omega}}}v_s df. \end{aligned}$$
We now write this equality as $$\label{I12345}
I_1 + I_2 = I_3 + I_4 + I_5 +I_6,$$ where $I_i$, $i\in\{1, \dots ,6\}$, are the corresponding terms.
[**Estimate for $I_1$:**]{} Since $\nabla w_s = e^{\mu |u_s|} \nabla u_s$, using the coercivity condition , we see that $$\begin{aligned}
\label{I1}
I_1 & =& {\int_{{\Omega}}}\aa(x, u_s, \nabla u_s) \cdot \nabla w_s e^{{\delta}|u_s|} \chi_{\{|u|\leq s\}} \ dx \\
& =& {\int_{{\Omega}}}\aa(x, u_s, \nabla u_s) \cdot \nabla u_s e^{(\mu+{\delta}) |u_s|} \ dx \nonumber\\
& \geq& \alpha_0 {\int_{{\Omega}}}|\nabla w_s|^p \ dx, \nonumber
\end{aligned}$$ where we used the fact $\mu + {\delta}= p\mu$.
[**Estimate for $I_2$:**]{} We have $$\label{I1prime}
I_2= \varepsilon {\int_{{\Omega}}}|u|^{p-1} e^{\delta |u_s|} \frac{e^{\mu |u_s|}-1}{\mu} \ dx \geq \varepsilon \, s^{p-1} {\int_{{\Omega}}}e^{\delta s} \frac{e^{\mu s}-1}{\mu} \chi_{\{|u|>s\}}\geq 0.$$
[**Estimate for $I_3 + I_4$:**]{} By we have $$\begin{aligned}
I_3 + I_4 &=& - {\int_{{\Omega}}}{\delta}|w| e^{{\delta}|u|} \aa(x, u, \nabla u)\cdot \nabla u \chi_{\{|u|\leq s\}} dx + {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) e^{{\delta}|u_s|} w_s dx \\
&\leq& - {\int_{{\Omega}}}{\delta}{\alpha}_0 |w_s| e^{{\delta}|u_s|} |\nabla u_s|^p \ dx + {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) {\rm sign}(u) e^{{\delta}|u_s|} |w_s| dx \\
&=& - {\int_{{\Omega}}}{\delta}{\alpha}_0 |w_s| e^{{\delta}|u_s|} |\nabla u_s|^p \ dx + \\
&& +\ {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) {\rm sign}(u) e^{{\delta}|u_s|} |w_s| [\chi_{\{|u|\leq s\}} + \chi_{\{|u|> s\}}] dx.
\end{aligned}$$
Since we assume ${\delta}\geq \gamma_0$, this and the second condition in imply that $$\begin{aligned}
\label{I34B}
I_3 + I_4 &\leq & {\int_{{\Omega}}}(-{\delta}+\gamma_0) {\alpha}_0 |w_s| e^{{\delta}|u_s|} |\nabla u_s|^p \ dx \\
&& +\ {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) {\rm sign}(u) e^{{\delta}|u_s|} |w_s| \chi_{\{|u|> s\}} dx\nonumber\\
&\leq& {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) {\rm sign}(u) e^{{\delta}|u_s|} |w_s| \chi_{\{|u|> s\}} dx.\nonumber\end{aligned}$$
Thus by the first inequality on and the fact that $$\begin{aligned}
\label{exptopo}
|v_s|&=& e^{{\delta}|u_s|} |w_s|=(1+\mu |w_s|)^{\delta/\mu} |w_s|=(1+\mu |w_s|)^{p-1} |w_s|\\
&\leq& \frac{1}{\mu}(1+\mu |w_s|)^{p}\leq \frac{1}{\mu} e^{p\mu |u_s|}\leq \frac{1}{\mu} e^{p\mu |u|},\nonumber\end{aligned}$$ we find $$\begin{aligned}
\label{I2I3}
I_3 + I_4 &\leq& {\int_{{\Omega}}}(b_0 |\nabla u|^p + b_1 |u|^{m}) \frac{1}{\mu}e^{p\mu |u|} \chi_{\{|u|> s\}} dx\\
&=& \frac{1}{\mu}{\int_{{\Omega}}}b_0 |\nabla w|^p \chi_{\{|u|> s\}} dx + \frac{1}{\mu}{\int_{{\Omega}}}b_1 |u|^{m} e^{p\mu |u|} \chi_{\{|u|> s\}} dx. \nonumber\end{aligned}$$
[**Estimate for $I_5 +I_6$:**]{} Using again and Lemma \[dis-mea\] we have $$\begin{aligned}
\label{realize-sig}
I_5 +I_6 &=& {\int_{{\Omega}}}F \cdot \nabla[(1+\mu |w_s|)^{p-1} w_s] \ dx + {\int_{{\Omega}}}v_s d f\\
&=& {\int_{{\Omega}}}F \cdot [(p-1)(1+\mu |w_s|)^{p-2}\nabla w_s \, {\rm sign}(w_s)\, \mu w_s] dx+ \nonumber\\
&& +\, {\int_{{\Omega}}}F \cdot [(1+\mu |w_s|)^{p-1} \nabla w_s] dx + {\int_{{\Omega}}}v_s df\nonumber\\
&\leq & p{\int_{{\Omega}}}|F| (1+\mu |w_s|)^{p-1} |\nabla w_s| dx + {\int_{{\Omega}}}(1+\mu |w_s|)^{p-1}|w_s| d|f|.\nonumber\end{aligned}$$
Using the inequality $$(1+\mu |w_s|)^{p-1}\leq (1+{\color{black}\tilde{\varepsilon}}) \mu^{p-1}|w_s|^{p-1} + C({\color{black}\tilde{\varepsilon}}, p),\qquad {\color{black}\tilde{\varepsilon}}>0,$$ and Hölder’s inequality we have
$$\begin{aligned}
I_5+ I_6&\leq& (1+{\color{black}\tilde{\varepsilon}}) \mu^{p-1} \, p {\int_{{\Omega}}}|F| |w_s|^{p-1} |\nabla w_s|dx + (1+{\color{black}\tilde{\varepsilon}}) \mu^{p-1} {\int_{{\Omega}}}|w_s|^p d|f| \\
&&+ \ C({\color{black}\tilde{\varepsilon}},p) \Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big) \norm{\nabla w_s}_{L^{p}({\Omega})}.
\end{aligned}$$
We recall that by approximation and Fatou’s lemma holds for all $\varphi\in {W_0^{1,p}({\Omega})}$. Then by we get $$\begin{aligned}
\label{I4I5}
I_5+ I_6&\leq& (1+{\color{black}\tilde{\varepsilon}}) \mu^{p-1} \lambda \norm{\nabla w_s}^{p}_{L^{p}({\Omega})}+\\
&& +\ C({\color{black}\tilde{\varepsilon}},p ) \Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big) \norm{\nabla w_s}_{L^{p}({\Omega})}.\nonumber\end{aligned}$$
We now use estimates , , and in equality to obtain the following bound $$\begin{aligned}
\kappa(\varepsilon)\norm{\nabla w_s}^{p}_{L^{p}({\Omega})}&\leq& \frac{1}{\mu}{\int_{{\Omega}}}b_0 |\nabla w|^p \chi_{\{|u|> s\}} dx + \frac{1}{\mu}{\int_{{\Omega}}}b_1 |u|^{m} e^{p\mu|u|} \chi_{\{|u|> s\}} dx +\\
&& +\ C({\color{black}\tilde{\varepsilon}}, p)\Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big) \norm{\nabla w_s}_{L^{p}({\Omega})},\end{aligned}$$ where $\kappa(\varepsilon)=\alpha_0 - (1+\varepsilon) \mu^{p-1} \lambda$. Observe that when $\delta<\delta_0=(p-1) (\alpha_{0}/\lambda)^{\frac{1}{p-1}}$ we have $$\mu^{p-1}\lambda= ({\delta}/(p-1))^{p-1}\lambda < ({\delta}_0/(p-1))^{p-1}\lambda = \alpha_0$$ and thus we can choose $\varepsilon>0$ small enough so that $\kappa(\varepsilon)>0$.
Since [$(e^{\mu |u|}-1) \in {W_0^{1,p}({\Omega})}$]{}, by Sobolev’s embedding theorem it holds that $e^{p\mu |u|}\in L^{\frac{n}{n-p}}({\Omega})$ [if $1<p<n$ and $e^{p\mu |u|}\in L^{2}({\Omega})$, say, if $p\geq n$]{}. Thus we have $|u|^m e^{p\mu |u|} \in L^{1}({\Omega})$. Now letting $s\textcolor{black}{\nearrow}\infty$ in the last we find $$\norm{\nabla w}^{p}_{L^{p}({\Omega})} \leq C\Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big) \norm{\nabla w}_{L^{p}({\Omega})},$$ which yields $$\norm{e^{{\delta}|u|/(p-1)}-1}_{{W_0^{1,p}({\Omega})}} \leq C(\delta, \lambda, p)\Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big)^{\frac{1}{p-1}}.$$
Finally, note that $$\norm{u}_{{W_0^{1,p}({\Omega})}}=\norm{\nabla u}_{L^p({\Omega})}\leq \frac{p-1}{{\delta}}\norm{\nabla(e^{{\delta}|u|/(p-1)}-1)}_{L^p({\Omega})}$$ and we also have $$\norm{u}_{{W_0^{1,p}({\Omega})}} \leq C(\delta, \lambda, p)\Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big)^{\frac{1}{p-1}}.$$ This proves inequality for all $\delta\in [\gamma_0, \delta_0)$.
To prove inequality for ${\delta}_{1}$, we first define $\mu_1=\frac{{\delta}_{1}}{p-1}$ and redefine $$\label{redefinedw}
w= {\rm sign}(u) [e^{\mu_1 |u|}-1]/\mu_{1}, \qquad w_s= {\rm sign}(u) [e^{\mu_1 |u_s|}-1]/\mu_{1}.$$
Observe that $$(e^{\mu_1 |u_s|} -1) e^{{\delta}_1 |u_s|}\geq (1-\varepsilon) e^{ (\delta_1 +\mu_1)|u_s|} -C(\varepsilon,\delta_1) \qquad \text{~for~all~} \varepsilon\in (0,1),$$ and thus by the first inequality in , with $({\delta}_1, \mu_1)$ in place of $({\delta},\mu)$, we have $$\begin{aligned}
I_3 + I_4 &\leq & {\int_{{\Omega}}}(-{\delta}_1+\gamma_0) \frac{{\alpha}_0}{\mu_1} (e^{\mu_1 |u_s|} -1) e^{{\delta}_1 |u_s|} |\nabla u_s|^p \ dx \\
&& +\ {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) {\rm sign}(u) e^{{\delta}_1 |u_s|} |w_s| \chi_{\{|u|> s\}} dx\nonumber\\
&\leq& {\int_{{\Omega}}}(1-\varepsilon)(-{\delta}_1+\gamma_0) \frac{{\alpha}_0}{\mu_1} |\nabla w_s|^p dx + {\int_{{\Omega}}}C(\varepsilon,\delta_1)({\delta}_1-\gamma_0) \frac{{\alpha}_0}{\mu_1} |\nabla u_s|^p \ dx \nonumber\\
&& +\ {\int_{{\Omega}}}\mathcal{B}(x, u, \nabla u) {\rm sign}(u) e^{{\delta}_1 |u_s|} |w_s| \chi_{\{|u|> s\}} dx.\nonumber\end{aligned}$$ Here in the last inequality we used that ${\delta}_1 > \gamma_0$ and $|\nabla w_s|^p= e^{ (\delta_1 +\mu_1)|u_s|} |\nabla u_s|^p$.
Thus arguing as in for the last term we find $$\begin{aligned}
\label{I34second}
I_3 + I_4 &\leq& {\int_{{\Omega}}}(1-\varepsilon)(-{\delta}_1+\gamma_0) \frac{{\alpha}_0}{\mu_1} |\nabla w_s|^p dx + C(\varepsilon) {\int_{{\Omega}}}|\nabla u_s|^p \ dx \\
&& +\, \frac{1}{\mu_1}{\int_{{\Omega}}}b_0 |\nabla w|^p \chi_{\{|u|> s\}} dx + \frac{1}{\mu_1}{\int_{{\Omega}}}b_1 |u|^{m} e^{p\mu_1 |u|} \chi_{\{|u|> s\}} dx. \nonumber\end{aligned}$$
Using estimates , , (with $({\delta}_1, \mu_1)$ in place of $({\delta},\mu)$) and in equality we then get $$\begin{aligned}
\kappa_1(\varepsilon)\norm{\nabla w_s}^{p}_{L^{p}({\Omega})}&\leq& \frac{1}{\mu_1}{\int_{{\Omega}}}b_0 |\nabla w|^p \chi_{\{|u|> s\}} dx + \frac{1}{\mu_1}{\int_{{\Omega}}}b_1 |u|^{m} e^{p\mu_1|u|} \chi_{\{|u|> s\}} dx +\\
&& +\ C(\varepsilon)\Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big) \norm{\nabla w_s}_{L^{p}({\Omega})} +\\
&& +\ C(\varepsilon) {\int_{{\Omega}}}|\nabla u_s|^p \ dx,\end{aligned}$$ where $\kappa_1(\varepsilon)=\alpha_0 + (1-\varepsilon)({\delta}_1-\gamma_0) \frac{{\alpha}_0}{\mu_1} - (1+\varepsilon) \mu_{1}^{p-1} \lambda$, with $\varepsilon\in (0,1)$. Thus when holds we can find $\varepsilon\in (0,1)$ such that $\kappa_1(\varepsilon)>0$. Then using Young’s inequality and letting $s\rightarrow\infty$ we eventually obtain $$\begin{aligned}
\norm{\nabla w}^{p}_{L^{p}({\Omega})}&\leq& C \Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big)^{\frac{1}{p-1}} + C {\int_{{\Omega}}}|\nabla u|^p \ dx.\end{aligned}$$
This proves inequality for all $\delta_1 > \gamma_0$ such holds.
Existence of solutions to an [approximate]{} equation {#ApproxExist}
=====================================================
[For $k>0$, we now define a function $\mathcal{H}_k(x,s,\xi)$ by letting $$\label{HK}
\textcolor{black}{\mathcal{H}_k(x,s,\xi):= \frac{\mathcal{B}(x,s,\xi)}{1+ \frac{1}{k}|\, \mathcal{B}(x,s,\xi)| }.}$$ Note $|\mathcal{H}_k(x,s,\xi)|\leq k$, and also holds with $\mathcal{H}_k(x,s,\xi)$ in place of $\mathcal{B}(x,s,\xi)$. Moreover, $$\lim_{k\rightarrow\infty}\mathcal{H}_k(x,s,\xi)=\mathcal{B}(x,s,\xi).$$]{}
[ The goal of this section is to obtain existence results for the [approximate]{} equation $$\label{kequ}
-\operatorname{div}\aa(x, u, \nabla u) = \mathcal{H}_k(x, u, \nabla u) + \sigma {\quad \textrm{in}\quad } \Omega.$$ ]{}
\[approx-k\] Let $\sigma={\rm div}\, F + f$ where and $f$ is a locally finite signed measure in ${\Omega}$ with $|f|\in (W^{1,p}_0({\Omega}))^*$ such that holds for all $\varphi\in C_c^\infty({\Omega})$, with $\lambda\in (0, \gamma_0^{1-p}\alpha_0(p-1)^{p-1}).$ Then for there exists a solution $u_{k}\in W_0^{1,p}({\Omega})$ to such that $e^{\frac{\delta|u_k|}{p-1}}-1\in W^{1,p}_0({\Omega})$ for all ${\delta}\in [\gamma_0, {\delta}_0)$, with $\delta_0=(p-1) (\alpha_{0}/\lambda)^{\frac{1}{p-1}}$, and $$\label{kexp}
\|u_k\|_{{W_0^{1,p}({\Omega})}} + \|e^{\frac{\delta|u_k|}{p-1}}-1\|_{{W_0^{1,p}({\Omega})}} \leq M_{{\delta}}.$$
Moreover, for any $\delta_1 > \gamma_0$ such that holds then we have $$\label{apri2-k}
\|e^{\frac{\delta_1|u_k|}{p-1}}-1\|_{{W_0^{1,p}({\Omega})}} \leq M_{{\delta}_{1}},$$ Here the constants $M_\delta$ and $M_{{\delta}_{1}}$ are independent of $k$.
Since $\sigma\in ({W_0^{1,p}({\Omega})})^{*}$ and by the theory of pseudomonotone operators (see, e.g., [@Li], [@MZ Chapter 6], and [@Bre]), for any $\varepsilon>0$ there exists a solution $u_{k, \varepsilon}\in {W_0^{1,p}({\Omega})}$ to the equation $$\label{approx-ep}
-\operatorname{div}\aa(x, u, \nabla u) +\varepsilon |u|^{p-2} u= {\mathcal{H}}_k(x, u, \nabla u) + \sigma {\quad \textrm{in}\quad } \Omega.$$
[The next step is to obtain uniform bounds of the form - for $\{u_{k, \varepsilon}\}$. However, we cannot directly apply Theorem \[regularity\_Murat-Ferone\] here since we do not know if $e^{\frac{\delta|u_{k, \varepsilon}|}{p-1}}-1\in W^{1,p}_0({\Omega})$. The strategy here is to follow the proof of Theorem \[regularity\_Murat-Ferone\]. For simplicity let us write $u=u_{k, \varepsilon}$,]{} and for each $s>0$, we set $v_s=e^{{\delta}|u_s|} w_s,$ where $u_s= T_s(u)$ and $w_s={\rm sign}(u) [e^{\mu |u_s|} -1]/\mu$ with $\mu=\delta/(p-1)$. Then using $v_s$ as a test function for we obtain the following equality $$\label{I12345prime}
I_1+I_2=I_3+I_4'+I_5+I_6,$$ where the expressions for $I_1, I_2, I_3, I_5, I_6$ are as in the proof of Theorem \[regularity\_Murat-Ferone\]. The term $I_4'$ is similar to $I_4$ given in the proof of Theorem \[regularity\_Murat-Ferone\] except that $\mathcal{B}(x, u, \nabla u)$ is now replaced by ${\mathcal{H}}_k(x, u, \nabla u)$. That is, $$I_4':= {\int_{{\Omega}}}{\mathcal{H}}_k(x, u, \nabla u) e^{{\delta}|u_s|} w_s \ dx.$$
Thus lower estimates for $I_1$, $I_2$ and upper estimates for $I_5+I_6$ are unchanged; see , , and . As in we have the following upper estimate for $I_3+I_4'$: $$\begin{aligned}
I_3 + I_4' &\leq & {\int_{{\Omega}}}(-{\delta}+\gamma_0) {\alpha}_0 |w_s| e^{{\delta}|u_s|} |\nabla u_s|^p \ dx \\
&& +\ {\int_{{\Omega}}}{\mathcal{H}}_k(x, u, \nabla u) {\rm sign}(u) e^{{\delta}|u_s|} |w_s| \chi_{\{|u|> s\}} dx\nonumber\\
&\leq& {\int_{{\Omega}}}{\mathcal{H}}_k(x, u, \nabla u) {\rm sign}(u) e^{{\delta}|u_s|} |w_s| \chi_{\{|u|> s\}} dx.\nonumber\end{aligned}$$
Thus, instead of , we now get $$\label{I34prime-k}
I_3 + I_4' \leq k {\int_{{\Omega}}}e^{{\delta}s} \frac{e^{\mu s}-1}{\mu} \chi_{\{|u|> s\}} dx.$$
Similarly, instead of , we now obtain $$\begin{aligned}
\label{I34second-k}
I_3 + I_4' &\leq& {\int_{{\Omega}}}(1-\varepsilon)(-{\delta}_1+\gamma_0) \frac{{\alpha}_0}{\mu_1} |\nabla w_s|^p dx + C(\varepsilon) {\int_{{\Omega}}}|\nabla u_s|^p \ dx \\
&& +\, k {\int_{{\Omega}}}e^{{\delta}s} \frac{e^{\mu s}-1}{\mu} \chi_{\{|u|> s\}} dx. \nonumber\end{aligned}$$ We recall that in , $\mu_1=\frac{{\delta}_1}{p-1}$ with $w, w_s$ to be understood as in .
When $\varepsilon>0$ and $s$ is such that $\varepsilon s^{p-1}\geq k$ by and we have $$\label{ep-absorb}
I_3 + I_4' -I_2\leq 0$$ and thus it follows from that $$I_1 \leq I_5+ I_6.$$
With this, employing and we find $$\norm{\nabla w_s}_{L^p({\Omega})} \leq C(\delta, \lambda,p)\Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big)^{\frac{1}{p-1}}.$$ At this point we let $s\textcolor{black}{\nearrow}\infty$ to obtain that any solution $u=u_{k,\varepsilon}$ to satisfies the bound for every $\delta\in [\gamma_0, \delta_0)$.
For ${\delta}_1>\gamma_0$ such that holds, using and arguing similarly we obtain $$\begin{aligned}
\label{de1-k}
\norm{e^{\frac{{\delta}_1 |u_{k, \varepsilon}|}{p-1}}-1}_{{W_0^{1,p}({\Omega})}} &\leq& C \Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big)^{\frac{1}{p-1}}\\
&& +\, C {\int_{{\Omega}}}|\nabla u_{k, {\varepsilon}}|^p \ dx\nonumber\\
&\leq& C \Big(\norm{F}_{L^{\frac{p}{p-1}}({\Omega})}+\norm{|f|}_{({W_0^{1,p}({\Omega})})^{*}}\Big)^{\frac{1}{p-1}}. \nonumber\end{aligned}$$
As the bound is uniform in $\varepsilon$, we can extract a subsequence, still denoted by $\varepsilon$, such that $$u_{k,\varepsilon} \rightarrow u_k \text{~weakly~in~} {W_0^{1,p}({\Omega})}, \text{~strongly~in~} L^p({\Omega}), \text{~and~a.e.~in~} {\Omega},$$ as $\varepsilon\textcolor{black}{\searrow} 0^{+}$ for a function $u_k\in {W_0^{1,p}({\Omega})}$. Due to the pointwise a.e. convergence, we see that $u_k$ also satisfies - for every $\delta\in [\gamma_0, \delta_0)$ and every ${\delta}_1>\gamma_0$ such that holds.
Recall that we have $$\label{ekequn}
-\operatorname{div}\aa(x, u_{k,\varepsilon},\nabla u_{k,\varepsilon}) +\varepsilon |u_{k,\varepsilon}|^{p-2} u_{k,\varepsilon}= {\mathcal{H}}_k(x, u_{k,\varepsilon}, \nabla u_{k,\varepsilon}) + \sigma \quad {\rm in~} \mathcal{D}'(\Omega).$$
For each fixed $k>0$, we know ${\mathcal{H}}_k(x, u_{k,\varepsilon}, \nabla u_{k,\varepsilon})-\varepsilon |u_{k,\varepsilon}|^{p-2} u_{k,\varepsilon}$ is uniformly bounded in $\varepsilon\in (0,1)$ as finite measures in ${\Omega}$. Thus by a convergence result shown in , we may further assume that $$\nabla u_{k,\varepsilon}\rightarrow \nabla u_k \quad \text{a.e.~in~} {\Omega}, \text{~as~} \varepsilon\textcolor{black}{\searrow} 0^+.$$
This allows us to pass to the limit in as $\varepsilon\textcolor{black}{\searrow} 0^+$ to see that $u_k$ solves and satisfies the bounds -.
Proof of Theorem \[MainExistence\] {#ActualExistence}
==================================
This section is devoted to the proof of Theorem \[MainExistence\].
For each $k>0$, let $u_k$ be a solution of the [approximate]{} equation as obtained in Proposition \[approx-k\]. Recall that $\mathcal{H}_k(x, s, \xi)$ is defined in . By - and Rellich’s compactness theorem, there is a subsequence, still denoted by $k$, such that $$u_{k} \xrightarrow{k} u \text{~weakly~in~} {W_0^{1,p}({\Omega})}, \text{~strongly~in~} L^p({\Omega}), \text{~and~a.e.~in~} {\Omega},$$ for some function $ u \in {W_0^{1,p}({\Omega})}$ such that $e^{\frac{\delta|u|}{p-1}}-1\in W^{1,p}_0({\Omega})$ for each ${\delta}\in [\gamma_0, {\delta}_0)$, and $e^{\frac{\delta_1|u|}{p-1}}-1\in W^{1,p}_0({\Omega})$ for any ${\delta}_1 >\gamma_0$ such that holds.
As $u_k$ solves , we have $$\label{kequ-k}
-\operatorname{div}\aa(x, u_k, \nabla u_k) = {\mathcal{H}}_k(x, u_k, \nabla u_k) + \sigma {\quad \textrm{in}\quad } \Omega.$$
Thus to show that $u$ is a solution of it is enough to show that $$\label{strongconv}
u_k\rightarrow u \quad \text{strongly~in~} {W_0^{1,p}({\Omega})}\text{~as~} k\textcolor{black}{\nearrow}\infty,$$ so that we can pass to the limit in using , , and Vitali’s [Convergence]{} Theorem.
For each $s>0$ we can write $$\nabla u_k-\nabla u= \nabla T_s(u_k)- \nabla T_s(u) + \nabla G_s(u_k)- \nabla G_s(u),$$ where $$G_s(r):=r-T_s(r), \qquad r\in{\mathbb{R}}.$$
In order to show we shall show that the following hold: $$\begin{gathered}
\lim_{s\rightarrow\infty}\ \sup_{k>0}\norm{\nabla G_s(u_k)- \nabla G_s(u)}_{L^p({\Omega})} =0 \label{tail} \\
\lim_{k\rightarrow\infty}\norm{\nabla T_s(u_k)- \nabla T_s(u)}_{L^p({\Omega})} =0 \quad \text{for each } s>0.\label{head}\end{gathered}$$
The rest of the proof will be devoted to the verification of these limits.
[**Proof of .**]{} Define $w_k = [e^{\frac{{\delta}}{p-1} |u_k| }-1]\tfrac{p-1}{\delta}$ and hence we get $$\begin{aligned}
{\int_{{\Omega}}}|\nabla G_s(u_k)|^p \ dx&=&{\int_{\{|u_k|>s\}}} |\nabla u_k|^p \ dx \\
&=& {\int_{\{|u_k|>s\}}} e^{-\frac{{\delta}p}{p-1}|u_k|}|\nabla w_k|^p \ dx \\
&\leq& e^{-\frac{{\delta}p}{p-1}s}{\int_{\{|u_k|>s\}}} |\nabla w_k|^p \ dx.
\end{aligned}$$
Using the estimate , we then find $$\label{taildelta}
{\int_{{\Omega}}}|\nabla G_s(u_k)|^p \ dx\leq C(\delta) e^{-\frac{{\delta}p}{p-1}s},$$ which yields .
[**Proof of .**]{} Following [@FM3] (see also the earlier works [@FM2; @BBM]), we shall make use of the following test function in : $$v_k = {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k), \quad {\rm with~} j\geq s,$$ where $z_k = T_s(u_k) - T_s(u)$ and $\psi$ is a $C^1$ and increasing function from ${\mathbb{R}}$ to ${\mathbb{R}}$ satisfying $$\label{psicond}
\psi(0) = 0 \quad \text{and} \quad \psi' - H_0 |\psi| \geq 1,$$ where $H_0=\frac{b_0+ (a_0+a_1) {\delta}}{\alpha_0}$. For example, the function $\psi(r)= 2re^{\frac{H_0^2 r^2}{4}}$ will do. We then have $$\begin{aligned}
\lefteqn{{\int_{{\Omega}}}{\mathcal{A}(x,\ifblank{u_k, \nabla u_k}{\nabla u\:}{u_k, \nabla u_k})} \cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \nabla z_k {\, dx}}\\
&=& {\int_{{\Omega}}}\Big[{\mathcal{H}}_k(x, u_k, \nabla u_k) - {\delta}\mathcal{A}(x,\nabla u_k) \cdot \nabla T_j(u_k) {\rm sign}(u_k)\Big] {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) {\, dx}\\
&& + \ \langle\sigma, {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) \rangle. \end{aligned}$$
Note that the term on the left-hand side in the above equality can be written as $$\begin{aligned}
\lefteqn{{\int_{{\Omega}}}{\mathcal{A}(x,\ifblank{u_k, \nabla u_k}{\nabla u\:}{u_k, \nabla u_k})} \cdot (\nabla T_s(u_k) - \nabla T_s(u)) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k){\, dx}} \\
& =& {\int_{\{{\lvert u_k \rvert}\leq s\}}} ({\mathcal{A}(x,\ifblank{T_s(u_k), \nabla T_s(u_k)}{\nabla u\:}{T_s(u_k), \nabla T_s(u_k)})} - {\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s (u)}{\nabla u\:}{ T_s(u_k), \nabla T_s (u)})} )\cdot \\
&& \qquad \qquad \qquad \cdot\, (\nabla T_s(u_k) - \nabla T_s(u)) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k){\, dx}\\
& &\quad + {\int_{\{{\lvert u_k \rvert}\leq s\}}}{\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s (u)}{\nabla u\:}{ T_s(u_k), \nabla T_s (u)})} \cdot (\nabla T_s(u_k) - \nabla T_s(u)) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k){\, dx}\\
& &\quad + {\int_{\{{\lvert u_k \rvert}>s\}}} {\mathcal{A}(x,\ifblank{ u_k, \nabla u_k}{\nabla u\:}{ u_k, \nabla u_k})} \cdot (- \nabla T_s(u)) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k){\, dx}.\end{aligned}$$
Thus combining the last two equalities we obtain $$\label{First5}
I_1=-I_2-I_3 + I_4 +I_5,$$ $$\begin{aligned}
I_1&=&{\int_{\{{\lvert u_k \rvert}\leq s\}}} ({\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s(u_k)}{\nabla u\:}{ T_s(u_k), \nabla T_s(u_k)})} - {\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s (u)}{\nabla u\:}{ T_s(u_k), \nabla T_s (u)})} )\cdot\\
&& \qquad \qquad \qquad \cdot\, (\nabla T_s(u_k) - \nabla T_s(u)) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k){\, dx},\end{aligned}$$ $$I_2={\int_{\{{\lvert u_k \rvert}\leq s\}}}{\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s (u)}{\nabla u\:}{ T_s(u_k), \nabla T_s (u)})} \cdot (\nabla T_s(u_k) - \nabla T_s(u)) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k){\, dx},$$ $$I_3={\int_{\{{\lvert u_k \rvert}>s\}}} {\mathcal{A}(x,\ifblank{ u_k, \nabla u_k}{\nabla u\:}{ u_k, \nabla u_k})} \cdot (- \nabla T_s(u)) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k){\, dx},$$ $$I_4={\int_{{\Omega}}}\Big[{\mathcal{H}}_k(x, u_k, \nabla u_k) - {\delta}\mathcal{A}(x, u_k, \nabla u_k) \cdot \nabla T_j(u_k) {\rm sign}(u_k)\Big] {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) {\, dx},$$ and $$I_5=\langle\sigma, {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) \rangle.$$
We now write $I_4$ as $$\label{Secondprime}
I_4= I_4' + I_4'',$$ where $$I_4':= {\int_{\{{\lvert u_k \rvert}> s\}}} H_{k,j}(x) \, {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) {\, dx},$$ $$I_4'':={\int_{\{{\lvert u_k \rvert}\leq s\}}} H_{k,j}(x) \, {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) {\, dx},$$ with $$H_{k,j}(x):= {\mathcal{H}}_k(x, u_k, \nabla u_k) - {\delta}\mathcal{A}(x, u_k,\nabla u_k) \cdot \nabla T_j(u_k) {\rm sign}(u_k).$$
Note that ${\lvert \nabla T_j(u_k) \rvert}\leq {\lvert \nabla u_k \rvert}$ and hence using the growth conditions in and we get $$\begin{aligned}
|I_4''| &\leq& {\int_{\{{\lvert u_k \rvert}\leq s\}}} \Big(b_0 {\lvert \nabla u_k \rvert}^p + b_1 |u_k|^m + {\delta}a_0 {\lvert \nabla u_k \rvert}^{p} +\delta a_1 |u_k|^{p-1} |\nabla u_k|\Big)\times\\
&& \qquad \qquad \qquad \times {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert}\ {\, dx}\\
&\leq& {\int_{\{{\lvert u_k \rvert}\leq s\}}} (b_0 + {\delta}(a_0+a_1) ){\lvert \nabla u_k \rvert}^p {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert}\ {\, dx}\\
&& +\ {\int_{\{{\lvert u_k \rvert}\leq s\}}} \Big(b_1 |u_k|^m +c(p)\delta a_1 |u_k|^{p} \Big) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert}\ {\, dx}\\
&\leq& \frac{b_0 + {\delta}(a_0+a_1)}{\alpha_0} {\int_{\{{\lvert u_k \rvert}\leq s\}}} \mathcal{A}(x, T_s(u_k), \nabla T_s(u_k)) \cdot \nabla T_s(u_k) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert}\ {\, dx}\\
&& +\ {\int_{\{{\lvert u_k \rvert}\leq s\}}} \Big(b_1 |u_k|^m + c(p)\delta a_1 |u_k|^{p} \Big) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert}\ {\, dx},\end{aligned}$$ where we used Young’s inequality in the second inequality and the coercivity condition in the last inequality. we find $$\begin{aligned}
|I_4''|
& \leq & H_0 {\int_{\{{\lvert u_k \rvert}\leq s\}}} \left[ {\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s(u_k)}{\nabla u\:}{ T_s(u_k), \nabla T_s(u_k)})} -{\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s(u)}{\nabla u\:}{ T_s(u_k), \nabla T_s(u)})} \right] \cdot \\
&& \qquad \qquad \qquad \cdot \left[ \nabla T_s(u_k) - \nabla T_s(u) \right] {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert} dx \\
&& + \ H_0{\int_{\{{\lvert u_k \rvert}\leq s\}}}{\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s(u)}{\nabla u\:}{ T_s(u_k), \nabla T_s(u)})} \cdot \left[ \nabla T_s(u_k) - \nabla T_s(u) \right] {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert} dx \\
&& + \ H_0{\int_{\{{\lvert u_k \rvert}\leq s\}}}{\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s(u_k)}{\nabla u\:}{ T_s(u_k), \nabla T_s(u_k)})} \cdot \nabla T_s(u) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert} dx \\
&& +\ {\int_{\{{\lvert u_k \rvert}\leq s\}}} \Big(b_1 |u_k|^m + c(p)\delta a_1 |u_k|^{p-1} |\nabla T_s(u_k)|\Big) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert} dx.\end{aligned}$$
Using this bound, equalities -, and the inequality in , we now obtain $$\label{I8}
I_1'\leq -I_2-I_3 + I_4' +I_5 +I_6 + I_7 +I_8,$$ where $$I_1'={\int_{\{{\lvert u_k \rvert}\leq s\}}} ({\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s(u_k)}{\nabla u\:}{ T_s(u_k), \nabla T_s(u_k)})} - {\mathcal{A}(x,\ifblank{T_s (u), \nabla T_s (u)}{\nabla u\:}{T_s (u), \nabla T_s (u)})} )\cdot (\nabla T_s(u_k) - \nabla T_s(u)) \ dx,$$ $$I_6=H_0 {\int_{\{{\lvert u_k \rvert}\leq s\}}}{\mathcal{A}(x,\ifblank{T_s(u_k), \nabla T_s(u)}{\nabla u\:}{T_s(u_k), \nabla T_s(u)})} \cdot \left[ \nabla T_s(u_k) - \nabla T_s(u) \right] {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert}\ {\, dx},$$ $$I_7=H_0 {\int_{\{{\lvert u_k \rvert}\leq s\}}}{\mathcal{A}(x,\ifblank{T_s(u_k), \nabla T_s(u_k)}{\nabla u\:}{T_s(u_k), \nabla T_s(u_k)})} \cdot \nabla T_s(u) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert}\ {\, dx},$$ and $$I_8={\int_{\{{\lvert u_k \rvert}\leq s\}}} \Big(b_1 |u_k|^m + c(p)\delta a_1 |u_k|^{p-1} |\nabla T_s(u_k)|\Big) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\ {\lvert \psi(z_k) \rvert}\ {\, dx}.$$
We shall next treat each term on the right-hand side of .
[**The term $I_2$:**]{} We know that $u_k \xrightarrow{k} u$ a.e., from which we see that $z_k \xrightarrow{k} 0$ a.e. and hence $${\mathcal{A}(x,\ifblank{T_s (u_k), \nabla T_s (u)}{\nabla u\:}{T_s (u_k), \nabla T_s (u)})} {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \xrightarrow{k} {\mathcal{A}(x,\ifblank{T_s (u), \nabla T_s (u)}{\nabla u\:}{T_s (u), \nabla T_s (u)})} {e^{{\delta}{\lvert T_j(u) \rvert}}} \psi'(0) \quad {\rm a.e.}$$
Thus using the pointwise estimate, which follows from , $$|{\mathcal{A}(x,\ifblank{T_s(u_k), \nabla T_s(u)}{\nabla u\:}{T_s(u_k), \nabla T_s(u)})} e^{{\delta}|T_j(u_k)|} \psi'(z_k)| \leq e^{{\delta}j} \max_{r \in [-2s,2s]}{\lvert \psi'(r) \rvert} \Big[ a_0|\nabla T_s(u)|^{p-1} + a_1 s^{p-1}\Big]$$ and the fact that , it follows from Lebesgue’s Dominated Convergence Theorem that $${\mathcal{A}(x,\ifblank{T_s (u_k), \nabla T_s (u)}{\nabla u\:}{T_s (u_k), \nabla T_s (u)})} {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \xrightarrow{k} {\mathcal{A}(x,\ifblank{T_s (u), \nabla T_s (u)}{\nabla u\:}{T_s (u), \nabla T_s (u)})} {e^{{\delta}{\lvert T_j(u) \rvert}}} \psi'(0)$$ strongly in
Since $ \norm {T_s(u_k)}_{{W_0^{1,p}({\Omega})}}$ is uniformly bounded in $k$ and $T_s(u_k) \xrightarrow{k} T_s(u)$ a.e. we get that $\nabla T_s(u_k) \xrightharpoonup{k} \nabla T_s(u)$ weakly in Also, since $$\label{chiconv}
\chi_{\{{\lvert u_k \rvert} \leq s\}} \xrightarrow{k} \chi_{\{{\lvert u \rvert} \leq s\}} {\rm~ a.e. ~in~} {\Omega}\setminus\{{\lvert u \rvert} = s\} {\rm ~while~} |\nabla T_s(u)| = 0 {\rm~ a.e.~ on~} \{{\lvert u \rvert} = s\},$$ we have from Lebesgue’s Dominated Convergence Theorem that $$\nabla T_s(u)\chi_{\{{\lvert u_k \rvert} \leq s\}} \xrightarrow{k} \nabla T_s(u)\chi_{\{{\lvert u \rvert} \leq s\}}=\nabla T_s(u) {\quad \textrm{strongly in}\quad } \textcolor{black}{L^p({\Omega},{\mathbb{R}}^n). }$$ Thus with the observation $\chi_{\{{\lvert u_k \rvert} \leq s\}} (\nabla T_s(u_k) - \nabla T_s(u)) = \nabla T_s(u_k) - \nabla T_s(u)\chi_{\{{\lvert u_k \rvert} \leq s\}}$, we see that $$\label{weaknablaz}
\chi_{\{{\lvert u_k \rvert} \leq s\}} (\nabla T_s(u_k) - \nabla T_s(u)) \xrightharpoonup{k} 0 {\quad \textrm{weakly in}\quad } \textcolor{black}{L^p({\Omega},{\mathbb{R}}^n).}$$
The above calculations imply that $\lim_{k\rightarrow\infty} I_2 =0$.
[**The term $I_3$:**]{} By , $|{\mathcal{A}(x,\ifblank{ u_k, \nabla u_k}{\nabla u\:}{ u_k, \nabla u_k})}|$ is uniformly bounded in $L^{\frac{p}{p-1}}({\Omega})$. On the other hand, again by and Lebesgue’s Dominated Convergence Theorem we have $$|\chi_{\{{\lvert u_k \rvert}>s\}} (-\nabla T_s(u)) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k)| \xrightarrow{k} 0 {\quad \textrm{strongly in}\quad } L^p({\Omega}).$$
Thus we see that $\lim_{k\rightarrow\infty}I_3=0$.
[**The term $I_4'$:**]{} We have the inequalities ${\mathcal{A}(x,\ifblank{u_k, \nabla u_k}{\nabla u\:}{u_k, \nabla u_k})}\cdot \nabla T_j(u_k) \geq \alpha_0 {\lvert \nabla u_k \rvert}^p \chi_{\{{\lvert u_k \rvert}\leq j\}}$ and $\chi_{\{{\lvert u_k \rvert}>s\}} {\rm sign}(u_k) \psi(z_k) \geq 0$. Thus using the second inequality in we see that $$\begin{aligned}
I_4' &=& {\int_{\{{\lvert u_k \rvert}>s\}}} \left[ {\rm sign}(u_k) {\mathcal{H}}_k (x,u_k, \nabla u_k) -{\delta}{\mathcal{A}(x,\ifblank{u_k, \nabla u_k}{\nabla u\:}{u_k, \nabla u_k})} \cdot \nabla T_j(u_k) \right] \times \\
&& \qquad \qquad \qquad \qquad \times \ {\rm sign}(u_k)
{e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) {\, dx}\\
&\leq& {\int_{\{{\lvert u_k \rvert}>s\}}} \left[ \gamma_0\alpha_0 {\lvert \nabla u_k \rvert}^p -{\delta}\alpha_0{\lvert \nabla u_k \rvert}^{p} \chi_{\{{\lvert u_k \rvert}\leq j\}} \right] {\rm sign}(u_k) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) {\, dx}\\
& \leq & {\int_{\{{\lvert u_k \rvert}>j\}}} \gamma_0\alpha_0 {\lvert \nabla u_k \rvert}^p {\rm sign}(u_k) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) {\, dx},\end{aligned}$$ where we used that $\delta\geq \gamma_0$ [and $j\geq s$]{} in the last inequality. At this point, using with $j$ in place of $s$, we get $$\begin{aligned}
I_4' & \leq & \gamma_0\alpha_0 \max_{r\in[-2s,2s]}|\psi(r)| \ e^{\delta j}{\int_{\{{\lvert u_k \rvert}>j\}}} {\lvert \nabla u_k \rvert}^p {\, dx}\\
& \leq & C(\delta) \gamma_0\alpha_0 \max_{r\in[-2s,2s]}|\psi(r)| \ e^{\delta j}\ e^{-\frac{\delta p}{p-1} j}.\end{aligned}$$
This yields that [$\limsup_{j\rightarrow\infty} \sup_{k>0} I_4'=0.$]{}
[**The term $I_5$:**]{} Since $f\in (W_0^{1,p}({\Omega}))^*$, there is a vector field such that ${\rm div}\, F_1 =f$ in $\mathcal{D}'({\Omega})$. Thus $\sigma= {\rm div}\, (F+F_1)$ which yields $$\begin{aligned}
\label{I5term}
I_5&=&{\delta}{\int_{{\Omega}}}(F+F_1) \cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) \nabla T_j(u_k) {\rm sign}(u_k)\ dx\\
&& +{\int_{{\Omega}}}(F+F_1)\cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \nabla z_k \ dx.\nonumber\end{aligned}$$
As $\psi(0)=0$ we have $ (F+F_1) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) \xrightarrow{k} (0, \dots,0)$ a.e. in ${\Omega}$. Thus by Lebesgue’s Dominated Convergence Theorem we find $$(F+F_1) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) \xrightarrow{k} (0,\dots, 0) {\quad \textrm{strongly in}\quad } \textcolor{black}{L^{\frac{p}{p-1}}({\Omega},{\mathbb{R}}^n)}.$$ Since $\nabla T_j(u_k) {\rm sign}(u_k)$ is uniformly bounded in $L^p({\Omega},{\mathbb{R}}^n)$, we then conclude that $$\label{de1}
{\delta}{\int_{{\Omega}}}(F+F_1) \cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi(z_k) \nabla T_j(u_k) {\rm sign}(u_k)\ dx \xrightarrow{k} 0.$$
We now write $$\label{splitF}
{\int_{{\Omega}}}(F+F_1)\cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \nabla z_k \ dx= {\color{black}R_1 + R_2}, $$ [where $$\begin{gathered}
R_1:= {\int_{\{{\lvert u_k \rvert}\leq s\}}}(F+F_1)\cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \nabla z_k \ dx \\R_2:= {\int_{\{{\lvert u_k \rvert}> s\}}}(F+F_1)\cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \nabla z_k \ dx.
\end{gathered}$$]{}
Again by Lebesgue’s Dominated Convergence Theorem we have $$(F+F_1) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \xrightarrow{k} {(F+F_1)} {e^{{\delta}{\lvert T_j(u) \rvert}}} \psi'(0) {\quad \textrm{strongly in}\quad } L^{\frac{p}{p-1}}({\Omega},{\mathbb{R}}^n).$$
Thus using (recall that $\nabla z_k= \nabla T_s(u_k) - \nabla T_s(u)$) we obtain that [$$R_1\xrightarrow{k} 0.$$]{}
On the other hand, from the definition of $z_k$ we have [$$R_2={\int_{{\Omega}}}{(F+F_1)}\cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) (-\nabla T_s(u)) \chi_{\{{\lvert u_k \rvert}> s\}}\ dx.$$]{}
Then by , Hölder’s inequality, and Lebesgue’s Dominated Convergence Theorem, it follows that [$$R_2 \xrightarrow{k} 0.$$]{}
Now recalling we get $$\label{de2}
{\int_{{\Omega}}}{(F+F_1)}\cdot {e^{{\delta}{\lvert T_j (u_k) \rvert}}}\psi'(z_k) \nabla z_k \ dx \xrightarrow{k} 0.$$
Hence using and in we conclude that $\lim_{k\rightarrow\infty} I_5=0$.
[**The terms $I_6$, $I_7$, and $I_8$:**]{} Since $\psi(0)=0$, by Lebesgue’s Dominated Convergence Theorem we find $$\chi_{\{{\lvert u_k \rvert}\leq s\}} {\mathcal{A}(x,\ifblank{T_s(u_k), \nabla T_s(u)}{\nabla u\:}{T_s(u_k), \nabla T_s(u)})} {e^{{\delta}{\lvert T_j (u_k) \rvert}}}{\lvert \psi(z_k) \rvert} \xrightarrow{k} 0 {\quad \textrm{strongly in }\quad } \textcolor{black}{L^{\frac{p}{p-1}}({\Omega},{\mathbb{R}}^n)}$$ and $$\chi_{\{{\lvert u_k \rvert}\leq s\}} \nabla T_s(u) {e^{{\delta}{\lvert T_j (u_k) \rvert}}}{\lvert \psi(z_k) \rvert} \xrightarrow{k} 0 {\quad \textrm{strongly in }\quad } \textcolor{black}{L^{p}({\Omega},{\mathbb{R}}^n).}$$
On the other hand, $\nabla T_s(u_k) - \nabla T_s(u)$ and ${\mathcal{A}(x,\ifblank{T_s(u_k), \nabla T_s(u_k)}{\nabla u\:}{T_s(u_k), \nabla T_s(u_k)})}$ are uniformly bounded in , respectively. Thus we obtain that $$\lim_{k\rightarrow\infty} I_6=\lim_{k\rightarrow\infty} I_7=0.$$
As for the term $I_8$, we estimate $$I_8\leq {\int_{\{{\lvert u_k \rvert}\leq s\}}} (b_1 |s|^m + c(p)\delta a_1 s^{p-1}) e^{\delta s} {\lvert \psi(z_k) \rvert} dx,$$ which also converges to zero, as $k\textcolor{black}{\nearrow}\infty$, by Lebesgue’s Dominated Convergence Theorem.
We have shown that $\lim_{k\rightarrow\infty} (-I_2-I_3 +I_5 +I_6 + I_7 +I_8)=0$ and [$\limsup_{j\rightarrow\infty} \sup_{k>0}I_4'=0$]{}. For each fixed $s>0$, we now let $$D_k=({\mathcal{A}(x,\ifblank{T_s(u_k), \nabla T_s(u_k)}{\nabla u\:}{T_s(u_k), \nabla T_s(u_k)})} - {\mathcal{A}(x,\ifblank{T_s(u_k), \nabla T_s (u)}{\nabla u\:}{T_s(u_k), \nabla T_s (u)})} )\cdot (\nabla T_s(u_k) - \nabla T_s(u)).$$ [As $D_k\geq 0$ (by ), in view of we find that]{} $$\label{Brow1}
{\int_{\{{\lvert u_k \rvert}\leq s\}}} D_k \ dx \xrightarrow{k} 0.$$
On the other hand, by , $$\begin{aligned}
\chi_{\{{\lvert u_k \rvert}> s\}} D_k
&=&\chi_{\{{\lvert u_k \rvert}> s\}} [{\mathcal{A}(x,\ifblank{ T_s(u_k), 0}{\nabla u\:}{ T_s(u_k), 0})} - {\mathcal{A}(x,\ifblank{ T_s(u_k), \nabla T_s (u)}{\nabla u\:}{ T_s(u_k), \nabla T_s (u)})}]\cdot (-\nabla T_s(u))\\
&\rightarrow& 0 \quad \text{a.e. as } k\textcolor{black}{\nearrow}\infty.\end{aligned}$$
It then follows from Lebesgue’s Dominated Convergence Theorem that $$\label{Brow2}
{\int_{\{{\lvert u_k \rvert} > s\}}} D_k \ dx \xrightarrow{k} 0.$$
Combining - we obtain $${\int_{{\Omega}}} D_k \ dx \xrightarrow{k} 0.$$
At this point we use the conditions - and a result of F. E. Browder (see [@Bro] or [@BMP Lemma 5]) to the proof of .
Proof of Theorems \[weakzero\] and \[Schro-type\] {#main-proofs}
=================================================
We are now ready to prove Theorem \[weakzero\].
\(i) Suppose that has a solution in $u\in W^{1,p}_0({\Omega})$ such that holds for some $A>0$. Then letting $F=|\nabla u|^{p-2}\nabla u$, we immediately have the desired representation for $\sigma$.
\(ii) Suppose that $\sigma={\rm div}\, F + f$ where and $f$ is a locally finite signed measure in ${\Omega}$ with $|f|\in (W^{1,p}_0({\Omega}))^*$ such that holds for some $\lambda\in (0, (p-1)^{p-1})$. Applying Theorem \[MainExistence\] we obtain a solution to that satisfies all of the properties stated in Theorem \[weakzero\](ii) except the Poincaré-Sobolev inequality . To verify it, we use $|\varphi|^p$, $\varphi\in C_c^\infty({\Omega})$, as a test function in to get $$\int_{{\Omega}} |\varphi|^p|\nabla u|^p dx = p \int_{\Omega}|\nabla u|^{p-2}\nabla u \cdot \nabla |\varphi| |\varphi|^{p-1} dx + \langle\sigma, |\varphi|^p \rangle.$$
Thus by Hölder’s inequality and condition we find $$\int_{{\Omega}} |\varphi|^p|\nabla u|^p dx \leq p \left(\int_{\Omega}|\nabla u|^{p} |\varphi|^p dx\right)^{\frac{p-1}{p}} \left(\int_{\Omega}|\nabla \varphi|^p dx\right)^{\frac{1}{p}} + (p-1)^{p-1} \int_{\Omega}|\nabla \varphi|^p dx.$$
At this point applying Young’s inequality we obtain the Poincaré-Sobolev inequality with some $A=A(p)>0$.
Finally, we prove Theorem \[Schro-type\].
By Theorem \[weakzero\](ii) we can find a solution $u\in W^{1,p}_0({\Omega}) $ to such that both $e^u-1$ and $e^{\frac{u}{p-1}}-1\in W^{1,p}_0({\Omega})$. Thus if we define $v=e^{\frac{u}{p-1}}$ then it holds that $v-1\in W^{1,p}_0({\Omega})$ and $v^{p-1}=e^{u}\in W^{1,p}({\Omega})$. We will show that $v$ is indeed a solution of .
We first observe that the function $e^u |\nabla u|^p$ belongs to $L^1({\Omega})$. Indeed, $$\begin{aligned}
\int_{\Omega}e^u |\nabla u|^pdx &=&\int_{\{u\geq 0\}\cap{\Omega}}e^u |\nabla u|^pdx + \int_{\{u< 0\}\cap{\Omega}} e^u |\nabla u|^pdx\\
&\leq& \int_{\{u\geq 0\}\cap{\Omega}} e^{p u} |\nabla u|^pdx + \int_{\{u< 0\}\cap{\Omega}} |\nabla u|^pdx\\
&\leq& \int_{{\Omega}} |\nabla (e^{u})|^pdx + \int_{{\Omega}} |\nabla u|^pdx <+\infty.\end{aligned}$$
Let $\varphi\in C_c^\infty({\Omega})$. Using $\phi_j:=\varphi \min\{ e^u, j\}$, $j>0$, as a test function for we have $$\label{j-test}
\int_{{\Omega}} |\nabla u|^{p-2}\nabla u \cdot \nabla\phi_jdx=\int_{{\Omega}} |\nabla u|^p \phi_j dx + \langle\sigma, \phi_j \rangle.$$ We now send $j\textcolor{black}{\nearrow}\infty$ in to obtain $$\int_{{\Omega}} |\nabla u|^{p-2}\nabla u \cdot \nabla(\varphi e^u)dx=\int_{{\Omega}} |\nabla u|^p \varphi e^u dx + \langle\sigma, \varphi e^u\rangle.$$ Here we use $e^u |\nabla u|^p \in L^1({\Omega})$ and Lebesgue’s Dominated Convergence Theorem. We note that actually by Lemma \[dis-mea\] we can immediately use $\varphi e^u$ as a test function. Thus after expanding and simplifying we get $$\int_{{\Omega}} [|\nabla u|^{p-2}\nabla u\cdot \nabla\varphi ] e^udx= \langle\sigma, \varphi e^u\rangle=\langle\sigma, \varphi v^{p-1}\rangle.$$
Note that $\nabla v=(p-1)^{-1} e^{\frac{u}{p-1}}\nabla u$ and thus $\nabla u= (p-1) e^{-\frac{u}{p-1}}\nabla v$. This yields that $$(|\nabla u|^{p-2}\nabla u) e^{u}= (p-1)^{p-1} |\nabla v|^{p-2} \nabla v,$$ and hence $$\int_{{\Omega}} |\nabla v|^{p-2}\nabla v\cdot \nabla\varphi dx= (p-1)^{1-p} \langle\sigma, \varphi v^{p-1}\rangle$$ for all $\varphi\in C_c^\infty({\Omega})$. This shows that $v$ is a solution of as claimed.
Finally, inequality follows from and the equality $|\frac{\nabla v}{v}|^p=(p-1)^{-p}|\nabla u|^p$.
\[mea-coeff\] [The above argument also works for the more general equation $$-{\rm div}\, \mathcal{A}(x, \nabla v) = (p-1)^{1-p}\, \sigma\, v^{p-1} \text{ in } \Omega, \qquad v \geq 0 \text{ in } \Omega, \qquad v = 1 \text{ on } \partial \Omega,$$ where $\mathcal{A}(x,\xi)$ satisfies - with $0<\alpha_0\leq a_0$ and the homogeneity condition $$\mathcal{A}(x, t\xi)= t^{p-1} \mathcal{A}(x, \xi) \qquad \text{for all } t>0.$$]{} [In this case $v=e^{\frac{u}{p-1}}$, where $u\in W^{1,p}_0({\Omega})$ solves the equation $$-{\rm div}\, \mathcal{A}(x, \nabla u)= \mathcal{A}(x, \nabla u)\cdot \nabla u + \sigma.$$ By Theorem \[MainExistence\], to guarantee that both $e^{u}-1$ and $e^{\frac{u}{p-1}} -1\in W^{1,p}_0({\Omega})$, we also need to assume $$\lambda\in \Big(0,\ a_0^{1-p} \alpha_0^{p}\, (p-1)^{ p-1}\Big) \quad \text{if} \quad \frac{a_0}{\alpha_0}\geq p-1$$ and $$\lambda\in (0,\ \alpha_0 p-a_0) \quad \text{if} \quad \frac{a_0}{\alpha_0} < p-1.$$ However, note that no regularity assumption in the $x$-variable of $\mathcal{A}(x,\xi)$ is needed here.]{}
[xx]{}
B. Abdellaoui, A. Dall’Aglio, and I. Peral, [*Some remarks on elliptic problems with critical growth in the gradient*]{}, J. Differential Equations [**222**]{} (2006) 21–62.
D. R. Adams and L. I. Hedberg, [*Function Spaces and Potential Theory*]{}, Springer-Verlag, Berlin, 1996.
A. Ancona, [*On strong barriers and an inequality of Hardy for domains in ${\mathbb{R}}^n$*]{}, J. London Math. Soc. [**34**]{} (1986), 274–290.
A. Bensoussan, L. Boccardo, and F. Murat, [*On a nonlinear partial differential equation having natural growth terms and unbounded solution*]{}, Ann. Inst. H. Poincaré Anal. Non Linéaire, [**5**]{} (1988), 347-–364.
L. Boccardo, T. Gallouët, and L. Orsina, [*Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data*]{}, Ann. Inst. H. Poincaré Anal. Non Linéaire [**13**]{} (1996) 539–551.
L. Boccardo and F. Murat, [*Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations*]{}, Nonlinear Anal. [**19**]{} (1992), 581–597.
L. Boccardo, F. Murat, and J.-P. Puel, [*Existence of bounded solutions for nonlinear elliptic unilateral problems*]{}, Ann. Mat. Pura Appl. (4) [**152**]{} (1988), 183–196.
H. Brézis, [*Équations et inéquations non-linéaires dans les espaces vectoriel en dualité*]{}, Ann. Inst. Fourier [**18**]{} (1968), 115–176.
H. Brézis and F. E. Browder, [*Some properties of higher order Sobolev spaces*]{}, J. Math. Pures Appl. [**61**]{} (1982), 245–259.
F. E. Browder, [*Existence theorems for nonlinear partial differential equations*]{}, in S.-S. Chern, S. Smale (Eds), Proc. Sympos. Pure Math., Vol. XVI, pp. 1–60, Amer. Math. Soc., Providence, R.I. 1970.
D. M. Duc, N. C. Phuc, and T. V. Nguyen, [*Weighted Sobolev’s inequalities for bounded domains and singular elliptic equations*]{}, Indiana Univ. Math. J. [**56**]{} (2007), 615–642.
S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff, [*Some weighted norm inequalities concerning the Schrödinger operators*]{}, Comment. Math. Helv. [**60**]{} (1985), 217–246.
C. Fefferman, [*The uncertainty principle*]{}, Bull. Amer. Math. Soc. [**9**]{} (1983), 129–206.
V. Ferone and F. Murat, [*Quasilinear problems having natural growth in the gradient: an existence result when the source term is small*]{}, in: Équations aux dérivées partielles et applications, Articles dédiés à Jacques-Louis Lions, Gauthier-Villars, Paris, 1998, pp. 497–515.
V. Ferone and F. Murat, [*Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small*]{}, Nonlinear Anal., [**42**]{} (2000), 1309–1326.
V. Ferone and F. Murat, [*Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz Spaces*]{}, J. Differential Equations [**256**]{} (2014), 577–608.
M. Frazier and I. E. Verbitsky, [*Positive solutions to Schrödinger’s equation and the exponential integrability of the balayage*]{}, Preprint 2015, arXiv:1509.09005.
M. Fukushima, K. Sato, and S. Taniguchi, [*On the closable part of pre-Dirichlet forms and the fine support of the underlying measures*]{}, Osaka J. Math. [**28**]{} (1991) 517–535.
H. A. Hamid and M. F. Bidaut-Veron, [*On the connection between two quasilinear elliptic problems with source terms of order $0$ or $1$*]{}. Commun. Contemp. Math. [**12**]{} (2010), 727–788.
K. Hansson, V. G. Maz’ya, and I. E. Verbitsky, [*Criteria of solvability for multidimensional Riccati equations*]{}, Ark. Mat. [**37**]{} (1999), 87-–120.
B. Jaye, V. G. Maz’ya, and I. E. Verbitsky, [*Existence and regularity of positive solutions of elliptic equations of Schrödinger type*]{}, J. Anal. Math. [**118**]{} (2012), 577–621.
B. Jaye, V. G. Maz’ya, and I. E. Verbitsky, [*Quasilinear elliptic equations and weighted Sobolev-Poincaré inequalities with distributional weights*]{}, Adv. Math. [**232**]{} (2013), 513–542.
M. Kardar, G. Parisi, and Y.-C. Zhang, [*Dynamic scaling of growing interfaces*]{}, Phys. Rev. Lett. [**56**]{} (1986) 889–892.
J. Krug and H. Spohn, [*Universality classes for deterministic surface growth*]{}, Phys. Rev. A (3) [**38**]{} (1988) 4271–4283.
J. L. Lewis, [*Uniformly fat sets*]{}, Trans. Amer. Math. Soc. [**308**]{} (1988), 177–196.
J.-L. Lions, [*Quelques méthodes de résolution des problèmes aux limites non linéaires*]{}, Dunod; Gauthier-Villars, Paris 1969. xx+554 pp.
J. Malý and W. P. Ziemer, [*Fine regularity of solutions of elliptic partial differential equations*]{}, Mathematical Surveys and Monographs, [**51**]{}. American Mathematical Society, Providence, RI, 1997. xiv+291 pp.
V. G. Maz’ya, [*Sobolev Spaces with Applications to Elliptic Partial Differential Equations*]{}, second, revised and augmented ed., in: Grundlehren der math. Wissenschaften, vol. 342, Springer, Heidelberg, 2011.
T. Mengesha and N. C. Phuc [*Quasilinear Ricatti type equations with distributional data in Morrey space framework*]{}, J. Differential Equations [**260**]{} (2016), 5421–5449.
P. Mikkonen, [*On the Wolff potential and quasilinear elliptic equations involving measures*]{}, Ann. Acad. Sci. Fenn., Ser AI, Math. Dissert. [**104**]{} 1996, 1–71. C. Pérez, [*Two weighted inequalities for potential and fractional type maximal operators*]{}, Indiana Univ. Math. J. [**43**]{} (1994), 663–683.
E. T. Sawyer and R. L. Wheeden, [*Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces*]{}, Amer. J. Math. [**114**]{} (1992), 813–874.
[^1]: $^{1}$ Supported in part by National Research Foundation of Korea grant funded by the Korean government (MEST) (NRF-2015R1A2A1A15053024)
[^2]: $^{2}$ Supported in part by Simons Foundation, award number 426071
|
---
abstract: 'We consider the Kaluza–Klein (KK) scenario in which only gravity exists in the bulk. Without the [*assumption*]{} of symmetric connection, the presence of brane fermions induces torsion. The result is a universal axial contact interaction that dominates those induced by KK gravitons. This enhancement arises from a large spin density on the brane. Using a global fit to $Z$–pole observables, we find the 3$\sigma$ bound on the scale of quantum gravity to be 28 TeV for $n=2$. If Dirac or light sterile neutrinos are present, the data from SN1987A increase the bound to $\sqrt{n}M_S \geq 210\; {\rm TeV}$.'
address: |
${}^{(1)}$Institute for Particle Physics and Astrophysics, Physics Department, Virginia Tech, Blacksburg, VA 24061\
${}^{(2)}$Department of Physics, Amherst College, Amherst MA 01002
author:
- 'Lay Nam Chang${}^{(1)}$[^1], Oleg Lebedev${}^{(1)}$[^2], Will Loinaz${}^{(1,2)}$[^3], and Tatsu Takeuchi${}^{(1)}$[^4]'
title: 'Universal Torsion–Induced Interaction from Large Extra Dimensions'
---
Consistent string theories require dimensions beyond the usual four, that could in principle be as large as a millimeter. These large extra dimensions provide new avenues for solution to the hierarchy problem [@Arkani-Hamed:1998rs] and have significant phenomenological consequences as well. Current collider experiments set a lower bound of about 1 TeV on the fundamental Planck scale, while cosmological considerations require it to be around 50 TeV for two extra dimensions. Future colliders will be able to probe up to 5$\sim$8 TeV [@junk].
This paper considers the minimal Kaluza–Klein scenario (KK) wherein only gravity exists in the bulk while all Standard Model fields are localized on a 4D brane. Without [*a priori*]{} assumptions on its symmetry, fermions always induce antisymmetric pieces, or torsion, in the gravity connection. We show that this implies a universal $U(45)$–invariant contact interaction which is suppressed only by the square of the [*fundamental*]{} Planck scale. This interaction therefore dominates four–fermion interactions induced by KK graviton exchange at current collider energies and provides important information concerning the viability of extra dimensions. The enhancement is distinct from the KK mechanism. It originates from a large $4+n$ dimensional spin density due to the spinor fields confined to a 4 dimensional brane. We obtain a limit on this interaction from electroweak precision data and compare it to other constraints available from particle physics and astrophysics. While the metric is coupled to the energy–momentum tensor in gravity, torsion is coupled directly to the spin density of matter systems[@Kibble:1961ba; @Hehl:1976kj]. It appears in any description of gravity where Lorentz transformations are treated as local symmetries and is a feature of string theory and other variations of general relativity. The basic quantity is the torsion tensor $T^{\alpha}{}_{\beta\gamma}$, defined as the antisymmetric part of the connection $\tilde{\Gamma}^{\alpha}{}_{\beta\gamma}$: $$T^{\alpha}{}_{\beta\gamma} =
\tilde{\Gamma}^{\alpha}{}_{\beta\gamma}-
\tilde{\Gamma}^{\alpha}{}_{\gamma\beta}.$$ Torsion violates the equivalence principle in its [*very strong*]{} form [@wheeler], as it cannot be removed by an appropriate choice of coordinates. It does not directly affect the propagation of light and test particles, and thus cannot be probed by standard tests of general relativity [@wheeler].
We denote with a tilde quantities derived from a non–symmetric connection $\tilde{\Gamma}^{\alpha}{}_{\beta\gamma}$, while those without a tilde refer to quantities derived from Christoffel symbols $\Gamma^{\alpha}{}_{\beta\gamma}$.[^5] Via the metric condition $\tilde{\nabla}_{\alpha} g_{\mu\nu}=0$, the torsion tensor can be related to the contorsion tensor $K^{\alpha}{}_{\beta\gamma}$ defined by $$\tilde{\Gamma}^{\alpha}{}_{\beta\gamma}=
\Gamma^{\alpha}{}_{\beta\gamma} + K^{\alpha}{}_{\beta\gamma},$$ such that $K_{\alpha\beta\gamma} = \frac{1}{2}\left( T_{\alpha\beta\gamma}
-T_{\beta\alpha\gamma}-T_{\gamma\alpha\beta} \right)$. ${T^{\alpha}}_{\beta\gamma}$ in 4D contains 24 independent components. However, only its totally antisymmetric part, expressible as an axial vector $S^\sigma$, $$S^\sigma = i\,\epsilon^{\mu\nu\rho\sigma} T_{\mu\nu\rho}
\label{pseudovector}$$ is relevant for spin 1/2 fermions [@buch]. In what follows we assume that torsion is completely antisymmetric. The relations above (except for Eq. (\[pseudovector\])) can be generalized for the case of $4+n$ dimensions in a straightforward manner. We will use the following convention for the indices [@Han:1999sg]: $\mu=1,\cdots,4$, $\hat{\mu}=1,\cdots,4+n$, $i=5,\cdots,4+n$. The signature of the metric is $(1,-1,\cdots,-1)$. To begin, note that torsion minimally couples to fermions only [@buch]. The minimal action for $4+n$ dimensional gravity coupled to fermions localized on a 4–dimensional brane is: $$\begin{aligned}
\lefteqn{
S = -\frac{1}{\hat{\kappa}^2} \int d^{4+n}x \;
\sqrt{\vert \hat{g}_{4+n}\vert} \;\tilde{R}
} & & \label{action}
\\
& + & \int d^{4}x \;\sqrt{\vert \hat{g}_4 \vert}\; \frac{i}{2}
\left[ \bar\Psi\gamma^\mu\tilde\nabla_\mu\Psi
- \left( \tilde\nabla_\mu\bar\Psi \right)\gamma^\mu\Psi
+ 2 i M \bar\Psi \Psi
\right],
\nonumber
$$ Here $\hat{\kappa}^2=16\pi G_N^{(4+n)}$, $\tilde{R}$ is the $4+n$ dimensional scalar curvature, and $\hat{g}_{4+n}$ and $\hat{g}_4$ are respectively the $4+n$ and $4$–dimensional (induced) metric determinants.
The covariant derivative $\tilde\nabla_{\mu}$ is defined by $\tilde\nabla_{\mu}\Psi=\partial_{\mu}\Psi
+\frac{i}{2}\tilde\omega^{ab}_{\mu}\sigma_{ab}\Psi$, where $\tilde\omega^{ab}_{\mu}$ is the spin–connection, $\sigma_{ab}=\frac{i}{2}\left[ \gamma_a, \gamma_b \right]$, with $a$, $b$ the local Lorentz indices. A general spin connection $\tilde\omega^{ab}_{\mu}$ can be expressed in terms of a torsion-free spin-connection $\omega^{ab}_{\mu}$, the contorsion tensor and the vierbein $e^a_\mu$: $$\tilde\omega^{ab}_{\mu} = \omega^{ab}_{\mu}
+ \frac{1}{4} K^{\nu}{}_{\lambda\mu}
\left( e^{\lambda a} e^b_\nu - e^{\lambda b} e^a_\nu \right).$$ Upon substitution, the action in the case of completely antisymmetric torsion becomes $$\begin{aligned}
\lefteqn{
S = -\frac{1}{\hat{\kappa}^2} \int d^{4+n}x \;\sqrt{\vert \hat{g}_{4+n} \vert}
\;\left( R - K^{\hat{\mu}\hat{\nu}\hat{\rho}}
K_{\hat{\mu}\hat{\nu}\hat{\rho}}
\right)
} \cr
& + & \int d^{4}x \;\sqrt{\vert \hat{g}_4 \vert}\;i
\bar\Psi
\left( \gamma^{\mu}\nabla_{\mu}
- \frac{1}{8}S^\mu \gamma_\mu \gamma_5
+ iM
\right) \Psi\;.
\label{action1}\end{aligned}$$ Here $R$ is the $4+n$ dimensional [*metric*]{} curvature, $K^{\hat{\mu}\hat{\nu}\hat{\rho}} =
\frac{1}{2}\;T^{\hat{\mu}\hat{\nu}\hat{\rho}}$, and $\nabla_{\mu}$ is the conventional covariant derivative without torsion. We use $\gamma_5=-i\gamma^0\gamma^1\gamma^2\gamma^3$. The resultant equations of motion are $$\begin{aligned}
&& K_{i\hat{\mu}\hat{\nu}} = 0\;,\cr
&& S_{\mu} = i\frac{3}{2}
\frac{ \sqrt{\vert \hat{g}_4 \vert } }{ \sqrt{ \vert \hat{g}_{4+n} \vert} }\;
\hat{\kappa}^2\;
\bar\Psi\gamma_{\mu}\gamma_5\Psi\;
\delta^{(n)}(x)\;, \cr
&& R_{\hat{\mu}\hat{\nu}} - \frac{1}{2} g_{\hat{\mu}\hat{\nu}} R
= \frac{ \hat{\kappa}^2 }{ 2 }\;T_{\hat{\mu}\hat{\nu}}
+ {\cal O}\left( \hat{\kappa}^4 \right) \;.
\label{eqmotion}\end{aligned}$$ $T_{\hat{\mu}\hat{\nu}}$ is the torsion–free energy–momentum tensor for the matter fields. The first two of these relations are algebraic constraints. Classically, torsion does not propagate and is zero outside the matter distribution [@Hehl:1976kj]. Its source is the [*spin density*]{} of fermions confined on the brane. Elimination of $S_\mu$ from the action via Eq. (\[eqmotion\]) produces a fermion contact interaction: $$\begin{aligned}
S & = & -\frac{1}{\hat{\kappa}^2} \int d^{4+n}x
\;\sqrt{\vert \hat{g}_{4+n} \vert}\;R
\\ \label{action2}
& & + \int d^{4}x \;\sqrt{\vert \hat{g}_4 \vert}\;
\left[ \bar\Psi\left( i\gamma^{\mu}\nabla_{\mu} - M \right)\Psi
+ \frac{3}{32}
\frac{ \sqrt{ \vert \hat{g}_4 \vert } }{ \sqrt{\vert \hat{g}_{4+n} \vert} }
\hat{\kappa}^2\,
\left( \bar\Psi\gamma_{\mu}\gamma_5\Psi \right)^2\,
\delta^{(n)}(0)
\right]\;. \nonumber\end{aligned}$$ The delta–function appearing in this expression should be regularized to account for a finite brane width: $$\begin{aligned}
\delta^{(n)}(0) \rightarrow \frac{1}{(2\pi)^n} \int_{0}^{M_S} d^n k =
{\frac{\displaystyle M_S^n }{\displaystyle 2^{n-1} \pi^{n/2}\,n\,\Gamma\!\left({\frac{\displaystyle n}{\displaystyle 2}}\right) }}\;,\end{aligned}$$ $M_S$ is the cutoff scale of the effective theory, here taken to be of the order of the inverse brane width. The $4+n$ dimensional coupling constant $\hat{\kappa}$ is related to the 4–dimensional coupling $\kappa$ and the volume of the extra dimensions compactified on a torus via $\hat{\kappa}^2=
\kappa^2 V_n=16\pi(4\pi)^{n/2}\Gamma(n/2)M_S^{-(n+2)}$ [@Han:1999sg][^6]. As a result, the leading ${\cal O}\left( \hat{\kappa}^2 \right)$ torsion contribution to the action is given by $$\Delta S
= \int d^4x\;\frac{3\pi}{n M_S^2}
\left[ \sum_j \bar\Psi_j\gamma_{\mu}\gamma_5\Psi_j
\right]^2\;,
\label{action3}$$ where $j$ runs over all fermions existing on the brane. The expansion in $\hat{\kappa}$ is expected to be valid provided the typical energy $E$ of a physical process is below the cutoff scale $M_S$ (see also Ref. [@Giudice:1999ck]). In this case a typical “size” of torsion is considerably below $E$. Note that the exact coefficient in Eq.(\[action3\]) depends on the assumptions about the short-distance physics; in particular, it depends on the regularization of the delta-function.
The interaction (\[action3\]) is the unique contact interaction possessing the maximal approximate global symmetry of the minimal Standard Model, [*i.e.*]{} the group $U(45)$ acting on the 45 Weyl spinors $\Psi_L=(q_L, u^c_R,d^c_R,l_L,e^c_R)$ [@Buchmuller:1997hn][^7]. The effect is truly universal for all fermions, in contrast to the four–fermion operators induced by KK graviton exchanges. Graviton couplings are mass and energy dependent, leading to different strengths for different fields. Finally, the KK–induced interactions have two additional suppression factors: $s/M_S^2$ [@Han:1999sg] and $f^2/M_S^2$ [@Bando:1999di]. The former follows from the graviton coupling to the energy–momentum tensor, the latter from the brane recoil effects (note that the rigidity of the brane $f$ plays no role in our argument). Consequently, at typical accelerator energies the interaction (\[action3\]) is enhanced over the KK–induced interactions by [*orders of magnitude*]{} ([*e.g.*]{} at LEP energies the enhancement factor is about $10^2$) and will completely dominate. This enhancement results from a large $4+n$ dimensional [*spin density*]{} on the brane and is present whenever fermions are localized. Note that the interaction (\[action3\]) is repulsive for aligned spins [@Hehl:1976kj].
Let us briefly discuss how this result is modified by quantum corrections [@cs]. Generally, fermion loops will induce propagation of torsion along the brane, with $S_\mu \bar\Psi\gamma^{\mu}\gamma_5\Psi$ generating the relevant kinetic terms. The associated quantum of torsion will have a mass of order $M_S$. Its propagation effects are therefore irrelevant at typical accelerator energies. Loop corrections to the tree level torsion coupling and mass evaluated using an explicit cutoff amount to a rescaling of $M_S$ in Eq. (\[action3\]) by a factor of 1$\sim$2. However, if we use dimensional regularization, the result is largely insensitive to radiative corrections due to the absence of quadratic divergences.
The universal interaction (\[action3\]) will affect $Z$–pole electroweak observables. Corrections to the oblique parameters appear at the two–loop level and can be neglected [[^8]]{}. Two types of vertex corrections are shown in Fig. 1. The combined contribution of the diagram in Fig. 1a and the corresponding wave function renormalization diagrams is suppressed by $1/M_S^2$ and leads only to a [*universal*]{} multiplicative correction to the couplings (neglecting light fermion masses). Since the observables we will consider are ratios of couplings, such corrections will cancel. The diagram in Fig. 1b is significant. Note that the corresponding wave function renormalization diagram vanishes. Summing contributions of all of the fermions and taking into account $\sum I_3=0$ , we write the leading correction to the $Z$–couplings as $$\delta h_L=-\delta h_R={3N_c m_t^2 \over 4\pi n M_S^2}\;
\label{delta}
\ln {M_S^2 \over m_t^2}\;,$$ where the Z–vertex is defined as $-i{g\over \cos \theta_W}
Z_\mu \bar\Psi \gamma^\mu (P_Lh_L+P_Rh_R)\Psi$. The contribution of torsion to $\delta h_L$ is strictly positive.
We perform a global fit to the LEP/SLD electroweak observables including $R_{\nu/\ell} = \Gamma(Z\rightarrow \nu\bar{\nu})/
\Gamma(Z\rightarrow \ell^+\ell^-)$, $R_{b,c}$, $A_{FB}(i)$ and $A_i$ ($i=e,\mu,\tau,b,c$) [[^9]]{}. The data used and a detailed description of the technique (applied to a different model) can be found in [@Lebedev:2000vc]. Note that the KK graviton radiative corrections are suppressed as discussed above; moreover, the KK vertex corrections will largely cancel in our set of observables since, for the case of light fermions, they modify the couplings multiplicatively. The KK graviton corrections to the oblique parameters may not be negligible [@oblique] and can affect our fit results through $\sin^2\theta_W$. In the fit, we leave $\delta s^2 \equiv \sin^2\theta_W - [\sin^2\theta_W]_{\rm SM}$ as a free parameter to account for a variation in the Higgs mass and KK graviton radiative corrections. From a two–parameter fit we obtain $$\begin{aligned}
\delta h_L & = & -0.00049 \pm 0.00021 \cr
\delta s^2 & = & -0.00068 \pm 0.00018 \; .\end{aligned}$$ The $\chi^2/d.o.f.$ of the fit is 17/12. Fig. 2 shows that $\delta h_L$ is most strongly constrained by $R_{\nu/\ell}$. Since the experimental value for $R_{\nu/\ell}$ is about $2\sigma$ below the SM prediction, the preferred value of $\delta h_L$ is about $2\sigma$ below zero. The value of $\delta h_L$ is almost uncorrelated with $\delta s^2$ and thus with the Higgs mass. As a result of this classical statistical analysis, the model is excluded at the $2\sigma$ level since it generates only a positive $\delta h_L$. Using Eq. (\[delta\]), we obtain the $3\sigma$ bound on $M_S$: $$M_S \geq 28 \; {\rm TeV}$$ for $n=2$. For $n=4 (6)$ the bound weakens to 19 (15) TeV. This implies that we do not expect deviations from Newton’s law at distances above $9 \times 10^{-4}$ mm for $n=2$.
We next consider other constraints on the universal interaction[[^10]]{}. $A\times A^{+}$ contact interaction (\[action3\]) affects at the tree level the differential cross sections for $e^+e^- \rightarrow f\bar f$ measured at LEP. The OPAL measurements [@opal] imply $$\sqrt{n} M_S \geq 10.3\;{\rm TeV}$$ at the 95% confidence level. Electron–quark contact interactions can also be constrained via HERA DIS data, Drell–Yan production at the Tevatron, [*etc.*]{} The global analysis [@cheung] yields $$\sqrt{n} M_S \geq 5.3\;{\rm TeV}\;.$$ Another potentially strong constraint can come from the measurement of the invisible width of the $\Upsilon$ and $J/\Psi$ resonances at B and $\tau$–charm factories [@Chang:1998tq].
A powerful astrophysical constraint can be derived if we admit existence of Dirac or light sterile neutrinos. For the case of Dirac neutrinos, the torsion–induced interaction containes a term $$\Delta {\cal{L}}=
-\frac{6\pi}{n M_S^2}\;
\bar q\gamma^{\mu}\gamma_5 q \; \bar \nu_R\gamma_{\mu} \nu_R\;.$$ This contact interaction provides a new channel of energy drain during neutron star collapse, since right handed neutrinos produced by nucleon interactions leave the core without rescattering [@grifols]. This would affect neutron star evolution; in particular, it would modify the duration of the standard neutrino burst. From observations of SN 1987A one infers [@grifols] $$\sqrt{n} M_S \geq 210\;{\rm TeV}\;.
\label{SN}$$ Similar considerations apply to the case of light sterile neutrinos: if $m_{\nu_s} \ll 50\;{\rm MeV}$, the core temperature, the analysis is completely analogous to that of Dirac neutrinos and the bound (\[SN\]) holds. This bound translates into an upper bound on the compactification radius of $3 \times 10^{-5}$ mm for $n=2$.
Since $M_S$ controls all gravity effects in extra dimensions, the limits on $M_S$ being larger than tens of TeV reported here imply weaker KK graviton couplings than those considered in the literature. The limits were obtained under a minimal set of assumptions in the context of physics of extra dimensions. We first consider the general set of connections consistent with general covariance and local Lorentz symmetries. We fix all matter fields to be on a brane, consistent with the most conservative scenario. The consequence is that tree–level effects from a minimal action are sufficiently strong to produce the bounds reported above. Even for less conservative scenarios the universal interaction (\[action3\]) can be expected to dominate possible KK gauge effects as long as the fermions are confined to the brane. We emphasize that this interaction is generic unless we impose the additional condition that the connection be symmetric.[^11] Finally, it is interesting to note that, in this context, particle physics places a constraint on violation of the strong equivalence principle.
This research is supported in part by a grant from the U.S. Department of Energy, DE–FG05–92ER40709.
(300,100)(10,-10) (50,50)[1.5]{} (99,58)[1.5]{} (99,42)[1.5]{} (25,50)(50,50)[5]{}[4]{} (75,50)(25,20,180) (75,50)(25,180,340) (110,70)(99,58) (99,42)(110,30) (18,50)\[\][$Z$]{} (75,86)\[\][$f$]{} (75,15)\[\][$f$]{} (117,72)\[\][$f$]{} (117,30)\[\][$f$]{} (170,50)[1.5]{} (220,50)[1.5]{} (230,50)[1.5]{} (145,50)(170,50)[5]{}[4]{} (195,50)(25,0,180) (195,50)(25,180,360) (245,65)(230,50) (230,50)(245,35) (138,50)\[\][$Z$]{} (195,86)\[\][$f'$]{} (195,15)\[\][$f'$]{} (253,72)\[\][$f$]{} (253,30)\[\][$f$]{} (75,-5)\[\][(a)]{} (195,-5)\[\][(b)]{}
(240,220)(0,0) (47,4)[$\delta s^2$]{} (0,42)[$\delta h_L$]{} (58,31.5)[$R_{\nu/\ell}$]{} (69,25)[$R_b$]{} (28,58)[$A_{\rm LR}$]{} (58,57)[$A_{\rm FB}(b)$]{}
[99]{}
N. Arkani–Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. [**B429**]{}, 263 (1998); Phys. Rev. [**D59**]{}, 086004 (1999); I. Antoniadis, N. Arkani–Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. [**B436**]{}, 257 (1998).
E. A. Mirabelli, M. Perelstein, M. Peskin, Phys. Rev. Lett. [**82**]{}, 2236 (1999); J. L. Hewett, Phys. Rev. Lett. [**82**]{}, 4765 (1999); S. Cullen, M. Perelstein, Phys. Rev. Lett. [**83**]{}, 268 (1999); S. Nussinov, R. E. Shrock, Phys. Rev. [**D59**]{}, 105002 (1999); T. G. Rizzo, Phys. Rev. [**D59**]{}, 115010 (1999); C. Balazs, H. He, W. W. Repko, C. P. Yuan and D. A. Dicus, Phys. Rev. Lett. [**83**]{}, 2112 (1999).
T. W. Kibble, J. Math. Phys. [**2**]{}, 212 (1961).
F. W. Hehl, P. Von Der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. [**48**]{}, 393 (1976).
I. Ciufolini, J. A. Wheeler, [*Gravitation and Inertia*]{} (Princeton University, New Jersey, 1995).
I. L. Buchbinder, S. D. Odintsov, I. L. Shapiro, [*Effective Action in Quantum Gravity*]{} (IPP, Bristol and Philadelphia, 1992).
L. N. Chang, C. Soo, hep–th/9905001; Phys. Rev. [**D53**]{}, 5682 (1996).
T. Han, J. D. Lykken and R. Zhang, Phys. Rev. [**D59**]{}, 105006 (1999).
G. F. Giudice, R. Rattazzi and J. D. Wells, Nucl. Phys. [**B544**]{}, 3 (1999).
W. Buchmüller and D. Wyler, Phys. Lett. [**B407**]{}, 147 (1997).
M. Bando, T. Kugo, T. Noguchi and K. Yoshioka, Phys. Rev. Lett. [**83**]{}, 3601 (1999).
O. Lebedev, W. Loinaz and T. Takeuchi, Phys. Rev. [**D61**]{}, 115005 (2000); Phys. Rev. [**D62**]{}, 015003 (2000); Phys. Rev. [**D62**]{}, 055014 (2000).
T. Han, D. Marfatia, R.–J. Zhang, hep-ph/0001320; P. Das, S. Raychaudhuri, hep-ph/9908205.
A. S. Belyaev, I. L. Shapiro, Nucl. Phys. [**B543**]{}, 20 (1999).
OPAL Collaboration, G. Alexander [*et al.*]{}, Phys. Lett. [**B387**]{}, 432 (1996); G. Abbiendi [*et al.*]{}, Eur. Phys. J. [**C6**]{}, 1 (1999).
K. Cheung, hep-ph/9807483; V. Barger, K. Cheung, K. Hagiwara, D. Zeppenfeld, Phys. Rev. [**D57**]{}, 391 (1998).
L. N. Chang, O. Lebedev and J. N. Ng, Phys. Lett. [**B441**]{}, 419 (1998).
J. A. Grifols, E. Massó, R. Toldrà, Phys. Rev. [**D57**]{}, 2005 (1998).
[^1]: electronic address: laynam@vt.edu
[^2]: electronic address: lebedev@quasar.phys.vt.edu
[^3]: electronic address: loinaz@alumni.princeton.edu
[^4]: electronic address: takeuchi@vt.edu
[^5]: We follow the conventions and definitions of Ref. [@buch] except for the definition of $\gamma_5$.
[^6]: For simplicity we set the string scale and the $4+n$ dimensional Planck mass equal.
[^7]: Other contact interactions possessing the same symmetry can be brought into the form of Eq. (\[action3\]).
[^8]: Throughout this analysis we retain only leading tree or one–loop contributions. This approximation is valid to leading order in $1/n$.
[^9]: We omit $R_\ell$ from our fit since a universal correction to $R_\ell$ only shifts the value of $\alpha_s(M_Z)$.
[^10]: Phenomenological implications of 4–d gravity with torsion were also considered in [@belyaev].
[^11]: The analogous four–fermion interaction involving gauginos in supergravity models needs a separate discussion because of their connections to the corresponding coupling among gauge bosons. However these terms do not affect the kinds of phenomenology addressed in the present paper.
|
---
abstract: 'Radiatively driven transfer flow perpendicular to a luminous disk was examined under a fully special relativistic treatment, taking into account radiation transfer. The flow was assumed to be vertical, and the gravity, the gas pressure, and the viscous heating were ignored. In order to construct the boundary condition at the flow top, the magic speed above the flat source was re-examined, and it was found that the magic speed above a moving source can exceed that above a static source ($\sim 0.45~c$). Then, the radiatively driven flow in a luminous disk was numerically solved, from the flow base (disk “inside”), where the flow speed is zero, to the flow top (disk “surface”), where the optical depth is zero. For a given optical depth and appropriate initial conditions at the flow base, where the flow starts, a loaded mass in the flow was obtained as an eigenvalue of the boundary condition at the flow top. Furthermore, a loaded mass and the flow final speed at the flow top were obtained as a function of the radiation pressure at the flow base; the flow final speed increases as the loaded mass decreases. Moreover, the flow velocity and radiation fields along the flow were obtained as a function of the optical depth. Within the present treatment, the flow three velocity $v$ is restricted to be within the range of $v < c/\sqrt{3}$, which is the relativistic sound speed, due to the relativistic effect.'
author:
- 'Jun <span style="font-variant:small-caps;">Fukue</span>'
title: Relativistic Radiative Flow in a Luminous Disk
---
Introduction
============
Accretion disks are tremendous energy sources in the active universe (see Kato et al. 1998 for a review). In particular, when the mass-accretion rate highly exceeds the critical rate, the disk local luminosity exceeds the Eddington one, and mass loss from the disk surface driven by radiation pressure takes place.
Such a radiatively driven outflow from a luminous disk has been extensively studied in the context of models for astrophysical jets by many researchers (Bisnovatyi-Kogan, Blinnikov 1977; Katz 1980; Icke 1980; Melia, Königl 1989; Misra, Melia 1993; Tajima, Fukue 1996, 1998; Watarai, Fukue 1999; Hirai, Fukue 2001; Fukue et al. 2001; Orihara, Fukue 2003), as on-axis jets (Icke 1989; Sikora et al. 1996; Renaud, Henri 1998; Luo, Protheroe 1999; Fukue 2005a), as outflows confined by a gaseous torus (Lynden-Bell 1978; Davidson, McCray 1980; Sikora, Wilson 1981; Fukue 1982), or as jets confined by the outer flow or corona (Sol et al. 1989; Marcowith et al. 1995; Fukue 1999), and as numerical simulations (Eggum et al. 1985, 1988). In almost all of these studies, however, the disk radiation fields were treated as external fields, and the radiation transfer was not solved.
The radiation transfer in the disk, on the other hand, was investigated in relation to the structure of a static disk atmosphere and the spectral energy distribution from the disk surface (e.g., Meyer, Meyer-Hofmeister 1982; Cannizzo, Wheeler 1984; Shaviv, Wehrse 1986; Adam et al. 1988; Hubeny 1990; Mineshige, Wood 1990; Ross et al. 1992; Artemova et al. 1996; Hubeny, Hubeny 1997, 1998; Hubeny et al. 2000, 2001; Davis et al. 2005; Hui et al. 2005). In these studies, however, the vertical movement and mass loss were not considered. Moreover, their treatments were restricted in the non-relativistic regime, and the relativistic effects were not considered.
In order to break such a situation, where radiation transfer has not been considered in the radiatively driven wind from the disk, we recently examined the radiatively driven vertical outflow – [*moving photosphere*]{} – in a luminous flat disk (Fukue 2005b, c). In these papers, the radiative transfer flow was analytically or numerically solved, but the obtained flow speed was limited in the subrelativistic regime, since the problem was treated up to the order of $(v/c)^1$. In some astrophysical jet sources, however, the jet speed is mildly or highly relativistic. In gamma-ray bursts, the flow speed is supposed to be extremely relativistic. Hence, in the next step, we should consider the transfer flow up to the full order of $(v/c)^2$ (cf. Fukue 1999; Hirai, Fukue 2001; Fukue et al. 2001; Orihara, Fukue 2003; Fukue 2005a without transfer). Moreover, although it was incorporated in the previous paper (Fukue 2005b), which were up to the first order of the flow velocity, the effect of radiation drag must become much more important (Phinney 1987; Icke 1989). Other relativistic effects would further appear in the fully relativistic regime.
In this paper, we thus examine the radiatively driven transfer outflow in a luminous flat disk within the fully relativistic regime of $(v/c)^2$. At the preliminary stage, we ignore the gravity of the central object, the gas pressure, and the viscous heating, and we treat the wind as a vertical one-dimensional flow without rotational motion.
In the next section we describe basic equations in the vertical direction. In section 3 we examine the boundary condition at the flow top, and show the magic speed above the moving source. In section 4 we then solve and examine a radiative flow under the appropriate boundary conditions at the flow base and top. The final section is devoted to concluding remarks.
Basic Equations
===============
Let us suppose a luminous flat disk, deep inside which gravitational or nuclear energy is released via viscous heating or other processes. The radiation energy is transported in the vertical direction, and the disk gas, itself, also [*moves*]{} in the vertical direction due to the action of radiation pressure (i.e., plane-parallel approximation). For the sake of simplicity, in the present paper, the radiation field is considered to be sufficiently intense that both the gravitational field of, e.g., the central object and the gas pressure can be ignored: tenuous cold normal plasmas in the super-Eddington disk, cold pair plasmas in the sub-Eddington disk, or dusty plasmas in the sub-Eddington disk. Internal heating is also ignored: the flow in or near to the surface envelope of the disk. As for the order of the flow velocity $v$, we consider the fully special relativistic regime, where the terms are retained up to the second order of $(v/c)$. Under these assumptions, the radiation hydrodynamic equations for steady vertical ($z$) flows are described as follows (Kato et al. 1998).
The continuity equation is $$\rho cu = J ~(={\rm const.}),
\label{rho1}$$ where $\rho$ is the proper gas density, $u$ the vertical four velocity, $J$ the mass-loss rate per unit area, and $c$ the speed of light. The four velocity $u$ is related to the three velocity $v$ by $u=\gamma v/c$, where $\gamma$ is the Lorentz factor: $\gamma=\sqrt{1+u^2}=1/\sqrt{1-(v/c)^2}$.
The equation of motion is $$c^2u\frac{du}{dz} = \frac{\kappa_{\rm abs}+\kappa_{\rm sca}}{c}
\left[ F \gamma (1+2u^2) - c(E+P)\gamma^2 u \right],
\label{u1}$$ where $\kappa_{\rm abs}$ and $\kappa_{\rm sca}$ are the absorption and scattering opacities (gray), defined in the comoving frame, $E$ the radiation energy density, $F$ the radiative flux, and $P$ the radiation pressure observed in the inertial frame. The first term in the brackets on the right-hand side of equation (\[u1\]) means the radiatively driven force, which is modified to the order of $u^2$, whereas the second term is the radiation drag force, which is also modified, but roughly proportional to the velocity.
In the no-gas pressure approximation and without heating, the energy equation is reduced to a radiative equilibrium relation, $$0 = j - c\kappa_{\rm abs} E \gamma^2 - c\kappa_{\rm abs} P u^2
+ 2 \kappa_{\rm abs} F \gamma u,
\label{j1}$$ where $j$ is the emissivity defined in the comoving frame. In this equation (\[j1\]), the third and fourth terms on the right-hand side appear in the relativistic regime.
For radiation fields, the zeroth-moment equation becomes $$\begin{aligned}
\frac{dF}{dz} &=& \rho \gamma
\left[ j - c\kappa_{\rm abs} E
+ c\kappa_{\rm sca}(E+P)u^2 \right.
\nonumber
\\
&& \left. + \kappa_{\rm abs}Fu/\gamma
-\kappa_{\rm sca}F ( 1+v^2/c^2 )\gamma u \right].
\label{F1}\end{aligned}$$ The first-moment equation is $$\begin{aligned}
\frac{dP}{dz} &=& \frac{\rho \gamma}{c}
\left[ ju/\gamma - \kappa_{\rm abs} F
+ c\kappa_{\rm abs}Pu/\gamma \right.
\nonumber
\\
&& \left. -\kappa_{\rm sca}F(1+2u^2)
+c\kappa_{\rm sca}(E+P)\gamma u \right].
\label{P1}\end{aligned}$$
Finally, the closure relation [*in the inertial frame*]{} is $$cP \left( 1 + \frac{2}{3}u^2 \right) =
cE \left( \frac{1}{3} - \frac{2}{3} u^2 \right)
+ \frac{4}{3} F \gamma u.
\label{E}$$ As a closure relation, the usual Eddington approximation [*in the comoving frame*]{} is adopted. Radiative quantities are then transformed from the comoving frame to the inertial frame, and we have the closure relation (\[E\]) in the inertial frame (see Kato et al. 1998 for details).
Eliminating $j$ with the help of equations (\[j1\]), and using continuity equation (\[rho1\]), equations (\[u1\]), (\[F1\]), and (\[P1\]) are rearranged as $$\begin{aligned}
cJ\frac{du}{dz} &=& (\kappa_{\rm abs}+\kappa_{\rm sca})
\rho \frac{\gamma}{c}
\left[ F (1+2u^2) - c(E+P)\gamma u \right],
\label{u2}
\\
\frac{dF}{dz} &=& (\kappa_{\rm abs}+\kappa_{\rm sca})
\rho u
\left[ c(E+P)\gamma u - F (1+2u^2) \right],
\label{F2}
\\
\frac{dP}{dz} &=& (\kappa_{\rm abs}+\kappa_{\rm sca})
\rho \frac{\gamma}{c}
\left[ c(E+P)\gamma u - F (1+2u^2) \right].
\label{P2}\end{aligned}$$
Integrating the sum of equations (\[u2\]) and (\[P2\]) yields to the momentum flux conservation along the flow, $$cJ u + P = K ~(={\rm const.}).
\label{K}$$ In the subrelativistic regime, this relation is reduced to that derived in Fukue (2005b). Similarly, after some manipulations, integrating the sum of equations (\[u2\]) and (\[F2\]) gives the energy flux conservation along the flow, $$c^2 J \gamma + F = L ~(={\rm const.}).
\label{L}$$ In the subrelativistic regime, this relation means that the flux $F$ is constant.
At this stage, the basic equations are the equation of motion (\[u2\]), the mass flux (\[rho1\]), the momentum flux (\[K\]), the energy flux (\[L\]), and the closure relation (\[E\]).
Next, by introducing the optical depth $\tau$ by $$d\tau = - ( \kappa_{\rm abs}+\kappa_{\rm sca} ) \rho dz,$$ the equation of motion (\[u2\]) is rewritten as $$cJ\frac{du}{d\tau} = - \frac{\gamma}{c}
\left[ F (1+2u^2) - c(E+P)\gamma u \right].
\label{u3}$$ Furthermore, eliminating $E$ with the help of equation (\[E\]), this equation (\[u3\]) can be finally rewritten as $$cJ\frac{du}{d\tau} = -\frac{\gamma}{c}
\frac{ F(1+4u^2) - 4cP \gamma u}{1-2u^2},
\label{u}$$ or $$c^2 J \gamma^2 \frac{d\beta}{d\tau} =
- \frac{ F(1+3\beta^2) - 4cP \beta}{1-3\beta^2},
\label{beta}$$ where $\beta=v/c$.
We shall solve equations (\[u\]), (\[K\]), and (\[L\]) for appropriate boundary conditions. Before this, we discuss the boundary conditions at the flow top in the next section.
Magic Speed Above a Moving Photosphere
======================================
When there is no motion in a luminous flat disk (“static photosphere”), the radiation fields above the disk are easily obtained. Namely, just above the disk with surface intensity $I_0$, the radiation energy density $E_{\rm s}$, the radiative flux $F_{\rm s}$, and the radiation pressure $P_{\rm s}$ are $(2/c)\pi I_0$, $\pi I_0$, and $(2/3c)\pi I_0$, respectively, where subscript s denotes the values at the disk surface. In the problem of radiation transfer in the accretion disk, these values are usually adopted as boundary conditions (e.g., Artemova et al. 1996). As will be shown below, however, the radiation fields above the luminous disk are changed when the disk gas itself does move upward (“moving photosphere”). Even in such a case, however, if the flow speed is small compared with the speed of light, the conditions for a static photosphere would be approximately adopted, and we can use these conditions in a previous paper (Fukue 2005b), where the flow speed is limited in the subrelativistic regime. When the flow speed is of the order of the speed of light, on the other hand, we should carefully treat the boundary condition for the moving photosphere. Thus, in the present paper, where the flow is treated in a fully relativistic manner, we must derive the exact boundary conditions above the moving photosphere.
In addition, for the radiatively driven flow in the relativistic regime, it becomes important the effect of radiation drag, which suppresses the jet speed (Phinney 1987; Icke 1989). For the flow above the static photosphere without gravity, Icke (1989) found that the [*magic speed*]{} of jets becomes $[(4-\sqrt{7})/3]~c \sim 0.45~c$. If the photophere is moving, however, such a magic speed will be also changed (cf. Fukue 2000).
Hence, before we can examine the relativistic radiative flow, we must derive the radiation fields above the moving photosphere and consider the boundary condition at the flow top (disk “surface”).
Let us suppose the situation that a flat infinite photosphere is not static, but moving upward with a speed $v_{\rm s}$ ($=c\beta_{\rm s}$, and the corresponding Lorentz factor is $\gamma_{\rm s}$). Then, the direction and intensity of radiation are changed due to aberration and Doppler effects (cf. Kato et al. 1998; Fukue 2000).
The transformations of the photon frequency $\nu$ and photon direction $\theta$ between the inertial and comoving frames become $$\begin{aligned}
\frac{\nu_0}{\nu} &=&
{\gamma_{\rm s} \left( 1- \beta_{\rm s} \cos\theta \right)}
= \frac{1}{\gamma_{\rm s} \left( 1+ \beta_{\rm s} \cos\theta_0 \right)},
\label{doppler}
\\
\cos\theta &=&
\frac{\cos\theta_0 + \beta_{\rm s}}{1 + \beta_{\rm s} \cos\theta_0},
\label{aberration}\end{aligned}$$ where subscript 0 means the values measured in the comoving frame and $\gamma_{\rm s} = 1/\sqrt{1-\beta_{\rm s}^2}$. For incident radiation with $\theta_0=\pi/2$ in the comoving frame, the direction cosine in the inertial frame is $\cos\theta=\beta_{\rm s}$.
Furthermore, the transformation of the intensity $I$ between the inertial and comoving frames is $$\begin{aligned}
I_0 = \left( \frac{\nu_0}{\nu} \right)^4 I
&=&
{\left[\gamma_{\rm s} \left( 1- \beta_{\rm s} \cos\theta \right)\right]^4}I
\nonumber \\
&=&
\frac{1}
{\left[\gamma_{\rm s} \left( 1+\beta_{\rm s} \cos\theta_0 \right)\right]^4}I.
\label{intensity}\end{aligned}$$
(80mm,80mm)[figure01.eps]{}
Considering the Doppler effect (\[doppler\]) and aberration (\[aberration\]), the radiation fields above the moving photosphere in the inertial frame are calculated as follows: $$\begin{aligned}
cE_{\rm s} &=& \int_0^{\cos^{-1}\beta_{\rm s}} I d\Omega
\nonumber \\
&=& \frac{2\pi I_0}{\gamma_{\rm s}^4}
\int_0^{\cos^{-1}\beta_{\rm s}}
\frac{\sin\theta d\theta}
{\left( 1-\beta_{\rm s} \cos\theta \right)^4}
\nonumber \\
&=& {2\pi I_0}{\gamma_{\rm s}^2}
\frac{3+3\beta_{\rm s}+\beta_{\rm s}^2}{3}
\nonumber \\
&=& {2\pi I_0}
\frac{3\gamma_{\rm s}^2+3\gamma_{\rm s}u_{\rm s}+u_{\rm s}^2}{3},
\label{Es}\end{aligned}$$ $$\begin{aligned}
F_{\rm s} &=& \int_0^{\cos^{-1}\beta_{\rm s}} I \cos\theta d\Omega
\nonumber \\
&=& \frac{2\pi I_0}{\gamma_{\rm s}^4}
\int_0^{\cos^{-1}\beta_{\rm s}}
\frac{\sin\theta \cos\theta d\theta}
{\left( 1-\beta_{\rm s} \cos\theta \right)^4}
\nonumber \\
&=& {2\pi I_0}{\gamma_{\rm s}^2}
\frac{3+8\beta_{\rm s}+3\beta_{\rm s}^2}{6}
\nonumber \\
&=& {2\pi I_0}
\frac{3\gamma_{\rm s}^2+8\gamma_{\rm s}u_{\rm s}+3u_{\rm s}^2}{6},
\label{Fs}\end{aligned}$$ $$\begin{aligned}
cP_{\rm s} &=& \int_0^{\cos^{-1}\beta_{\rm s}} I \cos^2\theta d\Omega
\nonumber \\
&=& \frac{2\pi I_0}{\gamma_{\rm s}^4}
\int_0^{\cos^{-1}\beta_{\rm s}}
\frac{\sin\theta \cos^2\theta d\theta}
{\left( 1-\beta_{\rm s} \cos\theta \right)^4}
\nonumber \\
&=& {2\pi I_0}{\gamma_{\rm s}^2}
\frac{1+3\beta_{\rm s}+3\beta_{\rm s}^2}{3}
\nonumber \\
&=& {2\pi I_0}
\frac{\gamma_{\rm s}^2+3\gamma_{\rm s}u_{\rm s}+3u_{\rm s}^2}{3}.
\label{Ps}\end{aligned}$$ That is, the radiation fields depend on the speed $v_{\rm s}$ of the photosphere, and every component increases as the speed increases (see figure 1).
We must use these values of radiation fields as boundary conditions above the moving photosphere.
(80mm,80mm)[figure02.eps]{}
In addition, the magic speed above the luminous infinite disk is obtained by the condition where the radiative force is balanced with the radiation drag force. Hence, from equation (\[u1\]), we have $$\begin{aligned}
0 &=& F_{\rm s} (1+2u^2) - c(E_{\rm s}+P_{\rm s})\gamma u
\nonumber \\
&=& [F_{\rm s} (1+\beta^2) - c(E_{\rm s}+P_{\rm s})\beta]\gamma^2,
\label{magic}\end{aligned}$$ for the magic speed $\beta$ ($=u/\gamma$).
Inserting equations (\[Es\])–(\[Ps\]) into this equation (\[magic\]), we can obtain the [*magic speed above a moving photosphere*]{} as a function of the speed $\beta_{\rm s}$ of the photosphere, $$\beta=\frac{4-\sqrt{7}+6\beta_{\rm s}+(4+\sqrt{7})\beta_{\rm s}^2}
{3+8\beta_{\rm s}+3\beta_{\rm s}^2}.$$ When the disk is static ($\beta_{\rm s}=0$), this relation is reduced to that obtained by Icke (1989), $\beta=(4-\sqrt{7})/3 \sim 0.45$.
In figure 2, the magic speed above a moving photosphere is shown as a function of the speed of the photosphere. When the luminous photosphere is not static, but moving, the magic speed can exceed the limit of $\sim 0.45c$, which Icke (1989) obtained for a static photosphere (cf. Fukue 2000).
Relativistic Radiative Transfer Flow
====================================
Now, we discuss our numerical solution of equations (\[u\]), (\[K\]), and (\[L\]) for appropriate boundary conditions.
Boundary Conditions and Singularity
-----------------------------------
As for boundary conditions, we impose the following cases. At the flow base (disk “inside”) with an arbitrary optical depth $\tau_0$ (which relates to the disk surface density), the flow velocity $u$ is zero, the radiative flux is $F_0$ (which is a measure of the strength of radiation field), and the radiation pressure is $P_0$ (which connects with the radiation pressure gradient and relates to the disk internal structure), where subscript 0 denotes the values at the flow base. At the flow top (disk “surface”) where the optical depth is zero, the radiation fields should satisfy the values above a moving photosphere derived in the previous section. Namely, just above the disk with surface intensity $I_{\rm s}$, the radiation energy density $E_{\rm s}$, the radiative flux $F_{\rm s}$, and the radiation pressure $P_{\rm s}$ are, respectively, $$\begin{aligned}
cE_{\rm s}
&=& {2\pi I_{\rm s}}
\frac{3\gamma_{\rm s}^2+3\gamma_{\rm s}u_{\rm s}+u_{\rm s}^2}{3},
\label{Es2}
\\
F_{\rm s}
&=& {2\pi I_{\rm s}}
\frac{3\gamma_{\rm s}^2+8\gamma_{\rm s}u_{\rm s}+3u_{\rm s}^2}{6},
\label{Fs2}
\\
cP_{\rm s}
&=& {2\pi I_{\rm s}}
\frac{\gamma_{\rm s}^2+3\gamma_{\rm s}u_{\rm s}+3u_{\rm s}^2}{3},
\label{Ps2}\end{aligned}$$ where $u_{\rm s}$ ($=\gamma_{\rm s}v_{\rm s}/c$) is the flow four velocity at the flow top and subscript s denotes the values at the flow top.
Applying these boundary conditions to equations (\[K\]) and (\[L\]), we have two relations on the boundary values and mass-loss rate: $$\begin{aligned}
Jc^2 u_{\rm s} + cP_{\rm s} &=& cP_0,
\label{bc1}
\\
Jc^2 \gamma_{\rm s} + F_{\rm s} &=& Jc^2 + F_0.
\label{bc2}\end{aligned}$$
Physically speaking, in the radiative flow starting from the flow base with an arbitrary optical depth $\tau_0$, for initial values of $F_0$ and $P_0$ at the flow base, the final values of the radiation fields $E_{\rm s}$, $F_{\rm s}$, $P_{\rm s}$, and the flow velocity $u_{\rm s}$ at the flow top can be obtained by solving basic equations. Furthermore, the mass-loss rate $J$ is determined as an eigenvalue so as to satisfy the bondary condition at the flow top (cf. Fukue 2005b in the subrelativistic regime).
In the present full relativistic case, however, the final values of the radiation fields at the flow top depend on the flow velocity there, and the final values at the flow top cannot be analytically expressed by the initial values at the flow base. Hence, in this paper we determine the mass-loss rate as follows.
In radiative flow with optical depth $\tau_0$, we first give the final flow velocity $u_{\rm s}$ (and $\gamma_{\rm s}$), instead of the initial value of $P_0$. Then, the final values of radiation fields $E_{\rm s}$, $F_{\rm s}$, and $P_{\rm s}$ can be fixed by equations (\[Es2\])–(\[Ps2\]). Next, we give a trial value for the mass-loss rate $J$, and the initial values of $P_0$ and $F_0$ can be fixed by equations (\[bc1\]) and (\[bc2\]). Since all the parameters are temporarily fixed, we solve equation (\[u\]) from $\tau=\tau_0$ to $\tau=0$. Generally, however, the obtained final velocity at $\tau=0$ is different from a given $u_{\rm s}$. Thus, we vary the value of $J$ and follow iterative processes, so that the calculated final velocity coincides with a given final velocity $u_{\rm s}$.
Another point to be noticed is the [*singularity*]{} in equation (\[u\]): i.e., the denominator of equation (\[u\]) vanishes when $u=\pm 1/\sqrt{2}$ or $\beta=\pm 1/\sqrt{3}$. This singularity originates from the closure relation (\[E\]), and coincides with the [*sound speed*]{} of the relativistic (photon) gas. In other words, equation (\[u\]) has a form of the transonic wind equation for the relativistic (photon) gas with sound speed $c/\sqrt{3}$. In usual wind equations from a gravitating source, there exist transonic (critical) points, where both the numerator and denominator of wind equations vanish simultaneously. In the present case, there also exist [*critical points*]{}, which yield that $u_{\rm c}=\pm 1/\sqrt{2}$ ($\beta_{\rm c}=\pm 1/\sqrt{3}$) and $P_{\rm c}={\rm sgn}(u) (\sqrt{3}/2) F_{\rm c}$, where subscript c denotes the critical point. In addition, from the linear analysis around a critical point, the velocity gradient near to the singularity is found to be $du/d\tau |_{\rm c} = {\rm sgn} (u) (3/4)$.
We show an example of “critical solutions” in figure 3, where the parameters are $\tau_{\rm c}=1$, $F_{\rm c}/(\pi I_{\rm s})=1$, and $J/(\pi I_{\rm s}/c^2)=1$. As can be seen in figure 3, one of the critical solutions (a solid curve) has a positive velocity, and is an outward breeze solution, which decelerates toward low optical depth. Another (a dashed curve) has a negative velocity, and is an inward settling solution, which decelerates toward a high optical depth. Both solutions decelerate in the direction of the flow due to the radiation drag force. However, neither satisfies the present boundary conditions at the flow base and the flow top. Indeed, in order for flow to be accelerated, the velocity gradient $du/d\tau$ should always be negative, but it is positive around a critical point for an outward solution. Hence, the flow in the present framework cannot pass through the critical point. Thus, for the present purpose, such “critical solutions” are inadequate, and the flow is always subsonic or supersonic in the sense that the flow speed is always less than or greater than $c/\sqrt{3}$. And since we assume that the flow starts with $u=0$ at $\tau=\tau_0$, we suppose a [*subsonic*]{} flow in the present paper: $u_s < 1/\sqrt{2} \sim 0.707$. We shall discuss this singularity problem later.
(80mm,80mm)[figure03.eps]{}
(80mm,80mm)[figure04a.eps]{}
(80mm,80mm)[figure04b.eps]{}
(80mm,80mm)[figure04c.eps]{}
(80mm,80mm)[figure05a.eps]{}
(80mm,80mm)[figure05b.eps]{}
(80mm,80mm)[figure05c.eps]{}
Subsonic Solutions
------------------
Examples of the results for relativistic radiative flows in a luminous disk under the present boundary conditions are shown in figures 4 and 5.
In figure 4 we show the final velocity $v_{\rm s}$ at the flow top (solid curves), the radiative flux $F_0$ at the flow base (upper dashed ones), the radiative flux $F_{\rm s}$ at the flow top (lower dashed ones), the radiation pressure $P_{\rm s}$ at the flow top (dotted ones), and the mass-loss rate $J$ (chain-dotted ones), as a function of $P_0$ for several values of $\tau_0$ at the flow base. The quantities are normalized in units of $c$ and $\pi I_{\rm s}$. For example, the unit of $J$ is $\pi I_{\rm s}/c^2$.
As can be seen in figure 4, as the radiative flux increases, the final flow velocity at the flow top increases, but the mass-loss rate decreases. Moreover, as can be seen in figure 4, and similar to the subrelativistic case (Fukue 2005b, c), in order for flow to exist, the radiation pressure $P_0$ at the flow base is restricted to be within some range. In the subrelativistic case without gravity and heating (Fukue 2005b), the initial pressure $P_0$ is proved to be restricted within the range of $2/3 < {cP_0}/{F_{\rm s}} < 2/3 + \tau_0$. In the present case, the initial pressure is also restricted to be within a similar range, but somewhat modified due to the relativistic effect: i.e., the radiative flux $F_{\rm s}$ is no longer constant, but depends on the flow final speed. At the one limit of $P_0$, the pressure gradient between the flow base and the top is maximum, the loaded mass diverges, and the flow final speed becomes zero. In the other limit of $P_0$, the pressure gradient vanishes, the loaded mass becomes zero, and the flow final speed becomes high.
As already stated, in the present model, the flow final speed is supposed to be within the range of $0< u_{\rm s} < 1/\sqrt{2}$ (i.e., $0 < v_{\rm s}/c < 1/\sqrt{3}$). As a result, using equations (\[Fs2\]) and (\[Ps2\]), the radiative flux $F_{\rm s}$ and the radiation pressure $P_{\rm s}$ are restricted to be within the following ranges: $1 < F_{\rm s}/(\pi I_{\rm s}) < 4.31$ and $2/3 < cP_{\rm s}/(\pi I_{\rm s}) < 3.73$. Finally, although the flow final speed is less than the photon sound speed ($v_{\rm s}/c < 1/\sqrt{3} \sim 0.577$), the flow final speed can exceed the magic speed above the luminous infinite disk \[$v_{\rm magic}/c = (4-\sqrt{7})/3 \sim 0.451$\].
In figure 5 we show the flow three velocity $v$ (solid curves), the radiative flux $F$ (dashed curves), and the radiation pressure $P$ (dotted curves) as a function of the optical depth $\tau$ for several values of $u_{\rm s}$ at the flow top in a few cases of $\tau_0$. The quantities are normalized in units of $c$ and $\pi I_{\rm s}$.
When the initial radiative flux $F_0$ at the flow base is large, the flow is effectively accelerated, and the flow final speed becomes large. On the other hand, when the pressure gradient between the flow base and the flow top is large, the loaded mass $J$ becomes large (cf. figure 4). The latter properties are somewhat complicated, and should be explained much more.
When the optical depth $\tau_0$ is not so large (figures 4a and 5a), the pressure gradient between the flow base and top is not so large. In this case, the pressure gradient becomes large, as the pressure $P_0$ at the flow base becomes small. This is the reason that the loaded mass becomes large, since $P_0$ is small (figure 4a). When the optical depth is large (figures 4b, 4c and 5b, 5c), on the other hand, the pressure gradient generally becomes large. Hence, the pressure gradient is large, as the pressure $P_0$ at the flow base becomes large. This is the reason that the loaded mass becomes large, since $P_0$ is large (figures 4b and 4c).
Alternatively, in the less-luminous, sub/non-relativistic limit with small $u_{\rm s}$, the radiation fields are roughly expressed as $$\begin{aligned}
F &\sim& F_{\rm s},
\\
cP &\sim& F_{\rm s}(2/3+\tau),\end{aligned}$$ which are the usual Milne approximations (cf. Fukue 2005b, c). Hence, in the less-luminous, small velocity case, the pressure gradient between the flow top and base becomes large, as the optical depth is large. In the highly-luminous, relativistic limit with large $u_{\rm s}$, on the other hand, the flow velocity quickly reaches the equilibrium velocity, where the right-hand side of equation (\[u\]) or (\[beta\]) vanishes, and becomes constant as $$\beta \sim \frac{2}{3}\frac{cP}{F}
- \sqrt{ \left( \frac{2}{3}\frac{cP}{F} \right)^2 - \frac{1}{3} },$$ where the radiative flux $F$ and the radiation pressure $P$ are also approximately constant according to equations (\[K\]) and (\[L\]). For example, in the limiting case of $\beta=1/\sqrt{3}$, $cP/F=\sqrt{3}/2$. As a result, the pressure gradient is small even for large optical depth.
In any case, the mass-loss rate $J$ is not given arbitrarily, but is determined as an eigenvalue, similar to the subrelativistic case (Fukue 2005b).
Concluding Remarks
==================
In this paper we have examined radiative flow in a luminous disk, while taking into account radiative transfer, in a fully relativistic manner (cf. Fukue 2005b, c for a subrelativistic regime). The vertical velocity $v$, the radiative flux $F$, and the radiation pressure $P$ are numerically solved as a function of the optical depth $\tau$ for the cases without gravity and heating. At the flow base (disk “inside”) where the flow speed is zero, the initial optical depth, the initial radiative flux, and the initial radiation pressure are $\tau_0$, $F_0$, and $P_0$, respectively; in the usual accretion disk these quantities are determined in terms of the central mass, the mass-accretion rate, and the viscous process as a function of radius. At the flow top (disk “surface”) where the optical depth $\tau$ is zero, the radiation fields ($E_{\rm s}$, $F_{\rm s}$, and $P_{\rm s}$) should coincide with those above a [*moving*]{} photosphere with the final speed $v_{\rm s}$ with uniform intensity. In order to match this boundary condition, the mass-loss rate $J$ is determined as an eigenvalue.
One of the relativistic manifestations is this boundary condition at the flow top. The radiation fields are not for the static flat source, but should be for the moving flat source. As a result, it is found that the magic speed above a moving photosphere can exceed that above a static photosphere ($\sim 0.45~c$).
Another relativistic manifestation is the existence of the singular point in the equation, which is related to the sound speed of the relativistic (photon) gas. Since the “critical solutions” for the present problem do not satisfy the present boundary conditions, we obtained [*subsonic*]{} solutions, which is always less than the relativistic sound speed of $c/\sqrt{3}$. Nevertheless, we can find the relativistic solution, whose final speed is greater than the magic speed above a static source.
The appearance of this singularity may come from the closure relation for the quantities of radiation fields, which assumes the usual Eddington approximation in the comoving frame (Kato et al. 1998). That is, in the comoving frame we assume $$P^{ij}_0 = \frac{\delta^{ij}}{3}E_0,
\label{P0E0}$$ where subscript 0 denotes the quantities in the comoving frame. The appearance of the singularity suggests that the Eddington approximation would be violated in the relativistic flow, whose velocity is greater than $c/\sqrt{3}$. This may be because the diffusion speed in the comoving frame cannot exceed $c/\sqrt{3}$, and/or the diffusion would not be isotropic in the comoving frame. Indeed, if we assume that the diffusion in the comoving frame is no longer isotropic, but the factor $1/3$ in equation (\[P0E0\]) is variable, similar to the usual variable Eddington factor, we can formally extinguish the singularity.
The plane-parallel assumption may also play a role in the existence of the singularity, as well as the existence of the radiation drag force, which is related to the closure approximation transformed to the inertial frame. The nature of critical points should be examined more carefully. In any case, the existence of the singularity does not mean that the flow velocity cannot exceed $c/\sqrt{3}$, but merely suggests a possible failure of the formalism adopted in the present analysis.
The relativistic radiative flow investigated in the present paper must be quite [*fundamental*]{} to accretion disk physics, astrophysical jet formation, gamma-ray bursts, etc., although the present paper is only the first step and there are many simplifications at the present stage.
At first, in a rigorous sense, the present situations, such as a plane-parallel assumption, are valid only in very specific circumstances. Indeed, when the accretion rate exceeds the critical one, the disk would puff up to be geometrically thick, and the plane-parallel approximation would be violated. In a rough sense, however, the present situations could be approximately valid in such a thick disk, as long as the gas motion and the radiative flux are almost vertical. If the gas motion is not vertical, but expands like a spherical flow, the present treatment must be reexamined. In contrast to the plane-parallel case considered in the present paper, the spherically expanding case is also of great interest to us, and should be examined in the future.
In the present paper, we have assumed that the opacity is frequency-independent (gray), and therefore the radiation pressure force is independent of the velocity. In a cold gas, however, atomic absorption opacities become important. Such opacities are frequency-dependent, and therefore the radiation pressure force would be a function of the Doppler shift between the main absorption troughs and the peak of the black body spectrum from the disk. Hence, the radiation energy flux $L$ would generally be a function of the velocity $u$. In addition, in such a case of atomic opacities, the super-Eddington radiation-pressure force on the flow may arise from a larger absorption opacity. Since the acceleration nature would be drastically changed, such a [*line-driven radiative flow*]{} must be carefully examined.
Furthermore, we ignored the gravitational field produced by the central object. This means that the flow considered in the present paper would correspond to normal plasmas in the super-Eddington disk, pair plasmas in the sub-Eddington disk, or dusty plasmas in the luminous disk. In other cases, or even for normal plasmas in the super-Eddington disk, the gravitational field would affect the flow properties (cf. Fukue 2005c for a subrelativistic case). In particular, since the gravitational field in the vertical direction is somewhat complicated (e.g., Fukue 2002, 2004), the influence of the gravitational field is important.
We also ignored the gas pressure. This means that the radiation field is sufficiently intense. In general cases, where the gas pressure is considered, there usually appear sonic points (e.g., Fukue 2002, 2004), and the flow is accelerated from subsonic to supersonic.
In the high-energy regime, including relativistic flow and hot gas, the Compton heating and cooling would be important, although we have dropped such an effect. If the Compton heating of the gas by intense radiation fields works, the gas might be radiatively heated to very high temperatures, and the atomic absorption opacity could be significantly reduced. Such a radiative effect in the energy equation should also be examined in future work.
Finally, it is left as an open question whether the singularity appearing in the present paper can be passed through or whether the relativistic radiation transfer formalism should be improved.
The authors would like to thank Professors S. Kato and S. Inagaki for their useful comments and discussions. Valuable comments from an anonymous referee are also gratefully acknowledged. This work has been supported in part by a Grant-in-Aid for Scientific Research (15540235 J.F.) of the Ministry of Education, Culture, Sports, Science and Technology.
Adam, J., Störzer, H., Shaviv, G., & Wehrse, R. 1988, A&A, 193, L1
Artemova, I. V., Bisnovatyi-Kogan, G. S., Björnsson, G., & Novikov, I. D. 1996, ApJ, 456, 119
Bisnovatyi-Kogan, G. S., & Blinnikov, S. I. 1977, A&A, 59, 111
Cannizzo, J. K., & Wheeler, J. C. 1984, ApJS, 55, 367
Davidson, K., & McCray, R. 1980, ApJ, 241, 1082
Davis, S. W., Blaes, O. M., Hubeny, I., & Turner, N. J. 2005, ApJ, 621, 372
Eggum, G. E., Coroniti, F. V., & Katz, J. I. 1985, ApJ, 298, L41
Eggum, G. E., Coroniti, F. V., & Katz, J. I. 1988, ApJ, 330, 142
Fukue, J. 1982, PASJ, 34, 163
Fukue, J. 1999, PASJ, 51, 703
Fukue, J. 2000, PASJ, 52, 613
Fukue, J. 2002, PASJ, 54, 415
Fukue, J. 2004, PASJ, 56, 181
Fukue, J. 2005a, PASJ, 57, 691
Fukue, J. 2005b, PASJ, 57, 841
Fukue, J. 2005c, PASJ submitted
Fukue, J., Tojyo, M., & Hirai, Y. 2001, PASJ, 53, 555
Hirai, Y., & Fukue, J. 2001, PASJ, 53, 285
Hubeny, I. 1990, ApJ, 351, 632
Hubeny, I., Agol, E., Blaes, O., & Krolik, J. H. 2000, ApJ, 533, 710
Hubeny, I., Blaes, O., Krolik, J. H., & Agol, E. 2001, ApJ, 559, 680
Hubeny, I., & Hubeny, V. 1997, ApJ, 484, L37
Hubeny, I., & Hubeny, V. 1998, ApJ, 505, 558
Hui, Y., Krolik, J. H., & Hubeny, I. 2005, ApJ, 625, 913
Icke, V. 1980, AJ, 85, 329
Icke, V. 1989, A&A, 216, 294
Kato, S., Fukue, J., & Mineshige, S. 1998, Black-Hole Accretion Disks (Kyoto: Kyoto University Press)
Katz, J. I. 1980, ApJ, 236, L127
Luo, Q., & Protheroe, R. J. 1999, MNRAS, 304, 800
Lynden-Bell, D. 1978, Phys. Scr., 17, 185
Marcowith, A., Henri, G., & Pelletier, G. 1995, MNRAS, 277, 681
Melia, F., & Königl, A. 1989, ApJ, 340, 162
Meyer, F., & Meyer-Hofmeister, E. 1982, A&A, 106, 34
Mineshige, S., & Wood, J. H. 1990, MNRAS, 247, 43
Misra, R., & Melia, F. 1993, ApJ, 419, L25
Orihara, S., & Fukue, J. 2003, PASJ, 55, 953
Phinney, E. S. 1987, in Superluminal Radio Sources, ed. J. A. Zensuz & T. J. Pearson (Cambridge: Cambridge University Press), 301
Renaud, N., & Henri, G. 1998, MNRAS, 300, 1047
Ross, R. R., Fabian, A. C., & Mineshige, S. 1992, MNRAS, 258, 189
Shaviv, G., & Wehrse, R. 1986, A&A, 159, L5
Sikora, M., Sol, H., Begelman, M. C., & Madejski, G. M. 1996, MNRAS, 280, 781
Sikora, M., & Wilson, D. B. 1981, MNRAS, 197, 529
Sol, H., Pelletier, G., & Asséo, E. 1989, MNRAS, 237, 411
Tajima, Y., & Fukue, J. 1996, PASJ, 48, 529
Tajima, Y., & Fukue, J. 1998, PASJ, 50, 483
Watarai, K., & Fukue, J. 1999, PASJ, 51, 725
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abstract: 'Driving a two-dimensional Mott insulator with circularly polarized light breaks time-reversal and inversion symmetry, which induces an optically-tunable synthetic scalar spin chirality interaction in the effective low-energy spin Hamiltonian. Here, we show that this mechanism can stabilize topological magnon excitations in honeycomb ferromagnets such as CrI$_3$ and in optical lattices. We find that the irradiated quantum magnet is described by a Haldane model for magnons that hosts topologically-protected edge modes. We study the evolution of the magnon spectrum in the Floquet regime and via time propagation of the magnon Hamiltonian for a slowly varying pulse envelope. Compared to similar but conceptually distinct driving schemes based on the Aharanov-Casher effect, the dimensionless light-matter coupling parameter $\lambda = eEa/\hbar\omega$ at fixed electric field strength is enhanced by a factor $\sim 10^5$. This increase of the coupling parameter allows to induce a topological gap of the order of $\Delta \approx 2$ meV with realistic laser pulses, bringing an experimental realization of light-induced topological magnon edge states within reach.'
author:
- Emil Viñas Boström
- Martin Claassen
- 'James W. McIver'
- Gregor Jotzu
- Angel Rubio
- 'Michael A. Sentef'
bibliography:
- 'references\_magnon.bib'
title: 'Light-induced topological magnons in two-dimensional van der Waals magnets'
---
The experimental realization of magnetic van der Waals (vdW) materials with a thickness down to the monolayer limit has sparked a new interest in fundamental aspects of two-dimensional magnetism [@Gong17; @Huang17; @Burch18; @Gong19]. Due to a competition of strong anisotropy, fluctuations, and spin-orbit effects, two-dimensional vdW materials are known to exhibit diverse magnetic orders ranging between semiconducting ferromagnetism, itinerant ferromagnetism, and insulating antiferromagnetism [@Williams15; @Bonilla18; @Deng18; @Lee16]. However, these properties also make them prime candidates to host topological phenomena such as Berezinskii-Kosterlitz-Thouless phase transitions [@Kosterlitz73], quantum spin liquids [@Banerjee17; @Claassen17], magnetic skyrmions [@Ding19], and fractional excitations [@Nasu16].
In addition to the intrinsic topological properties of vdW magnets, the tremendous progress in functionalization of materials through light-matter coupling [@Kirilyuk10; @Stojchevska14; @McIver19; @Shin18; @Sato19; @Shin19] shows that it is possible to manipulate the magnetic and topological order of such materials using laser fields. In recent theoretical studies it has been shown that driving a two-dimensional Mott insulator with circularly polarized light breaks both time-reversal and inversion symmetries. This is reflected by an induced scalar spin chirality interaction that governs the transient dynamics of low-energy spin excitations [@Claassen17; @Kitamura17]. Remarkably, optical irradiation red-detuned from the Mott gap can limit heating and absorption to enable a controlled realization of such Floquet-engineered spin dynamics, and it has been argued for a Kagomé lattice antiferromagnet that the spin chirality term leads to a chiral spin liquid ground state in herbertsmithite and kapellasite [@Claassen17]. Experimental realizations of Floquet-engineered spin Hamiltonians have also been demonstrated for both classical [@Struck11] and quantum magnetism [@Gorg18] using ultracold atoms in driven optical lattices [@Eckardt17].
In this work, we demonstrate that the photo-induced scalar spin chirality has consequences for the low-energy magnetic excitations of ferromagnetic systems. In particular, for honeycomb ferromagnets such as monolayer CrI$_3$, it leads to a magnon Haldane model [@Haldane88] with a topological gap and chiral magnon edge states [@Owerre17a]. To this end, we first derive the magnitude of the induced time-reversal symmetry breaking contribution for a honeycomb Mott insulator. We then show that application of the effective spin Hamiltonian to the prototypical monolayer vdW magnet CrI$_3$ [@Huang17; @Chen18; @Li20; @Cenker20] can lead to a gap $\Delta \approx 2$ meV in the magnon spectrum for a realistic field strength $E = 10^9$ V/m and photon energy $\hbar\omega = 1$ eV, inducing non-zero Chern numbers and leading to chiral magnon edge states. Importantly, we find that the dimensionless Floquet parameter that describes the magnitude of light-matter interaction is enhanced by a factor $\sim 10^5$ compared to similar but conceptually distinct driving schemes based on the Aharanov-Casher effect for pure spin models [@Owerre17a; @Elyasi19], since the electric field couples to the charge instead of the magnetic moment. This amplification is shown to be crucial for a potential experimental realization of a topological magnon phase in monolayer vdW magnets.
[**Model.–**]{} To assess the magnitude of photo-induced time-reversal symmetry breaking for honeycomb Mott insulators, we commence by deriving an effective transient spin-1/2 Hamiltonian from a single-band Mott insulator $$\begin{aligned}
H = &-t\sum_{\langle ij\rangle\sigma} e^{i\theta_{ij}(t)} c_{i\sigma}^\dagger c_{j\sigma} + U_0 \sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} \\
&+ \frac{V}{2} \sum_{\langle ij\rangle} \hat{n}_i \hat{n}_j - J_D \sum_{\langle ij\rangle} \hat{\bf S}_i \cdot \hat{\bf S}_j, \nonumber\end{aligned}$$ where $c_{i\sigma}^\dagger$ creates an electron at site $i$ with spin projection $\sigma$, $t$ is the hopping amplitude between nearest neighbor sites $i$ and $j$, and $U_0$ is a local interaction. We also consider nearest neighbor direct and exchange interactions $V$ and $J_D$, the later being expressed in terms of the spin operator $\hat{{\bf S}}_i = c_{i\sigma}^\dagger \boldsymbol\tau_{\sigma\sigma'} c_{i\sigma'}$ [^1] where $\boldsymbol\tau$ is the vector of Pauli matrices. We use the Einstein summation convention for repeated spin indexes.
The electrons interact with an external electromagnetic field described via the Peierls phases $$\begin{aligned}
\theta_{ij}(t) &= -\frac{e}{\hbar}\int_{{\bf r}_j}^{{\bf r}_i} d{\bf r}\cdot {\bf A}({\bf r},t).\end{aligned}$$ To break time-reversal symmetry and induce a scalar spin chirality, we use a circularly polarized laser in the dipole approximation with vector potential $\partial_t {\bf A}({\bf r},t) = - E(t)(\cos\omega t, \zeta\sin\omega t)$, where $\zeta = \pm1$ for right/left-handed polarization. Assuming a constant envelope $E(t) = E$ and writing $\boldsymbol\delta_{ij} = {\bf r}_i - {\bf r}_j = a (\cos\phi_{ij}, \sin\phi_{ij})$ with $a$ the lattice constant, the Peierls phases are $\theta_{ij}(t) = -\lambda \sin(\omega t - \zeta\phi_{ij})$. The dimensionless quantity $\lambda = eEa/\hbar\omega$ determines the effective field strength of the laser. In an optical lattice, $eE$ is replaced by the driving force $F$, which may result from an acceleration of the lattice [@Lignier07] or a magnetic field gradient [@Jotzu15].
Although the above model provides a simplified description of realistic monolayer vdW magnets, neglecting both the multi-orbital structure of the transition metals ions and the superexchange processes induced by interactions with the surrounding halides [@Stavropoulos19], it provides a starting point for more advanced treatments. Further, since the topological properties of honeycomb ferromagnets are determined by the lattice structure and the presence or absence of time-reversal symmetry [@Haldane88], we expect the model to give a correct description of the topological features of the magnon excitations.
[**Effective spin Hamiltonian.–**]{} We now construct an effective spin Hamiltonian for driving frequencies $J \ll \hbar\omega \ll U$, where $J \sim t^2/U$ is the leading order Heisenberg exchange in equilibrium. We have followed the method of Ref. [@Claassen17] to obtain the effective Hamiltonian to fourth order in $t/U$ for a periodic external field. For a slowly varying envelope $E(t)$ the Hamiltonian is almost periodic with the period $H(t + 2\pi/\omega) = H(t)$. This allows us to employ Floquet theory and rewrite the electronic Hamiltonian exactly using a Fourier expansion $$\begin{aligned}
H =& -t\sum_{\langle ij\rangle\sigma} \sum_{mm'} \mathcal{J}_{m-m'}(\lambda) e^{i(m-m')\zeta\phi_{ij}} c_{i\sigma}^\dagger c_{j\sigma} \otimes |m \rangle \langle m'| \nonumber \\
&+ H_I \otimes \mathbf{1} - \sum_m m\omega \otimes |m \rangle \langle m|, \end{aligned}$$ expressed in the product space of the electronic Hamiltonian and the space of periodic functions [@Sambe73] denoted by Fourier modes $|m\rangle$, which can be identified with the classical limit of $m$ absorbed or emitted virtual photons. Here, $\mathcal{J}_m(x)$ is the Bessel function of the first kind of order $m$. The interaction Hamiltonian is $H_I = U \sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow} - J_D \sum_{\langle ij\rangle} \hat{\bf S}_i \cdot \hat{\bf S}_j$, where the nearest neighbor direct interaction has been absorbed by a renormalization of the Hubbard $U$ [@Schuler13]. Using quasi-degenerate perturbation theory to simultaneously integrate out the doubly occupied states and the $m \neq 0$ Floquet states [@Mentink15; @Claassen17], the effective honeycomb lattice spin Hamiltonian corresponding to the electronic system is given to fourth order in $t/U$ by $$\begin{aligned}
\label{eq:floquet}
\mathcal{H} &= \sum_{\langle ij\rangle} J_{ij} \hat{\bf S}_i \cdot \hat{\bf S}_j + \sum_{\langle\langle ik\rangle\rangle} J'_{ik} \hat{\bf S}_i \cdot \hat{\bf S}_k \\
&+ \sum_{\langle\langle ik\rangle\rangle} \chi_{ik} \hat{\bf S}_j \cdot (\hat{\bf S}_i \times \hat{\bf S}_k). \nonumber\end{aligned}$$ Here $J$ and $J'$ are respectively the nearest and next-nearest neighbor light-induced Heisenberg exchanges, and $\chi$ is a synthetic scalar spin chirality. A non-zero value of $\chi$ signals a non-coplanar spin texture and can appear in equilibrium due to e.g. Dzyaloshinskii-Moriya interactions or geometric frustration [@Taguchi01; @Grytsiuk20]. For electrons hopping around closed loops in such a spin texture the spin chirality acts as an effective magnetic field that can give rise to the topological Hall effect [@Kanazawa11]. The full expressions for the spin parameters are given in the Supplemental Material (SM).
We note that $J$ has contributions from all even orders in $t/U$, while $J'$ and $\chi$ appear only at fourth order. On the honeycomb lattice, a non-zero spin chirality arises due hopping processes that enclose an isosceles triangle within the hexagons, as indicated schematically in Fig. \[fig:ferro\]a (and discussed further in the SM). Such processes lead to a net phase accumulation in analogy with electrons moving in closed loops in an external magnetic field, and lead to time-reversal symmetry breaking. However, in contrast to using an external magnetic field, driving with a circularly polarized electric field conserves the $SU(2)$ spin symmetry. In the non-interacting limit the corresponding complex next-nearest neighbor tunneling has already been implemented in optical lattices using circular driving [@Jotzu14].
[**Justifying the Hamiltonian for CrI$_3$ and optical lattices.–**]{} Below we use the spin Hamiltonian to study the magnon excitations of driven monolayer CrI$_3$. Since our effective Hamiltonian was derived for $S = 1/2$, it gives a simplified description of $S = 3/2$ ferromagnets such as CrI$_3$. However, similar spin Hamiltonians have been used to successfully describe the magnon excitations in CrI$_3$ [@Chen18; @Costa20].
The main effects of including the $t_{2g}$ orbitals of Cr$^{3+}$ via a Kanamori-Hubbard model (except for an obvious renormalization of the spin parameters), are spin-orbit coupling and the appearance of biquadratic exchange terms $(\hat{\bf S}_i \cdot \hat{\bf S}_j)^2$ [@anderson_new_1959; @kittel_model_1960]. Biquadratic exchange can generate nematic instabilities [@Lauchli06; @Fridman11] and break the $C_6$ rotation symmetry down to the $C_3$ subgroup, generating a trivial mass term that competes with Haldane mass generated from the breaking of time-reversal symmetry and can trivialize the magnon band topology. To estimate this effect, we performed density functional theory calculations in the DFT$+U$ formalism with the Octopus code [@TancogneDejean17; @TancogneDejean20] to estimate the size of the trivial mass term in monolayer CrI$_3$ (with the value of $U$ self-consistently determined via the ACBN0 hybrid functional [@Agapito15]). We find a ground state with $C_6$ symmetry to a numerical accuracy $10^{-6}$, and thus conclude that the effects of the biquadratic terms are negligible in CrI$_3$.
Ultracold fermions in optical lattices naturally realize the Hubbard Hamiltonian, and $V$ and $J_D$ are typically negligible [@Esslinger10]. Nevertheless, ferromagnetic spin models can be implemented using near-resonant periodic driving [@Gorg18]. For most systems, $S=1/2$ (where we expect our results to still hold approximately), but magnetic correlations for larger $S$ have also been observed using alkali-earth-like atoms [@Ozawa18].
[**Antiferromagnetic systems.–**]{} In the following we assume $|\chi| \ll |J|$, so that depending on the sign of $J$ the system is either ferromagnetic ($J < 0$) or antiferromagnetic ($J > 0$). It has previously been shown that topological magnon edge states can be induced by a constant electric field gradient that splits the magnon bands into Landau levels and leads to a magnon version of the quantum (spin) Hall effect in (anti-) ferromagnets [@Nakata17a; @Nakata17b]. In the present work the homogeneous but time-dependent electric field instead opens a gap at magnon band crossings, leading to a magnon analog of the quantum anomalous Hall effect. In the antiferromagnetic regime we find that the bands are nearly degenerate with no crossings, and the system remains in a topologically trivial phase. This agrees with previous work where the edge modes of the Néel state were shown to be topologically trivial [@Owerre17b]. However, by adding an in-plane magnetic field [@Owerre17b] or considering an antiferromagnetically coupled bilayer [@Owerre19], topological magnon edge states can be induced. Although we focus below on the ferromagnetic state, we expect that an application of our formalism to the non-collinear and bilayer antiferromagnetic cases would lead to similar conclusions.
[**Magnons on the honeycomb lattice.–**]{} We denote the lattice vectors of the honeycomb lattice by ${\bf b}_i$ and the vectors between nearest neighbor sites by $\boldsymbol\delta_i$ (see Fig. \[fig:ferro\]b). On the honeycomb lattice the angles $\Phi_{ik} = \zeta(\phi_{ij}-\phi_{kj})$ between next-nearest neighbor sites are given by $\Phi_{ik} = 2\pi \zeta\nu_{ik}/3$, where $\nu_{ik} = 1$ ($\nu_{ik} = -1$) for hopping in a clockwise (anti-clockwise) direction (see Fig. \[fig:ferro\]b). This leads to a spin chirality of the form $\chi_{ik} = \zeta\nu_{ik} \chi$ where the sign alternates depending on the bond direction.
For a ferromagnetic ground state we can solve the system to leading order in $S^{-1}$ using the Holstein-Primakoff transformation $\hat{S}_i^z = S - a_i^\dagger a_i$, $\hat{S}_i^- \approx \sqrt{2S} a_i^\dagger$ and $\hat{S}_i^+ \approx \sqrt{2S} a_i$ for the spins on sublattice $A$, and similarly but with $a_i \to b_i$ for the spins on sublattice $B$. In terms of its Fourier components the Hamiltonian can be written as $\mathcal{H} = \sum_{\bf k} \Psi_{\bf k}^\dagger \mathcal{H}_{\bf k} \Psi_{\bf k}$, where $\Psi_{\bf k} = (a_{\bf k}, b_{\bf k})^T$, $\mathcal{H}_{\bf k} = h_0 {\bf 1} + {\bf h} \cdot \boldsymbol\tau$, ${\bf h} = (h_x, h_y, h_z)$ and $\boldsymbol\tau$ is the vector of Pauli matrices. The eigenvalues of this matrix are $\epsilon_{\pm}({\bf k}) = h_0({\bf k}) \pm \sqrt{{\bf h}({\bf k}) \cdot {\bf h}({\bf k})}$, where $h_0 = 3JS + 6J'S + 2J'S\xi_{\bf k}$, $h_x + ih_y = -JS \rho_{\bf k}$ and $h_z = 2\zeta \chi S \sigma_{\bf k}$. To simplify the notation we have defined the quantities $\rho_{\bf k} = \sum_i e^{-i {\bf k} \cdot \boldsymbol\delta_i}$, $\xi_{\bf k} = \sum_i \cos({\bf k} \cdot {\bf b}_i)$ and $\sigma_{\bf k} = \sum_i \sin({\bf k} \cdot {\bf b}_i)$.
In Fig. \[fig:ferro\]c we show the magnon band structure for equilibrium spin parameters of bulk CrI$_3$ [@Chen18]. For $\chi = 0$ the system has Dirac points at ${\bf K}_\eta = \eta (4\pi/3\sqrt{3},0)$ (where $\eta = \pm$), while for $\chi > 0$ a gap is opened of magnitude $\Delta =6\sqrt{3} \chi S$. As shown below, the gap opening is associated with a transition to a non-trivial topological state.
at (0,0) [![[**Energy scales of light-induced chirality.**]{} $(a)$ Ratio of the synthetic scalar spin chirality $\chi$ to the field-renormalized Heisenberg exchange $J$ as a function of field strength $\lambda$ and photon energy $\hbar\omega$. The results are for a system with $t = 50$ meV, $U = 1.25$ eV and $J_D = 12$ meV, giving an effective ferromagnetic exchange in equilibrium of $J = 4.0$ meV and $J' = 13$ $\mu$eV. $(b)$ $\chi$ as a function of photon energy $\hbar\omega$ for different values of the effective field strength $\lambda$. $(c)$ $\chi$ as a function of electric field strength $E$ for different values of the photon energy $\hbar\omega$.[]{data-label="fig:chi"}](chi_density_large.pdf "fig:"){width="0.57\columnwidth"}]{}; at (-1.35,-1.2) [$(a)$]{}; at (4.3,1.0) [![[**Energy scales of light-induced chirality.**]{} $(a)$ Ratio of the synthetic scalar spin chirality $\chi$ to the field-renormalized Heisenberg exchange $J$ as a function of field strength $\lambda$ and photon energy $\hbar\omega$. The results are for a system with $t = 50$ meV, $U = 1.25$ eV and $J_D = 12$ meV, giving an effective ferromagnetic exchange in equilibrium of $J = 4.0$ meV and $J' = 13$ $\mu$eV. $(b)$ $\chi$ as a function of photon energy $\hbar\omega$ for different values of the effective field strength $\lambda$. $(c)$ $\chi$ as a function of electric field strength $E$ for different values of the photon energy $\hbar\omega$.[]{data-label="fig:chi"}](chi.pdf "fig:"){width="0.42\columnwidth"}]{}; at (5.5,2.0) [$(b)$]{}; at (4.35,-1.4) [![[**Energy scales of light-induced chirality.**]{} $(a)$ Ratio of the synthetic scalar spin chirality $\chi$ to the field-renormalized Heisenberg exchange $J$ as a function of field strength $\lambda$ and photon energy $\hbar\omega$. The results are for a system with $t = 50$ meV, $U = 1.25$ eV and $J_D = 12$ meV, giving an effective ferromagnetic exchange in equilibrium of $J = 4.0$ meV and $J' = 13$ $\mu$eV. $(b)$ $\chi$ as a function of photon energy $\hbar\omega$ for different values of the effective field strength $\lambda$. $(c)$ $\chi$ as a function of electric field strength $E$ for different values of the photon energy $\hbar\omega$.[]{data-label="fig:chi"}](chi_efield.pdf "fig:"){width="0.42\columnwidth"}]{}; at (5.95,-0.4) [$(c)$]{};
[**Chern numbers and edge states.–**]{} To determine the topological structure of the system for $\chi > 0$, we calculate the Chern numbers of the dressed magnon bands. Since the dominant contribution to the Berry curvature comes from the regions around the Dirac points, we expand the Hamiltonian around ${\bf K}_\eta$. To linear order in $\boldsymbol\kappa = {\bf k} - {\bf K}$ we find the Hamiltonian $\mathcal{H}^\eta = v\eta\kappa_x \tau_x - v\kappa_y \tau_y + w\eta \tau_z$, where $v = 3JS/2$, $w = - 3\sqrt{3}\zeta\chi S$, and we have neglected constant terms proportional to $J$ and $J'$.
The Berry potential for the quasi-stationary state is obtained from the expression [@Bernevig13] $$\begin{aligned}
v_s^\eta ({\bf k}) = \operatorname{Im}\frac{\langle\psi_{ks}|\nabla \mathcal{H}^\eta |\psi_{k,-s}\rangle \times \langle\psi_{k,-s}|\nabla \mathcal{H}^\eta |\psi_{ks}\rangle}{(\epsilon_+({\bf k}) - \epsilon_+({\bf k}))^2},\end{aligned}$$ where $|\psi_{ks}\rangle $ are the eigenstates of $\mathcal{H}^\eta$. It is clear from the cross product that $v_{-s} = -v_s$ and so it is sufficient to compute $v_+$. We calculate the matrix elements by noting that $\partial_{\kappa_x} \mathcal{H}^\eta = v\eta\tau_x$ and $\partial_{\kappa_y} \mathcal{H}^\eta = -v\tau_y$, and defining $d^2 = {\bf h} \cdot {\bf h} = \tfrac{1}{4}(\epsilon_+({\bf k}) - \epsilon_-({\bf k}))^2$ we find the Berry potential $v_s^\eta = -s\eta v^2 h_z/(2d^3)$. Since the Chern number is the integral over the Berry potential we have $$\begin{aligned}
\mathcal{C}_s &= \frac{1}{2\pi} \sum_\eta \int d^2\kappa v_s^\eta ({\bf k}) = s\zeta\operatorname{sgn}(\chi).\end{aligned}$$ For positive $\chi$ the upper (lower) band has a Chern number $\mathcal{C} = \zeta$ ($\mathcal{C} = -\zeta$).
The non-zero Chern numbers imply the existence of topological magnon edge states. We verify this explicitly for a ribbon geometry with zig-zag edges, periodic in the $x$-direction and with $n_y$ sites in the $y$-direction. In Fig. \[fig:ferro\]d we show the band structure of the ribbon for $n_y = 100$, where chiral edge states are situated in the bulk band gap and connect the Dirac points at ${\bf K}_+$ and ${\bf K}_-$.
We note that in contrast to the edge states of a quantum spin Hall insulator, the edge magnons of different chirality are located on opposite edges of the sample. Since the sign of the Chern numbers and thereby the chirality of the edge states are determined by the polarization of the optical field, this allows to control the propagation direction of the edge magnons by changing the helicity of the field.
[**Parameter dependence of the scalar spin chirality.–**]{} We have seen that a non-zero value of the scalar spin chirality $\chi$ leads to a topological magnon state. We now discuss the values of the frequency and electric field strength needed to induce this state in monolayer CrI$_3$.
We start by considering the ratio $\chi/J$ as a function of the photon energy $\hbar\omega$ and effective field strength $\lambda$, which is a measure of the ratio between the bandgap and the bandwidth. The results are shown in Fig. \[fig:chi\]a for the electronic parameters $t = 50$ meV, $U = 1.25$ eV and $J_D = 12$ meV corresponding approximately to monolayer CrI$_3$ [^2]. We find a resonant behavior in $\chi/J$ when the frequency $\hbar\omega = U/n$ with $n$ integer, which corresponds to the thresholds for $n$-photon excitation across the Mott gap. In addition, the diagonal feature extending across Fig. \[fig:chi\]a indicates the transition from a ferromagnetic to an antiferromagnet effective exchange parameter [@Mentink15]. Approaching this transition while simultaneously ensuring $|\chi| > |J'|$ will bring the system into a state dominated by the spin chirality term. This could potentially lead to new exotic physics such as a skyrmion lattice [@Nagaosa13] or chiral spin liquid ground state [@Claassen17; @Kalmeyer87].
Naively these results suggest employing a sub-gap driving protocol that exploits the resonant enhancement of $\chi$ for $\hbar\omega \approx U/n$ while simultaneously minimizing electronic interband transitions. However, numerical studies have shown that for driving frequencies close to the multi-photon resonances the system heats immediately and the spin description becomes invalid [@Claassen17]. In addition, since the real Mott gap is not at $U$ but at the slightly smaller value $U - xt$ (with $x$ a numerical factor of order unity), the frequency has to be chosen below this gap to avoid heating. In the following we therefore focus on photon energies $\hbar\omega/U \approx 0.8$, which is below the Mott gap for $x < 5$.
Assuming a realistic field strength $E \approx 10^9$ V/m, $U = 1.25$ eV, $a = 5$ Åand $\hbar\omega \approx 1$ eV, an effective field strength $\lambda \approx 0.5$ can be achieved. Because interband transitions are avoided in this driving protocol, larger field strengths may still yet be applied without inflicting material damage or other detrimental effects that would disrupt the induced scalar spin chirality. However, even for $\lambda \approx 0.5$ it is possible to open a bandgap of magnitude $\Delta \approx 2$ meV (see Fig. \[fig:chi\]b). In contrast, a treatment based on the Aharanov-Casher effect in pure spin systems leads to a field strength $\lambda_m = (g\mu_B Ea)/(\hbar c^2)$ [@Owerre17a], which is smaller than the electronic equivalent by a factor $\lambda_m/\lambda_e = (g\mu_B \omega)/(ec^2) \approx 10^{-5}$. Since $\chi \sim \lambda^2$ for small $\lambda$, this leads to a reduction of the gap size by about $10^{-10}$ making an experimental realization of topological magnon systems based on the Aharanov-Casher effect highly challenging. In contrast, the driving protocol proposed here opens a topological gap well within reach of experimental probes.
In optical lattices, heating rates have been shown to be manageable even for $\lambda >1$ [@Messer18]. The magnon band gap can hence be enhanced to values above the currently accessible temperature scales [@Mazurenko17].
[**Validating the Floquet treatment.–**]{} To validate the Floquet treatment we compare the results obtained via the static Floquet Hamiltonian with numerical results from time-propagating the system with a quasi-periodic spin Hamiltonian (for details see SM). We take the external field to be switched on adiabatically over approximately $350$ periods $T = 2\pi/\omega$, after which we propagate the system for an additional $1750$ periods.
at (0,0) [![[**Spectral functions for a ribbon geometry.**]{} $(a)$ Floquet spectral function $A_{\bf k}^F(\epsilon)$ for a ribbon with $n_y = 20$ and spin parameters $S = 3/2$, $J = 2.26$ meV, $J' = 0.05$ meV and $\chi = 0.13$ meV. $(b)$ Non-equilibrium spectral function $A_{\bf k}(t,\epsilon)$ at $t = 8.2$ ps for a zig-zag ribbon with $n_y = 20$ and electronic parameters $t = 50$ meV, $U = 1.25$ eV and $J_D = 12$ meV. The optical field has a frequency $\hbar\omega = 1$ eV and field strength $E = 10^9$ V/m.[]{data-label="fig:spectral"}](spectral_ribbon.pdf "fig:"){width="0.5\columnwidth"}]{}; at (-0.9,1.6) [$(a)$]{}; at (4.4,0.1) [![[**Spectral functions for a ribbon geometry.**]{} $(a)$ Floquet spectral function $A_{\bf k}^F(\epsilon)$ for a ribbon with $n_y = 20$ and spin parameters $S = 3/2$, $J = 2.26$ meV, $J' = 0.05$ meV and $\chi = 0.13$ meV. $(b)$ Non-equilibrium spectral function $A_{\bf k}(t,\epsilon)$ at $t = 8.2$ ps for a zig-zag ribbon with $n_y = 20$ and electronic parameters $t = 50$ meV, $U = 1.25$ eV and $J_D = 12$ meV. The optical field has a frequency $\hbar\omega = 1$ eV and field strength $E = 10^9$ V/m.[]{data-label="fig:spectral"}](td_spectral_ribbon.png "fig:"){width="0.5\columnwidth"}]{}; at (3.3,1.6) [$(b)$]{};
To visualize the magnon edge states we consider the spectral function $A_{\bf k}(\epsilon,t)$. For a non-equilibrium system the time-dependent spectral function is defined by $$\begin{aligned}
A_{\bf k}(\epsilon,t) = i \int \frac{d\tau}{2\pi} e^{i\epsilon\tau} [G_{\bf k}^> - G_{\bf k}^<](t+\frac{\tau}{2},t-\frac{\tau}{2}).\end{aligned}$$ The lesser Green’s function is proportional to the distribution function $f$ of the initial state, and therefore $G_{\bf k}^<(t,t') = 0$ since we start the time-evolution from the magnon ground state. The greater Green’s function is given by $$\begin{aligned}
G_{\bf k}^>(t,t') &= -i \sum_{s} \operatorname{Tr}\left( |s{\bf k} (t) \rangle \langle s{\bf k}(t')| \right),\end{aligned}$$ where $|s{\bf k}(t)\rangle = U(t)|s{\bf k}\rangle$ are the time-evolved single-magnon eigenstates $|s{\bf k}\rangle$ of the equilibrium Hamiltonian, $U(t) = \mathcal{T} \{ e^{-i\int_0^t d\tau H(\tau)} \}$ is the time-ordered evolution operator, and the trace is over all single-magnon states. Since $G_{\bf k}$ is diagonal in ${\bf k}$, we can calculate the spectral function by separately time-propagating the states $|s{\bf k}\rangle$ for each ${\bf k}$. In equilibrium $|s{\bf k}(t)\rangle = e^{-i\epsilon_{s\bf k} t}|s{\bf k}\rangle$, and we find the Floquet spectral function $$\begin{aligned}
A_{\bf k}^F(\epsilon) = 2\pi \sum_s \delta (\epsilon - \epsilon_{s\bf k}).\end{aligned}$$
In Fig. \[fig:spectral\] we compare the Floquet spectral function $A_{\bf k}^F(\epsilon)$ and the non-equilibrium spectral function $A_{\bf k}(\epsilon,t)$ for a ribbon with $n_y = 20$. We find a very good agreement between the Floquet and non-equilibrium spectral functions, indicating that for the given parameters the static Floquet Hamiltonian provides a good description of the non-equilibrium magnon dynamics.
[**Suggested experiments.–**]{} We end the paper with a discussion of possible experiments that would support the presence of a topological magnon band structure in a driven system. We note that so far, there has been no experiments that address topological magnons in a non-equilibrium setting. However, in equilibrium studies on ferromagnetic bulk CrI$_3$ and antiferromagnetic Cu$_3$TeO$_6$ have found a magnon band structure consistent with a non-trivial topology attributed to either next-nearest neighbor Dzyaloshinskii-Moriya interactions [@Chen18], nearest neighbor Kitaev interactions [@Lee20], or the lattice structure [@Yao18].
As shown above, a non-zero scalar spin chirality leads to a topological gap, and can generally be probed by Faraday or Kerr rotation measurements [@Kitamura17]. The associated gap opening at the ${\bf K}$-point in the magnon dispersion will affect the two-magnon optical excitation spectra, as probed by THz spectroscopy [@Richards67] or Raman and Brillouin spectroscopy [@Shen66]. However, the details of the optical spectra in these types of experiments would require dedicated calculations. The magnon edge states could potentially be probed directly using non-local magnon transport techniques, where magnons can be (detected) injected via the (inverse) spin Hall effect in platinum strips [@Cornelissen15]. Finally, resonant inelastic X-ray scattering can be used to probe the magnon dispersion [@Ament11].
In optical lattices, a spectroscopic probe could be implemented using oscillating magnetic field gradients. In addition, static gradients can be used to imprint magnons with specific wavenumbers. Their subsequent dynamics gives access to the magnon dispersion and can be probed using spin- and site-resolved detection [@Hild14].
[**Conclusions.–**]{} To summarize, we have demonstrated that non-equilibrium driving based on periodic laser fields coupling to charge degrees of freedom can induce topological magnon edge states in the spin sector of prototypical two-dimensional quantum magnets. Specifically, for the recently discovered monolayer van der Waals magnet CrI$_3$, we predict that a scalar spin chirality term can be induced leading to a sizeable magnon bandgap under realistic driving conditions. This opens the door for potential all-optical topological spintronics applications.
However, an important open problem for future studies is the question of how magnon edge states can be populated in a controlled fashion. Here we note that the situation is different compared to optical engineering of electronic systems, where the generation of dressed Floquet bands, their population, as well as the associated material heating, are intimately linked [@Sato19]. In the present work the separation between photon and magnon energy scales means that driving does not automatically populate the magnon bands, and as discussed above heating is largely avoided by adopting a sub-gap driving protocol. Populating the magnon states thus becomes a separate issue to be dealt with in addition to the generation of the non-trivial Floquet bands. We note that population by direct optical pumping have been discussed and experimentally verified for chiral edge states in topological exciton-polariton systems [@karzig_topological_2015; @nalitov_polariton_2015; @yuen-zhou_plexciton_2016; @klembt_exciton-polariton_2018; @hofmann_resonant_2020]. However, direct optical population of chiral magnon edge states through dipolar excitation is usually not possible and one should rather explore indirect mechanisms, for instance through two-magnon Raman scattering.
As an alternative route towards engineering topological magnon edge states with light, non-classical photon fields in cavities can be employed to control magnetic exchange interactions [@kiffner_manipulating_2019; @sentef_quantum_2020] and induce nontrivial topology with chiral light modes [@wang_cavity_2019; @Hubener20]. We also envisage the possibility to combine optical engineering with the control offered by bilayer Moiré systems [@Kennes20] to induce and control topological magnons.
*Acknowledgments.-* We acknowledge inspiring discussions with Abhisek Kole, Jin Zhang, Lede Xian and Claudio Verdozzi. We acknowledge support by the Max Planck Institute - New York City Center for Non-Equilibrium Quantum Phenomena. MAS acknowledges support by the DFG through the Emmy Noether programme (SE 2558/2-1). This work was supported by the European Research Council (ERC-2015-AdG694097), the Cluster of Excellence “Advanced Imaging of Matter” (AIM), and Grupos Consolidados (IT1249-19). The Flatiron Institute is a Division of the Simons Foundation.
[^1]: The Heisenberg term arises by writing the exchange interaction as $c_{i\sigma}^\dagger c_{i\sigma} c_{j\sigma'}^\dagger c_{j\sigma} = \frac{1}{2}n_i n_j + 2\hat{\bf S}_i \cdot \hat{\bf S}_j$, and absorbing the first term into the direct interaction by a renormalization of $V$.
[^2]: We have calculated the value of $U$ by DFT+U simulations of monolayer CrI$_3$ using the Octopus TD-DFT code. The values of $t$ and $J_D$ were then chosen by comparison to the equilibrium values of $J$ and $J'$ reported in Ref. [@Chen18].
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---
abstract: 'We discuss predictions for the total inelastic $\gamma \gamma$ cross-section and their model dependence on the input parameters. We compare results from a simple extension of the Regge Pomeron exchange model as well as predictions from the eikonalized mini-jet model with recent LEP data.'
author:
- |
A. CORSETTI\
Physics Department, Northeastern University, Boston, USA\
R. M. GODBOLE\
Center for Theoretical Studies, Indian Institute of Science, Bangalore, India\
and\
G. PANCHERI\
INFN - Laboratori Nazionali di Frascati, Frascati, Italy
title: 'PHOTON-PHOTON TOTAL INELASTIC CROSS-SECTION[^1]'
---
It is by now established that all total cross-sections, including photoproduction, rise as the c.m. energy of the colliding particles increases. So far a successful description of total cross-sections is obtained in the Regge/Pomeron exchange model [@DL], in which a Regge pole and a Pomeron are exchanged and total cross-sections are seen to first decrease and subsequently rise according to the expression
$$\sigma^{tot}_{ab}=Y_{ab} s^{-\eta}+X_{ab}s^{\epsilon}$$ where $\epsilon $ and $\eta$ are related to the intercept at zero of the leading Regge trajectory and of the Pomeron, respectively $\eta\approx 0.5$ and $\epsilon \approx 0.08$. This parametrization applies successfully [@DL] to photoproduction, as shown in Fig. 1, and to the lower energy data on $\gamma \gamma$[@SJOS]. Assuming the hypothesis of factorization at the poles, one can make a prediction for $\gamma \gamma$ total inelastic cross-section, using $$Y_{ab}^2=Y_{aa} Y_{bb}\ \ \ \ \ \ X_{ab}^2=X_{aa} X_{bb}$$ and extracting the coefficients X and Y from those for the fit to photo-production and hadron-hadron data. In particular, using for $\eta$ and $\epsilon$ the average values from the Particle Data Group compilation [@PDG] and averaging among the $pp$ and $\bar p p$ coefficients, one can have a first check of the factorization hypothesis. Noticing that the coefficient Y from photoproduction data has a large error and that prediction from the Regge/Pomeron exchange model refer to total cross-sections rather than the inelastic ones, these predictions can be enlarged into a band as shown in Fig.2.
An alternative model for the rise of all total cross-sections, relies on hard parton-parton scattering. It was suggested [@CLINE] that hard collisions between elementary constituents of the colliding hadrons, the partons, could be responsible for this rise which starts around $\sqrt{s} \ge 10\div 20 \ GeV$. This suggestion has subsequently evolved into mini-jet models [@minijet], whose eikonal formulation satisfies unitarity while embodying the concepts of rising total cross-sections with rising jet cross-sections. For processes involving photons, the model has to incorporate [@ladinsky] the hadronisation probability $P_\gamma^{had}$ for the photon to fluctuate itself into a hadronic state. The eikonalised mini–jet cross-section is then $$\label{eikonal}
\sigma^{inel}_{ab} = P^{had}_{ab}\int d^2\vec{b}[1-e^{n(b,s)}]$$ with the average number of collisions at a given impact parameter $\vec{b}$ given by $$\label{av_n}
n(b,s)=A_{ab} (b) (\sigma^{soft}_{ab} + {{1}\over{P^{had}_{ab}}}
\sigma^{jet}_{ab})$$ In eqs.(\[eikonal\], \[av\_n\]), $P^{had}_{ab}$ is the probability that the colliding particles $a,b$ are both in a hadronic state, $A_{ab} (b)$ describes the transverse overlap of the partons in the two projectiles normalised to 1, $\sigma^{soft}_{ab}$ is the non-perturbative part of the cross-section from which the factor of $P_{ab}^{had} $ has already been factored out and $\sigma^{jet}_{ab} $ is the hard part of the cross–section. The basic statement of the mini-jet model for total cross-sections is that the rise in $\sigma^{jet}_{ab} $ drives the rise of $\sigma_{ab}^{inel}$ with energy. Letting $$\label{phad}
P_{\gamma p}^{had} = P_{\gamma}^{had} \ \ \ \ \ and \ \ \
P_{\gamma \gamma}^{had} \approx (P_{\gamma}^{had})^2$$ one can extrapolate the model from photoproduction to photon-photon collisions. The issue of total $\gamma \gamma$ cross-sections assumes an additional significance in view of the large potential backgrounds that Beamstrahlung photons could cause at future Linear Colliders [@messy]. Because the hadronic structure of the photon involves both a perturbative and nonperturbative part, it has been proposed [@SJOS; @SARC] to use a sum of eikonalized functions instead of eq.(\[eikonal\]) in processes involving photons.
The predictions of the eikonalised mini-jet model for photon induced processes thus depend on 1) the assumption of one or more eikonals 2) the hard jet cross-section $\sigma_{jet}=\int_{p_{tmin}} {{d^2\hat{\sigma}}\over{dp_t^2}} dp_t^2$ which in turn depends on the minimum $p_t$ above which one can expect perturbative QCD to hold viz. $ p_{tmin}$ and the parton densities in the colliding particles $a$ and $b$, 3) the soft cross–section $\sigma^{soft}_{ab}$ 4) the overlap function $ A_{ab}(b) $, defined as $$\label{aob}
A(b)={{1}\over{(2\pi)^2}}\int d^2\vec{q}{\cal F}_1(q) {\cal F}_2(q)
e^{i\vec{q}\cdot \vec{b}}$$ where ${\cal F}$ is the Fourier transform of the b-distribution of partons in the colliding particles and 5) last, but not the least, $P_{ab}^{had}$.
In this note we shall restrict ourselves to a single eikonal. The hard jet cross-sections are calculated in LO perturbative QCD and use photonic parton densities GRV [@GRV] calculated to the leading order. We determine $\sigma_{\gamma \gamma}^{soft}$ from $\sigma_{\gamma p}^{soft}$ which in turn is determined by a fit to the photoproduction data. From inspection of the photoproduction data, one can assume that $\sigma_{soft}$ should contain both a constant and an energy decreasing term. Following the suggestion[@SARC] $$\label{soft}
\sigma^{soft}_{\gamma p} =\sigma^0 +
{{A}\over{\sqrt{s}}}+{{B}\over{s}}$$ we then calculate values for $\sigma^0, A$ and $B$ from a best fit [@thesis] to the low energy photoproduction data, starting with the Quark Parton Model ansatz $\sigma^0_{\gamma p}\approx {{2}\over{3}}\sigma^0_{pp}$. For $\gamma \gamma$ collisions, we repeat the QPM suggestion and propose $$\sigma^{soft}_{\gamma \gamma}={{2}\over{3}} \sigma^{soft}_{\gamma p},\ i.e.\
\sigma^0_{\gamma\gamma}=20.8 mb,A_{\gamma \gamma}=6.7\ mb\ GeV^{3/2},
B_{\gamma \gamma}= 25.3 \ mb\ GeV$$ Whereas the effect of the uncertainties in the above three quantities on the predictions of the inelastic photoproduction and $\gamma \gamma$ cross-sections has been studied in literature to a fair extent [@SJOS; @SARC; @FS] the effect of the other two has not been much discussed. In the original use of the eikonal model, the overlap function $A_{ab} (b)$ of eq.(\[aob\]) is obtained using for ${\cal F}$ the electromagnetic form factors. For protons this is given by the dipole expression $$\label{dipole}
{\cal F}_{prot}(q)=[{{\nu^2}\over{q^2+\nu^2}}]^2$$ with $\nu^2=0.71\ GeV^2$. For photons a number of authors [@SARC; @FLETCHER], on the basis of Vector Meson Dominance, have assumed the same functional form as for pion, i.e. the pole expression $$\label{pole}
{\cal F}_{pion}(q)={{k_0^2}\over{q^2+k_0^2}}\ \ \ with \ \ \ k_0=0.735\ GeV.$$ There also exists another possibility, i.e. that the b-space distribution of partons is the Fourier transform of their intrinsic transverse momentum distributions [@BN]. While for the proton this would correspond to use a Gaussian distribution instead of the dipole expression, eq.(\[dipole\]), for the photon one can argue that the intrinsic transverse momentum ansatz [@rohini] would imply the use of a different value of the parameter $k_o$[@ZEUS] in the pole expression for the form factor. By varying $k_o$ one can then explore both the intrinsic transverse distribution case and the form factor cum VMD hypothesis. Notice that the region most important to this calculation is for large values of the parameter b, where the overlap function changes trend, and is larger for smaller $k_o$ values.
Let us now look at $P^{had}_\gamma$. This is clearly expected to be ${\cal O} (\alpha_{em})$. Based on Vector Meson Dominance one expects, $$\label{PVMD}
P_\gamma^{had} = P_{VMD}=\sum_{V=\rho,\omega,\phi} {{4\pi \alpha}
\over{f^2_V}}= {{1}\over{250}}$$ Although in principle, $P^{had}_\gamma$ is not a constant, for simplicity, we adopt here a fixed value[@FLETCHER] of 1/204, which includes a non-VMD contribution of $\approx 20\%$. Notice that a fixed value of $P_{had}$ can be absorbed into a redefinition of the parameter $k_o$ through a simple change of variables [@review].
Having thus established the range of variability of the quantities involved in the calculation of total inelastic photonic cross sections, we can proceed to compare the predictions of the eikonalized minijet model with data. We use GRV (LO) densities and show the mini-jet result in Fig.1, using the form factor model for A(b), i.e. eq.(\[aob\]) with $k_o=0.735\ GeV$. In the figures, we have not added the direct contribution, which will slightly increase the cross-section in the 10 GeV region. We observe that it is possible to include the high energy points using GRV densities and $p_{tmin}=2\ GeV$, but the low energy region would be better described by a smaller $p_{tmin}$. This is the region where the rise, according to some authors, notably within the framework of the Dual Parton Model, is attributed to the so-called [*soft Pomeron*]{}.
We now apply the same criteria and parameter set used in $\gamma p$ collisions to the case of photon-photon collisions, i.e. $P_{h/\gamma}=1/204$, $p_{tmin}=2\ GeV$ and A(b) from eq.(\[aob\]). A comparison with $\gamma \gamma$ data shows that although the value $k_o=0.735$, corresponding to the pion-factor, is compatible with the low energy data up to $10\ GeV$ [@desy96] within the limits established by the large errors involved, at higher energies [@thisconf] the best fit is obtained using a slightly larger value, i.e. $k_0=1\ GeV$, and this is the one used in Fig.2. For comparison, we have also added mini-jet model predictions with SAS1 photon densities [@SAS1]and predictions (Pomeron/SaS) based on a Pomeron/Regge type parametrization[@SJOS].
[99]{} A. Donnachie and P.V. Landshoff, Phys. Lett. B296 (1992) 227. \[DL\] G. Schuler and T. Sjöstrand, Phys. Lett. [**B 300**]{} (1993) 169, Nucl. Phys. [**B 407**]{} (1993) 539, CERN-TH/95-62. \[SJOS\] Particle Data Group, Physical Review D54 (1996) 191. \[PDG\] D.Cline, F.Halzen and J. Luthe, Phys. Rev. Lett. [**31**]{} (1973) 491. \[CLINE\] A. Capella and J. Tran Thanh Van, Z. Phys. [**C23**]{} (1984)168, T.Gaisser and F.Halzen, Phys. Rev. Lett. [**54**]{} (1985) 1754, G.Pancheri and Y.N.Srivastava, Phys. Letters [**B182**]{} (1985), P. l‘Heureux, B. Margolis and P. Valin, Phys. Rev. [**D 32**]{} (1985) 1681, L. Durand and H. Pi, Phys. Rev. Lett. [**58**]{} (1987) 58. \[minijets\] J.C. Collins and G.A. Ladinsky, Phys. Rev. [**D 43**]{} (1991) 2847. \[ladinsky\] M. Drees and R.M. Godbole, Phys. Rev. Lett. [**67**]{} (1991) 1189, P. Chen, T.L. Barklow and M. E. Peskin, Phys. Rev. [**D 49**]{} (1994) 3209. \[messy\]
K. Honjo, L. Durand, R. Gandhi, H. Pi and I. Sarcevic, Phys. Rev. [**D 48**]{} (1993) 1048. \[SARC\] M. Glück, E. Reya and A. Vogt, Phys. Rev. [**D 46**]{} (1992) 1973. \[GRV\] A. Corsetti, September 1994 Laurea Thesis, University of Rome La Sapienza. \[thesis\] J.R. Foreshaw and J.K. Storrow, Phys. Lett. [**B 278**]{} (1992) 193; Phys. Rev. [**D46**]{} (1992) 3279. \[FS\] R.S. Fletcher , T.K. Gaisser and F.Halzen, Phys. Rev. [**D 45**]{} (1992) 377; erratum Phys. Rev. [**D 45**]{} (1992) 3279. \[FLETCHER\] A. Corsetti, Grau, G. Pancheri and Y.N. Srivastava, [**PLB 382**]{} (1996) 282. \[BN\] J. Field, E. Pietarinen and K. Kajantie, Nucl. Phys. [**B 171**]{} (1980) 377; M. Drees, [*Proceedings of 23rd International Symposium on Multiparticle Dynamics*]{}, Aspen, Colo., Sep. 1993. Eds. M.M. Block and A.R. White. M. Derrick et al., ZEUS collaboration, PLB 354 (1995) 163. \[ZEUS\] M. Drees, Univ. Wisconsin report [**MAD/PH-95-867**]{},[*Proceedings of the 4th workshop on TRISTAN physics at High Luminosities*]{}, KEK, Tsukuba, Japan, Nov. 1994; M. Drees and R. Godbole, J. Phys. G 21 (1995) 1559. \[review\] A. Corsetti, R. Godbole and G. Pancheri, in Proceedings of the Workshop on $e^+e^- $ Collisions at TeV Energies, Annecy, Gran Sasso, Hamburg, DESY 96-123D, June 1996, page. 495. \[desy96\] W. van Rossum, these propceedings. \[thisconf\] G. Shuler and T. Sjostrand, Zeit Phys. C68 (1995) 607: Phys., Lett. B376 (1996) 193.
[^1]: Talk presented at Photon’97, Egmond aan Zee, May 1997
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---
abstract: 'Non-radial pulsations (NRPs) are a proposed mechanism for the formation of decretion disks around Be stars and are important tools to study the internal structure of stars. NGC 3766 has an unusually large fraction of transient Be stars, so it is an excellent location to study the formation mechanism of Be star disks. High resolution spectroscopy can reveal line profile variations from NRPs, allowing measurements of both the degree, $l$, and azimuthal order, $m$. However, spectroscopic studies require large amounts of time with large telescopes to achieve the necessary high S/N and time domain coverage. On the other hand, multi-color photometry can be performed more easily with small telescopes to measure $l$ only. Here, we present representative light curves of Be stars and non-emitting B stars in NGC 3766 from the CTIO 0.9m telescope in an effort to study NRPs in this cluster.'
date: 'xxxx ?? and in revised form ??'
title: 'Non-radial Pulsations in the Open Cluster NGC 3766'
---
Introduction
============
Be stars are a class of non-supergiant B-type stars with Balmer and other line emission features due to an equatorial decretion disk. The disk is likely the result of a combination of the star’s rapid rotation (near the critical limit) and non-radial pulsations (NRPs; Porter & Rivinius 2003).
NRPs are spherical harmonic waves traversing the surface of a star. These pulsations can be found in multiple frequencies on the surface simultaneously (Rivinius, Baade, & Stefl 2003). There are two primary classes of NRP modes: $g$- and $p$-modes. $g$-modes are described by a low frequency pulsation that has gravity as its restoring force. The dominant oscillation in this mode is transverse across the surface. $p$-modes are dominated by high frequency, radial oscillations with a pressure restoring force (De Ridder 2001). These modes in main-sequence, pulsating B stars are driven by the $\kappa$ mechanism (Gutiérrez-Soto et al. 2007).
Temperature and flux gradients are established between the dimmer, cooler material on the peaks of the pulsations and the brighter, warmer material in the troughs. The flux variations over the stellar surface are then observed as either ripples within photospheric absorption line profiles or as periodic variations in magnitude. A large, high-resolution spectroscopic study would reveal both the degree, $l$, and the azimuthal order, $m$, but such studies are challenging due to the need for large amounts of time on a large telescope. Photometry, which is easily performed with data gathered by small telescopes, only measures $l$ (Rivinius, Baade, & Stefl 2003).
McSwain et al. (2008) previously showed that NGC 3766, an open cluster in Centaurus, is rich with transient Be stars. In an effort to detect NRPs and study the formation of these transient disks, we are currently performing a long-term photometric study of the cluster. Here we present preliminary differential light curves that reveal magnitude variations of several Be stars that are consistent with NRPs.
Observations and Data Analysis
==============================
We observed the cluster NGC 3766 using the CTIO 0.9m telescope and SITe 2048 CCD from 2008 March 19–24. The CCD was used in the quad readout mode without binning. We used Strömgren $uvby$ filters and exposure times of $120$, $20$, $10$, and $10$ s, respectively. No additional standard stars were measured. Sky flats were used to calibrate the $u$ and $v$ filters, and dome flats were used for the $b$ and $y$ filters.
The Strömgren $uvby$ photometric data were zero-corrected and flat field-corrected using standard routines in IRAF using the $quadred$ package. The $daofind$ and $phot$ procedures were used to automatically identify the stars and perform aperture photometry. We used the numbering scheme established by Ahmed (1962) and found in the WEBDA database to identify the stars. Differential magnitudes were determined for the variable Be stars identified in Balona & Engelbrecht (1986) and McSwain et al. (2008) and four check stars, Nos. 16, 95, 111, and 147. The check stars have constant differential magnitudes within the measured errors, shown in Figure \[WEBDA01lc\] (left). Figures \[WEBDA01lc\] (right) and \[WEBDA63lc\] show differential light curves for the Be stars Nos. 1, 20, and 63 with variations in magnitude consistent with NRPs.
![On the left, Strömgren $uvby$ differential magnitudes are plotted for two check stars, Nos. 16 and 95, in NGC 3766. On the right, differential magnitudes for No. 1 in the same format. Representative error bars are shown in the upper right of each plot. []{data-label="WEBDA01lc"}](WEBDA1695zoomebnum.pdf "fig:") \[checklc\] ![On the left, Strömgren $uvby$ differential magnitudes are plotted for two check stars, Nos. 16 and 95, in NGC 3766. On the right, differential magnitudes for No. 1 in the same format. Representative error bars are shown in the upper right of each plot. []{data-label="WEBDA01lc"}](WEBDA0195zoomebnum.pdf "fig:")
![Differential photometry for Nos. 20 and 63 in the same format as Figure \[WEBDA01lc\].[]{data-label="WEBDA63lc"}](WEBDA2095zoomebnum.pdf "fig:") \[WEBDA20lc\] ![Differential photometry for Nos. 20 and 63 in the same format as Figure \[WEBDA01lc\].[]{data-label="WEBDA63lc"}](WEBDA6395zoomebnum.pdf "fig:")
Results and Further Work
========================
Nineteen of the 25 Be stars measured at this preliminary stage have magnitude variations consistent with NRPs with periods on the order of several hours. For example, the $u$-band light curve of No. 20 was folded using a period of about $6.98$ hours, shown in Figure \[foldedlc\].
![The folded $u$-band light curve for No. 20 using a period of $6.98$ hours.[]{data-label="foldedlc"}](WEBDA20periodbtnum.pdf)
![A pole-on and an inclined model of a sphere distorted by non-radial pulsations, with $l = m = 8$. This is a highly exaggerated model of both the amplitude and high order of pulsations for display purposes.[]{data-label="shi"}](NRPlm8.pdf "fig:") \[shp\] ![A pole-on and an inclined model of a sphere distorted by non-radial pulsations, with $l = m = 8$. This is a highly exaggerated model of both the amplitude and high order of pulsations for display purposes.[]{data-label="shi"}](NRPlm8tilt.pdf "fig:")
Additional data collected at the CTIO 0.9m telescope in 2008 June, 2009 February, and 2009 May on NGC 3766 will be analyzed in conjunction with these data to perform a period search and identify the various periods that are present. In the future, we will construct a model for the non-radial pulsations that will allow theoretical light curves to be generated and compared to the observed light curves. Assuming slowly rotating stars, the non-radial pulsations can be modeled by spherical harmonic perturbations of a spherical stellar surface, given by Buta & Smith (1979) as $$\frac{\delta R}{R} \thicksim A^m_{\ell} Y^m_{\ell}(\theta , \phi ) e^{\imath \sigma^m_{\ell} t} ,$$ where $\delta R/R$ is the fractional change in radius, $A^m_{\ell}$ is the amplitude associated with spherical harmonic $Y^m_{\ell} (\theta, \phi )$, and $\sigma ^m_{\ell}$ is the frequency of the mode. Pole-on and inclined examples of spherical harmonics (with greatly exaggerated amplitudes) are presented in Figure \[shi\]. The most commonly observed mode of pulsation in Be stars is $l = | m | = 2$ (Rivinius, Baade, & Stefl 2003), but for illustrative purposes, $l = m = 8$ is presented.
Acknowledgements
================
We gratefully acknowledge travel support from the International Astronomical Union, the American Astronomical Society, and NASA DPR number NNX08AV70G. We also thank Charles Bailyn and the SMARTS Consortium for their help in scheduling these observations. We are also grateful for an institutional grant from Lehigh University and the U.S. Department of Education GAANN Fellowship.
1962, *Pub. Roy. Obs. Edinburgh*, 3, 57
1986, *MNRAS*, 219, 131
1979, *ApJ*, 232, 213
2001, Ph.D. thesis, *Katholieke Univ. Leuven*
2007, *A&A*, 476, 927
2008, *ApJ*, 672, 590
2003, *PASP*, 115, 1153
2003, *A&A*, 411, 229
|
---
abstract: 'This paper focuses on the challenging task of learning 3D object surface reconstructions from single RGB images. Existing methods achieve varying degrees of success by using different geometric representations. However, they all have their own drawbacks, and cannot well reconstruct those surfaces of complex topologies. To this end, we propose in this paper a skeleton-bridged, stage-wise learning approach to address the challenge. Our use of skeleton is due to its nice property of topology preservation, while being of lower complexity to learn. To learn skeleton from an input image, we design a deep architecture whose decoder is based on a novel design of parallel streams respectively for synthesis of curve- and surface-like skeleton points. We use different shape representations of point cloud, volume, and mesh in our stage-wise learning, in order to take their respective advantages. We also propose multi-stage use of the input image to correct prediction errors that are possibly accumulated in each stage. We conduct intensive experiments to investigate the efficacy of our proposed approach. Qualitative and quantitative results on representative object categories of both simple and complex topologies demonstrate the superiority of our approach over existing ones. We will make our ShapeNet-Skeleton dataset publicly available.'
author:
- 'Jiapeng Tang $^{*}$'
- 'Xiaoguang Han [^1]'
- Junyi Pan
- 'Kui Jia [^2]'
- Xin Tong
bibliography:
- 'egbib.bib'
title: |
A Skeleton-bridged Deep Learning Approach for Generating Meshes\
of Complex Topologies from Single RGB Images
---
Introduction {#SecIntro}
============
![Our proposed approach can generate a closed surface mesh from a single view RGB image, by correctly recovering the complex topology.[]{data-label="fig:show"}](show.pdf){width="0.8\linewidth"}
Learning 3D surface reconstructions of objects from single RGB images is an important topic from both the academic and practical perspectives. It plays fundamental roles in applications such as augmented reality and image editing. It also connects with the more traditional research of 3D vision [@hartley2000multiple; @fuentes-pacheco2015visual; @häming2010structure]. This inverse problem is extremely challenging due to the arbitrary shapes of different object instances and their possibly complex topologies. Recent methods [@choy20163d; @girdhar2016learning; @tatarchenko2017octree; @wu2016learning; @groueix2018atlasnet; @kato2018neural; @sinha2017surfnet; @fan2017point; @lin2017learning; @wang2018pixel2mesh; @pan2018residual] leverage the powerful learning capacities of deep networks, and achieve varying degrees of success by using different shape representations, e.g., volume, point cloud, or mesh. These methods have their own merits but also have their respective drawbacks. For example, volume-based methods [@choy20163d; @girdhar2016learning; @wu2016learning] exploit the establishment of Convolutional Neural Networks (CNNs) [@simonyan2015very; @krizhevsky2012imagenet; @szegedy2015going; @he2016deep], and simply extend CNNs its 3D versions to generate volume representations of 3D shapes; however, both of their computational and memory complexities are high enough which prohibit them to be deployed to generate high-resolution outputs. On the other hand, point cloud based methods [@fan2017point; @lin2017learning] are by nature difficult to generate smooth and clean surfaces.
Given the fact that mesh representation is a more efficient, discrete approximation of the continuous manifold of an object surface, a few recent methods [@fan2017point; @lin2017learning] attempt to directly learn mesh reconstructions from single input images. These methods are inherently of mesh deformation, since they assume that an initial meshing over point cloud is available; for example, they typically assume unit square/sphere as the initial mesh. In spite of the success achieved by these recent methods, they still suffer from generating surface meshes of complex topologies, e.g., those with thin structures as shown in Fig \[fig:show\].
To this end, we propose in this paper a *skeleton-bridged, stage-wise* deep learning approach for generating mesh reconstructions of object surfaces from single RGB images. We particularly focus on those object surfaces with complex topologies, e.g., chairs or tables that have local, long and thin structures. Our choice of the *meso-skeleton* [^3] is due to its nice property of topology preservation, while being of lower complexity to learn when compared with learning the surface meshes directly. Our proposed approach is composed of three stages. The first stage learns to generate skeleton points from the input image, for which we design a deep architecture whose decoder is based on a novel, parallel design of CurSkeNet and SurSkeNet, which are respectively responsible for the synthesis of curve- and surface-like skeleton points. To train CurSkeNet and SurSkeNet, we compute skeletal shape representations for instances of ShapeNet [@chang2015shapenet]. *We will make our ShapeNet-Skeleton dataset publicly available.* In the second stage, we produce a base mesh by firstly converting the obtained skeleton to its coarse volumetric representation, and then refining the coarse volume using a learned 3D CNN, where we adopt a strategy of independent sub-volume synthesis with regularization of global structure, in order to reduce the complexity of producing high-resolution volumes. In the last stage, we generate our final mesh result by extracting a base mesh from the obtained volume [@lorensen1987marching], and deforming vertices of the base mesh using a learned Graph CNN (GCNN) [@kipf2017semi; @defferrard2016convolutional; @boscaini2016learning; @scarselli2009graph]. Learning and inference in three stages of our approach are based on different shape representations, which take the respective advantages of point cloud, volume, and mesh. We also propose multi-stage use of the input image to correct prediction errors that are possibly accumulated in each stage. We conduct intensive ablation studies which show the efficacy of stage-wise designs of our proposed approach.
We summarize our main contributions as follows.
- Our approach is based on an integrated stage-wise learning, where learning and inference in different stages are based on different shape representations by taking the respective advantages of point cloud, volume, and mesh. We also propose multi-stage use of the input image to correct prediction errors that are possibly accumulated in each stage.
- We propose in this paper a skeleton-bridged approach for learning object surface meshes of complex topologies from single RGB images. Our use of skeleton is due to its nice property of topology preservation, while being of lower complexity to learn. We design a deep architecture for skeleton learning, whose decoder is based on a novel design of parallel streams respectively for the synthesis of curve- and surface-like skeleton points. To train the network, we prepare ShapeNet-Skeleton dataset and will make it publicly available.
- We conduct intensive ablation studies to investigate the efficacy of our proposed approach. Qualitative and quantitative results on representative object categories of both simple and complex topologies demonstrate the superiority of our approach over existing ones, especially for those objects with local thin structures.
Related Works
=============
In this section, we only focus on the related works about deepnets-based algorithms for fully object reconstruction. The literature reviews are studied in the following three aspects.
**Volume-based Generator** Voxels, extended from pixels, are usually used in the form of binary values or signed distances to represent a 3D shape. Because of its regularity, most of existing deepnets-based shape analysis [@wu20153d; @brock2016generative] or shape generation [@choy20163d; @girdhar2016learning; @wu2016learning] methods adopt it as the primary representation. For example, the work of [@choy20163d] combines 3D convolutions with long short-term memory (LSTM) units to achieve volumetric grid reconstruction from single-view or multi-view RGB images. A 3D auto-encoder is trained in [@girdhar2016learning], whose decoder part is used to construct the mapping from a 2D image to a 3D occupancy grid. These methods tend to predict a low-resolution volumetric grid due to the high computational cost of 3D convolution operators. Based on the observation that only a small portion of regions around the boundary surface contain the shape information, the Octree representation has been adopted in recent shape analysis works [@riegler2017octnet; @wang2017cnn]. A convolutional Octree decoder is also designed in [@tatarchenko2017octree] to support high-resolution reconstruction with a limited time and memory cost. In our work, we aim to generate the surface mesh of the object instead of its solid volume. As its efficiency and topology-insensitivity, we also leverage volumetric-based generator to convert the inferred skeletal point cloud to a solid volume, effectively bridging the gap between the skeleton and the surface mesh.
**Surface-based Generator** Point cloud, sampled from the object’ surface and formed by a set of points, is one of the most popular representations of 3D shapes. Fan et al. [@fan2017point] proposes the first point could generation neural network, which is built upon a deep regression model trained with the loss functions that evaluate the similarity of two unordered point set, such as chamfer distance. Although the rough shape can be captured, the generated points are placed sparse and scattered. Multi-view depth maps are used as the output representation in [@lin2017learning], which are generated with image generative models and then fused to give rise to a dense point cloud. Nevertheless, the predicted points are still of low accuracy. Mesh, as the most natural discretization of a manifold surface, has been widely used in many graphics applications. Due to its irregular structure, CNN is difficult to be directly applied to mesh generation. To alleviate this challenge, the methods of [@kato2018neural; @wang2018pixel2mesh] take an extra template mesh as input and attempt to learn the deformations to approximate the target surfaces. Limited to the requirement of an initial mesh, they cannot deal with topology-free reconstruction. Another recent method, called Atlasnet [@groueix2018atlasnet], proposes to deform multiple 2D planar patches to cover the surface of the object. Residual prediction and progressive deformation are adopted in [@pan2018residual], which decrease the complexity of learning and make more details added. It is free of complex topology yet causes severe patch overlaps and holes. In our work, we aim not only to generate a clean mesh but also to capture the correct topology. To do so, we firstly borrow the idea in [@groueix2018atlasnet] to infer the meso-skeleton points, which are then converted to a base mesh. Finally, the method of [@wang2018pixel2mesh] is further adopted for generating geometric details.
**Structure Inference** Instead of estimating geometric shapes, many recent works attempt to recover the 3D structures of objects. From a single image, Zou et al. [@Zou_2017_ICCV] presents a primitive recurrent neural network to sequentially predict a set of cuboid primitives to approximate the shape structure. A recursive decoder is proposed in [@li_sig17] to generate shape parts and infer reasonable high-level structure information including part connectivity, adjacency and symmetry relation. This is further exploited in [@niu2018im2struct] for image-based structure inference. However, the cuboids are hard to be used for fitting curved shapes. In addition, these methods also require a large human-labeled dataset. We use meso-skeleton, a point cloud, to represent the shape structure which is easier to be obtained from the ground truth. The usage of parametric line and square elements also eases the approximation of the diverse local structures.
The Proposed Approach {#SecFramework}
=====================
We first overview our proposed skeleton-bridged approach for generating a surface mesh from an input RGB image, before explaining the details of stage-wise learning. Given an input image $I$ of an object, our goal is to recover a surface mesh $M$ that ideally captures the *possibly complex* topology of 3D shape of the object. This is an extremely challenging inverse task; existing methods [@kato2018neural; @wang2018pixel2mesh; @groueix2018atlasnet] may only achieve partial success for objects with relatively simple topologies. To address the challenge, our key idea in this work is to bridge the mesh generation of object surface via learning of meso-skeleton. As discussed in Section \[SecIntro\], the rationale is that skeletal shape representation preserves the main topological structure of a shape, while being of lower complexity to learn.
More specifically, our mesh generation process is composed of the following three stages. In the first stage, we learn an encoder-decoder architecture that maps $I$ to its meso-skeleton $K$, represented as a compact point cloud. In the second stage, we produce a volume $V$ from $K$ by firstly converting $K$ to its coarse volumetric representation $V_k$, and then refining $V_k$ using a learned 3D CNN (e.g., of the style [@han2017high]). In the last stage, we generate the final output mesh $M$ by extracting a base mesh $M_b$ from $V$, and further optimizing vertices of $M_b$ using a learned graph CNN [@scarselli2009graph]. Each stage owns its own image encoder, and thus inferences in all the three stages are guided by the input image $I$. Fig \[fig:overall\] illustrates the whole pipeline of our approach.
Learning of Meso-Skeleton {#SecSkeletonLearning}
-------------------------
As defined in Section \[SecIntro\], the meso-skeleton of a shape is represented as its medial axis, and the medial axis of a 3D model is made up of curve skeletons and median sheets, which are adaptively generated from local regions of the shape. In this work, we utilize the skeleton representation introduced in [@wu2015deep], i.e., a compact point cloud. Fig \[fig:com\_ske\] shows an example of skeleton that we aim to recover.
**The ShapeNet-Skeleton dataset** Training skeletons are necessary in order to learn to generate a skeleton from an input image. In this work, we prepare training data of skeleton for ShapeNet [@chang2015shapenet] as follows: 1) for each 3D polygonal model in ShapeNet, we convert it into a point cloud; 2) we extract meso-skeleton points using the method of [@wu2015deep]; 3) we classify each skeleton point as either curve-like or surface-like categories, based on principle component analysis of its neighbor points. *We will make our ShapeNet-Skeleton dataset publicly available.*
**CurSkeNet and SurSkeNet** Given the training skeleton points for the object in each image, we design an encoder-decoder architecture for skeleton learning, where the input $I$ is firstly encoded to a latent vector that is then decoded to a point cloud of skeleton. Our encoder is similar to those in existing methods of point set generation, such as [@fan2017point; @groueix2018atlasnet]. In this work, we use ResNet-18 [@he2016deep] as our image encoder. Our key contribution is a novel design of decoder architecture that will be presented shortly. We note that one may think of using existing methods [@fan2017point; @groueix2018atlasnet] to generate $K$ from $I$; however, they tend to fail due to the complex, especially thin, structures of skeletons, as shown in Fig \[fig:com\_ske\]. Our decoder is based on two parallel streams of CurSkeNet and SurSkeNet, which are designed to synthesize the points at curve-shaped and surface-shaped regions respectively. Both CurSkeNet and SurSkeNet are based on multilayer perceptrons (MLPs) with the same settings as in AtlasNet [@groueix2018atlasnet], including 4 fully-connected layers with the respective sizes of 1024, 512, 256, and 3, where the non-linear activation functions are ReLU for the first 3 layers and tanh for the last layer. Our SurSkeNet learns to deform a set of 2D primitives defined on the open unit square $[0,1]^2$, producing a local approximation of the desired sheet skeleton points. Our CurSkeNet learns to deform a set of 1D primitives defined on the open unit line $[0,1]$; it thus conducts affine transformations on them to form curves, and learns to assemble generated curves to approximate the curve-like skeleton part. In our current implementation, we use 20 line primitives in CurSkeNet and 20 square primitives in SurSkeNet. In Section \[SecCompSke\], we conduct ablation studies that verify the efficacy of our design of CurSkeNet and SurSkeNet.
**Network Training** We use training data of curve-like and surface-like skeleton points to train CurSkeNet and SurSkeNet. The learning task is essentially of point set generation. Similar to [@groueix2018atlasnet], we use the Chamfer Distance (CD) as one of our loss functions. The CD loss is defined as: $$\begin{aligned}
{\cal{L}}_{cd} = \sum_{x\in K} \min_{y\in K^{*}}\|x-y\|_2^2 + \sum_{y\in K^{*}}\min_{x\in K}\|x-y\|_2^2 ,
\end{aligned}$$ where $\{x \in K\}$ and $\{y \in K^{*}\}$ are respectively the sets of predicted and training points. Besides, to ensure local consistency, regularizer of Laplacian smoothness is also used for generation of both curve- and surface-like points. It is defined as: $$\begin{aligned}
{\cal{L}}_{lap} = \sum_{x \in K}\Big\|x -\frac{1}{|\mathcal{N}(x)|}\sum_{p\in \mathcal{N}(x)} p\Big\|_{2} ,
\end{aligned}$$ where $\mathcal{N}(x)$ is the neighbor of point $x$.
From Skeleton to Base Mesh {#SecBaseMesh}
--------------------------
We present in this section how to generate a base mesh $M_b$ from the obtained skeleton $K$. To do so, a straightforward approach is to coarsen $K$ to a volume directly with hand-crafted methods, and then to produce the base mesh using the method of Marching Cubes [@lorensen1987marching]. However, such an approach may accumulate stage-wise prediction errors. Instead, we rely on the original input $I$ to correct the possible stage-wise errors, by firstly converting $K$ to its volumetric representation $V_k$, and then using a trained 3D CNN for a finer and more accurate volumetric shape synthesis, resulting in a volume $V$. Base mesh $M_b$ can then be obtained by applying Marching Cubes to the finer $V$.
![The pipeline of our high-resolution skeletal volume synthesis method. We convert the inferred skeletal points $K$ to low-resolution volume $V_{64}$ and high-resolution volume $V_{128}$ in parallel. Given $V_{64}$, $V_{128}$ paired with the input image $I$, a global-guided sub-volume synthesis network is proposed to output a refined volume of $V_{128}$. It consists of two subnetworks: one network generates a coarse skeletal volume from $I$ and $V_{64}$ while the other enhances $V_{128}$ locally patch by patch under the guidance of the output from the first network.[]{data-label="fig:local_global_alg"}](local_global_alg_final.pdf)
**Sub-volume Synthesis with Global Guidance** To preserve the topology captured by $K$, a high-resolution volume representation is required. However, this is not easy to satisfy due to the expensive computational cost of 3D convolution operations. OctNet [@riegler2017octnet] may alleviate the computational burden, it is however complex and difficult to implement. We instead partition the volume space into overlapped sub-volumes, and conduct refinement on them in parallel. We also follow [@han2017high] and employ a global guidance to preserve spatial consistency across sub-volumes. More specifically, we firstly convert $K$ to two volumes of varying scales, denoted as $V^{l}_k$ and $V^{h}_k$. We set $| V^{l}_k | = 64^3$ and $| V^{h}_k | = 128^3$ in this work. We use two networks of 3D CNNs for global and local synthesis of skeletal volumes. The global network is trained to refine $V^{l}_k$ and generate a skeletal volume $V^{l}$ of the size $64^3$. The local network takes as inputs sub-volumes of the size $64^3$, which are uniformly cropped from $V^{h}_k$, and then conduct their refinement individually. Both of our global and local refinement networks are based on 3D U-Net architecture [@ronneberger2015u]. When refining each sub-volume of $V^{h}_k$, the corresponding $32^3$-sized sub-volume of $V^{l}$ is concatenated to provide structural regularization. The overall pipeline of our method is shown in Fig \[fig:local\_global\_alg\]. As seen in Fig \[fig:local\_global\_comp\], our method not only supports high-resolution synthesis but also preserves global structure.
![(a)Input images; (b)Inferred skeletal points; (c)sub-volume synthesis only; (d) adding global guidance; (e) adding image guidance.[]{data-label="fig:local_global_comp"}](local_global.pdf "fig:")\
**Image-guided Volume Correction** To correct the possiblely accumulated prediction errors from the stage of skeleton generation, we reuse the original input $I$ by learning an independent encoder-decoder network, which is trained to map $I$ to a $32^3$-sized volume. We use ResNet-18 as the encoder and several 3D de-convolution layers as the decoder. The output of the decoder is incorporated into the aforementioned global synthesis network, aiming for a more accurate $V^{l}$, which ultimately contributes to the generation of a better $V$. From the perspective of learning task for generating 3D volumes from single images [@choy20163d; @girdhar2016learning; @wu2016learning; @tatarchenko2017octree], our method is superior to existing ones by augmenting with an additional path of skeleton inference. As shown in Fig \[fig:local\_global\_comp\], our usage of $I$ for error correction greatly improves the synthesis results.
**Base Mesh extraction** Given $V$, we use Marching Cubes [@lorensen1987marching] to produce the base mesh $M_b$, which ideally preserves the same topology as that of the skeleton $K$. Because $V$ is in high resolution, $M_b$ would contain a large number of vertices and faces. To reduce the computational burden of the last stage, we apply QEM algorithm [@kobbelt1998general] on $M_b$ to get a simplified mesh for subsequent processing.
Mesh Refinement
---------------
![Our mesh refinement network. Given an image $I$ and an initial mesh $M_b$, we concatenate pixel-wise features of $I$ (extracted by VGG-16) to vertices’ coordinates and form vertex-wise features which are followed by a graph-CNN to generate the geometric details. []{data-label="fig:mesh_refine_alg"}](mesh_refine_alg_final.pdf){width="0.90\linewidth"}
We have up to now the base mesh $M_b$ that captures the topology of the underlying object surface, but may lack surface details. To compensate $M_b$ with surface details, we take the approach of mesh deformation using graph CNNs [@kipf2017semi; @defferrard2016convolutional; @boscaini2016learning; @scarselli2009graph].
{width="0.93\linewidth"}\
**Mesh Deformation using Graph CNNs** Take $M_b$ as the input, our graph CNN is simply composed of a few graph convolutional layers, each of which apply spatial filtering operation to local neighborhood associated with each vertex point of $M_b$. The graph-based covolutional layer is defined as: $$\begin{aligned}
h_{p}^{l+1} = w_{0}h_{p}^{l} + \sum_{q\in\mathcal{N}(p)}w_{1}h_{q}^{l} ,
\end{aligned}$$ where $h_{p}^{l}$, $h_{p}^{l+1}$ are the feature vectors on the vertex $p$ before and after applying a convolution operation, and $\mathcal{N}(p)$ is the neighbor of $p$. $w_{0}$ and $w_{1}$ are the learnable parameter matrices that are applied to all vertices.
Similar to [@wang2018pixel2mesh], we also concatenate pixel-wise VGG features extracted from $I$ with coordinates of the corresponding vertices to enhance learning. We again use CD loss to train our graph CNN. Several smoothness terms are also added to regularize the mesh deformation. One is edge regularization, used to avoid large deformations, by restricting the length of output edges. Another one is normal loss, used to guarantee the smoothness of the output surface. The geometric details commonly exist at the regions where the normals are changed obviously. Regarding this fact, to guide the GCNN to better learn the surface in those areas, we accordingly construct weighted loss functions. Fig \[fig:mesh\_refine\_alg\] shows the efficacy of this weighting strategy, where the sharp edges are better synthesized.
Experiments
===========
**Dataset** To support the training and testing of our proposed approach, we collect 17705 3D shapes from five categories in ShapeNet [@chang2015shapenet]: plane(1000), bench(1816), chair(5380), table(8509), firearm(1000). The dataset is split into two parts, $80\%$ shapes are used for training and the other for testing. We take as the inputs the rendered images provided by [@choy20163d], where each 3D model is rendered into 24 RGB images. Each shape in the dataset is converted to a point cloud ($10,000$ points are sampled on the surface) as the ground truth for mesh refinement network.
**Implementation details** The input images are all in the size of 224\*224. We train CurSkeNet and SurSkeNet using a batch size of 32 with a learning rate of 1e-3 (dropped to 3e-4 after 80 epochs) for 120 epochs. The skeletal volume refinement network is trained in three steps: 1) the global volume inference network is trained alone with learning rate 1e-4 for 50 epochs(dropped to 1e-5 after 35 epochs); 2) we train the sub-volume synthesis network with learning rate 1e-5 for 10 epochs; 3) the entire network is fine-tuned. The mesh refinement network is trained with learning rate 3e-5 for 50 epochs(dropped to 1e-5 after 20 epochs) using a batch size of 1.
[\*[11]{}[c]{}]{} \*[Category]{} & &\
(lr)[2-6]{}(lr)[7-11]{} & R2N2 & PSG & AtlasNet & Pixel2Mesh & Ours & R2N2 & PSG & AtlasNet & Pixel2Mesh & Ours\
plane & 10.434 & 3.824 & 1.529 & 1.890 & **1.364** & 11.060 & 13.945 & 8.981 & 7.728 & **6.026**\
bench & 10.511 & 3.504 & 2.264 & 1.774 & **1.639** & 10.555 & 8.053 & 9.143 & 7.083 & **6.059**\
chair & 4.723 & 2.553 & 1.342 & 1.923 & **1.002** & 7.762 & 10.222 & 7.866 & 8.312 & **5.484**\
firearm & 10.176 & **1.473** & 2.276 & 1.793 & 1.784 & 9.760 & 12.555 & 9.825 & 6.887 & **6.413**\
table & 12.230 & 5.466 & 1.751 & 2.109 & **1.321** & 11.160 & 9.561 & 9.053 & 7.442 & **5.688**\
mean & 9.615 & 3.364 & 1.832 & 1.898 & **1.422** & 10.059 & 10.867 & 8.974 & 7.490 & **5.934**\
Comparisons against State-of-the-Arts {#SecCompRecon}
-------------------------------------
We first evaluated our overall pipeline against existing methods on singe-view reconstruction. 3D-R2N2 [@choy20163d], PSG [@fan2017point], AtlasNet [@groueix2018atlasnet], Pixel2Mesh [@wang2018pixel2mesh] are chosen for their popularity: 3D-R2N2 is one of the most famous volumetric shape generators, PSG is the first point set generator based on a deep regression model, and both AtlasNet and Pixel2Mesh are current state-of-the-art mesh generator. For fair comparison, these models are retrained under our preprocessed dataset.
**Qualitative results** The visual comparisons are shown in Fig \[fig:comp\_recon\]. As seen, 3D-R2N2 always produces low-resolution volumes which cause broken structures. Their results show no surface details either. The point sets regressed by PSG are sparse and scattered, leading to the difficulty of extracting triangular meshes from them. AtlasNet is capable of generating mesh representations without a strong restriction on the shape’s topology. Yet, the outputs are of non-closed and suffer from surface self-penetration, which also gives rise to a challenge to convert it to a manifold mesh. Limited to the requirement of a genus-0 template mesh input, Pixel2Mesh is difficult to reach an accurate reconstruction for the objects with complex topologies, as the chairs shown. Our method shows great superiority than the others from the visual appearances, as it generates closed meshes with accurate topologies and more details. For the examples of firearm as shown, our approach also outperforms Pixel2Mesh, which in another aspect, indicates the proposed approach is also good at recovering the shapes with complex structures no mention to topology.
**Quantitative results** Similar to Pixel2Mesh [@wang2018pixel2mesh], we adopt Chamfer Distance(CD) and Earth Mover’s Distance(EMD) to evaluate the reconstruction quality. Both of them are calculated between the point set ($10,000$ points) sampled on the predicted mesh surface and the ground truth point cloud. The quantitative comparison results are reported in Tab \[Tab:cd\_emd\]. Notably, on both metrics, our approach outperforms all the other methods across almost all listed categories, especial on the models with complex topologies like chairs and tables.
**Generalization on real images** Fig \[fig:pix3d\] illustrates 3D shapes reconstructed by our method on three real photographs from Pix3D [@sun2018pix3d], where the chairs and tables in the images are manually segmented. The results’ quality is similar to the results obtained from synthetic images. As seen in Fig \[fig:pix3d\] (a), the real-world images has no relation with ShapeNet, while the chair rod can still be well reconstructed. This validates the generalization ability of our method.
{width="0.98\linewidth"}
\[fig:pix3d\]
Ablation Studies on Mesh Generation {#SecAblaMesh}
-----------------------------------
Our whole framework contains multiple stages. In this section, we conduct the ablation studies by alternatively removing one of them, to verify the necessity of each stage.
**w/o skeleton inference** Based on our pipeline, an alternative solution without using skeleton inference is firstly generating a volume directly from the image and then applying our mesh refinement model to output the final result. Then, we implement this approach by using OGN [@tatarchenko2017octree] as the image-based volume generator, for high-resolution($128^3$) reconstruction. This method is compared with ours visually in Fig \[fig:wo\_ske\]. As seen, the OGN-based mesh generation method fails to capture the thin structures which causes incorrect topologies. In contrast, our approach gives rise to much better performance.
{width="0.80\linewidth"}
![(a)Input images; (b)Final meshes whose base meshes are generated using OGN; (c)The generated meshes of our method; (d)Ground truth.[]{data-label="fig:wo_ske"}](wo_ske.pdf){width="0.8\linewidth"}
![(a)Input images; (b)Inferred skeleton points; (c)The sythesized meshes whose base meshes are extracted from the coarsened skeletal volume using the corrosion techinique; (d) The generated meshes of our method; (e)Ground truth.[]{data-label="fig:wo_voxel"}](wo_voxel2.pdf){width="0.85\linewidth"}
**w/o voxel-based correction** After inferring skeleton from our first stage, it is a straightforward approach to acquire a base mesh by directly applying the corrosion technique for volume generation, and the base mesh can be extracted. The visual comparisons of this method against ours are shown in Fig \[fig:wo\_voxel\]. It can be seen, without volume correction, the wrong predictions caused by skeleton inference will be transferred to the mesh refinement stage, affecting the final output. Our proposed voxel-based correction network addresses this issue effectively.
Evaluation on Skeleton Inference {#SecCompSke}
--------------------------------
In this section, we conduct comparisons with several variants of our skeleton inference approach, to verify our final model is the optimal choice. These variants include: “Point-only fitting” method directly adopts PSG [@fan2017point] to regress the skeletal points; “Line-only fitting” method removes the square stream of our model and only deforms multiple lines to approximate the skeleton; “Square-only fitting” removes the line stream of our model and deforms multiple squares to fit the skeleton; “Line-and-Square fitting” method learns the deformation of multiple lines and squares together using a single MLP to approximate the skeleton; “Ours w/o laplacian” stands for our model without laplacian smoothness term. Note that, laplacian smoothness loss is also used for the training of “Line-only fitting”, “Square-only fitting” and “Line-and-Square fitting”.
Methods CD
------------------------- -----------
Point-only fitting 1.185
Line-only fitting 1.649
Square-only fitting 1.185
Line-and-Square fitting 1.252
Ours w/o laplacian 1.621
Ours **1.103**
: The quantitative comparisons on the variants of our skeleton inference method. The Chamfer Distance($\times$ $10^3$) are reported.[]{data-label="tab:ske_metric"}
**Quantitative results** All of these methods are evaluated on CD metric and the results are shown in Tab \[tab:ske\_metric\]. It can be seen that our final model outperforms all the others. Another discovery is that laplacian regularizer is very helpful to reach better accuracy.
**Qualitative results** We then report the visual comparisons of these methods on a sampled example in Fig \[fig:com\_ske\]. As shown, point-only fitting results in scattered points no mention to the structures. Line only fitting fails to recover the surface-shaped skeleton parts. Square-only fitting can not capture the long and thin rods and legs. The method of Line-and-Square fitting causes messy outputs since a single MLP is difficult to approximate diverse local structures. As observed, the involvement of laplacian loss effectively improves the visual appearance of the results.
Conclusion
==========
Recovering the 3D shape of an object from one of its perspectives is a very fundamental yet challenging task in computer vision field. The proposed framework splits this challenge task into three stages. It firstly recovers a 3D meso-skeleton represented as points, these skeletal points are then converted to its volumetric representation and passed to a 3DCNN for a solid volume synthesis. From which, a coarse mesh can be extracted. A GCNN is finally trained to learn the mesh deformation for producing geometric details. As demonstrated in our experiments both qualitatively and quantitatively, the proposed pipeline outperforms all existing methods. There are two directions worth being explored in the future: 1)how to change the whole pipeline to be an end-to-end network; 2) trying to apply adversarial learning on skeletal point inference, volume generation, and mesh refinement, for further improving the quality of final output mesh.
Acknowledge
===========
This work is supported in part by the National Natural Science Foundation of China (Grant No.: 61771201), the Program for Guangdong Introducing Innovative and Enterpreneurial Teams (Grant No.: 2017ZT07X183), the Pearl River Talent Recruitment Program Innovative and Entrepreneurial Teams in 2017 (Grant No.: 2017ZT07X152), and the Shenzhen Fundamental Research Fund (Grants No.: KQTD2015033114415450 and ZDSYS201707251409055).
**Supplementary Material**\
[c|\*[5]{}[c]{}]{} Method & car& cab.& cou.& lam.& wat.\
AtlasNet &1.320 &0.848 & 1.358 & 3.999 & 2.560\
Pixel2Mesh & 1.408 & 0.893 & 1.376 & **3.561** & 1.867\
Ours & **0.717** & **0.708** & **1.350** & 3.639 &**1.597**\
Method & car & cab. & cou. & lam. & wat.\
AtlasNet &8.035 &6.366 & 7.390 & 10.750 & 9.385\
Pixel2Mesh &5.706 &4.307 & **5.490** & 10.176 & 6.671\
Ours & **4.765** & **4.277** & 5.770 & **9.990** &**6.408**\
In this supplementary material, we conduct comparisons with state-of-art methods on more categories in Tab \[tab:cd\_emd\_sup\] and provide additional metrics defined on meshes in Tab \[tab:metro\]. We present more experimental results: Fig \[fig:more\_recon\] shows more comparisons of our method against state-of-the-arts. A set of input-output pairs are also shown in Fig \[fig:gallery\].
[c|\*[5]{}[c]{}]{} Methods & pla. & ben. & cha. & tab. & fir.\
AtlasNet & 1.856 & 1.258 & 1.093 & 1.441 & 1.635\
Pixel2Mesh & **1.477** & 1.267 & 1.246 & 1.360 & 1.117\
Ours & 1.643 & **1.157** & **0.948** & **0.980** & **0.988**\
Methods & car & cab. & cou. & lam. & wat.\
AtlasNet & 1.528 & 1.105 & **1.145** & 1.865 & 1.633\
Pixel2Mesh & 1.230 & 1.113 & 1.209 & **1.172** & 1.315\
Ours & **1.190** &**1.028** & 1.270 & **1.172** & **1.245**\
More evaluations on other categories {#SecMoreCat}
====================================
To demonstrate the superiority of method against the state-of-art methods, we conduct more quantitative comparisons by CD and EMD on other five categories. We select other five categories in ShapeNet [@chang2015shapenet]: car(4000), cabinet(1572), couch(3172), lamp(1956), watercraft(1658). The training and testing paradigms are consistent with details described in the previous text. For simplicity, only the results of AtlasNet [@groueix2018atlasnet] and Pixel2Mesh [@wang2018pixel2mesh] are reported.
Comparisons on metro distance {#SecComMet}
=============================
Metro distance [@cignoni1998metro] is defined as the hausdorff distance between point clouds sampled from the true and generated meshes. Tab \[tab:metro\] lists the quantitative comparisons measured by Metro distance (we exactly follow the method presented in AtlasNet [@groueix2018atlasnet] for the evaluation) on all 10 categories. Note that in both evaluations, our method outperforms the others in almost all categories.
More Qualitative comparisons {#SecMoreQua}
============================
In this section, we show more qualitative comparison results against state-of-the-arts. As shown in Fig \[fig:more\_recon\], for objects with more complex topology (i.e. non-zero genus), our method can better reconstruct the holes and loops of these 3D objects than other state-of-arts.
{height="0.92\textheight"}\
{width="0.95\linewidth"}\
Results gallery {#SecGal}
===============
A set of input-output pairs are also shown in Fig \[fig:gallery\].
[^1]: Equal contributions
[^2]: Corresponding author
[^3]: Skeletal shape representation is a kind of medial axis transform (MAT). While the MAT of a 2D shape is a 1D skeleton, for a 3D model, the MAT is generally composed of 2D surface sheets. The skeleton composed of skeletal curves and skeletal sheets (i.e., medial axes) is generally called meso-skeleton.
|
---
abstract: 'The Euler’s form of odd perfect numbers, if any, is $n=\pi^\alpha N^2$, where $\pi$ is prime, $(\pi,N)=1$ and $\pi\equiv \alpha\equiv1\pmod{4}$. Dris conjecture states that $N>\pi^{\alpha}$. We find that $N^2>\frac{1}{2}\pi^{\gamma}$, with $\gamma=max\{\omega(n)-1,\alpha\}$; $\omega(n)\geq 9$ is the number of distinct prime factors of $n$.'
author:
- Paolo Starni
title: On Dris Conjecture about Odd Perfect Numbers
---
INTRODUCTION
============
Without explicit definitions all the numbers considered in what follows must be taken as strictly positive integers. Let $\sigma(n)$ be the sum of the divisors of a number $n$; $n$ is said to be perfect if and only if $\sigma(n)=2n$. The multiplicative structure of odd perfect numbers, if any, is $$n=\pi^\alpha N^2
\label{eq:1}$$ where $\pi$ is prime, $\pi\equiv \alpha\equiv1\pmod{4}$ and $(\pi , N)=1$ (Euler, cited in [@Dickson p. 19]); $\pi^{\alpha}$ is called the Euler’s factor. From equation (1) and from the fact that the $\sigma$ is multiplicative, it results also $$n=\frac{\sigma(\pi^{\alpha})}{2}\sigma(N^2)
\label{eq:2}$$ where $\sigma(N^2)$ is odd and $2\Vert \sigma(\pi^{\alpha})$. Many details concerning the Euler’s factor and $N^2$ are given, for example, in [@Starni1][@Starni2][@Starni3][@MacDaniel][@Shi]. Regarding the relation between the magnitudo of $N^2$ and $\pi^{\alpha}$ it has been conjectured by Dris that $N>\pi^{\alpha}$ [@Dris]. The result obtained in this paper is *a necessary condition for odd perfection* (Theorem 2.1) which provides an indication about Dris conjecture. Indicating with $\omega(n)$ the number of distinct prime factors of $n$, we prove that (Corollary 2.3):\
$(i)$$N^2>\frac{1}{2}\pi^{\gamma}$, where $\gamma=max\{\omega(n)-1, \alpha\}$\
Since $\omega(n)\geq9$ (Nielsen, [@Nielsen]), it follows:\
$(i)_{1}$$N^2>\frac{1}{2}\pi^8$; this improves the result $N>\pi$ claimed in [@Brown] by Brown in his approach to Dris conjecture.\
Besides\
$(i)_2$ If $\omega(n)-1 >2\alpha$, then $N>\pi^{\alpha}$\
so that\
$(i)_3$ If $\omega(n)-1 > 2\alpha$ for each odd perfect number $n$, then Dris conjecture is true.\
Now, some questions arise: $\omega(n)$ depends on $\alpha$? Is there a maximum value of $\alpha$? The minimum value of $\alpha$ is $1$? The only possible value of $\alpha$ is 1 (Sorli, [@Sorli conjecture 2]) so that Dris conjecture is true? Without ever forgetting the main question: do odd perfect numbers exist?
THE PROOF
=========
Referring to an odd perfect number $n$ with the symbols used in equation , we obtain:
If $n$ is an odd perfect number, then $$N^2=A\frac{\sigma(\pi^{\alpha})}{2} \hspace{0.2cm}and\hspace{0.2cm} \sigma(N^2)= A\pi^{\alpha}$$ \[lemma\]
From equation and from the fact that $(\sigma(\pi^{\alpha}),\pi^{\alpha})=1$, it follows $$N^2=A\frac{\sigma(\pi^{\alpha})}{2}$$ where $A$ is an odd positive integer given by $$A=\frac{\sigma(N^2)}{\pi^{\alpha}}$$
In relation to the odd parameter $A$ in Lemma 2.1, we give two further lemmas:
If $A=1$, then $\alpha\geq \omega(n)-1 \hspace{0.1cm}and\hspace{0.1cm} N^2>\frac{1}{2}\pi^{\alpha}$
Let $q_k, k=1, 2, ...,\omega(N)=\omega (N^2)$, are the prime factors of $N^2$; from hypothesis and from (4) we have $$\pi^{\alpha}=\sigma(N^2)=\sigma(\prod_{k=1}^{\omega(N)}q_k^{2\beta_k})=\prod_{k=1}^{\omega(N)}\sigma(q_k^{2\beta_k})=\prod_{k=1}^{\omega(N)}\pi^{\delta_k}$$ in which $\alpha=\sum_{k=1}^{\omega(N)}\delta_k\geq \sum_{k=1}^{\omega(N)}1_k=\omega(N)$.\
\
Since $\omega(n)=\omega(N)+1$, it results $$\alpha\geq \omega(n)-1$$ Besides, from Equation (3) it follows $$N^2=\frac{1}{2}\sigma(\pi^{\alpha})>\frac{1}{2}\pi^{\alpha}$$
If $A>1$, then $N^2>\frac{3}{2}\pi^{\alpha}$
From Equation (3) it results $A\geq3$. Thus $$N^2\geq\frac{3}{2}\sigma(\pi^{\alpha})>\frac{3}{2}\pi^{\alpha}$$
The following theorem summarizes a necessary condition for odd perfection.
If $n$ is an odd perfect number, then $$(\neg a\land d)\lor(a\land b\land c)\lor (b\land c\land d)$$ \[th1\] where: $a\cong (A=1)$,$\neg a\cong (A>1)$, $b\cong (\alpha\geq \omega(n) -1)$, $c\cong (N^2>\frac{1}{2}\pi^{\alpha})$, $d\cong (N^2>\frac{3}{2}\pi^{\alpha})$
We combine Lemmas 2.2 and 2.3 setting $$\left\{
\begin{array}
[c]{c}
lemma\hspace{0.1cm}2.2: (a\implies b\land c)\\
\hspace{-0.35cm}lemma\hspace{0.1cm} 2.3: (\neg a\implies d)
\end{array}
\right.$$ where, since it cannot be $A<1$, it is $ (a)\cong (A=1)$ and $(\neg a)\cong (A>1)$. One obtains from (5) $$[\neg a\lor(b\land c)]\land (a\lor d)$$ which is equivalent to $$(\neg a\land d)\lor (a\land b \land c)\lor (b\land c \land d)$$
Considering cases in which the necessary condition for odd perfection (6) is false, we obtain the following corollaries:
If $n$ is an odd perfect number, then $N^2>\frac{1}{2}\pi^{\alpha}$
We have\
(7) $(\neg c\land \neg d) (\cong\ N^2<\frac{1}{2}\pi^{\alpha})\implies n$ is not an odd perfect number\
From the contrapositive formulation of (7) it follows the proof.
If $n$ is an odd perfect number, then $$N^2>\frac{3}{2}\pi^{\omega(n)-1}>\frac{1}{2}\pi^{\omega(n)-1}$$
We have\
(8) $(\neg b\land \neg d) (\cong\ N^2<\frac{3}{2}\pi^{\omega(n)-1})\implies n$ is not an odd perfect number\
From the contrapositive formulation of (8) it follows the proof.
Combining these two corollaries, we have
If $n$ is an odd perfect number, then $$N^2>\frac{1}{2}\pi^{\gamma}, where\hspace{0.1cm} \gamma=max\{ \omega(n)-1,\alpha\}$$
Immediate.
[10]{} P. Brown, *A partial proof of a conjecture of Dris*, arXiv:1602.01591v1, 2016. Shi-Cao Chen and Hao Luo, *Odd multiperfect numbers*, arXiv:1102.4396, 2011. L.E. Dickson, *History of the theory of numbers*, vol. 1, Dover, 2003. J.A.B. Dris, *Solving the odd perfect number problem: some old and new approaches*, M.Sc. thesis, De La Salle University, Manila, 2008 W.L. MacDaniel and P. Hagis, *Some results concerning the non-existence of odd perfect numbers of the form $\pi^{\alpha}M^{2\beta}$*, Fibonacci Quart. **131** (1975), 25-28. P.P. Nielsen, *Odd perfect numbers have at least nine distinct prime factors*, Math. Comp. **76** (2007), 2109-2120. R.M. Sorli, *Algorithms in the study of multiperfect and odd perfect numbers*, Ph.D. thesis, University of Technology, Sidney, 2003. P. Starni, *On the Euler’s factor of an odd perfect number*, J. Number Theory **37** (1991), 366-369. P. Starni, *Odd perfect numbers: a divisor related to the Euler’s factor*, J. Number Theory **44** (1993), 58-59. P. Starni, *On some properties of the Euler’s factor of certain odd perfect numbers*, J. Number Theory **116** (2006), 483-486.
|
---
abstract: 'Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${{\mathbb N}_0^r}$. Some sufficient conditions for the existence of a not necessarily regular isometric dilation of a completely contractive representation of ${\mathbb E}$ are established and difference between regular and $^*$-regular dilations discussed. It is in particular shown that a minimal isometric dilation is $^*$-regular if and only if it is doubly commuting. The case of product systems associated with higher-rank graphs is analysed in detail.'
address: 'Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom '
author:
- Adam Skalski
title: '**On isometric dilations of product systems of $C^*$-correspondences and applications to families of contractions associated to higher-rank graphs**'
---
[^1]
Classical multi-dimensional dilation theory ([@Nagy]) for Hilbert space operators is concerned with dilating tuples of contractions to tuples of isometries or unitaries, preserving as many properties of the original family as possible. In particular if the tuple with which we start consists of mutually commuting operators, it is desirable to obtain a commuting dilation. Celebrated examples of S.Parrott, N.Varopoulos and others show that a joint dilation of three or more commuting contractions to commuting isometries need not exist. In general it is difficult to decide whether a given commuting tuple has a commuting isometric dilation. On the other hand the existence of so-called regular or $^*$-regular dilations (i.e. dilations satisfying additional conditions with respect to products of the original contractions and their adjoints, see for example [@Timot]) can be detected via simple conditions corresponding to positive-definiteness of certain operator-valued functions associated with the initial tuple.
In a recent paper [@graphdil] together with J.Zacharias we considered dilations of ${\Lambda}$-contractions, that is tuples of operators satisfying commutation relations encoded by a (higher rank) graph ${\Lambda}$. It has now become clear that using the constructions provided by I.Raeburn and A.Sims in [@toep] some of the results of [@graphdil] can be viewed as statements on completely contractive representations of the canonical product system of $C^*$-correspondences associated to ${\Lambda}$. Product systems of $C^*$-correspondences were first defined in [@fowl] as generalisations of product systems of Hilbert spaces and quickly proved to provide a natural framework for extensions of the classical multi-dimensional dilation theory to more complicated objects (see [@Solk] and references therein). The questions about the existence of a joint dilation of a family of contractions satisfying certain commutativity relations to an analogous family of isometries translates here into a question on the existence of an isometric dilation of a (completely) contractive representation of a given product system.
Motivated by the observations above we show in this paper that the generalised Poisson transform constructed in [@graphdil] (see also [@PPois], [@MSker]) can be associated to a completely contractive representation of a product system of $C^*$-correspondences over ${{\mathbb N}_0^r}$, if only the system enjoys what we call a normal ordering property and the representation satisfies a so-called ‘Popescu condition’. This implies in turn that any such representation admits an isometric dilation. These sufficient conditions for the existence of an isometric dilation should be compared with recent results of [@Solk], where sufficient and necessary conditions for the existence of a *regular* dilation were established. The dilations constructed via the Poisson transform in the case of product systems related to graphs are of a *$^*$-regular* type. It is shown that in general a minimal (not necessarily regular) isometric dilation of a contractive representation is doubly commuting if and only if it satisfies the $^*$-regularity property. Contrary to the classical case of commuting Hilbert space contractions, here the difference between the regular and $^*$-regular dilations is fundamental, as there is no natural adjoint operation on a class of representations of a given product system (moreover we cannot always assume that the dilations have natural ‘unitary’ extensions, see [@Wold] and references therein).
In the second part of the paper we consider the case of certain families of contractions associated with a higher-rank graph ${\Lambda}$ and formalise heuristic observations listed in the second paragraph of this introduction. It is shown that the dilations of [@graphdil] can indeed be viewed as dilations of representations of the canonical product system ${\mathbb E}({\Lambda})$. General results of the first part of the paper specialised to this context can be interpreted as giving sufficient conditions on existence of regular or $^*$-regular dilations of certain tuples of contractions satisfying the commutation relations encoded by a higher-rank graph. In particular one can deduce immediately from [@Soltwo] that any ${\Lambda}$-contraction associated with a rank-2 graph has a dilation to a Toeplitz-type family.
The detailed plan of the paper is as follows: after listing some general notations we proceed to introduce in Section 1 basic notions of $C^*$-correspondences, their product systems over ${{\mathbb N}_0^r}$ and (covariant completely) contractive representations of such objects. In Section 2 we proceed to define isometric dilations of contractive representations and to quote fundamental results of B.Solel ([@Soltwo], [@Solk]) on the existence of dilations in the two-dimensional case and on sufficient and necessary conditions for the existence of regular dilations. A notion of a $^*$-regular isometric dilation is also introduced and a fact that a minimal isometric dilation is $^*$-regular if and only if it is doubly commuting established. Section 3 is devoted to the construction of a generalised Poisson transform associated to a representation of a product system with the normal ordering property satisfying the Popescu condition and to applications of the transform to isometric dilations. In Section 4 we recall the canonical product system of $C^*$-correspondences associated to a higher-rank graph ${\Lambda}$ ([@toep]) and describe its representations in terms of the ${\Lambda}$-families of operators on a Hilbert space. Finally Section 5 presents the general results of Sections 2 and 3 specified and adapted to the case of dilations of ${\Lambda}$-families described in Section 4.
Let ${\mathbb N}_0= {\mathbb N}\cup \{0\}$. Fix now and for the rest of the paper $r \in {\mathbb N}$. The canonical ‘basis’ in ${{\mathbb N}_0^r}$ will be denoted by $(e_1, \ldots, e_r)$, with $e:=\sum_{i=1}^r e_i$. The componentwise maximum (respectively, minimum) of $n,m \in {{\Bbb Z}^r}$ is denoted by $n \vee m$ (respectively, $n \wedge m$) and we write $|n|= n_1 + \cdots + n_r$, $n_+=n \vee 0$, $n_- = - (n \wedge 0)$.
Product systems of $C^*$-correspondences and their representations {#prodsys}
==================================================================
Let ${\mathsf{A}}$ be a $C^*$-algebra. By a $C^*$-correspondence $E$ over ${\mathsf{A}}$ is meant a Hilbert $C^*$-module over ${\mathsf{A}}$, equipped with the structure of a left ${\mathsf{A}}$-module (via a nonzero $*$-homomorphism $\phi$ mapping ${\mathsf{A}}$ into the $C^*$-algebra of adjointable operators on $E$). $E$ is *essential* as a left ${\mathsf{A}}$-module if the closed linear span of $\phi({\mathsf{A}}) E$ is equal to $E$. Each $C^*$-correspondence is considered with the usual operator space structure (i.e. the one coming from viewing it as a corner in the appropriate linking algebra). The $C^*$-algebra of adjointable operators on $E$ is denoted by $\mathcal{L}(E)$. Further details can be found in [@Lance] or [@Morita]; note that we will often use the concept of internal tensor products in the category of Hilbert $C^*$-modules equipped with left actions. In particular any representation $\sigma$ of ${\mathsf{A}}$ on a Hilbert space ${\mathsf{H}}$ allows us to consider a new Hilbert space $E {\otimes}_{\sigma} {\mathsf{H}}$ equipped with the representation of ${\mathsf{A}}$ arising from the left action of ${\mathsf{A}}$ on $E$.
Fix now and for the rest of the paper $r \in {\mathbb N}$. As explained in \[So$_{1-2}$\] a *product system ${\mathbb E}$ of $C^*$-correspondences over ${\mathbb N}_0^r$* (formally introduced in [@fowl] for a general countable semigroup with a neutral element) can be thought of as a family of $r$ $C^*$-correspondences $\{E_1, \ldots,E_r\}$ over the same $C^*$-algebra together with the unitary isomorphisms $t_{i,j}: E_i {\otimes}E_j \to E_j {\otimes}E_i$ ($i>j$) satisfying the natural associativity relations: $$({{\textup{id}}}_{E_l} {\otimes}t_{i,j}) (t_{i,l} {\otimes}{{\textup{id}}}_{E_j}) ({{\textup{id}}}_{E_i} {\otimes}t_{j,l}) =
(t_{j,l} {\otimes}{{\textup{id}}}_{E_i}) ({{\textup{id}}}_{E_j} {\otimes}t_{i,l}) (t_{i,j} {\otimes}{{\textup{id}}}_{E_l})$$ for all $1\leq i<j<l\leq k$. This point of view entails identifying for all $n=(n_1, \ldots, n_r) \in {\mathbb N}_0^r$ the correspondence ${\mathbb E}(n)$ with $E_1^{{\otimes}^{
n_1}} {\otimes}\cdots {\otimes}E_r^{{\otimes}^{ n_r}}$. We write $t_{i,i} = {{\textup{id}}}_{E_i {\otimes}E_i}$, $t_{i,j} = t_{j,i}^{-1}$ for $i<j$ and also define unitary isomorphisms $t_{m,n}:{\mathbb E}(m) {\otimes}{\mathbb E}(n) \to {\mathbb E}(n) {\otimes}{\mathbb E}(m)$ ($m,n \in {{\mathbb N}_0^r}$) by obvious compositions of tensor extensions of appropriate $t_{i,j}$’s.
Let ${\mathcal{F}}_{{\mathbb E}}:= \bigoplus_{n \in {{\mathbb N}_0^r}} {\mathbb E}(n)$ denote the Fock module of ${\mathbb E}$ (see [@fowl] for the details of the construction). It is a $C^*$-correspondence over ${\mathsf{A}}$. For each $n \in {{\mathbb N}_0^r}$ and $e \in {\mathbb E}(n)$ define the creation operator $L_e: {\mathcal{F}}_{{\mathbb E}} \to {\mathcal{F}}_{{\mathbb E}}$ by the formula $$L_e (f) = e {\otimes}f, \;\; f \in {\mathcal{F}}_{{\mathbb E}}.$$ The Toeplitz algebra associated with ${\mathbb E}$ is a concrete $C^*$-algebra in $\mathcal{L}({\mathcal{F}}_{{\mathbb E}})$ generated by all creation operators as above. It will be denoted by ${\mathcal{T}_{{\mathbb E}}}$.
A product system ${\mathbb E}$ of $C^*$-correspondences over ${{\mathbb N}_0^r}$ is called compactly aligned if given $n,m \in {{\mathbb N}_0^r}$ and two operators $S\in \mathcal{K}({\mathbb E}(n))$, $T\in \mathcal{K}({\mathbb E}(m))$ the operator $S_n^{n \vee m} T_m^{n \vee m} \in \mathcal{K}({\mathbb E}(n \vee m))$, where $S_n^{n \vee m}:= S {\otimes}I_{{\mathbb E}(n\vee m- n)}$ and $T_m^{n \vee m}:= T {\otimes}I_{{\mathbb E}(n\vee m- m)}$.
The notion of compact alignment may seem rather technical, but it proved to be very useful ([@fowl]). For the product system associated with a higher-rank graph it is equivalent to the graph in question being *finitely aligned* (see Sections \[graphrep\] and \[dilgraph\]). Examples coming from graphs suggest also that compact alignment of a product system is closely related to a form of ‘normal ordering’ in the Toeplitz algebra. As we have not been able to determine whether these two properties coincide in general, we introduce the following definition:
A product system ${\mathbb E}$ of $C^*$-correspondences over ${{\mathbb N}_0^r}$ is said to have a normal ordering property if ${\mathcal{T}_{{\mathbb E}}}={\overline}{\textrm{Lin}}\{L_e L_f^*:e,f \in \bigcup_{n \in {{\mathbb N}_0^r}}{\mathbb E}(n)\}$.
The normal ordering property may be thought of as a strong form of ‘double commutativity’ of the creation operators in the Toeplitz algebra. This is naturally seen when we work with product systems associated with higher-rank graphs in Sections \[graphrep\] and \[dilgraph\]. Note that if ${\mathsf{A}}={\mathbb C}$ then each $E_j$ is a Hilbert space and the structure of a product system is determined by the Hilbert space unitaries $t_{i,j}:E_i \to E_j$ (precise description can be found in [@Solk] or in [@Wold]). If each $E_j$ is additionally assumed to be finite-dimensional we are in the situation analysed in [@SteveBaruch] and it is easy to see that the corresponding product system has a normal ordering property (and is compactly aligned).
Representations of $C^*$-correspondences {#representations-of-c-correspondences .unnumbered}
----------------------------------------
The notions presented in this subsection have been introduced and developed in the series of papers by P.Muhly and B.Solel (see [@MSgen] and references therein).
Let ${\mathsf{H}}$ be a Hilbert space. By a (completely contractive covariant) representation of a $C^*$-correspondence $E$ over a $C^*$-algebra ${\mathsf{A}}$ on ${\mathsf{H}}$ is meant a pair $(\sigma, T)$, where $(\sigma,{\mathsf{H}})$ is a representation of ${\mathsf{A}}$ on ${\mathsf{H}}$, and $T:E \to B({\mathsf{H}})$ is a linear completely contractive map such that $$T(a \xi b) = \sigma(a) T(\xi) \sigma(b), \;\;\; a,b \in {\mathsf{A}}, \xi \in E.$$ It is called isometric if for each $\xi, \eta \in E$ $$T(\xi)^* T(\eta) = \sigma(\langle \xi, \eta {\rangle}).$$
A representation $(T, \sigma)$ determines a contraction ${{\widetilde}{T}}: E {\otimes}_{\sigma} {\mathsf{H}}\to {\mathsf{H}}$ given by ${{\widetilde}{T}}(\xi {\otimes}h) = T(\xi) h$ ($\xi \in E, h \in {\mathsf{H}}$). This satisfies: $$\label{tilde} {{\widetilde}{T}}(\phi(a) {\otimes}I_{{\mathsf{H}}}) = \sigma(a) {{\widetilde}{T}}, \;\;\; a \in {\mathsf{A}}$$ ($\phi$ denoting the left action of ${\mathsf{A}}$ on $E$), and one can in fact show that, given a representation $\sigma$, there is a 1-1 correspondence between contractions satisfying and representations of $E$ ([@MSgen] Lemma 2.1). The isometric representations are exactly those for which ${{\widetilde}{T}}$ is an isometry. The representation $(\sigma, T)$ is said to be *(fully) coisometric* if ${\widetilde}{T}{\widetilde}{T}^* = I_{{\mathsf{H}}}$.
It is easy to see how the notion of a representation of a $C^*$-correspondence extends to a product system.
\[prodrep\] Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${{\mathbb N}_0^r}$. By a (completely contractive covariant) representation of ${\mathbb E}$ on a Hilbert space ${\mathsf{H}}$ is meant a tuple $(\sigma, T^{(1)}, \ldots, T^{(r)})$, where $(\sigma,T^{(i)})$ is a representation of $E_i$ on ${\mathsf{H}}$ and $$\label{rep} {{\widetilde}{T}}^{(i)} (I_{E_i} {\otimes}{{\widetilde}{T}}^{(j)}) = {{\widetilde}{T}}^{(j)} (I_{E_j} {\otimes}{{\widetilde}{T}}^{(i)}) (t_{i,j} {\otimes}I_{{\mathsf{H}}})$$ for $i,j\in {\{1,\ldots,r\}}$. Such a representation is called isometric if each $(\sigma, T^{(i)})$ is isometric, and coisometric if each $(\sigma, T^{(i)})$ is coisometric.
To lighten the notation we will occasionally write ${\overrightarrow{T}}$ for $(\sigma, T^{(1)}, \ldots, T^{(r)})$. We will also exploit the inductively defined maps $T(n)(e)\in B({\mathsf{H}})$ ($n \in {{\mathbb N}_0^r}$, $e \in {\mathbb E}(n)$) (see [@MSgen] or [@Wold]) and their natural partners ${\widetilde}{T}(n):{\mathbb E}(n) {\otimes}_{\sigma} {\mathsf{H}}\to {\mathsf{H}}$. It is important to note that because of the condition operators ${\widetilde}{T}(n) {\widetilde}{T}(n)^*$ belong to $\sigma({\mathsf{A}})'$.
If we represent the Toeplitz algebra faithfully on a Hilbert space, then the map $\bigcup_{n \in {{\mathbb N}_0^r}}{\mathbb E}(n)\ni e \to L_e \in
{\mathcal{T}_{{\mathbb E}}}$ yields in a natural way a representation of ${\mathbb E}$, called further the Fock-Toeplitz representation. It is easily seen to be isometric.
In what follows we will often consider doubly commuting representations; these have especially good properties in terms of the dilations or Wold decompositions (see respectively [@Solk] and [@Wold]).
\[dcom\] A representation $(\sigma, T^{(1)}, \ldots, T^{(r)})$ of ${\mathbb E}$ on a Hilbert space ${\mathsf{H}}$ is called doubly commuting if for each $i,j \in \{1,\ldots,r\}$, $i \neq j$ implies $${{\widetilde}{T}}^{(j)^*} {{\widetilde}{T}}^{(i)} =
(I_{E_j} {\otimes}{{\widetilde}{T}}^{(i)}) (t_{i,j} {\otimes}I_{{\mathsf{H}}}) (I_{E_i} {\otimes}{{\widetilde}{T}}^{(j)^*}).
\label{doubly}$$
For isometric representations of product systems over ${{\mathbb N}_0^r}$ double commutativity is exactly the same as Nica-covariance considered in [@toep] ([@Solk], Remark 3.12). It also has the following equivalent characterisation:
\[kerlem\] An isometric representation $(\sigma, T^{(1)}, \ldots, T^{(r)})$ of ${\mathbb E}$ on a Hilbert space ${\mathsf{H}}$ is doubly commuting if and only if for each $i,j \in \{1,\ldots,r\}$, $i \neq j$ $$\label{kernels} {\widetilde}{T}^{(i)} ({{\textup{Ker}}}(I_{E_i} {\otimes}{\widetilde}{T}^{(j)^*})) \subset
{{\textup{Ker}}}({\widetilde}{T}^{(j)^*}).$$
Let $i,j$ be as above and denote the operator $I_{E_i} {\otimes}{\widetilde}{T}^{(j)}:E_i {\otimes}E_j {\otimes}_{\sigma} {\mathsf{H}}\to E_i {\otimes}_{\sigma} {\mathsf{H}}$ by $\Gamma_{ij}$. Note that the ${{\textup{Ker}}}(\Gamma_{ij}^*) = E_i {\otimes}_{\sigma} {{\textup{Ker}}}({\widetilde}{T}^{(j)^*})$. It can be proved exactly in the same way as the well known statement for kernel of the operator $I_{{\mathsf{K}}_1} {\otimes}S$, where ${\mathsf{K}}_1, {\mathsf{K}}_2$ are Hilbert spaces and $S \in B({\mathsf{K}}_2)$ (at least if you know how to show the latter without using an orthonormal basis in ${\mathsf{K}}_1$).
If ${\overrightarrow{T}}$ is doubly commuting, then is easily seen to be satisfied. Suppose then that holds. Any vector in $E_i {\otimes}_{\sigma} {\mathsf{H}}$ can be decomposed as a sum of an element in ${{\textup{Ker}}}(\Gamma_{ij}^*)$ and in ${{\textup{Ker}}}(\Gamma_{ij}^*)^{\perp} = {\overline}{ {{\textrm{Ran}}}(\Gamma_{ij})}$. It is therefore enough to show that the both sides of the equation hold on ${{\textrm{Ran}}}(\Gamma_{ij})$. Let then $z \in E_i {\otimes}E_j {\otimes}_{\sigma} {\mathsf{H}}$. Then $$\begin{aligned}
{{\widetilde}{T}}^{(j)^*} {{\widetilde}{T}}^{(i)}
\Gamma_{ij} (z) &= {{\widetilde}{T}}^{(j)^*} {{\widetilde}{T}}^{(i)} (I_{E_i} {\otimes}{\widetilde}{T}^{(j)}) (z) = {{\widetilde}{T}}^{(j)^*} {{\widetilde}{T}}^{(j)} (I_{E_j} {\otimes}{{\widetilde}{T}}^{(i)}) (t_{i,j}
{\otimes}I_{{\mathsf{H}}})(z) \\&= (I_{E_j} {\otimes}{{\widetilde}{T}}^{(i)}) (t_{i,j} {\otimes}I_{{\mathsf{H}}})(I_{E_i} {\otimes}{{\widetilde}{T}}^{(j)^*} {{\widetilde}{T}}^{(j)} )(z)
\\&= (I_{E_j} {\otimes}{{\widetilde}{T}}^{(i)}) (t_{i,j} {\otimes}I_{{\mathsf{H}}})(I_{E_i} {\otimes}{{\widetilde}{T}}^{(j)^*}) (\Gamma_{ij}(z)).\end{aligned}$$ This ends the proof.
General properties of isometric dilations of representations of product systems of $C^*$-correspondences {#dilgen}
========================================================================================================
In this section we discuss several classes of isometric dilations of a representation of a product system of $C^*$-correspondences.
Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${\mathbb N}_0^r$ and let $(\sigma,
T^{(1)},\ldots, T^{(r)})$ be a representation of ${\mathbb E}$ on a Hilbert space ${\mathsf{K}}$. We say that $(\pi, V^{(1)},\ldots, V^{(r)})$, an isometric representation of ${\mathbb E}$ on a Hilbert space ${\mathsf{K}}\supset {\mathsf{H}}$, is an isometric dilation of $(\sigma, T^{(1)},\ldots, T^{(r)})$ if
$\forall_{a \in {\mathsf{A}}}\;\;\; \pi(a)|_{{\mathsf{H}}} = \sigma(a)$;
$\forall_{i \in {\{1,\ldots,r\}}}\; \forall_{\xi \in {\mathsf{H}}} \;\; ({\widetilde}{V}^{(i)})^* (\xi) = ({\widetilde}{T}^{(i)})^* (\xi)$.
The dilation is called minimal if ${\mathsf{K}}= {\overline}{{\textrm{Lin}}}\{V(n)(e) \xi: n \in {{\mathbb N}_0^r}, e \in {\mathbb E}(n), \xi \in
{\mathsf{H}}\}$.
Note that the condition (ii) above in particular exploits the identification of $E_i {\otimes}_{\sigma} {\mathsf{H}}$ with a subspace of $E_i
{\otimes}_{\pi} {\mathsf{K}}$. Moreover $P_{{\mathsf{H}}} \in \pi({\mathsf{A}})'$, so that the operators of the form $T {\otimes}P_{{\mathsf{H}}}$ ($T \in
\mathcal{L}({\mathbb E}(n))$) are well defined as operators in $\mathcal{L}({\mathbb E}(n) {\otimes}_{\pi} {\mathsf{K}})$. This is used in Definition \[regular\] below.
It is known that two commuting contractions can be always jointly dilated to commuting isometries. The following result established by B.Solel shows that this phenomenon persists in the category of representations of $C^*$-correspondences.
\[sol1\] Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${\mathbb N}_0^2$. Every (completely contractive covariant) representation $(\sigma, T^{(1)}, T^{(2)})$ of ${\mathbb E}$ on a Hilbert space has a minimal isometric dilation $(\pi, V^{(1)}, V^{(2)})$. If $\sigma$ is nondegenerate and $E_1, E_2$ are essential then $\rho$ is nondegenerate.
Regular isometric dilations (after B.Solel) {#regular-isometric-dilations-after-b.solel .unnumbered}
-------------------------------------------
To formulate the next result we need a few more definitions. For $u=\{u_1,\ldots, u_k\} \subset
{\{1,\ldots,r\}}$ write $e(u) = e_{u_1} + \cdots + e_{u_k}$.
\[Solelcond\] Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${\mathbb N}_0^r$. A representation $(\sigma, T^{(1)}, \ldots, T^{(r)})$ of ${\mathbb E}$ on a Hilbert space is said to satisfy the Brehmer-Solel condition if for each $v \subset {\{1,\ldots,r\}}$ $$\sum_{u\subset v} (-1)^{|u|} (I_{{\mathbb E}(e(v) - e(u))} {\otimes}{\widetilde}{T}(e(u))^* {\widetilde}{T}(e(u))) \geq 0.$$
The condition above first appeared in the context of commuting families of contractions in [@Brehmer]; recently it was exploited in the context of product systems of $C^*$-correspondences in [@Solk].
\[regular\] Let $(\sigma, T^{(1)}, \ldots, T^{(r)})$ be a representation of ${\mathbb E}$ on a Hilbert space ${\mathsf{H}}$. An isometric dilation ${\overrightarrow{V}}$ of ${\overrightarrow{T}}$ is said to be regular if for all $n \in {{\Bbb Z}^r}$ $$\label{reg} (I_{{\mathbb E}(n_{-})} {\otimes}P_{{\mathsf{H}}}) {\widetilde}{V}(n_{-})^*
{\widetilde}{V}(n_+)|_{{\mathbb E}(n_+) {\otimes}_{\sigma} {\mathsf{H}}} = {\widetilde}{T}(n_{-})^* {\widetilde}{T}(n_+).$$
B.Solel showed that the condition described in Definition \[Solelcond\] characterises these representations which allow *regular* isometric dilations.
\[sol2\] Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${\mathbb N}_0^r$. A (completely contractive covariant) representation $(\sigma, T^{(1)},\ldots, T^{(r)})$ of ${\mathbb E}$ on a Hilbert space has a regular isometric dilation if and only if it satisfies the Brehmer-Solel condition.
Note that minimal regular dilations are necessarily unique, in the sense that any two such dilations respectively on Hilbert spaces ${\mathsf{K}}$ and ${\mathsf{K}}'$ are intertwined by a unitary $U: {\mathsf{K}}\to {\mathsf{K}}'$.
$^*$-regular dilations {#regular-dilations .unnumbered}
----------------------
Let us begin with a simple equivalent characterisation of regularity of an isometric dilation.
\[simpreg\] An isometric dilation $(\pi, V^{(1)},\ldots, V^{(r)})$ of a representation\
$(\sigma,
T^{(1)},\ldots, T^{(r)})$ of ${\mathbb E}$ is regular if and only if for all $n,m \in {{\mathbb N}_0^r}$ such that $n_j
\neq 0$ implies $m_j= 0$ ($j \in {\{1,\ldots,r\}}$) and all $e\in {\mathbb E}(n)$, $f \in {\mathbb E}(m)$ $$\label{regsimp} P_{{\mathsf{H}}} (V(n)(e))^* V(m)(f)|_{{\mathsf{H}}} = (T(n)(e))^* T(m)(f).$$
Note that condition is satisfied for all $n \in {\{1,\ldots,r\}}$ if and only if for all $n,m \in
{{\mathbb N}_0^r}$ such that $n_j \neq 0$ implies $m_j= 0$ ($j \in {\{1,\ldots,r\}}$) there is $$(I_{{\mathbb E}(n)} {\otimes}P_{{\mathsf{H}}}) {\widetilde}{V}(n)^* {\widetilde}{V}(m)|_{{\mathbb E}(m) {\otimes}{\mathsf{H}}} =
{\widetilde}{T}(n)^* {\widetilde}{T}(m).$$ The last condition is equivalent to the fact that for all $\xi\in {\mathsf{H}}$, $f \in {\mathbb E}(m)$ $$(I_{{\mathbb E}(n)} {\otimes}P_{{\mathsf{H}}}) {\widetilde}{V}(n)^* V(m)(f) \xi =
{\widetilde}{T}(n)^* T(m) (f) \xi,$$ and further to the fact that for all $e \in {\mathbb E}(n), \eta \in {\mathsf{H}}$ $$\langle e {\otimes}\eta, (I_{{\mathbb E}(n)} {\otimes}P_{{\mathsf{H}}}) {\widetilde}{V}(n)^* V(m)(f) \xi \rangle = \langle e {\otimes}\eta, {\widetilde}{T}(n)^* T(m) (f)
\xi \rangle.$$ This in turn holds if and only if $$\langle V(n) (e) \eta, V(m)(f) \xi \rangle = \langle T(n) (e) \eta, T(m)(f) \xi \rangle,$$ if and only if $$\langle \eta, P_{{\mathsf{H}}} (V(n) (e))^*V(m)(f) \xi \rangle = \langle \eta, (T(n) (e))^*T(m)(f) \xi \rangle.$$ This ends the proof.
Recall ([@Timot]) that if $(S,T)$ is a commuting pair of contractions on a Hilbert space ${\mathsf{H}}$ and $(U,V)$ is a commuting isometric dilation of $(S,T)$ then it is said to be $^*$-regular if for all $k,l \in {\mathbb N}$ $$P_{{\mathsf{H}}} (U^*)^l V^k|_{{\mathsf{H}}} = T^k (S^*)^l.$$
How should a corresponding definition look here? Note that if ${\overrightarrow{V}}=(\pi, V^{(1)},\ldots, V^{(r)})$ is an isometric dilation of a representation ${\overrightarrow{T}}=(\sigma, T^{(1)},\ldots, T^{(r)})$ of ${\mathbb E}$ then for all $n,m \in {{\mathbb N}_0^r}$ and all $e\in {\mathbb E}(n)$, $ f \in
{\mathbb E}(m)$ $$\label{correct0} P_{{\mathsf{H}}} V(n)(e) (V(m)(f))^*|_{{\mathsf{H}}} = T(n)(e) (T(m)(f))^*.$$ Moreover one can see that similarly for all $n \in
{{\Bbb Z}^r}$ $$\begin{aligned}
\label{correct} (I_{{\mathbb E}(n_{-})} {\otimes}P_{{\mathsf{H}}}) &(I_{{\mathbb E}(n_{-})} {\otimes}{\widetilde}{V}(n_+)) (t_{n_{+},
n_{-}} {\otimes}I_{{\mathsf{K}}}) (I_{{\mathbb E}(n_+)} {\otimes}{\widetilde}{V}(n_{-}))^*)|_{{\mathbb E}(n_+) {\otimes}{\mathsf{H}}} \\ \notag =&(I_{{\mathbb E}(n_{-})} {\otimes}{\widetilde}{T}(n_+))
(t_{n_{+}, n_{-}} {\otimes}I_{{\mathsf{H}}}) (I_{{\mathbb E}(n_+)} {\otimes}{\widetilde}{T}(n_{-}))^*);\end{aligned}$$ it is enough to observe that from the definition of the dilation it follows that for all $n,m \in {{\mathbb N}_0^r}$ $$(I_{{\mathbb E}(n)} {\otimes}{\widetilde}{V}(m))^*|_{{\mathbb E}(n) {\otimes}_{\sigma} {\mathsf{H}}} = (I_{{\mathbb E}(n)} {\otimes}{\widetilde}{T}(m))^*|_{{\mathbb E}(n) {\otimes}_{\sigma} {\mathsf{H}}}.$$
If the dilation ${\overrightarrow{V}}$ is doubly commuting then the condition reduces to $$(I_{{\mathbb E}(n_{-})} {\otimes}P_{{\mathsf{H}}})
{\widetilde}{V}(n_{-})^* {\widetilde}{V}(n_+) |_{{\mathbb E}(n_+) {\otimes}_{\sigma} {\mathsf{H}}} = (I_{{\mathbb E}(n_{-})} {\otimes}{\widetilde}{T}(n_+)) (t_{n_{+}, n_{-}} {\otimes}I_{{\mathsf{H}}})
(I_{{\mathbb E}(n_+)} {\otimes}{\widetilde}{T}(n_{-}))^*.$$ The latter can be seen as a natural generalisation of the notion of $^*$-regularity.
In the classical context of commuting contractions Theorem 2 of [@Timot] (see also [@GaSu]) shows that a minimal isometric dilation is $^*$-regular if and only if it is doubly commuting. The same remains true in our context, as the next theorem shows. The proof is a natural generalisation of that in [@Timot]. The basic idea is the following: as the last equation implies ‘double commutativity on ${\mathsf{H}}$’, we need to exploit minimality to deduce ‘double commutativity on ${\mathsf{K}}$’.
\[dc=streg\] A minimal isometric dilation $(\pi, V^{(1)},\ldots, V^{(r)})$ of a representation $(\sigma, T^{(1)},\ldots, T^{(r)})$ of ${\mathbb E}$ is doubly commuting if and only if it is $^*$-regular, that is if for all $n \in {{\Bbb Z}^r}$ $$\begin{aligned}
\label{streg}(I_{{\mathbb E}(n_{-})} {\otimes}P_{{\mathsf{H}}})& {\widetilde}{V}(n_{-})^* {\widetilde}{V}(n_+) |_{{\mathbb E}(n_+) {\otimes}_{\sigma} {\mathsf{H}}}
= \\& \notag (I_{{\mathbb E}(n_{-})} {\otimes}{\widetilde}{T}(n_+)) (t_{n_{+}, n_{-}} {\otimes}I_{{\mathsf{H}}}) (I_{{\mathbb E}(n_+)} {\otimes}{\widetilde}{T}(n_{-}))^*.\end{aligned}$$
The fact that a doubly commuting dilation is automatically $^*$-regular has been explained in the discussion before the theorem (and does not require minimality). Suppose then that ${\overrightarrow{V}}$ (acting on ${\mathsf{K}}$) is a minimal $^*$-regular dilation of ${\overrightarrow{T}}$ (acting on ${\mathsf{H}}$) and fix $i,j \in {\{1,\ldots,r\}}$, $i \neq j$. By Lemma \[kerlem\] it is enough to show that ${\widetilde}{V}^{(i)} ({{\textup{Ker}}}(\Gamma_{ij}^*)) \subset {{\textup{Ker}}}({\widetilde}{V}^{(j)^*})$, where $\Gamma_{ij}= I_{E_i} {\otimes}{\widetilde}{V}^{(j)}$. As ${\overrightarrow{V}}$ is minimal, $E_i {\otimes}E_j {\otimes}_{\pi} {\mathsf{K}}$ is generated by $\{e {\otimes}f {\otimes}{\widetilde}{V}(m) (g {\otimes}\xi): e \in E_i, f \in E_j, m \in {{\mathbb N}_0^r}, g \in {\mathbb E}(m), \xi
\in {\mathsf{H}}\}$. This implies that ${{\textrm{Ran}}}(\Gamma_{ij})$ is generated by $$\{e {\otimes}{\widetilde}{V}^{(j)}( f {\otimes}{\widetilde}{V}(m) (g {\otimes}\xi)) : e \in
E_i, f \in E_j, m \in {{\mathbb N}_0^r}, g \in {\mathbb E}(m),\xi \in {\mathsf{H}}\},$$ so also by $$\{e {\otimes}{\widetilde}{V}(m) (g {\otimes}\xi) : e \in
E_i, m \in {{\mathbb N}_0^r}, m_j \neq 0, g \in {\mathbb E}(m),\xi \in {\mathsf{H}}\}.$$ This in turn implies that ${\mathsf{K}}_0:={{\textup{Ker}}}(\Gamma_{ij}^*) =
{{\textrm{Ran}}}(\Gamma_{ij})^{\perp} \subset E_i {\otimes}_{\pi} {\mathsf{K}}$ is equal to the subspace generated by $$\left\{P_{{\mathsf{K}}_0} \left(e {\otimes}{\widetilde}{V}(m) (g {\otimes}\xi)\right) : e \in
E_i, m \in {{\mathbb N}_0^r}, m_j = 0, g \in {\mathbb E}(m),\xi \in {\mathsf{H}}\right\}.$$ Suppose then that $e \in E_i, m \in {{\mathbb N}_0^r}, m_j = 0, g \in {\mathbb E}(m),\xi \in {\mathsf{H}}$. We will show that $$P_{{\mathsf{K}}_0} (e {\otimes}{\widetilde}{V}(m) (g {\otimes}\xi) ) = e {\otimes}{\widetilde}{V}(m) (g {\otimes}\xi - g {\otimes}{\widetilde}{V}^{(j)}
{\widetilde}{T}^{(j)^*} \xi). \label{projection}$$ Note that as $e$ does not play any significant role here, remembering the remarks made in the beginning of the proof of Lemma \[kerlem\] it is enough to show that $$P_{{\mathsf{K}}_1} z = z - v,$$ where $ z={\widetilde}{V}(m) (g {\otimes}\xi)$, $ v = {\widetilde}{V}(m) (g {\otimes}{\widetilde}{V}^{(j)} {\widetilde}{T}^{(j)^*} \xi)$ and ${\mathsf{K}}_1 = {{\textup{Ker}}}({\widetilde}{V}^{(j)^*})$. As $v \in {{\textrm{Ran}}}({\widetilde}{V}^{(j)})$, it suffices if we can show that $z-v \perp {{\textrm{Ran}}}({\widetilde}{V}^{(j)})$. We are going to exploit minimality once more. Let $f \in E_j, n \in {{\mathbb N}_0^r}$, $h \in {\mathbb E}(n), \eta \in {\mathsf{H}}$ and compute $$A :=\langle {\widetilde}{V}^{(j)} (f {\otimes}{\widetilde}{V}(n) (h {\otimes}\eta)), z \rangle = \langle f {\otimes}h {\otimes}\eta, {\widetilde}{V}(n+e_j)^* {\widetilde}{V} (m) (g {\otimes}\xi)\rangle$$ Let now $l = (n+e_j) \wedge m$, $p = n+e_j -l$, $q=m-l$. Note that $l_j = 0$, $p_j \neq 0$. Then $$\begin{aligned}
A &= \langle f {\otimes}h {\otimes}\eta, (t_{l,j} {\otimes}I_{{\mathbb E}(p-e_j)} {\otimes}P_{{\mathsf{H}}}) ( I_{{\mathbb E}(l)} {\otimes}{\widetilde}{V} (p)^*) {\widetilde}{V}(l)^*{\widetilde}{V}(l) ( I_{{\mathbb E}(l)} {\otimes}{\widetilde}{V}
(q)) (g {\otimes}\xi)\rangle \\ &= \langle (t_{j,l} {\otimes}I_{{\mathbb E}(p-e_j){\otimes}_{\sigma} {\mathsf{H}}}) (f {\otimes}h {\otimes}\eta), (I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(p)} {\otimes}P_{{\mathsf{H}}}) ( I_{{\mathbb E}(l)} {\otimes}{\widetilde}{V} (p)^*) ( I_{{\mathbb E}(l)} {\otimes}{\widetilde}{V} (q)) (g {\otimes}\xi)\rangle\end{aligned}$$ The $^*$-regularity condition implies that for all $l \in {{\mathbb N}_0^r}$ and all $p,q \in {{\mathbb N}_0^r}$ such that $p \wedge q =0$ there is $$\begin{aligned}
(I_{{\mathbb E}(l)} & {\otimes}I_{{\mathbb E}(p)} {\otimes}P_{{\mathsf{H}}}) (I_{{\mathbb E}(l)} {\otimes}{\widetilde}{V}(p)^*) (I_{{\mathbb E}(l)} {\otimes}{\widetilde}{V}(q)) |_{{\mathbb E}(l+q) {\otimes}_{\sigma} {\mathsf{H}}}
\\&= (I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(p)} {\otimes}{\widetilde}{T}(q)) (I_{{\mathbb E}(l)} {\otimes}t_{q,p} {\otimes}I_{{\mathsf{H}}}) (I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(q)} {\otimes}{\widetilde}{T}(p))^*)\end{aligned}$$ Therefore $$\begin{aligned}
A =& \langle (t_{j,l} {\otimes}I_{{\mathbb E}(p-e_j){\otimes}_{\sigma} {\mathsf{H}}}) (f {\otimes}h {\otimes}\eta),
\\&
(I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(p)} {\otimes}{\widetilde}{T}(q)) (I_{{\mathbb E}(l)} {\otimes}t_{q,p} {\otimes}I_{{\mathsf{H}}}) (I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(q)} {\otimes}{\widetilde}{T}(p)^*) (g
{\otimes}\xi)\rangle.\end{aligned}$$ Similarly $$\begin{aligned}
B:= & \langle {\widetilde}{V}^{(j)} (f {\otimes}{\widetilde}{V}(n) (h {\otimes}\eta)), v \rangle = \langle f {\otimes}h {\otimes}\eta, {\widetilde}{V}(n+e_j)^* {\widetilde}{V} (m) (g
{\otimes}{\widetilde}{V}^{(j)} {\widetilde}{T}^{(j)^*} \xi )\rangle\\& =
\langle f {\otimes}h {\otimes}\eta, {\widetilde}{V}(n+e_j)^* {\widetilde}{V} (m+e_j) (g {\otimes}{\widetilde}{T}^{(j)^*} \xi )\rangle.\end{aligned}$$ Put now $l' = (n+e_j) \wedge (m+e_j)$, $p' = n+e_j -l'$, $q'=m+e_j-l'$. Note that $l'=l+e_j$, $p'=p-e_j$, $ q'= q$. Continuing as before we obtain (note that $t_{j,l}$ no longer features, as $l'_j \neq 0$) $$\begin{aligned}
B &= \langle f {\otimes}h {\otimes}\eta, (I_{l'} {\otimes}I_{{\mathbb E}(p')} {\otimes}P_{{\mathsf{H}}}) ( I_{{\mathbb E}(l')} {\otimes}{\widetilde}{V} (p')^*) {\widetilde}{V}(l')^*{\widetilde}{V}(l') ( I_{{\mathbb E}(l')} {\otimes}{\widetilde}{V} (q')) (g {\otimes}{\widetilde}{T}^{(j)^*} \xi)\rangle
\\ &= \langle f {\otimes}h {\otimes}\eta,
(I_{{\mathbb E}(l')} {\otimes}I_{{\mathbb E}(p')} {\otimes}{\widetilde}{T}(q')) (I_{{\mathbb E}(l')} {\otimes}t_{q',p'} {\otimes}I_{{\mathsf{H}}}) (I_{{\mathbb E}(l')} {\otimes}I_{{\mathbb E}(q')} {\otimes}{\widetilde}{T}(p')^*) (g
{\otimes}{\widetilde}{T}^{(j)^*} \xi)\rangle
\\&= \langle f {\otimes}h {\otimes}\eta, (I_{{\mathbb E}(l+e_j)} {\otimes}I_{{\mathbb E}(p-e_j)} {\otimes}{\widetilde}{T}(q)) \\
& \;\;\hspace*{2cm} (I_{{\mathbb E}(l+e_j)} {\otimes}t_{q,p-e_j} {\otimes}I_{{\mathsf{H}}}) (I_{{\mathbb E}(l)} {\otimes}t_{q,j} {\otimes}I_{{\mathbb E}(p-e_j)})
(I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(q)} {\otimes}{\widetilde}{T}(p)^*) (g
{\otimes}\xi)\rangle.\end{aligned}$$ The comparison of the formulas above shows that $A=B$ if only $$\begin{aligned}
(t_{l,j} & {\otimes}I_{{\mathbb E}(p-e_j)}{\otimes}I_{{\mathbb E}(q)} ) (I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(p)} {\otimes}I_{{\mathbb E}(q)}) (I_{{\mathbb E}(l)} {\otimes}t_{q,p} ) (I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(q)} {\otimes}I_{{\mathbb E}(p)})
\\&= (I_{{\mathbb E}(l+e_j)} {\otimes}I_{{\mathbb E}(p-e_j)} {\otimes}I_{{\mathbb E}(q)}) (I_{{\mathbb E}(l+e_j)} {\otimes}t_{q,p-e_j}) (I_{{\mathbb E}(l)} {\otimes}t_{q,j} {\otimes}I_{{\mathbb E}(p-e_j)}) (I_{{\mathbb E}(l)} {\otimes}I_{{\mathbb E}(q)} {\otimes}I_{{\mathbb E}(p)})\end{aligned}$$ This can be further reduced to checking two equalities $$t_{l,j} {\otimes}I_{{\mathbb E}(p-e_j)} =I_{{\mathbb E}(l+e_j)} {\otimes}I_{{\mathbb E}(p-e_j)}$$ and $$I_{{\mathbb E}(l)} {\otimes}t_{q,p} = (I_{{\mathbb E}(l+e_j)} {\otimes}t_{q,p-e_j}) (I_{{\mathbb E}(l)} {\otimes}t_{q,j} {\otimes}I_{{\mathbb E}(p-e_j)});$$ these finally are simple consequences of the definition of $t_{m,n}$ in the beginning of Section \[prodsys\].
The equality $A=B$ implies that $$\langle {\widetilde}{V}^{(j)} (f {\otimes}{\widetilde}{V}(n) (h {\otimes}\eta)), z-v \rangle = 0.$$ As $f \in E_j, n \in {{\mathbb N}_0^r}$, $h \in {\mathbb E}(n), \eta \in {\mathsf{H}}$ are arbitrary and ${\overrightarrow{V}}$ is minimal, $z-v \perp {{\textrm{Ran}}}({\widetilde}{V}^{(j)})$ and is proved. Note now that $$\begin{aligned}
{\widetilde}{V}^{(i)} ( P_{{\mathsf{K}}_0} (e {\otimes}z)) &= {\widetilde}{V}^{(i)} (e{\otimes}(z - v))\\&=
{\widetilde}{V}^{(i)} (e {\otimes}{\widetilde}{V}(m) (g {\otimes}\xi - g {\otimes}{\widetilde}{V}^{(j)} {\widetilde}{T}^{(j)^*} \xi)) \\&= {\widetilde}{V}^{(m+e_i)} (e {\otimes}g {\otimes}\xi - e
{\otimes}g {\otimes}{\widetilde}{V}^{(j)} {\widetilde}{T}^{(j)^*} \xi) \\&= P_{{\mathsf{K}}_0} {\widetilde}{V}^{(m+e_i)} (e {\otimes}g {\otimes}\xi) \in {\mathsf{K}}_0.\end{aligned}$$ This ends the proof.
It follows from the theorem above that a minimal isometric doubly commuting dilation of a representation of a product system is unique up to a unitary equivalence, as condition together with minimality determines scalar products between all vectors in ${\mathsf{K}}$. In general a minimal isometric dilation need not be unique. Concrete examples of this phenomenon can be found in [@DPY] (Examples 4.3 and 4.4).
The following result can be shown in a similar way to Proposition 2.6 in [@Wold].
If ${\overrightarrow{V}}$ is a minimal isometric doubly commuting dilation of a coisometric representation ${\overrightarrow{T}}$, then ${\overrightarrow{V}}$ is coisometric.
Generalised Poisson transform and isometric dilations
=====================================================
In this section we describe how to construct isometric dilations via the generalised Poisson transform associated with a given representation of a product system. In a similar multi-dimensional context it has been first introduced in [@PPois]; the one-dimensional counterpart for representations of $W^*$-correspondences has been recently investigated in [@MSker].
The next definition describes a natural variation on the type of conditions considered when one wants to construct isometric dilations of higher-rank objects (see [@Nagy] and references therein) and should be compared to the introduced earlier Brehmer-Solel condition.
\[PoTcond\] Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${\mathbb N}_0^r$. For a representation $(\sigma, T^{(1)}, \ldots, T^{(r)})$ of ${\mathbb E}$ on a Hilbert space ${\mathsf{H}}$ define the defect operator ($s \in (0,1)$) $$\label{DeltaT} {\Delta_s({\overrightarrow{T}})}=
\sum_{n \in {{\mathbb N}_0^r}, n\leq e} (-s^2)^{|n|} {\widetilde}{T}(n) {\widetilde}{T}(n)^*.$$ The representation ${\overrightarrow{T}}$ is said to satisfy the Popescu condition (or condition ‘P’) if there exists $\rho\in (0,1)$ such that for all $s \in (\rho,1)$ the operator ${\Delta_s({\overrightarrow{T}})}$ is positive.
The condition above in a similar form first appeared in [@PPois]. Its variant for families of contractions associated with higher-rank graphs was extensively studied in [@graphdil]. Because of the condition the defect operator ${\Delta_s({\overrightarrow{T}})}$ is in $\sigma({\mathsf{A}})'$. It is easy to see that if ${\overrightarrow{T}}$ is doubly commuting or coisometric then it satisfies the Popescu condition.
\[PoisE\] Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${\mathbb N}_0^r$ having a normal ordering property and let ${\overrightarrow{T}}=(\sigma, T^{(1)}, \ldots, T^{(r)})$ be a representation of ${\mathbb E}$ on a Hilbert space ${\mathsf{H}}$ satisfying the Popescu condition. Then there exists a unique continuous linear map $R_{{\overrightarrow{T}}}:{\mathcal{T}_{{\mathbb E}}}\to B({\mathsf{H}})$ satisfying $$R_{{\overrightarrow{T}}}(L_{e} L_{f}^*) = T(n)(e) (T(m)(f))^*, \;\;\;\; n,m \in {{\mathbb N}_0^r}, e\in {\mathbb E}(n),f \in {\mathbb E}(m).$$ The map $R_{{\overrightarrow{T}}}$ will be called the generalised ${\mathbb E}$-Poisson transform (associated with ${\overrightarrow{T}}$). It is completely positive and contractive, unital if ${\mathcal{T}_{{\mathbb E}}}$ is unital.
The proof is almost identical to the one given for the case of ${\Lambda}$-Poisson transforms associated with higher-rank graphs in [@graphdil]. We will therefore only indicate the main points and extra difficulties arising here. Let $s \in (0,1)$ and consider the operator ${\Gamma_s({\overrightarrow{T}})}\in B({\mathsf{H}})$ given by $${\Gamma_s({\overrightarrow{T}})}(\xi) = \sum_{n \in {{\mathbb N}_0^r}} s^{2|n|} {\widetilde}{T}(n) (I_{{\mathbb E}(n)} {\otimes}{\Delta_s({\overrightarrow{T}})}) {\widetilde}{T}(n)^* \xi$$ ($\xi \in {\mathsf{H}}$). It can be checked that ${\Gamma_s({\overrightarrow{T}})}=I_{{\mathsf{H}}}$ (see Lemma 2.1 of [@graphdil]). CHECK!!! Let $\rho\in (0,1)$ be such that for all $s \in (\rho,1)$ the operator ${\Delta_s({\overrightarrow{T}})}$ is positive. As ${\Delta_s({\overrightarrow{T}})}\in \sigma({\mathsf{A}})'$, also $({\Delta_s({\overrightarrow{T}})})^{\frac{1}{2}} \in
\sigma({\mathsf{A}})'$ and moreover for each $n \in {{\mathbb N}_0^r}$ the operator $I_{{\mathbb E}(n)} {\otimes}{\Delta_s({\overrightarrow{T}})}$ on ${\mathbb E}(n) {\otimes}_{\sigma} {\mathsf{H}}$ is positive, $$(I_{{\mathbb E}(n)} {\otimes}{\Delta_s({\overrightarrow{T}})})^{\frac{1}{2}} = I_{{\mathbb E}(n)} {\otimes}{\Delta_s({\overrightarrow{T}})}^{\frac{1}{2}}.$$ Similarly, if $n \in {\mathbb N}^r, T \in \mathcal{L}({\mathbb E}(n))$, $S \in
\sigma({\mathsf{A}})'$, then $$(T {\otimes}S)^* = T^* {\otimes}S^*.$$ These properties will be further used without any comments.
Define the isometry ${W_s({\overrightarrow{T}})}: {\mathsf{H}}\to {\mathcal{F}}_{{\mathbb E}} {\otimes}_{\sigma} {\mathsf{H}}$ by $${W_s({\overrightarrow{T}})}\xi = \bigoplus_{n \in {{\mathbb N}_0^r}} s^{|n|} (I_{{\mathbb E}(n)} {\otimes}{\Delta_s({\overrightarrow{T}})}^{\frac{1}{2}}) {\widetilde}{T}(n)^* \xi.$$ Let the map ${R_{s,{\overrightarrow{T}}}}:\mathcal{L}({\mathcal{F}}_{{\mathbb E}}) \to B({\mathsf{H}})$ be given by the formula $${R_{s,{\overrightarrow{T}}}}(x) = {W_s({\overrightarrow{T}})}^* (x {\otimes}I_{{\mathsf{H}}}) {W_s({\overrightarrow{T}})},$$ It is clear that ${R_{s,{\overrightarrow{T}}}}$ is completely positive and contractive. Moreover for any $e \in {\mathbb E}(n), f \in {\mathbb E}(m)$ ($n,m \in {{\mathbb N}_0^r}$) $$\label{TPois}
{R_{s,{\overrightarrow{T}}}}(L_e L_f^*) = s^{|n| + |m|}T(n)(e) T(m)(f)^*.$$ Indeed, let $e,f$ be as above. Note first that for all $n'\in {{\mathbb N}_0^r}$ $${\widetilde}{T}(n') (L_e {\otimes}I_{{\mathbb E}(n'-n){\otimes}_{\sigma} {\mathsf{H}}}) = T(n) (e) {\widetilde}{T}(n'-n),$$ so also $$(L_e^* {\otimes}I_{{\mathbb E}(n'-n){\otimes}_{\sigma}{\mathsf{H}}}) {\widetilde}{T}(n')^* = {\widetilde}{T}(n'-n)^* (T(n) (e))^*.$$ Compute further ($ \xi, \eta \in {\mathsf{H}}$): $$\begin{aligned}
{\left\langle}\eta, \right.& \left. {R_{s,{\overrightarrow{T}}}}(L_{e} L_{f}^*) \xi {\right\rangle}= {\left\langle}{W_s({\overrightarrow{T}})}\eta, \left( L_{e} L_{f}^* {\otimes}I_{{\mathsf{H}}} \right){W_s({\overrightarrow{T}})}\xi {\right\rangle}\\
&= {\left\langle}\sum_{n' \in {{\mathbb N}_0^r}} s^{|n'|} (I_{{\mathbb E}(n')} {\otimes}{\Delta_s({\overrightarrow{T}})}^{\frac{1}{2}}) {\widetilde}{T}(n')^* \eta, \right.\\&\;\;\;\;\;\;\; \left.
\left(L_{e} L_{f}^* {\otimes}I_{{\mathsf{H}}}
\right) \sum_{m' \in {{\mathbb N}_0^r}} s^{|m'|} (I_{{\mathbb E}(m')} {\otimes}{\Delta_s({\overrightarrow{T}})}^{\frac{1}{2}}) {\widetilde}{T}(m')^* \xi {\right\rangle}\\
=& {\left\langle}\sum_{n' \in {{\mathbb N}_0^r}, n' \geq n} s^{|n'|} (I_{{\mathbb E}(n'-n)} {\otimes}{\Delta_s({\overrightarrow{T}})}^{\frac{1}{2}}) (L_{e}^* {\otimes}I_{{\mathbb E}(n'-n){\otimes}_{\sigma}{\mathsf{H}}})
{\widetilde}{T}(n')^* \eta, \right. \\ & \left. \;\;\;\;\sum_{m' \in {{\mathbb N}_0^r}, m' \geq m} s^{|m'|} (I_{{\mathbb E}(m'-m)} {\otimes}{\Delta_s({\overrightarrow{T}})}^{\frac{1}{2}})
(L_{f}^* {\otimes}I_{{\mathbb E}(m'-m){\otimes}_{\sigma}{\mathsf{H}}}) {\widetilde}{T}(m')^* \xi {\right\rangle}\\
=&\sum_{p \in {{\mathbb N}_0^r}} {\left\langle}s^{|p|+|n|} (I_{{\mathbb E}(p)} {\otimes}{\Delta_s({\overrightarrow{T}})}^{\frac{1}{2}}) {\widetilde}{T}(p)^* (T(n) (e))^* \xi, \right. \\&
\;\;\;\;\;\;\;\;\; \left. s^{|p|+|m|} (I_{{\mathbb E}(p)} {\otimes}{\Delta_s({\overrightarrow{T}})}^{\frac{1}{2}}) {\widetilde}{T}(p)^* (T(m) (f))^* \eta {\right\rangle}\\
=& s^{|n|+|m|} {\left\langle}(T(n) (e))^* \xi, \left( \sum_{p \in {{\mathbb N}_0^r}} s^{2|p|} {\widetilde}{T}(p) (I_{{\mathbb E}(p)} {\otimes}{\Delta_s({\overrightarrow{T}})}) {\widetilde}{T}(p)^* \right) (T(m) (f))^* \eta {\right\rangle}\\
=& s^{|n|+|m|} {\left\langle}(T(n) (e))^* \xi, {\Gamma_s({\overrightarrow{T}})}(T(m) (f))^* \eta {\right\rangle}= {\left\langle}\xi, s^{|n| + |m|}T(n)(e) T(m)(f)^* \eta {\right\rangle}.\end{aligned}$$ It is now easy to see that by the normal ordering property the limit $\lim_{s \to 1^-} {R_{s,{\overrightarrow{T}}}}(x)$ in the norm topology exists for each $x \in {\mathcal{T}_{{\mathbb E}}}$ , and moreover the map ${R_{{\overrightarrow{T}}}}:{\mathcal{T}_{{\mathbb E}}}\to B({\mathsf{H}})$ defined by $${R_{{\overrightarrow{T}}}}(x) = \lim_{s
\to 1^-} {R_{s,{\overrightarrow{T}}}}(x), \;\;\; x \in {\mathcal{T}_{{\mathbb E}}}$$ satisfies all the requirements of the theorem. Uniqueness is another consequence of the normal ordering property.
\[main\] Let ${\mathbb E}$ be a product system of $C^*$-correspondences over ${\mathbb N}_0^r$ having a normal ordering property and let ${\overrightarrow{T}}=(\sigma,
T^{(1)}, \ldots, T^{(r)})$ be a representation of ${\mathbb E}$ on a Hilbert space ${\mathsf{H}}$ satisfying the Popescu condition. Then ${\overrightarrow{T}}$ has an isometric dilation.
Consider the minimal Stinespring dilation of the Poisson transform ${R_{{\overrightarrow{T}}}}$ constructed in Theorem \[PoisE\]. This provides us with a Hilbert space ${\mathsf{K}}$, a representation $\rho: {\mathcal{T}_{{\mathbb E}}}\to B({\mathsf{K}})$ and an operator $V\in B({\mathsf{H}};{\mathsf{K}})$ such that for all $x
\in {\mathcal{T}_{{\mathbb E}}}$ $${R_{{\mathcal{V}}}}(x) = V^* \rho(x) V$$ and ${\mathsf{K}}= \overline{{\textrm{Lin}}} \{\rho(x)V\xi: x \in {\mathcal{T}_{{\mathbb E}}}, \xi \in {\mathsf{H}}\}$. We may assume that $V$ is an isometry, if necessary extending ${R_{{\overrightarrow{T}}}}$ in the unital manner to the unitisation of ${\mathcal{T}_{{\mathbb E}}}$ in $\mathcal{L}({\mathcal{F}}_{{\mathbb E}})$. This allows us to view ${\mathsf{H}}$ as a subspace of ${\mathsf{K}}$. Define for each $i \in {{\mathbb N}_0^r}$, $e \in E_i$ $$V^{(i)}(e) = \rho(L_e);$$ and for $a \in {\mathsf{A}}$ (note that in our framework ${\mathsf{A}}= {\mathbb E}(0) \subset {\mathcal{F}}_{{\mathbb E}}$) $$\pi(a) = \rho(L_a).$$ It is clear that the tuple ${\overrightarrow{V}}=(\pi, V^{(1)},\ldots, V^{(r)})$ is an isometric representation of ${\mathbb E}$, as it is a $^*$-homomorphic image of the Fock-Toeplitz representation.
The fact that condition is satisfied follows directly from the definition of ${\overrightarrow{V}}$, so it remains to establish that each $(V(n)(e))^*$ ($n \in{{\mathbb N}_0^r}$, $e \in {\mathbb E}(n)$) leaves ${\mathsf{H}}$ invariant. By the minimality of the Stinespring dilation we know that $${\mathsf{K}}={\overline}{{\textrm{Lin}}}\{V(m) (f) (V(p)(g))^* \xi: m,p \in {{\mathbb N}_0^r}, f \in {\mathbb E}(m), g \in {\mathbb E}(p), \xi \in {\mathsf{H}}\}.$$ Further given $m,p \in {{\mathbb N}_0^r}, f \in {\mathbb E}(m), g \in {\mathbb E}(p)$ and $\xi, \eta \in {\mathsf{H}}$, $$\begin{aligned}
{\left\langle}\right.& \left. V(m)(f)(V(p)(g))^* \xi,(V(n) (e))^* \eta {\right\rangle}= {\left\langle}V(n) (e) V(m) (f) (V(p)(g))^*\xi, \eta {\right\rangle}\\&=
{\left\langle}V(n+m) (e{\otimes}f) (V(p)(g))^* \xi, \eta {\right\rangle}= {\left\langle}P_{{\mathsf{H}}} V(n+m) (e{\otimes}f) (V(p)(g))^* P_{{\mathsf{H}}}\xi, \eta {\right\rangle}\\&= {\left\langle}T(n+m) (e{\otimes}f) (T(p)(g))^* \xi, \eta {\right\rangle}= {\left\langle}T(m) (f) (T(p)(g))^* \xi, (T(n) (e))^* \eta {\right\rangle}\\&= {\left\langle}P_{{\mathsf{H}}} T(m) (f)
(T(p)(g))^* P_{{\mathsf{H}}} \xi, (T(n) (e))^* \eta {\right\rangle}\\&= {\left\langle}V(m) (f) (V(p)(g))^*\xi, (T(n) (e))^* \eta {\right\rangle}. \end{aligned}$$ This shows that $(V(n)(e))^*|_{{\mathsf{H}}} = (T(n) (e))^*$. In particular $${\mathsf{K}}={\overline}{{\textrm{Lin}}}\{V(m) (f) \xi: m \in {{\mathbb N}_0^r}, f \in {\mathbb E}(m), \xi \in {\mathsf{H}}\}.$$
The approach via a Poisson transform suggests that the constructed dilation should be $^*$-regular. If the creation operators in ${\mathcal{T}_{{\mathbb E}}}$ satisfy some variant of the double commutativity, this will be the case (see Theorem \[quoted\]). Recall that minimal $^*$-regular dilations are unique, as explained in the comments after Theorem \[dc=streg\]. Once again we see here potential analogies between the normal ordering property, compact alignment and double commutativity of the creation operators.
Product system of Hilbert bimodules associated to a higher rank graph {#graphrep}
=====================================================================
In this section we recall a construction of a product system of $C^*$-correspondences associated to a higher-rank graph ${\Lambda}$ introduced in [@toep] and describe its representations in terms of the ${\Lambda}$-families of operators on a Hilbert space.
A rank-$r$ graph ${\Lambda}$ is a small category with set of objects $\Lambda^0$ and shape functor $\sigma : \Lambda \to {\bf N}^r$ (where ${\bf
N}^r$ is viewed as the category with one object and morphisms ${{\mathbb N}_0^r}$) satisfying the *factorisation property* defined in [@kupa]. If $n \in {{\mathbb N}_0^r}$ the set of morphisms in ${\Lambda}$ of shape $n$ is denoted by ${\Lambda}^n$. Further for each $a \in {\Lambda}^0$ and $n \in {{\mathbb N}_0^r}$ write ${\Lambda}^n_a:=\{ {\lambda}\in {\Lambda}: s({\lambda}) = a, \sigma({\lambda}) =n\}$ and $|\lambda |=|\sigma(\lambda)|$. The morphisms in $\Lambda$ may be thought of as paths in a ‘multi-coloured’ graph with vertices indexed by the set ${\Lambda}^0$. The range and source maps are respectively denoted by $r:{\Lambda}\to {\Lambda}^0$ and $s:{\Lambda}\to {\Lambda}^0$. The factorisation property says that if $m,n \in {{\mathbb N}_0^r}$ then every morphism $\lambda \in \Lambda^{m+n}$ is a unique product $\lambda=\mu \nu$ of a $\mu \in \Lambda^m$ and $\nu \in \Lambda^n$, where $s(\mu)=r(\nu)$.
A rank-$r$ graph ${\Lambda}$ is called *finitely aligned* if for each ${\lambda},\mu \in {\Lambda}$ the set of minimal common extensions of ${\lambda}$ and $\mu$, that is $MCE({\lambda}, \mu):=\{ \nu \in {\Lambda}:
\exists_{\alpha ,\beta \in \Lambda} \; \nu = {\lambda}\alpha = \mu \beta, \: \sigma({\lambda}\alpha) =
\sigma({\lambda}) \vee \sigma(\mu)\}$, is finite.
In [@toep] it was shown that every higher-rank graph can be viewed as a product system of rank-1 graphs and this point of view leads to associating to such a graph a product system of $C^*$-correspondences. We rephrase this construction below - note that our conventions on the rank and source follow [@book] rather than [@toep] and we are solely interested in product systems over ${\mathbb N}_0^r$ (which leads to certain simplifications).
Let ${\mathsf{A}}_0= C_0 ({\Lambda}^0)$ denote the $C^*$-algebra of all complex-valued functions on ${\Lambda}^0$ vanishing at infinity. Let $j \in
{\{1,\ldots,r\}}$. Define the $C^*$-correspondence $E_j ({\Lambda})$ over ${\mathsf{A}}_0$ as follows: $E_j({\Lambda})$ consists of these functions $x:{\Lambda}^{e_j}\to{\mathbb C}$ which are ‘locally square integrable’, i.e. for each $a \in {\Lambda}^0$ $$x_a:=\sum_{{\lambda}\in {\Lambda}^{e_j}_a} |x({\lambda})|^2 < \infty$$ and the function $a \to x_a$ vanishes at infinity. The actions of ${\mathsf{A}}_0$ on $E_j({\Lambda})$ are defined via ($f \in {\mathsf{A}}_0, x \in E_j({\Lambda}), {\lambda}\in {\Lambda}^{e_j}$) $$(x \cdot f)
({\lambda}) = x ({\lambda}) f(s({\lambda})), \;\;\;(f \cdot x) ({\lambda}) = f(r({\lambda})) x({\lambda}), \label{action}$$ and the ${\mathsf{A}}_0$ valued scalar product by ($x,y \in E_j({\Lambda}), a \in {\Lambda}^0$) $$\langle x,
y {\rangle}(a) = \sum_{{\lambda}\in {\Lambda}^{e_j}_a} {\overline}{x({\lambda})} y ({\lambda}).\label{prod}$$ As finitely supported functions are dense in $E_j({\Lambda})$, it is easy to see that each of the $C^*$-correspondences $E_j({\Lambda})$ is essential: ${\overline}{{\mathsf{A}}_0 E_j({\Lambda})}=
E_j({\Lambda})$.
It will also be important at a certain point to consider the natural operator space structure of $E_j({\Lambda})$. Intuitively one should think of $E_j$ as a bundle of Hilbert spaces over ${\Lambda}^0$ and observe that in the linking algebra picture the Hilbert spaces in question act as columns. Therefore the natural operator space structure on $E_j$ is the one coming from viewing it as a bundle of operator spaces $({\mathsf{H}}_a)_{\textrm{c}}$. The explicit formula for the matricial norms is as follows: $$\label{opstruct} \left\|(x_{ij})_{i,j=1}^n
\right\| = \sup_{a\in {\Lambda}^0} \left\| \left (\sum_{l=1}^n \sum_{{\lambda}\in {\Lambda}^{e_j}_a}
{\overline}{x_{li}({\lambda})} x_{lk}({\lambda})\right)_{i,k=1}^n \right\|_{M_n}.$$
To introduce on $(E_1({\Lambda}), \ldots, E_r({\Lambda}))$ the structure of a product system we identify $E_i({\Lambda}) {\otimes}E_j ({\Lambda})$ with the space of all functions $z: {\Lambda}^{e_i + e_j}\to
{\mathbb C}$ such that for each $a \in {\Lambda}^0$ $$z_a:=\sum_{{\lambda}\in {\Lambda}^{e_i+ e_j}_a} |z({\lambda})|^2 < \infty$$ and the function $a \to z_a$ vanishes at infinity. The identification is implemented via the factorisation property: given $\nu \in {\Lambda}^{e_i+e_j}$ we can decompose it uniquely as $\nu = {\lambda}\mu$, $\nu \in {\Lambda}^{e_i}, \mu \in {\Lambda}^{e_j}$ and for $x \in E_i({\Lambda})$, $y \in E_j({\Lambda})$ define $$(x {\otimes}y) (\nu):= x({\lambda}) y (\mu).$$ In other words, if $x \in E_i({\Lambda})$, $y \in E_j({\Lambda})$ then for $\nu \in {\Lambda}^{e_i +e_j}$ $$t_{i,j} (x {\otimes}y) (\nu) = x({\lambda}) y (\mu), \;\; \textrm{where} \;\; {\lambda}\in {\Lambda}^{e_i}, \mu \in {\Lambda}^{e_j}, \nu = \mu
{\lambda}.$$ Note that this leads to natural identifications of ${\mathbb E}(n)$ with the spaces of ‘locally square integrable’ functions on ${\Lambda}^n$. Precisely speaking, if for each $n \in {{\mathbb N}_0^r}$ we define $E_n({\lambda})$ to be the space of all functions $x:{\Lambda}^n \to {\mathbb C}$ such that for each $a \in {\Lambda}^0$ $$x_a:=\sum_{{\lambda}\in {\Lambda}^{n}_a} |x({\lambda})|^2 < \infty$$ and the function $a \to x_a$ vanishes at infinity. The actions of ${\mathsf{A}}_0$ and the ${\mathsf{A}}_0$ valued scalar product on $E_n({\Lambda})$ are defined again via formulas and (this time $f \in {\mathsf{A}}_0, x,y \in E_n, {\lambda}\in {\Lambda}^n$). Define for all $n,m \in {\{1,\ldots,r\}}$ the map $U_{n,m}: E_n({\Lambda}) {\otimes}E_n({\Lambda}) \to E_{n+m}({\Lambda})$ via the continuous linear extension of the formula $$U_{n,m} (x {\otimes}y) ({\lambda})= x(\mu ) y (\nu),$$ where $x \in E_n({\Lambda}), y \in E_m({\Lambda}), {\lambda}\in {\Lambda}^{n+m}, {\lambda}= \mu \nu, \mu \in {\Lambda}^{n}, \nu \in {\Lambda}^m$. It can be checked that $U_{n,m}$ is an isomorphism in the category of $C^*$-correspondences. Because of that we will identify ${\mathbb E}(n)$ with $E_n ({\Lambda})$ without any further comments. The resulting product system of $C^*$-correspondences will be called the product system of the graph ${\Lambda}$ and denoted by ${\mathbb E}({\Lambda})$. In what follows we will often view the Dirac functions $\delta_a$ ($a \in {\Lambda}^0$) and $\delta_{{\lambda}}$ (${\lambda}\in {\Lambda}^{e_j}$) as elements respectively of ${\mathsf{A}}_0$ and of $E_{j}({\Lambda})$.
Representations of ${\mathbb E}({\Lambda})$ {#representations-of-mathbb-elambda .unnumbered}
-------------------------------------------
\[TCK\] Suppose that ${\Lambda}$ is a higher-rank graph. A family of partial isometries $\{x_{{\lambda}}:{\lambda}\in {\Lambda}\}$ in a $C^*$-algebra $B$ is called a Toeplitz ${\Lambda}$-family if the following are satisfied:
$\{x_a: a \in {\Lambda}^0\}$ is a family of mutually orthogonal projections;
$x_{{\lambda}} x_{\mu} = x_{{\lambda}\mu}$ if ${\lambda}, \mu \in {\Lambda}$, $s({\lambda}) = r(\mu)$;
$x_{{\lambda}}^* x_{{\lambda}} = x_{s({\lambda})}$ if ${\lambda}\in {\Lambda}$;
if $n \in {{\mathbb N}_0^r}\setminus\{0\}$, $a \in {\Lambda}^0$ and $F\subset\{{\lambda}\in{\Lambda}^n: r({\lambda})=a\}$ is finite then $x_a \geq \sum_{{\lambda}\in F} x_{{\lambda}} x_{{\lambda}}^*$;
If ${\Lambda}$ is finitely aligned and additionally the condition $$\textrm{(v)} \;\; x_{\mu}^* x _{\nu} = \sum_{\mu
\alpha = \nu \beta \in MCE(\mu, \nu)} x_{\alpha} x_{\beta}^*\hspace*{6.5 cm}$$ is satisfied for all $\mu, \nu \in {\Lambda}$, the family $\{x_{{\lambda}}:{\lambda}\in {\Lambda}\}$ is called a Toeplitz-Cuntz-Krieger family.
In [@toep] isometric representations of ${\mathbb E}({\Lambda})$ are called Toeplitz representations. They are given by Toeplitz families.
\[isorep\] Let ${\Lambda}$ be a higher-rank graph. There is a 1-1 correspondence between isometric representations of ${\mathbb E}({\Lambda})$ on a Hilbert space ${\mathsf{H}}$ and Toeplitz ${\Lambda}$ families in $B({\mathsf{H}})$. The correspondence is given by $$\sigma(\delta_{a}) = x_a,\;\;\; a \in {\Lambda}^0,$$ $$T^{(j)} (\delta_{{\lambda}}) = x_{{\lambda}}, \;\;\; j \in {\{1,\ldots,r\}}, {\lambda}\in {\Lambda}^{e_j}.$$
Note that $\sigma$ defined as above is nondegenerate if and only if $\sum_{a \in {\Lambda}_0} x_a =
I_{{\mathsf{H}}}$, where the sum is understood in the strong operator topology.
It is also possible to give an easy characterisation of those isometric representations which are doubly commuting (equivalently, Nica-covariant).
\[[[@toep]]{}, Proposition 6.4 \] \[Nicacov\] Let ${\Lambda}$ be a finitely aligned higher-rank graph. An isometric representation of ${\mathbb E}({\Lambda})$ on a Hilbert space ${\mathsf{H}}$ is doubly commuting if and only if the corresponding Toeplitz family is a Toeplitz-Cuntz-Krieger family.
It is easy to see that if ${\Lambda}$ is finitely aligned then ${\mathbb E}({\Lambda})$ satisfies the normal ordering condition. Note that by Theorem 5.4 of [@toep] ${\Lambda}$ is finitely aligned if and only if ${\mathbb E}({\Lambda})$ is compactly aligned.
We are now ready to define objects which were the main subject of investigation in [@graphdil].
\[Lcont\] Let ${\mathsf{H}}$ be a Hilbert space. A family ${\mathcal{V}}=\{V_{\lambda}:{\lambda}\in {\Lambda}\}$ of operators in $B({\mathsf{H}})$ is called a ${\Lambda}$-contraction if the following conditions are satisfied:
$\forall_{{\lambda},\mu \in \Lambda,\, s({\lambda})\neq r(\mu)} \; V_{{\lambda}} V_{\mu} =0$;
$\forall_{{\lambda},\mu \in \Lambda,\, s({\lambda}) = r(\mu)} \; V_{{\lambda}} V_{\mu} = V_{{\lambda}\mu}$;
$\forall_{n \in {{\mathbb N}_0^r}} \; \sum_{{\lambda}\in {\Lambda}^n} V_{{\lambda}} V_{{\lambda}}^* \leq I$;
each $V_a$ ($a \in {\Lambda}^0$) is an orthogonal projection.
All infinite sums here and in what follows are understood in the strong operator topology.
The definition in [@graphdil] was slightly different as we additionally requested that $\sum_{a\in {\Lambda}^0} V_a =I$. As explained in that paper the distinction is not very important: then conditions (ii) and (iii) imply that each $V_a$ for $a \in
{\Lambda}^0$ is a contractive idempotent, hence a projection (so that in particular (iv) is a consequence of (ii) and (iii)). Further (i) shows that $V_a V_b = 0$ if $b \in {\Lambda}^0$ and $a\neq b$. Denoting by $p$ the sum $\sum_{a \in {\Lambda}^0} V_a$ we see that $V_{{\lambda}} = pV_{{\lambda}}p$ (by (i) and (ii)). Therefore even if $\sum_{a\in {\Lambda}^0} V_a =I$ is not satisfied at the outset, it will be fulfilled by the obvious ${\Lambda}$-contraction on $p{\mathsf{H}}$. In condition (iii) above it is enough to assume that the inequalities hold only for $n$ of the form $e_j$, $j \in {\{1,\ldots,r\}}$.
We write $V_{\lambda \mu}:=0$ if $s({\lambda}) \neq r(\mu)$. Sometimes we will also write $V_{\emptyset}=
I_{{\mathsf{H}}}$.
The following observation is not very complicated but lies at the heart of this section; it was actually the motivating point for trying to extend the results of [@graphdil] to the framework of representations of product systems of $C^*$-correspondences.
\[ccrep\] Let ${\Lambda}$ be a higher-rank graph. There is a 1-1 correspondence between completely contractive representations of ${\mathbb E}({\Lambda})$ on a Hilbert space ${\mathsf{H}}$ and ${\Lambda}$-contractions in $B({\mathsf{H}})$. The correspondence is given by $$\sigma(\delta_{a}) = V_a,\;\;\; a \in
{\Lambda}^0,\label{rep1}$$ $$T^{(j)} (\delta_{{\lambda}}) = V_{{\lambda}}, \;\;\; j \in {\{1,\ldots,r\}}, {\lambda}\in
{\Lambda}^{e_j}.\label{rep2}$$
Let ${\mathcal{V}}$ be a ${\Lambda}$-contraction. Fix for a moment $j \in {\{1,\ldots,r\}}$ and write $E$ instead of $E_j({\Lambda})$. To show that $T:=T^{(j)}$ defined by the linear extension of the formula is completely contractive consider a matrix $(x_{il})_{i,l=1}^n$ of finitely supported functions in $E$ and let $\xi_1, \ldots, \xi_n$ be vectors in ${\mathsf{H}}$. Let $T_n$ denote the $n$-th matrix lifting of $T$ and write $\xi = [\xi_1, \cdots, \xi_n]^{\textrm{T}} \in {\mathsf{H}}^{\oplus n}$. Then $$\begin{aligned}
\|T_n \left((x_{i,k})_{i,k=1}^n\right) \xi\|^2 &= \sum_{i,l,k=1}^n \langle T(x_{ik}) \xi_k, T (x_{il})
\xi_l {\rangle}\\&= \sum_{i,k,l=1}^n \left\langle \sum_{{\lambda}\in {\Lambda}^{e_j}} x_{ik}({\lambda}) V_{{\lambda}} \xi_k ,
\sum_{{\lambda}' \in {\Lambda}^{e_j}} x_{il}({\lambda}') V_{{\lambda}'} \xi_l\right{\rangle}\\&= \sum_{i=1}^n \left\langle
\sum_{{\lambda}\in {\Lambda}^{e_j}} V_{{\lambda}} \sum_{k=1}^n x_{ik}({\lambda}) V_{s({\lambda})} \xi_k, \sum_{{\lambda}' \in
{\Lambda}^{e_j}} V_{{\lambda}'} \sum_{l=1}^n x_{il}({\lambda}') V_{s({\lambda}')} \xi_l\right{\rangle}.\end{aligned}$$ Define for each $i =1, \ldots, n$ and ${\lambda}\in {\Lambda}^{e_j}$ $$\zeta^{i}_{{\lambda}} = \sum_{k=1}^n x_{ik}({\lambda}) V_{s({\lambda})} \xi_k.$$ Then $$\begin{aligned}
\|T_n \left((x_{i,k})_{i,k=1}^n\right)\xi\|^2 = \sum_{i=1}^n \left\langle \sum_{{\lambda}\in
{\Lambda}^{e_j}} V_{{\lambda}} \zeta^i_{{\lambda}}, \sum_{{\lambda}' \in {\Lambda}^{e_j}} V_{{\lambda}'} \zeta^i_{{\lambda}'}\right{\rangle}=\sum_{i=1}^n \left\| \sum_{{\lambda}\in {\Lambda}^{e_j}}V_{{\lambda}} \zeta^i_{{\lambda}}\right\|^2.\end{aligned}$$ The condition (iii) in Definition \[Lcont\] implies that $\| \sum_{{\lambda}\in {\Lambda}^{e_j}}V_{{\lambda}}
\zeta^i_{{\lambda}}\|^2\leq \sum_{{\lambda}\in {\Lambda}^{e_j}} \|\zeta_{{\lambda}}^i\|^2$. Moreover for ${\lambda}\in
{\Lambda}^{e_j}_a$ $$\|\zeta^{i}_{{\lambda}}\|^2 = \|\sum_{k=1}^n x_{ik}({\lambda}) V_a \xi_k\|^2 = \sum_{k,l=1}^n
{\overline}{x_{ik}({\lambda})} x_{il}({\lambda}) \langle V_a \xi_k, V_a \xi_l \rangle$$ so that $$\|T_n \left((x_{i,k})_{i,k=1}^n\right)\xi\|^2 \leq \sum_{i=1}^n \sum_{{\lambda}\in {\Lambda}^{e_j}} \|\zeta_{{\lambda}}^i\|^2 =
\sum_{i=1}^n \sum_{a \in {\Lambda}^0}\sum_{{\lambda}\in {\Lambda}^{e_j}_a}\sum_{k,l=1}^n {\overline}{x_{ik}({\lambda})}
x_{il}({\lambda}) \langle V_a \xi_k, V_a \xi_l \rangle. \label{longlast}$$ Define for each $a \in {\Lambda}_0$ a matrix $A_a \in M_n$ by $$(A_a)_{k,l} = \sum_{i=1}^n \sum_{{\lambda}\in {\Lambda}^{e_j}_a} {\overline}{x_{ik}({\lambda})}
x_{il}({\lambda}).$$ Note that the implies that $$\|(x_{i,k})_{i,k=1}^n\|_{M_n(E)} = \sup_{a \in {\Lambda}^0} \|A_a\|.$$ On the other hand implies that $$\|T_n \left((x_{i,k})_{i,k=1}^n\right) \xi\|^2 \leq \sum_{a \in {\Lambda}^0} \langle P_a^{(n)} \xi, (A_a {\otimes}I_{{\mathsf{H}}}) P_a \xi\rangle,$$ where $P_a^{(n)}=P_a \oplus \cdots \oplus P_a \in B({\mathsf{H}}^{\oplus^n})$. Thus finally $$\|T_n \left((x_{i,k})_{i,k=1}^n\right) \xi\|^2 \leq \sup_{a\in {\Lambda}^0}\{\|A_a\|\} \sum_{a \in {\Lambda}^0} \| P_a^{(n)} \xi\|^2 =
\sup_{a\in {\Lambda}^0}\{\|A_a\|\} \| \xi\|^2,$$ so that $T$ extends to a complete contraction from $E$ to $B({\mathsf{H}})$. The fact that the continuous linear extension of yields a representation of ${\mathsf{A}}_0$ is immediate and then the routine check shows that $(\sigma, T^{(1)}, \ldots,
T^{(r)})$ is a representation of ${\mathbb E}({\Lambda})$.
Conversely, if $(\sigma, T^{(1)}, \ldots, T^{(r)})$ is a representation of ${\mathbb E}({\Lambda})$ we can use formulas and to define operators $V_{{\lambda}}$ for ${\lambda}\in {\Lambda}^0 \cup \bigcup_{j=1}^r {\Lambda}^{e_j}$. Given any $\mu \in {\Lambda}$ due to the factorisation property we can always write it as a concatenation of elements in $ {\Lambda}^0 \cup \bigcup_{j=1}^r {\Lambda}^{e_j}$ and define $V_{\mu}$ as a corresponding composition. The fact that this gives a unique prescription is a consequence of the fact that $(\sigma, T^{(1)}, \ldots, T^{(r)})$ is a representation of ${\mathbb E}({\Lambda})$; moreover it is easy to check that conditions (i), (ii) and (iv) of Definition \[Lcont\] are satisfied (contractive idempotents in $B({\mathsf{H}})$ are orthogonal projections). It remains to check (iii). By (i) and remarks after the definition of a ${\Lambda}$-contraction it is enough to do it for $n=e_j$ ($j\in {\{1,\ldots,r\}}$). Let then $n \in{\mathbb N}$ and let ${\lambda}_1,\cdots, {\lambda}_n$ be distinct elements in ${\Lambda}^{e_j}$. Then the row matrix $[V_{{\lambda}_1} \cdots
V_{{\lambda}_n}]$ is equal to $T^n((x_{1k})_{k=1}^n)$, where $x_{1k} = \delta_{{\lambda}_k}$. It follows easily from that $\|(x_{1k})_{k=1}^n\|_{M_n(E_j)} = 1$, and as $T$ is assumed to be a complete contraction we obtain $\|[V_{{\lambda}_1} \cdots
V_{{\lambda}_n}]\|\leq 1$ and the result follows.
Note that the representation $\sigma$ of ${\mathsf{A}}_0$ associated to a ${\Lambda}$-contraction ${\mathcal{V}}$ is nondegenerate if and only if $\sum_{a \in {\Lambda}^0} V_a = I_{{\mathsf{H}}}$.
Let ${\Lambda}$ be a rank-r graph and let ${\mathcal{V}}$ be a ${\Lambda}$-contraction on a Hilbert space ${\mathsf{H}}$. The representation ${\overrightarrow{T}}$ of ${\mathbb E}({\Lambda})$ associated with ${\mathcal{V}}$ is doubly commuting if and only if for all $i,j\in {\{1,\ldots,r\}}$, ${\lambda}\in {\Lambda}^{e_i}$, $\mu \in
{\Lambda}^{e_j}$, there is $$V_{{\lambda}}^* V_{\mu} = \sum_{\alpha \in {\Lambda}^{e_i}, \beta \in {\Lambda}^{e_j}, \mu \beta = {\lambda}\alpha} V_{\beta} V_{\alpha}^*.$$
Let $i\in {\{1,\ldots,r\}}$, ${\lambda}\in {\Lambda}^{e_i}$ and $\xi \in {\mathsf{H}}$. Then $$T^{(i)} (\delta_{{\lambda}} {\otimes}\xi) = V_{{\lambda}} \xi$$ and it follows that for $\eta \in {\mathsf{H}}$ $$(T^{(i)})^* \eta = \sum_{{\lambda}\in {\Lambda}^{e_i}} \delta_{{\lambda}} {\otimes}V_{{\lambda}}^*
\eta.\label{adjoint}$$ The equivalence of the conditions in the lemma follows from straightforward computations.
It would be interesting and nontrivial to analyse how the results of this section extend to topological higher-rank graphs as discussed for example in [@Trent].
Dilating graph-contractions via dilating representations of associated product systems of Hilbert $C^*$-correspondences {#dilgraph}
=======================================================================================================================
Here we apply the conclusions of the discussions of previous two sections to obtain the dilations of ${\Lambda}$-contractions to Toeplitz-type families.
Let ${\Lambda}$ be a rank-2 graph and let ${\mathcal{V}}$ be a ${\Lambda}$-contraction on a Hilbert space ${\mathsf{H}}$. There exists a Hilbert space ${\mathsf{K}}\supset {\mathsf{H}}$ and a ${\Lambda}$-contraction ${\mathcal{W}}$ on ${\mathsf{K}}$ consisting of partial isometries forming a Toeplitz family such that for each ${\lambda}\in {\Lambda}$ $$W_{{\lambda}}^*|_{{\mathsf{H}}} = V_{{\lambda}}^*.\label{dil}$$ One may assume that ${\mathsf{K}}=
\overline{\textup{Lin}} \{W_{{\lambda}} {\mathsf{H}}: {\lambda}\in {\Lambda}\}$. Under this assumption $\sum_{a \in {\Lambda}^0} W_a = I_{{\mathsf{K}}}$ if $\sum_{a \in
{\Lambda}^0} V_a = I_{{\mathsf{H}}}$.
Let ${\overrightarrow{T}}$ be the representation of ${\mathbb E}({\Lambda})$ associated with ${\mathcal{V}}$ by Lemma \[ccrep\]. From Theorem \[isorep\] it follows that any isometric dilation of ${\overrightarrow{T}}$ has to be given by a Toeplitz family ${\mathcal{W}}$ such that holds. The main statement therefore follows directly from Theorem \[sol1\].
Before we identify necessary conditions for the representation of ${\mathbb E}({\lambda})$ associated to a given ${\Lambda}$-contraction to satisfy the Brehmer-Solel condition we need to understand the Hilbert spaces involved. Observe that if $\sigma$ is a representation of ${\mathsf{A}}_0$ on a Hilbert space ${\mathsf{H}}$ then for each $n \in {{\mathbb N}_0^r}$ the Hilbert space ${\mathbb E}({\lambda})(n) {\otimes}_{\sigma} {\mathsf{H}}$ is isometrically isomorphic to the Hilbert space $ \bigoplus_{a \in {\Lambda}^0} l^2({\Lambda}^n_a;P_a {\mathsf{H}})$, where $P_a= \sigma(\delta_a)$ for each $a \in
{\Lambda}^0$.
Let $r \in {\mathbb N}$, let ${\Lambda}$ be a rank-r graph and let ${\mathcal{V}}$ be a ${\Lambda}$-contraction on a Hilbert space ${\mathsf{H}}$. Define for each $u\subset v \subset {\{1,\ldots,r\}}$ an operator $P_{u,v}$ on the Hilbert space $ \bigoplus_{a \in {\Lambda}^0} l^2({\Lambda}^{e(v)}_a;V_a {\mathsf{H}})$ via the continuous linear extension of the formula $$\label{Pdef} P_{u,v} (\delta_{\mu \nu} {\otimes}\xi) = \bigoplus_{a\in
{\Lambda}^0} \sum_{{\lambda}\in {\Lambda}^{e(v)}_a} \delta_{\mu{\lambda}} {\otimes}V_{{\lambda}}^* V_{\nu} \xi,$$ where $\mu \in {\Lambda}^{e(u)-e(v)}, \nu \in
{\Lambda}^{e(v)}, r(\nu) = s(\mu), \xi \in V_{s(\nu)} {\mathsf{H}}$. Then the associated representation of ${\mathbb E}({\Lambda})$ satisfies the Brehmer-Solel condition if and only if $$\forall_{v \subset {\{1,\ldots,r\}}} \;\;\;\sum_{u\subset v} (-1)^{|u|} P_{u,v} \geq 0.\label{VSolel}$$
Direct consequence of the remark before the lemma and the formula .
Let ${\Lambda}$ be a higher rank graph and let ${\mathcal{V}}$ be a ${\Lambda}$-contraction on a Hilbert space ${\mathsf{H}}$. Suppose that ${\mathcal{V}}$ satisfies the condition , where the operators $P_{u,v}$ are defined by . Then there exists a Hilbert space ${\mathsf{K}}\supset {\mathsf{H}}$ and a ${\Lambda}$-contraction ${\mathcal{W}}$ on ${\mathsf{K}}$ consisting of partial isometries forming a Toeplitz family such that each $W_{{\lambda}}^*$ leaves ${\mathsf{H}}$ invariant and for ${\lambda}\in {\Lambda}$, $\mu \in {\Lambda}$ such that $\sigma({\lambda})_j \neq 0$ implies $\sigma(\mu)_j= 0$ ($j \in {\{1,\ldots,r\}}$) $$P_{{\mathsf{H}}}W_{{\lambda}}^*W_{\mu}|_{{\mathsf{H}}} = V_{{\lambda}}^*V_{\mu}.$$ One may assume that ${\mathsf{K}}= \overline{\textup{Lin}} \{W_{{\lambda}} {\mathsf{H}}: {\lambda}\in {\Lambda}\}$; under this assumption the family ${\mathcal{W}}$ is unique up to unitary equivalence.
Let ${\overrightarrow{T}}$ be the representation of ${\mathbb E}({\Lambda})$ associated with ${\mathcal{V}}$ by Lemma \[ccrep\]. From Theorem \[isorep\] it follows that any isometric dilation of ${\overrightarrow{T}}$ has to be given by a Toeplitz family ${\mathcal{W}}$ such that holds. The existence of a regular dilation to a Toeplitz family is a consequence of Theorem \[sol2\]; Lemma \[simpreg\] implies that regularity of the dilation can be expressed by a simple formula above.
The next corollary is a consequence of Theorem 3.15 of [@Solk] and Lemma \[Nicacov\] above.
\[correg\] Let ${\Lambda}$ be a finitely-aligned higher rank graph and let ${\mathcal{V}}$ be a doubly commuting ${\Lambda}$-contraction on a Hilbert space ${\mathsf{H}}$. Then there exists a Hilbert space ${\mathsf{K}}\supset {\mathsf{H}}$ and a ${\Lambda}$-contraction ${\mathcal{W}}$ on ${\mathsf{K}}$ consisting of partial isometries forming a Toeplitz-Cuntz-Krieger family such that each $W_{{\lambda}}^*$ leaves ${\mathsf{H}}$ invariant and for ${\lambda}\in
{\Lambda}$, $\mu \in {\Lambda}$ such that $\sigma({\lambda})_j \neq 0$ implies $\sigma(\mu)_j= 0$ ($j \in {\{1,\ldots,r\}}$) $$P_{{\mathsf{H}}} W_{{\lambda}}^*W_{\mu}|_{{\mathsf{H}}} = V_{{\lambda}}^*V_{\mu}.$$ One may assume that ${\mathsf{K}}= \overline{\textup{Lin}} \{W_{{\lambda}} {\mathsf{H}}: {\lambda}\in {\Lambda}\}$; under this assumption the family ${\mathcal{W}}$ is unique up to unitary equivalence.
The following definition was introduced in [@graphdil] as a generalisation of the notion of condition ‘P’ suggested in [@PPois].
\[Popcond\] Let ${\mathcal{V}}$ be a ${\Lambda}$-contraction and define for $s \in (0,1)$ the defect operator $$\label{Delta} {\Delta_s({\mathcal{V}})}=
\sum_{\mu \in {\Lambda}, \, \sigma(\mu) \leq e} (-s^2)^{|\mu|} V_{\mu} V_{\mu}^*.$$ The family ${\mathcal{V}}$ is said to satisfy the Popescu condition (or condition ‘P’) if there exists $\rho\in (0,1)$ such that for all $s \in (\rho,1)$ the operator ${\Delta_s({\mathcal{V}})}$ is positive.
In this context Theorem \[main\] can be used to establish the following:
\[quoted\] Let ${\Lambda}$ be a finitely-aligned higher rank graph and let ${\mathcal{V}}$ be a ${\Lambda}$-contraction on a Hilbert space ${\mathsf{H}}$ which satisfies the Popescu condition. Then there exists a Hilbert space ${\mathsf{K}}\supset {\mathsf{H}}$ and a ${\Lambda}$-contraction ${\mathcal{W}}$ on ${\mathsf{K}}$ consisting of partial isometries forming a Toeplitz-Cuntz-Krieger family such that $W_{{\lambda}}^*|_{{\mathsf{H}}} =V_{{\lambda}}^*$ for each ${\lambda}\in {\Lambda}$. One may assume that ${\mathsf{K}}= \overline{\textup{Lin}} \{W_{{\lambda}} {\mathsf{H}}: {\lambda}\in {\Lambda}\}$; under this assumption the family ${\mathcal{W}}$ is unique up to unitary equivalence.
It follows from the remark stated after Lemma \[Nicacov\] that ${\mathbb E}({\Lambda})$ has the normal ordering property. Most of the statements in the theorem follow therefore immediately from Theorem \[main\] by now standard applications of the identifications obtained in Section \[graphrep\]. The only extra element is double commutativity and uniqueness of the dilation. The first follows from the fact that if the graph is finitely aligned, then the Fock-Toeplitz representation is automatically doubly commuting in the natural sense and therefore so is its $^*$-homomorphic image yielding the dilation in Theorem \[main\]. The uniqueness follows from the remarks after Theorem \[dc=streg\].
[fowl]{}
S.Brehmer, Über vetauschbare Kontraktionen des Hilbertschen Raumes, *Acta Sci.Math. Szeged* **22** (1961), 106–111.
K.Davidson, S.C.Power and D.Yang, Dilation theory for rank 2 graph algebras, *J. Operator Theory*, to appear.
N.Fowler, Discrete product systems of Hilbert bimodules, *Pacific J. Math.* **204** (2002), 335–375.
N.Fowler and I.Raeburn, The Toeplitz algebra of a Hilbert bimodule, *Indiana Univ. Math. J.* **48** (1999), 155–181.
D.Gaşpar and N.Suciu, On the intertwinings of regular dilations, *Ann.Pol.Math.* **LXVI** (1997), 105–121.
A.Kumjian and D.Pask, Higher rank graph $C^*$-algebras, *New York J.Math.* **6** (2000), 1–20.
E.C. Lance, “Hilbert $C^*$-modules", LMS Lecture Notes Series **210**, Cambridge University Press, Cambridge, 1995.
P.Muhly and B.Solel, Tensor algebras over $C^*$-correspondences (Representations, dilations and $C^*$-envelopes), *J.Funct.Anal.* **158** (1998), 389–457.
P.Muhly and B.Solel, The Poisson Kernel for Hardy Algebras, *Complex Analysis and Operator Theory*, to appear.
G.Popescu, Poisson transforms on some $C^*$-algebras generated by isometries, *J. Funct.Anal.* **161** (1999), no.1, 27–61.
S.C.Power and B.Solel, Operator algebras associated with unitary commutation relations, *preprint*, arXiv:0704.0079v1.
I.Raeburn, “Graph $C^*$-algebras", CBMS Regional Conference Series in Mathematics, 103, Providence, RI, 2005.
I.Raeburn and A.Sims, Product systems of graphs and the Toeplitz algebras of higher-rank graphs, *J.Operator Theory* **53** (2005), no. 2, 399–429.
I.Raeburn and D.Williams, “Morita Equivalence and Continuous-Trace-$C^*$-Algebras", Mathematical Surveys and Monographs, 60. American Mathematical Society, Providence, RI, 1998.
A.Skalski and J.Zacharias, Wold decomposition for representations of product systems of $C^*$-correspondences, *International Journal of Mathematics* **19** (2008), no.4, 455-479.
A.Skalski and J.Zacharias, Poisson transform for higher-rank graph algebras and its applications, *J.Operator Theory*, to appear.
B.Solel, Representations of product systems over semigroups and dilations of commuting CP maps, *J.Funct.Anal.* **235** (2006), no. 2, 593–618.
B.Solel, Regular dilations of representations of product systems, *Math.Proc.Royal Irish Soc.*, to appear.
B.Sz.-Nagy and C.Foias, “Harmonic analysis of operators on Hilbert space", North Holland, Amsterdam, 1970.
D.Timotin, Regular dilations and models for multicontractions, *Indiana Univ.Math.J.* **47** (1998), no. 2, 593–618.
T.Yeend, Topological higher-rank graphs and the $C^*$-algebras of topological 1-graphs, *in* Operator theory, operator algebras, and applications, 231–244, *Contemp. Math.* **414**, Amer. Math. Soc., Providence, RI, 2006.
[^1]: *Permanent address of the author:* Department of Mathematics, University of Łódź, ul. Banacha 22, 90-238 Łódź, Poland.
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---
abstract: 'Many complex systems have been shown to share universal properties of organization, such as *scale independence*, *modularity* and *self-similarity*. We borrow tools from statistical physics in order to study *structural preferential attachment* (SPA), a recently proposed growth principle for the emergence of the aforementioned properties. We study the corresponding stochastic process in terms of its time evolution, its asymptotic behavior and the scaling properties of its statistical steady state. Moreover, approximations are introduced to facilitate the modelling of real systems, mainly complex networks, using SPA. Finally, we investigate a particular behavior observed in the stochastic process, the *peloton dynamics*, and show how it predicts some features of real growing systems using prose samples as an example.'
author:
- 'Laurent Hébert-Dufresne'
- Antoine Allard
- Vincent Marceau
- 'Pierre-Andr[é]{} Noël'
- 'Louis J. Dubé'
title: |
Structural preferential attachment:\
Stochastic process for the growth of scale-free, modular and self-similar systems
---
Introduction
============
In a recent contribution, we have proposed a model of network organization [@SPA] based on a generalization of the classical preferential attachment principle (PA) [@Simon55; @barabasi99] to a higher order: structural preferential attachment (SPA). In this model, elements of the system join and create structures. In all attachment events, both the element and the structure involved are chosen proportionally to their past activities. Elements can represent money being invested, written words, individuals in a social network, proteins or websites, while the structures can be business firms, semantic fields, friendships and communities, protein complexes or types of activities and interest [@Simon55; @barabasi99; @palla05; @ahn].
SPA can be described by the following stochastic process (see Fig. \[SPA\_scheme\] for a visual aid). At every time step, an element joins a structure. With probability $q$, the element is a new one; or with probability $1-q$, it is chosen among existing elements proportionally to the current number of structures to which they belong (i.e., their *membership* number). Moreover, with probability $p$, the structure is a new one of size $s$; or with probability $1-p$, it is chosen among existing structures proportionally to the current number of elements they possess (i.e., their *size*). Whenever the structure is a new one, the remaining $s-1$ elements involved in its creation are once again preferentially chosen among existing nodes. The basic structure size $s$ is called the *system base* and refers to the smallest structural unit of the system. For example, if $s=1$, the system base is simply the elements themselves and we refer to this version as *node-based* SPA, while if $s=2$, the system base is a pair of elements resulting in *link-based* SPA.
This stochastic process can either be seen as a scheme of throwing balls (the elements) in bins (the structures) or as a process of network growth. In the latter, the elements are the *nodes* of the network while the structures represent significant topological patterns, motifs, modules or *communities*, within which elements are linked.
SPA results in the growth of *modular* systems, because modules (or structures) are the basic building blocks of the model. These systems are also *scale-free*, in the sense that their main statistical features (membership and size distributions) converge toward power laws (free of any characteristic scale) as a result of the preferential attachment principle [@Simon55; @barabasi99]. Finally, these systems are said to be *self-similar* as different levels of organization follow the same general behavior: elements are interconnected with one another by sharing structures in the same way the structures themselves are interconnected by sharing elements.
![(color online). A step of node-based SPA.[]{data-label="SPA_scheme"}](LHD_fig1){width="0.875\columnwidth"}
In this paper, we borrow tools from statistical physics to study SPA in detail. In Sec. \[sec:process\], an exact description of SPA is obtained by writing the corresponding discrete stochastic process. From this description, we obtain the statistical steady-state of the resulting system with asymptotic expressions for its scaling behaviors. In Sec. \[sec:approximations\], some useful approximations are introduced and studied in order to facilitate the comparison between systems produced by SPA and real-world systems, using the *cond-mat arXiv* co-author network as an example. In order to investigate the validity of these approximations, we then study the existence of correlations between elements and structures, in both the SPA process and in the *cond-mat arXiv*. Lastly, in Sec. \[sec:peloton\], we highlight an interesting behavior of discrete PA processes, which we call the *peloton dynamics*, by comparing the initial stochastic process with an explicit solution for the time evolution of the continuous time version (further details are presented in the Appendices A and B). We then seek empirical evidences of this behavior in growing prose samples. A conclusion summarizes our results.
Stochastic process \[sec:process\]
==================================
Time evolution
--------------
To follow the growth of a system as prescribed by the SPA process, we separate elements and structures. We distinguish nodes by their respective number of memberships, $m$, and structures by their respective size, $n$, as these are the only features relevant to their evolution. Let $\tilde{N}_m(t)$ be the mean number of elements (or *nodes* to use the network terminology) with $m$ memberships and $\tilde{S}_{\! n}(t)$ be the mean number of structures of size $n$. Throughout the paper, tildes are used in quantities describing absolute numbers. Also note that as we follow the mean distribution of these quantities, we restrict ourselves to a deterministic approximation of the process.
At each time step, the evolution of these quantities is twofold: first, a constant increment for potential new nodes and structures; second, an operation corresponding to the preferential growth of existing nodes and structures. More clearly, each time step corresponds to an iteration of the following rule: $$\begin{aligned}
\tilde{N}_m(t+1) = & \tilde{N}_m(t) + q\delta _{m1} \nonumber \\
& + \frac{1\! -\! q\! +\! p\left(s\! -\! 1\right)}{t\left[1+p\left(s\! -\! 1\right)\right]}\left[\left(m\! -\! 1\right)\tilde{N}_{m\! -\! 1}(t) -\! m\tilde{N}_m(t)\right] \label{d1} \\
\tilde{S}_{\! n}(t+1) = & \tilde{S}_{\! n}(t) + p\delta _{ns} \nonumber \\
& + \frac{1-p}{t\left[1+p\left(s\! -\! 1\right)\right]}\left[\left(n\! -\! 1\right)\tilde{S}_{\! n\! -\! 1}(t) -\! n\tilde{S}_{\! n}(t)\right] \; . \label{d2}\end{aligned}$$ The two increments $q\delta _{m1}$ and $p\delta _{ns}$, where $\delta _{ij}$ is the Kronecker delta, correspond to birth events for elements (with one membership) and structures (of size $s$), respectively. The last increments correspond to the growth of old entities, where a compartment has a negative effect on itself and a positive effect on its neighboring compartment (e.g., $\tilde{N}_m \rightarrow \tilde{N}_{m+1}$) at a given rate and the denominator $t\left[1+p(s-1)\right]$ normalizes the preferential attachment probabilities.
This iterative description is straightforward, yet we can define the system in closed form by using generating functions (GFs) [@wilf]. We define two functions whose power series coefficients correspond to the elements of our two ensembles: $$\widetilde{\mathcal{N}}(x; t) = \sum _m \tilde{N}_m(t) x^m \;\;\; \textrm{and} \;\;\; \widetilde{\mathcal{S}}(x; t) = \sum _n \tilde{S}_{\! n}(t)x^n$$ In terms of these GFs, Eqs. (\[d1\]) and (\[d2\]) can be rewritten as: $$\begin{aligned}
\widetilde{\mathcal{N}}(x;t+1) & = & \left(1 + \dfrac{\Gamma _s}{t}x\left(x-1\right)\frac{d}{d x}\right) \widetilde{\mathcal{N}}(x;t) + qx \; ; \label{abs1}\\
\widetilde{\mathcal{S}}(x;t+1) & = & \left(1 + \dfrac{\Omega _s}{t}x\left(x-1\right)\frac{d}{d x}\right) \widetilde{\mathcal{S}}(x;t) + px^s \; , \label{abs2}\end{aligned}$$ where we have also introduced $$\Gamma _s = \frac{1-q+p(s-1)}{1+p(s-1)} \;\;\; \textrm{and } \;\; \Omega _s = \frac{1-p}{1+p(s-1)} \; .
\label{gam_ome}$$
A similar description can be obtained in terms of the corresponding probability generating functions (PGFs), $\mathcal{N}(x;t)$ and $\mathcal{S}(x;t)$, which generate the distributions of memberships per element and size per structures respectively. To transform the previous description in terms of these PGFs, note that the mean numbers of elements, $\tilde{N}_m$, or structures, $\tilde{S}_{\! n}$, in a given state corresponds to the proportion of such elements, $N_m$, or structures, $S_{\! n}$, multiplied by the mean total number of elements, $qt$, or structures, $pt$, expected at time $t$. One can now rewrite Eqs. (\[abs1\]) and (\[abs2\]) in terms of $\mathcal{N}(x;t)$ and $\mathcal{S}(x;t)$ by multiplying these functions by $qt$ and $pt$, respectively: $$\begin{aligned}
\left(t+1\right)\mathcal{N}(x;t+1) & = & \left(t + \Gamma _s x\left(x-1\right)\frac{d}{d x}\right) \mathcal{N}(x;t) + x \label{rel1}\\
\left(t+1\right)\mathcal{S}(x;t+1) & = & \left(t + \Omega _s x\left(x-1\right)\frac{d}{d x}\right) \mathcal{S}(x;t) + x^s . \label{rel2}\end{aligned}$$
As we will see in what follows, the description in terms of PGFs is generally more useful and will hereafter be used in our results to validate the analytical description.
Degree distributions \[degdis\]
-------------------------------
PGFs provide simple ways to evaluate secondary properties of a given state. For example, the node degree distribution and the community degree distribution. The former describes how many elements can be reached from a randomly chosen element, in other words, the number of links connected to this node in the network representation. The latter refers to a similar concept, namely, the number of structures that overlap (by sharing elements) with one randomly chosen structure.
To illustrate how this calculation is performed, one can simply refer to the composition property of PGFs. We first pick a random element whose membership distribution is generated by $\mathcal{N}(x;t)$. For every possible value of its membership number $m$, we sum over all possible cases for the different sizes of these structures. However, we know that all of these $m$ structures have *at least* one element. It is thus $k$ times more likely that one of these $m$ structures is a structure of size $k$ than a structure of size one. Furthermore, we do not want to count the initial element we chose, and will thus reduce the size of each structure by one. Hence, their size distribution is not generated by $\mathcal{S}(x;t)$, but instead by $\mathcal{S}'(x;t) /\mathcal{S}'(1;t)$, where the denominator acts as a normalisation factor. Knowing that the convolution of two sequences is generated by the product of the corresponding PGFs, one can take the $m$-th power of the new size PGF to obtain the PGF for the sum of $m$ structures. Finally, we sum over all possible values of $m$ to obtain [@newman03]: $$\begin{aligned}
D(x;t) & = \sum _m N_m \left[ \mathcal{S}'(x;t) /\mathcal{S}'(1;t) \right]^m \nonumber \\
& = \mathcal{N}\left(\left[\mathcal{S}'(x;t)\bigg /\mathcal{S}'(1;t)\right], t\right) \; .
\label{node_deg}\end{aligned}$$ Using the same logic for structures and their community degree, one can write: $$C(x;t) = \mathcal{S}\left(\left[\mathcal{N}'(x;t)\bigg /\mathcal{N}'(1;t)\right], t\right) \; .
\label{comm_deg}$$ The self-similarity between different levels of organization in the systems created by SPA stems from the similarity between Eqs. (\[node\_deg\]) and (\[comm\_deg\]). As long as $\mathcal{N}(x;t)$ and $\mathcal{S}(x;t)$ are similar, the various possible compositions, which represent different organization properties, will also be similar.
The validation of our analytical description for the time evolution of SPA is presented on Fig. \[timeevo\] using Monte Carlo simulations. The initial conditions of all systems (i.e., the state of the system at $t=0$), in both numerical simulation and analytical integration, consist of a single structure containing a single element; this remains true throughout the paper. Note that our calculations for the degree distributions are merely approximations because they suppose *homogeneous mixing* between elements and structures, while an element with $m=i$ might not see exactly the same size distribution as an element with $m=j$. Such element-structure correlations are investigated in Sec. \[sec:correlations\].
Statistical equilibrium
-----------------------
![(color online). Validation of Eqs. (\[scale1\]) and (\[scale2\]) as predictions for the asymptotic scaling behaviors of the main statistical distributions (dashed lines: steady-state solutions, continuous line: scaling predictions) for node-based SPA ($s=1$) using $q=0.6$ ($\gamma _N = 7/2$) and $p=0.25$ ($\gamma _S = 7/3$).[]{data-label="scaling"}](LHD_fig4){width="35.00000%"}
The statistical equilibrium can be imposed by setting $\mathcal{N}(x;t+1) = \mathcal{N}(x,t) \equiv \mathcal{N}^*(x)$ and $\mathcal{S}(x,t+1) = \mathcal{S}(x,t) \equiv \mathcal{S}^*(x)$ in Eqs. (\[rel1\]) and (\[rel2\]), yielding: $$\begin{aligned}
\mathcal{N}^*(x) & = & \Gamma _s x\left(x-1\right)\frac{d}{d x}\mathcal{N}^*(x) + x \; ; \label{eq1}\\
\mathcal{S}^*(x) & = & \Omega _s x\left(x-1\right)\frac{d}{d x}\mathcal{S}^*(x) + x^s \; . \label{eq2}\end{aligned}$$ These ordinary differential equations can be solved straightforwardly to obtain their solutions in terms of hypergeometric functions of the form ${}_2F_1\left(a,b;c;x\right)$: $$\mathcal{N}^*(x) = \frac{x}{1+\Gamma _s} \; {}_2F_1\left(1,1;2+\frac{1}{\Gamma _s};x\right) \; ,$$ and: $$\mathcal{S}^*(x) = \frac{(s-1)!\Omega _s^{s-1}x^s}{1+s\Omega _s} \; {}_2F_1\left(1,s;(s+1)+\frac{1}{\Omega _s};x\right) \; .$$ The statistical equilibrium for the two distributions of interest can now be obtained through the power series coefficients of these two functions: $$\begin{aligned}
& N_m^*(s) = \dfrac{\prod _{k=1}^{m-1} k\Gamma _s}{\prod _{k=1}^{m} \left(1+k\Gamma _s\right)} \;, &\; S_n^*(s) = \dfrac{\prod _{k=s}^{n-1} k\Omega _s}{\prod _{k=s}^{n} \left(1+k\Omega _s\right)} \; .
\label{powers}\end{aligned}$$ These solutions for the asymptotic behavior of the statistical distributions can be validated through comparison with the long term behavior of our predicted time evolution, as done in Fig. \[asympto\].
Scaling behavior
----------------
From PA, it is well known that the $N^*_m$ and $S^*_n$ distributions will fall as power laws, i.e., $$N^*_m \propto m^{-\gamma _N} \;\;\; \textrm{and } \;\; S^*_n \propto n^{-\gamma _S} \; .$$ To calculate the scaling exponent $\gamma _N$, we can evaluate the following ratio using Eq. (\[powers\]) $$\lim _{m \rightarrow \infty} \frac{N^*_m}{N^*_{m-1}} = \lim _{m \rightarrow \infty} \left(\frac{m}{m-1}\right)^{-\gamma _N} = \lim _{m \rightarrow \infty} \frac{\left(m-1\right)\Gamma _s}{1 + m\Gamma _s}$$ from which it follows that $$\gamma _N = \lim _{m \rightarrow \infty} \dfrac{\log \left(\left(m-1\right)\Gamma _s \bigg / \left(1 + m\Gamma _s\right) \right)}{\log \left(\left(m-1\right)\bigg /m \right)} = 1 + \frac{1}{\Gamma _s} \; .
\label{scale1}$$ Similarly, one can directly write for structures: $$\gamma _S = 1 + \frac{1}{\Omega _s} \; .
\label{scale2}$$
The node and community degree distributions, as compositions of two power-law distributions, will fall as the slower of the two original distributions. Noting that $\mathcal{N}'(x,t)$ and $\mathcal{S}'(x,t)$ will follow $\gamma _{N'} = \gamma _N - 1$ and $\gamma _{S'} = \gamma _S - 1$ because of the derivative, we obtain: $$\gamma _D = \min \bigg\lbrace\gamma _N, \gamma _S - 1 \bigg\rbrace \;\;\; \textrm{and } \;\; \gamma _C = \min \bigg\lbrace\gamma _N - 1, \gamma _S \bigg\rbrace \; .$$
These results are validated on Fig. \[scaling\].
Approximations and limitations \[sec:approximations\]
=====================================================
To complete our description of the SPA process, this section examines some approximations that have either proven useful when reproducing empirical data with the SPA process or that correspond to limitations of the present formalism.
Correspondence between system bases
-----------------------------------
Some systems reproduced in [@SPA] with node-based SPA ($s=1$) are actually link-based, for example the author collaboration network of the *cond-mat arXiv*, where authors only appear once they have at least one collaboration. The link between node and link-based SPA is done by ignoring structures of size one when compiling the final system.
In [@SPA], we mention that the system base $s$ was not a parameter of the model per se, but depends on the information available or on the nature of the system. For instance, the World-Wide Web is mapped by following links between webpages, such that it is impossible to find a page with no links. The smallest structural unit is thus the link and not the webpage itself: it is a link-based system ($s=2$). Similarly, the author collaboration network of the *cond-mat arXiv* is built through collaborations and thus excludes authors without any links. Despite this fact, it can modelled through node-based SPA by ignoring structures of size one at the very end of the process. Furthermore, structures of size one can rarely be detected in network data if they are not completely disconnected from the rest of the systems. Hence, it is useful to be able to ignore these structures at the end of the stochastic growth process, independently of the system base.
For the size distribution, ignoring structures of size one simply implies a renormalization for structures of size two or greater. Noting the PGF for an approximate link-based SPA $\mathcal{S}^{\textrm{app}}_2(x)$ using the original node-based functions $\mathcal{S}_1(x)$, we can write: $$\mathcal{S}^{\textrm{app}}_2(x) = \frac{\mathcal{S}_1(x) - S_1x}{\mathcal{S}_1(1) - S_1} \; .
\label{approx1}$$ For the membership distribution, once again assuming homogeneous mixing, we must randomly remove the fraction of memberships which corresponds to the structures of size one. Using the composition of PGFs, this can be done by composing the membership PGF with the PGF for a binomial trial: $$\mathcal{N}^{\textrm{app}}_2(x) = \dfrac{\mathcal{N}_1\left(x\left(1-\epsilon\right)+\epsilon\right)-\mathcal{N}_1\left(\epsilon\right)}{1-\mathcal{N}_1\left(\epsilon\right)}
\label{approx2}$$ where $\mathcal{N}_1\left(\epsilon\right)$ corresponds to the elements left with no memberships and thus need to be removed from the system. This trial will remove a fraction $\epsilon$ of memberships, where $\epsilon$ corresponds to the fraction of memberships which are associated with structures of size one: $$\epsilon = \frac{S_1}{\sum _n nS_n} = \frac{\mathcal{S}'_1(0)}{\mathcal{S}'_1(1)} \; .$$ The validity of this approximate description and the effects of switching between system bases are illustrated on Fig. \[s\_compar\]. Note how changing the system base, while keeping the parameters constant, greatly modifies the produced system. This highlights both the validity of Eqs. (\[approx1\]) and (\[approx2\]) (which feature two levels of approximation of homogeneous mixing) and the importance of considering the influence of the system base on the scaling behavior.
To compare the results of approximated and actual link-based SPA for the same community structure, we first need to identify the relation between the parameter pairs $\lbrace q_1 , p_1 \rbrace$ and $\lbrace q_2 , p_2 \rbrace$ which is such that $\Gamma _1 = \Gamma _2$ and $\Omega _1 = \Omega _2$. From Eq. (\[gam\_ome\]), we obtain: $$p_2 = \frac{p_1}{2-p_1} \;\;\; \textrm{and } \;\; q_2 = \frac{2q_1}{2-p_1} \; .
\label{transition}$$
While it is easily verified that ignoring structures of size one in node-based SPA can result in statistical features similar to that of link-based SPA (see Fig. \[arXiv\]), there exists one particularly important structural difference between these two kinds of systems. Mainly, a true link-based system is necessarily fully connected as each new elements creates at least one link with the old elements, while node-based systems can create many disconnected components that may or may not end up interconnecting through new structures (depending on $q$ and $p$). In real link-based systems, there is no restriction on connectedness. For instance, the *cond-mat arXiv* network of co-authors has one giant component which consists of $\sim 93\%$ of the system, but other smaller satellite components still exist. While both SPA versions illustrated on Fig. \[arXiv\] create a similar community structure as the *cond-mat arXiv*, the node-based version is actually closer to reality.
![(color online). Size distribution of structures as seen from elements with different $m$ memberships. Markers represent empirical measures done on the *cond-mat arXiv* and numerical results on the two SPA processes (using the parameters of Fig. \[arXiv\]). The dashed line corresponds to what would be obtained through homogeneous pairing of memberships and structures.[]{data-label="correlations"}](LHD_fig7){width="35.00000%"}
Multiple memberships, multiple links and self-loops
---------------------------------------------------
In our description of the time evolution of SPA, we have never explicitly forbidden an element to join the same structure more than once. These multiple memberships, whose likelihood depends directly on the value of the $p$ or $q$ parameters, lead to multiple links between the same individuals and self-loops (where an element shares a structure with itself). Similarly, in our derivation of the degree distributions, we have supposed an infinite system where the probabilities that two structures overlap by more than one element fall to zero.
In empirical data, multiple links and self-loop are rarely considered. It can thus be useful to have an idea of the effect of such restrictions on SPA. Fig. \[multiple\] presents two snapshots of the same scenarios of SPA, with or without forbidding multiple memberships, multiple links and self-loops when analyzing the final stage of the system. The cutoffs in the distributions of the first system are not surprising, as large and old structures are very likely to have recruited the same element more than once, especially with a small $q$. Yet, this effect rapidly becomes negligible as the system grows and we enter the large size limit in accordance with the assumptions of our analytical description (see Fig. \[multiple2\]).
Element-structure correlations \[sec:correlations\]
---------------------------------------------------
Most of the approximations used throughout this paper are based on the assumption of homogeneous mixing: the elements belonging to a number $x$ of structures *see* the same size distribution as the elements belonging to $y$ structures. This implies that there is no correlations except for the fact that an element is $x$ times more likely to belong to a given structure of size $x$ than to a particular structure of size one (*natural correlations*). To investigate this matter, we compare the size distributions as seen from elements with different memberships in both the simulations done for Fig. \[arXiv\] and the corresponding *arXiv* data.
Figure \[correlations\] presents the results of this investigation. First, the similitude between SPA and homogeneous mixing explains why our approximations were accurate. The small difference between the node-based and link-based SPA processes is most likely due to the fact that the link-based version requires more elements for the birth of new structures, which are consequently more likely to be old elements than in the node-based version. Second, there is a major difference between element-structure correlations in real-systems and SPA: elements with few memberships are much more likely to belong to larger structures in the arXiv data than in our SPA simulations. This shows how other levels of organization have yet to be taken into account in our stochastic models. Depending on what one wants to model, these correlations could potentially be important.
\
Peloton dynamics \[sec:peloton\]
================================
One particularly interesting feature of the results presented in Fig. \[timeevo\] and \[asympto\] is the dynamics of the entities in the tail of the distributions. In fact, these groups of individuals or structures resulted in clearly identifiable *bulges* on their respective distributions. The dynamics of a system’s leader is well-documented in the context of growing networks [@Krapivsky; @Godreche] or word frequencies [@bernhardsson09], but can be applied to any problem where one is interested in the statistics of the extremes (i.e., the growth of the biggest business firm, of the most popular website, etc.). What we observe here is that averaging over multiple realizations of the same experiment will result in the creation of a *peloton* where one is significantly more likely to find entities than predicted by the asymptotic distribution (i.e., the leaders).
The clear distinction between the statistical distribution of leaders versus the rest of the system is a consequence of the maximal size of the system and of the limited growth resources available. To illustrate this claim, we can consider a continuous time version of PA in which there is no finite limitation to the number of growth events at every time step (see Appendix \[AA\] for explicit solution of this process). Comparing the results of the discrete and continuous versions of our stochastic process on Fig. \[DisVsCont\] illustrates how limiting growth resources results in the condensation of the leaders in a peloton. This draws a strong parallel between discrete preferential attachment and some sandpile models known to result in scale-free avalanche size distributions through *self-organized criticality*. In some cases, such as the Oslo model (see [@christensen] 3.9), the biggest avalanches are limited by the size of the considered sandpile and are thus condensed in bulges identical to our pelotons.
Also striking is the fact that this peloton conserves its shape on a log-log scale (see Fig. \[decay\]). To highlight this feature, Fig. \[superpose\] rescales the distributions to account for the scaling in size ($\gamma _s$) and the peloton growth through time ($t^{1-p}$, see Appendix \[AB\] for derivation). This rescaling method was borrowed from [@christensen] 3.9.8.
Leaders emerge in every single preferential growth realization, while the peloton dynamics can only manifest itself once we average over multiple systems or over many characteristic time scales of a single system (through the births and deaths of many different leaders). Consequently, empirical observations of this phenomenon are rare, because on the one hand we have only one Internet, one arXiv, and basically a unique copy of most complex systems, and on the other hand, we rarely have access to extensive data through long time scales. We can however find a solution if we go back to the first example used by Simon [@Simon55] to derive his model: the scale-free distribution of words by their number of occurrences in written text (i.e., Zipf’s law [@zipf]). In this context, $q$ equals zero and the $p$ parameter corresponds to the probability that each new written word has never been used before. We can therefore consider different samples of text of equal length written by the same author as different realizations of the same experiment.
With this in mind, we have picked different authors according to personal preferences and size of their body of work and divided their œuvres in samples of given lengths which we then used to evaluate Zipf’s law under averaging (see Fig. \[lovecraft\]). As predicted by PA, taking the average of multiple realizations of the same experiment results in a peloton which diverges from the traditional Zipf’s law. In this case, the peloton implies that the leaders of this system (i.e., the most frequent words) consistently fall in the same scale of occurrences.
Lastly, Fig. \[rescaledmel\] reproduces the scaling analysis of Fig. \[superpose\] for empirical results on prose samples. The varying surface of the peloton hints at a non-constant growth rate: a well-known feature of written text (see [@heap] 7.5).
Conclusion
==========
In this paper, several analytical results for *structural preferential attachment* have been obtained: solutions for its time evolution and asymptotic behavior as well as approximations for its different degree distributions. Those approximate descriptions are especially useful when it comes to using organization models as part of modelling efforts.
We have also highlighted one particular shortcoming of the model: element-structure correlations. That is, SPA lacks any modelling or predictive power when it comes to asking *who belongs to what structure*.
On the other hand, we have observed an interesting behavior of both the SPA and the classic PA models: *the peloton dynamics*. This particular feature is important in order to predict the position of the leaders of a PA growth process. More interestingly, we have been able to observe this behavior in the growth of prose samples, which differentiates the PA principle from the other models generating scale-free designs but failing to predict this property.
The presentation of shortcomings and successes of the SPA principle (in terms of predictive value) shows the importance and the need for further study in stochastic growth models.
The authors thank Yong-Yeol Ahn *et al.* for their link community algorithm and Gergely Palla for providing the arXiv dataset. We also wish to acknowledge the help of Jean-Gabriel Young and Sebastian Bernhardsson for useful comments and criticism. The research team is grateful to NSERC, FQRNT and CIHR for financial support.
Explicit solution to continuous time SPA \[AA\]
===============================================
Section \[sec:peloton\] has presented an explicit solution for the time evolution of SPA in continuous time. This Appendix summarizes its derivation, based on a recently proposed method [@morin].
Definition of a continuous time PA process
------------------------------------------
The transition to continuous time simply implies that $q$ and $p$ now refer to birth rates for both elements and structures. The corresponding rates $1-q$ and $1-p$ thereby correspond to the growth rates of existing elements and structures, respectively. This means that in a given time interval $[t, t+1]$, this new stochastic process could create an infinite number of elements with probability $\lim _{dt \rightarrow 0}\left(qdt\right)^{1/dt}$; whereas the discrete version could only create one element with probability $q$. While it is highly improbable that continuous time PA results in a system several orders of magnitude larger than $qt$ or $pt$, there is no maximal size per se.
This sort of continuous time dynamics is better described using simple ODEs, or master equations, as was done in [@SPA]. To this end, we once again follow $\tilde{N}_m$, the number of elements with $m$ memberships, and $\tilde{S}_n$, the number of structures enclosing $n$ elements. Using the same logic behind Eqs. (\[d1\]) and (\[d2\]), but considering infinitesimal time steps $dt$, one can write $$\begin{aligned}
\tilde{N}_m(t+dt)\! = \tilde{N}_m(t)\! + dt\mathbf{\bigg\lbrace}\frac{\Gamma _s}{t}\left((m\! -\! 1)\tilde{N}_{m-1}(t)-\! m\tilde{N}_m(t)\right) +\! q \, \delta _{m1}\mathbf{\bigg\rbrace} \nonumber\end{aligned}$$ and $$\tilde{S}_n(t+dt)\! = \tilde{S}_n(t)\! + dt\mathbf{\bigg\lbrace} \frac{\Omega _s}{t}\left((n\! -\! 1)\tilde{S}_{n-1}(t) -\! n\tilde{S}_n(t)\right) +\! p \, \delta _{ns} \mathbf{\bigg\rbrace} \; , \nonumber$$ which are straightforwardly rewritten as two ODEs: $$\frac{d}{dt}\tilde{N}_m(t) = \frac{\Gamma _s}{t}\left((m-1)\tilde{N}_{m-1}(t)-m\tilde{N}_m(t)\right) + q\, \delta _{m1} \; ;
\label{node_master}$$ $$\frac{d}{dt}\tilde{S}_n(t) = \frac{\Omega _s}{t}\left((n-1)\tilde{S}_{n-1}(t) - n\tilde{S}_n(t)\right) + p \, \delta _{ns} \; .
\label{struc_master}$$ Because these two last equations have the same form, we solve them separately using a general continuous time PA equation. Consider $$\frac{d}{dt}P_k(t) = \beta\, \delta _{km} + R_{k-1}(t)P_{k-1}(t) - R_k(t)P_k(t)
\label{gen_eq}$$ where $\beta$ is the birth rate, $m$ is the size of new entities and $R_i(t)$ is the attachment rate on entities of size $i$, which we define using a growth rate $\alpha$, an initial total size $m_0$ and a normalization rate $\lambda$: $$R_i(t) = \frac{\alpha i}{m_0 +\lambda t } \; .$$ It proves useful to rewrite (\[gen\_eq\]) in dimensionless form as $$\frac{d}{d\tau}P_k(\tau) = \overline{\beta}\, \delta _{km} + \overline{R}_{k-1}(\tau) P_{k-1}(\tau) - \overline{R}_k(\tau) P_k(\tau)
\label{nodim_gen_eq}$$ with dimensionless time $\tau = \alpha t$, parameters $\overline{\beta}= \beta/\alpha$, $\overline{\lambda}= \lambda/\alpha$, and attachment rate $\overline{R}_k(\tau) = k / (m_0 + \overline{\lambda} \tau)$ respectively. Table $\ref{table}$ gives the values of the different parameters for the classical PA models and for SPA.
[c|c|c|c|c|]{} & &\
& Simon & BA & elements & structures\
& $\; p/(1-p) \;$ & $\quad 1/m \quad$ & $q/\alpha$ & $p/\alpha$\
& $1-p$ & $m$ & $1-q+p(s-1)$ & $1-p$\
& $1/(1-p)$ & $2$ & $[1+p(s-1)]/\alpha$ & $[1+p(s-1)]/\alpha$\
& $1$ & $m$ & $1$ & $s$\
Explicit solution
-----------------
Let $$\overline{H}_k(t) = \textrm{exp}\left[ \int \overline{R}_k(\tau) d\tau \right] = \left( m_0 + \overline{\lambda} \tau \right)^{k / \overline{\lambda}} \; ,
\label{H}$$ so that Eq. (\[nodim\_gen\_eq\]) can be written as: $$\frac{d}{d\tau}\left[P_k(\tau) \overline{H}_k(\tau)\right] = \overline{\beta} \overline{H}_k(\tau) \delta _{km}
+ \overline{R}_{k-1}(\tau) \overline{H}_k(\tau) P_{k-1}(\tau) \; .$$ The general solution of this transformed equation is: $$\begin{aligned}
P_k(\tau) &=& \overline{\beta} \frac{ (m_0 + \overline{\lambda} \tau)}{k + \overline{\lambda}} \delta_{km} \nonumber \\
&+ & \frac{(1 - \delta_{km})}{\overline{H}_k(\tau)}\int \overline{R}_{k-1}(\tau) \overline{H}_k(\tau) P_{k-1}(\tau) d\tau +C_k \; ,\end{aligned}$$ where $\{C_k\}$ are constants of integration determined by the initial conditions. Solving for the first few values of $k$ ($m$, $m+1$, $m+2$, …) reveals the following pattern for the solutions: $$\begin{aligned}
P_{m+k}(\tau) &= & \overline{\beta} \frac{ (m)_k }{ (m+\overline{\lambda})_{k+1} } \left( m_0 + \overline{\lambda} \tau \right) \nonumber \\
&+& \sum_{i=0}^k \frac{ (m)_k }{ (m)_i } \frac{ C_{m+i} }{\left( k-i \right)!}
\left( m_0+ \overline{\lambda} \tau \right)^{-(m+i) / \overline{\lambda} }
\label{gen_soln}\end{aligned}$$ where $(\gamma)_j \equiv (\gamma) (\gamma +1) \ldots (\gamma +j -1)$ are Pochammer symbols. The last step towards a complete solution is to determine an explicit form of the constants of integrations $\{ C_{m+k}\}$ in terms of the initial conditions $\{ P_{m+k}(0)\}$. This is easily accomplished by writing (\[gen\_soln\]) in a matrix form for the vector of initial conditions $\boldsymbol{P}(0)$ $$\boldsymbol{P}(0) = \boldsymbol{A}(0) + \bf{L}(0) \boldsymbol{C}$$ in terms of the vector $\boldsymbol{C}$ of integration constants and a [*lower triangular*]{} matrix $\bf L$, followed by the observation that the inverse of a (lower/upper) triangular matrix is also a (lower/upper) triangular matrix whose elements can be constructed by forward substitution. Given that the elements of ${\bf L}(0)$ are $$L_{m+k,m+i}(0) = \binom{m+k-1}{m+i-1}\frac{1}{m_0^{m+i}}$$ we find that the elements of the inverse matrix, denoted $\bf M$, are simply $$M_{m+k,m+i} = (-1)^{k-i} \binom{m+k-1}{m+i-1}m_0^{m+i} \; .$$ Inserting this solution in (\[gen\_soln\]), we get $$\boldsymbol{P}(\tau) = [\boldsymbol{A}(\tau) - {\bf L}(\tau) {\bf M} \boldsymbol{A}(0) ]
+ \bf{L}(\tau) {\bf M} \boldsymbol{P}(0) \ ,$$ which nicely isolates the principal dynamics (the first 2 terms) from the initial conditions. Specifically, by imposing the usual initial conditions, $P_{m+k}(0) = \delta_{k0}$, it is straightforward, albeit somewhat lengthy, to obtain a closed-form expression for the complete dynamical elements as $$\begin{aligned}
P_{m+k}(\tau) &=& \overline{\beta} m_0 (m)_k
\left[ \frac{1}{(m+\overline{\lambda})_{k+1}} X(\tau ) \right. \nonumber \\
&-& \left. \frac{1}{(m+\overline{\lambda})} \frac{1}{\Gamma(k+1)} X(\tau)^m F_k(X(\tau))\right] \nonumber \\
&+& (m)_k \frac{1}{\Gamma(k+1)} X(\tau)^m (1 -X(\tau))^k\end{aligned}$$ with $X(\tau) = m_0 / (m_0 + \overline{\lambda} \tau)$ and where $F_k(X)= {\ }_2F_1(-k,m+\overline{\lambda}; m+\overline{\lambda} +1; X)\ $ represents a terminating hypergeometric series of degree $k$. One verifies that, by setting $\tau=0$ in the previous expression, one obtains $P_{m+k}(0) = \delta_{k0}$ as it should.
It can further be shown that the continuous and discrete time versions of PA converge toward the same asymptotic behavior.
Scaling exponents in the peloton dynamics \[AB\]
================================================
It has been seen in Fig. \[superpose\], that the probability distribution $P(x;t)$ follows the scaling relation $$\widetilde{P}(x) \propto x^{\gamma} P(x/f(t); t\gg 1) \ ,$$ where $\gamma$ is either equal to $\gamma_N$ for elements or $\gamma_S$ for structures. This Appendix derives the growth function, $f(t)$, describing the mean state of a single entity (e.g., its number of occurences or its size) at time $t$ within a system whose global growth is governed by PA. Once again, because we follow mean quantities, the process is deterministic.
Without loss of generality, we suppose that only one entity is present at time $t=1$, such that always exactly $t$ events will have occured by time $t$. This simplifies the normalization of transition probability and we can thus write the effect of a general PA step on a single entity as: $$\begin{aligned}
f(t+1) = \left[\beta + \alpha\frac{t-f(t)}{t}\right]f(t) + \alpha\frac{f(t)}{t}\bigg(f(t)+1\bigg) \; .\end{aligned}$$ For the node-based cases, a further simplification arises, $\alpha + \beta = 1$, yielding a recursive rule for the growth function $f(t)$: $$f(t+1) = \left(1+\frac{\alpha}{t}\right)f(t) \; ,$$ which directly fixes the derivative in the limit of large $t$: $$\frac{d}{dt}f(t) = \frac{\alpha}{t}f(t) \; .
\label{Bdiff}$$ The general solution to Eq. (\[Bdiff\]) is: $$f(t) = At^{\alpha} + B \; .$$ For the original entity, $f(1) = 1$, which is destined to be the leader of this deterministic process, one obtains the following mean position at time $t$: $$f(t) = t^{\alpha} \; .
\label{Blead}$$ Equation (\[Blead\]) dictates the evolution of the leader’s position and thus fixes the renormalization used in Fig. \[superpose\]. Once again, one can refer to Tab. \[table\] for the values of $\alpha$ in different PA models.
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ROME1/1377/04
DSFNA/14/04
**A model for next-to-leading order resummed form factors**\
**Ugo Aglietti**[^1]\
Dipartimento di Fisica,\
Universitá di Roma “La Sapienza”\
and I.N.F.N., Sezione di Roma, Italy.\
**Giulia Ricciardi**[^2]\
\[0pt\]
Dipartimento di Scienze Fisiche,\
Universitá di Napoli “Federico II”\
and I.N.F.N., Sezione di Napoli, Italy.\
\[1pt\]
$~~~$\
**Abstract**\
We present a model for next-to-leading order resummed threshold form factors based on a time-like coupling recently introduced in the framework of small $x$ physics. Improved expressions for the form factors in $N$-space are obtained which are not plagued by Landau-pole singularities, as the included absorptive effects – usually neglected — act as regulators. The physical reason is that, because of faster decay of gluon jets, there is not enough resolution time to observe the Landau pole. Our form factors reduce to the standard ones when the absorptive parts related to the coupling are neglected. The inverse transform from $N$-space to $x$-space can be done directly without any prescription and we obtain analytical expressions for the form factors, which are well defined in all $x$-space.
Introduction
============
Resummation of large infrared logarithms in form factors and shape variables is essential in order to predict accurate cross sections in many phenomenologically relevant processes [@parpet; @kodtren; @sterman; @cattren; @cattren2]. In this paper we present a model for next-to-leading order (NLO) resummed form factors based on the time-like coupling recently introduced by B. Ermolaev, M. Greco and S. Troyan in [@ermolaev] in the framework of small $x$ physics (see also [@pennington]). The usual expression for resummed threshold form factors in $N$-space is [@sterman; @cattren; @cattren2]: $$\begin{aligned}
\label{generale}
f_N(\alpha_S) &=& \exp \int_0^1 dz
\frac{z^{N-1}-1}{1-z} \Big\{ \int_{Q^2(1-z)^2}^{Q^2(1-z)}
\frac{dk_t^2}{k_t^2} \big[ A_1\,\alpha_S(k_t^2) +
A_2\,\alpha_S(k_t^2)^2 \,+\,\cdots \big]~+
\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~
+\,B_1\,\alpha_S(Q^2(1-z))\,+\,\cdots \,+\, D_1\,\alpha_S(Q^2(1-z)^2)+\cdots\Big\}.\end{aligned}$$ $Q$ is the hard scale of the process. $A_1,\,A_2,\cdots B_1,\cdots D_1,\cdots$ are the first coefficients of the functions $A\left(\alpha_S\right)$, $B\left(\alpha_S\right)$ and $D\left(\alpha_S\right)$: $$A\left(\alpha_S\right) =
A_{1}\alpha _S+A_{2}\alpha_S^{2}+\cdots, \quad
B\left( \alpha_S\right) = B_1\alpha_S+B_2\alpha_S^2+\cdots,
\quad D\left(\alpha_S\right) =D_1\alpha _S+D_2\alpha_S^2+\cdots .$$ $A\left(\alpha_S\right)$ describes the emission of partons which are both soft and collinear, $B\left(\alpha_S\right)$ describes hard and collinear partons while $D\left(\alpha_S\right)$ describe partons which are soft and at large angles. The knowledge of the quantities $A_1$, $A_{2}$, $B_1$ and $D_1$ is needed for resummation at next-to-leading order (see later for definition). For instance, in the case of the thrust distribution we have [@thrust]: $$A_1 = \frac{2 C_F}{\pi },\qquad
A_2 = \frac{2 C_F}{\pi^2}\left[C_{A}\left( \frac{67}{36}-\frac{\pi^2}{12}\right) -\frac{5}{9}n_F T_R\right],\qquad
B_1 = -\frac{3}{2}\frac{C_F}{\pi }, \qquad
D_1 = 0,$$ where $C_A=N_C$, $C_F=(N^2_C -1)/(2 N_C)$, $T_R=1/2$, $N_C=3$ is the number of colors and $n_F$ is the number of active quark flavors[^3]. For heavy flavor decays we have instead [@heavyfla; @ug3]: $$A_1 =\frac{ C_F }{ \pi },\qquad
A_2 =\frac{ C_F }{\pi^2}\left[C_A\left(
\frac{67}{36}-\frac{\pi^2}{12}\right) -\frac{5}{9} n_F T_R\right],\qquad
B_1 =-\frac{3}{4}\frac{C_F}{ \pi }, \qquad
D_1 =-\frac{C_F}{\pi}.$$ Generally, one assumes $\alpha_S\ll 1$ and uses truncated $\alpha_S$ expansions for the functions $A(\alpha_S)$, $B(\alpha_S)$ and $D(\alpha_S)$ in eq. (\[generale\]). Since the running coupling $\alpha_S\left(k_t^2\right)$ is integrated over all gluon transverse momenta $k_t$ from the hard scale $Q$ down to zero, the Landau pole is hit. Therefore $\alpha_S$ diverges inside the integration region — it is certainly not small there — indicating that a truncated expansion for the $A$, $B$ and $D$ functions is not correct. We have conceptually a breakdown of the scheme: resummed perturbation theory assumes $\alpha_S$ small at the beginning, while it ends up with a large $\alpha_S$. A prescription has to be assigned to give a meaning to the formal expression (\[generale\]).
The standard solution is to expand the exponent of eq. (\[generale\]) in a function series of the form [@cattren]: $$\label{series}
f_N\left(\alpha _S\right) = \exp \left[
L\,g_1\left(\lambda\right) +\sum_{n=0}^{\infty
}\alpha_S^n\,g_{n+2}\left(\lambda\right) \right] =\exp \left[
L\,g_1\left(\lambda\right) +\,g_2\left(\lambda\right)
+\alpha_S\,g_3\left(\lambda\right) +\cdots \right],$$ where $$\lambda~=~\beta_0~\alpha_S(Q^2)~L,~~~~~~~L~=~\log N$$ and $\beta_0 = (11/3\;N_C-2/3\; n_F)/(4\pi)$. The functions $g_i\left( \lambda \right)$ have a power-series expansion: $$g_i\left(\lambda \right) =\sum_{n=1}^{\infty }g_{i,n}\lambda^n.$$ The effects of the Landau pole in this framework are the following:
1. the series in eq. (\[series\]) is divergent as the higher order functions have factorially growing coefficients [@ug3; @gardi];
2. the functions $g_i(\lambda)$ have branch cuts starting at $\lambda=1/2$ and going up to infinity. The form factors are then formally well defined up to a critical value $N_{crit}\sim Q/\Lambda$ ($\Lambda$ is the QCD scale), above which they acquire a (completely unphysical) imaginary part.
The first problem is solved by means of a truncation of the series to its first few terms — typically two or three — which is effectively a prescription of the Landau pole. This is the so-called fixed logarithmic accuracy, which we will describe later. The resummation to leading order requires the knowledge of the function $g_1$, the resummation to next-to-leading order also requires the knowledge of $g_2$, and so on. As far as the second problem is concerned, one simply restricts himself to a fiducial region in $N$-space below $N_{crit}$. The inverse Mellin transform from $N$-space to $x$-space, at next-to-leading $\log(1-x)$ accuracy, is well defined up to $x_{crit}\sim 1-\Lambda/Q$, above which singularities occur. However, a form factor formally well defined in the whole $x$-space and containing all the requested $\log(1-x)$ terms can be obtained by means of an additional prescription for the contour integration in $N$-space, the so-called minimal-prescription [@minimalpre].
Another common solution to this problem involves the renormalon calculus [@gardi]. The latter uses the inconsistency discussed above to get information about non-perturbative effects, in the form of power-suppressed corrections to the cross sections: $$\delta \sigma \sim c \left(\frac{\Lambda}{Q}\right)^b.$$ The exponent $b$ of the power correction can be computed, but not its coefficient $c$. The resulting information is therefore of a qualitative kind and, ultimately, perturbation theory is deprived of predictive power. In fact, one can substantially modify the spectra coming out of a renormalon calculation by changing the prescriptions in the Borel plane, which are completely arbitrary.
Our perspective is different: we want to use resummed perturbation theory in a predictive way, by curing the “disease” of usual resummation formula. Therefore we begin by re-analyzing the derivation of the standard formula. $\alpha_S(k_t^2)$ occurs in eq. (\[generale\]) because we are computing a so-called inclusive-gluon-decay quantity, in which one does not observe the development of the jets originating from the gluons emitted by the hard partons. One then sums over all possible final states, i.e. formally over all cuts of dressed gluon propagators, reconstructing their discontinuity. The radiated gluons have a positive virtuality $k^2>0$, as the development of jets is intrinsically a time-like process. Absorptive effects, i.e. the $-i\pi$ terms in gluon polarization functions, are usually neglected in literature: by taking them into account, one enforces that all gluons participating to the cascade are unstable particles [@ermolaev]. They are analogous to the quasi-particles of statistical physics, possessing in lowest order — non interacting quasi-particles — a non-zero width [@quasiparticles]. The outcome of the inclusion of this physical effect is that $\alpha_S(k_t^2)$ does not occur any longer in the resummation formula, and it is replaced by a different function of $k_t^2$, $$\alpha_S(k_t^2)~\rightarrow~\tilde{\alpha}_S(k_t^2),$$ which does not possess the Landau pole. We call this new function “effective coupling”, as it specifies the effective strength of the interaction in the gluon cascade — not in a general QCD process. The main point is that the effective coupling never becomes large, as the typical expansion parameter is $$\frac{\tilde{\alpha}_S (k_t^2) } {\pi} <
\frac{1}{\beta_0\pi} < 1~~~~~~~~{\rm for~~any}~~~k_t^2$$ and one never leaves the perturbative domain. By including absorptive effects related to the coupling constant — usually neglected — we derive an improved expression for the resummation formula of the form factors. The resummation formula is free of Landau pole pathologies, it does not involve any new free parameter and is strictly predictive.
The improved resummation formula
================================
=0.40=0.25
By taking into account renormalization effects in the form factors, the tree-level, momentum-independent coupling is replaced by the integral on the discontinuity of the gluon propagator [@dokshitzer; @amati]: $$\label{partenza} \alpha_S~~~\rightarrow~~~\tilde{\alpha}_S(k_t^2)
= \frac{1}{2\pi i} \int_0^{k_t^2} ds~\mathrm{Disc}_s ~\frac{1}{
s~\beta_0 \log \frac{-s}{\Lambda^2} }.$$ The discontinuity is defined as usual: $\rm{Disc}_s
F(s)=F(s+i\epsilon)-F(s-i\epsilon),$ with $\epsilon$ being a positive infinitesimal number. We now close the integration contour with a circle of radius $k_t^2$ and a circle of infinitesimal radius. By using Cauchy’s theorem and neglecting the residue of the pole in $s=-\Lambda^2$ in order the preserve asymptotic freedom [^4], we obtain: $$\tilde{\alpha}_S(k_t^2) = \int_{-\pi}^{+\pi} \frac{d\varphi}{2\pi}
\frac{1}{ \beta_0 \left[ \log \frac{k_t^2}{\Lambda^2} + i\varphi
\right] }. \label{geometric}$$ The standard resummation formula (\[generale\]) is obtained by neglecting the imaginary term $i\varphi$ in the denominator. In hard processes $Q\,\gg\,\Lambda$ and one expects on physical grounds that $k_t\sim Q$, implying: $$\label{nonvera}
\log \frac{k_t^2}{\Lambda^2} \, \gg \, \pi.$$ As a consequence, one obtains the coupling evaluated at the gluon transverse momentum squared: $$\tilde{\alpha}_S(k_t^2)~\rightarrow~\frac{1}{\beta_0 \log \frac{k_t^2}{\Lambda^2} }
~=~\alpha_S(k_t^2).
\label{oldcoup}$$ However, the assumption (\[nonvera\]) is not correct, because the transverse momentum $k_t$ is integrated from $Q$ down to very small scales. We propose to modify the resummation formula (\[generale\]) according to this criticism: approximation (\[nonvera\]) disregards absorptive effects related to the gluon jet decay, which are important in the infrared region, and therefore has to be avoided. By performing the integration exactly, we obtain: $$\label{newcoup}
\tilde{\alpha}_S(k_t^2) =
\frac{1}{\beta_0} \left[ \frac{1}{2} -
\frac{1}{\pi} \arctan \frac{ \log \frac{k_t^2}{\Lambda^2} }{\pi} \right].$$ The effective coupling (\[newcoup\]) approaches the standard one in the asymptotic region $k_t^2\gg \Lambda^2$, but it does not contain the infrared pole in $k_t^2 = \Lambda^2$ [^5]. Furthermore, $\tilde{\alpha}_S(k_t^2)$ is positive definite, monotonically decreasing in all the $k_t$ range and has a finite limit at zero momentum (see fig. 1): $$\lim_{k_t^2\rightarrow 0}\tilde{\alpha}_S(k_t^2) ~=~ \frac{1}{\beta_0}.
\label{zerolimit}$$ According to eq. (\[newcoup\]), the effective coupling deviates from the standard one and saturates at a scale of order $$k_t ~\sim~ \Lambda \, e^{\pi/2} ~\sim~ 1 \; \rm{GeV},$$ for $\Lambda\sim 200$ MeV.
=0.40=0.25
Our improved expression for the form factor at one-loop approximation reads: $$\label{generale_new}
f_N(\alpha_S) = \exp \int_0^1 dz \frac{z^{N-1}-1}{1-z}
\Big\{ \int_{Q^2(1-z)^2}^{Q^2(1-z)} \frac{dk_t^2}{k_t^2} \, A_1\,\tilde{\alpha}_S(k_t^2)
\,+\,B_1\,\tilde{\alpha}_S(Q^2(1-z))\,+\, D_1\,\tilde{\alpha}_S(Q^2(1-z)^2)\Big\},$$ where the expression for the effective coupling $\tilde{\alpha}_S$ is given by eq. (\[newcoup\]). By comparing with the standard resummation formula (\[generale\]), we see that the only difference is the appearance of $\tilde{\alpha}_S$ in place of $\alpha_S$.
=0.40=0.25
Let us now consider two-loop effects. Note that eq. (\[partenza\]) can be written as: $$\label{rewrite}
\tilde{\alpha}_S(k_t^2) = \frac{1}{2\pi i} \int_0^{k_t^2} ds~\mathrm{Disc}_s
~\frac{\alpha_S(-s)^{1L}}{ s },$$ where $$\alpha_S(\mu^2)^{1L} = \frac{1}{\beta_0 \log \frac{\mu^2}{~\Lambda^2} }$$ is the one-loop coupling. We now assume that eq. (\[rewrite\]) generalizes to the two-loop coupling $$\label{coupling}
\alpha_S\left(\mu^2\right) = \frac{1}{\beta_0\log \mu^2/\Lambda^2}
-\frac{\beta_1}{\beta_0^3}\,\frac{\log
\log \mu^2/\Lambda^2}{\log^2\mu^2/\Lambda^2},$$ where $\beta_1 = (51-19/3\;n_F)/(8\pi^2)$. Therefore, the subleading contribution to the effective coupling $\tilde{\alpha}_S(k_t^2)$ is: $$\begin{aligned}
&& \frac{1}{2\pi i} \int_0^{k_t^2}\,d s \,\mathrm{Disc}_s \,
\frac{1}{s} \,\, \frac{-\beta_1\log \log
(-s/\Lambda^{2})}{\beta_0^3\,\log^2 (-s/\Lambda^2)\,}
\nonumber\\
&=&
-\,\frac{\beta_1}{\beta_0^3} \frac{1}{\pi^2+ \log^2
k_t^2/\Lambda^2}
\left\{ \left[ -\frac{1}{2} +
\frac{1}{\pi}\arctan \frac{\log k_t^2/\Lambda^2}{\pi } \right] \,
\log \frac{k_t^2}{{\Lambda}^2} + \frac{1}{2} \log \left[ \pi^2 +
\log^2 \frac{k_t^2}{\Lambda^2} \right] + 1 \right\}.\end{aligned}$$ It is not easy to derive the terms proportional to $A_2$. Let us make the following guess: $$\begin{aligned}
\label{coupling_doppio}
\alpha_S^2(k_t^2)~\rightarrow~\tilde{\alpha_S^2}(k_t^2)\,&=&\,
\frac{1}{2\pi i} \int_0^{k_t^2}d s \, \mathrm{Disc}_s
~\frac{\alpha^2_S(-s)^{1L}}{ s }
\nonumber \\
&=&
\frac{1}{2\pi i} \int_0^{k_t^2}d s \, \mathrm{Disc}_s
\frac{1}{s~\beta_0^2\log^2 (-s/\Lambda^2) }
\nonumber \\
&=& \frac{1}{\beta_0^2(\pi^2+
\log^2 k_t^2/\Lambda^2)}.\end{aligned}$$ An explicit verification of the validity of eq. (\[coupling\_doppio\]) would require a three loops computation, which is beyond the purpose of the present work. That is because $A_2$ first occurs at order $\alpha_S^2$ and running coupling effects involve at least one additional power of $\alpha_S$, i.e. $\alpha_S^3$ in total. Let us stress that also the standard replacement $\alpha_S^2 \rightarrow \alpha_S(k_t)^2$ has never been explicitly checked, as its verification requires a three loop computation as well. In eq. (\[coupling\_doppio\]) we just assumed that the $s$-discontinuity had to be taken after squaring the gluon polarization function; it is remarkable that our choice gives a simple analytic result. Other choices are possible at this level, such as squaring the effective coupling: $\alpha_S^2(k_t^2) \rightarrow \tilde{\alpha}_S(k_t^2)^2$ [^6]. Anyway, since two-loop terms are rather small, our choice seems acceptable in a model of soft gluon dynamics.
In conclusion, our expression for the resummed form factor at next-to-leading order reads: $$\begin{aligned}
\label{improved} f_N(\alpha_S) &=& \exp \int_0^1 dz
\frac{z^{N-1}-1}{1-z} \Big\{ \int_{Q^2(1-z)^2}^{Q^2(1-z)}
\frac{dk_t^2}{k_t^2} \big[ A_1 \tilde{\alpha}_S(k_t^2) + A_2
\tilde{\alpha_S^2}(k_t^2) + \cdots \big]\,+
\nonumber\\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~
+\,B_1 \tilde{\alpha}_S(Q^2(1-z))+\cdots \,
+\, D_1 \tilde{\alpha}_S(Q^2(1-z)^2)+\cdots\Big\},
\label{espansione}\end{aligned}$$ where: $$\begin{aligned}
\label{alphaS2L}
\tilde{\alpha}_S(k_t^2)&=&
\frac{1}{\beta_0} \left[ \frac{1}{2} -
\frac{1}{\pi} \arctan \frac{ \log \frac{k_t^2}{\Lambda^2} }{\pi} \right] +
\nonumber\\
&-&\frac{\beta_1}{\beta_0^3}
\frac{1}{\pi^2+ \log^2 k_t^2/\Lambda^2}
\left\{\left[ -\frac{1}{2} +
\frac{1}{\pi}\arctan \frac{\log k_t^2/\Lambda^2}{\pi } \right] \,
\log \frac{k_t^2}{{\Lambda}^2} + \frac{1}{2} \log \left[ \pi^2 +
\log^2 \frac{k_t^2}{\Lambda^2} \right] + 1 \right\}\end{aligned}$$ and $\tilde{\alpha_S^2}(k_t^2)$ is given by eq. (\[coupling\_doppio\]).
Eq. (\[improved\]) is the main result of this work and replaces eq. (\[generale\]). The main problem of eq. (\[generale\]), discussed before, does not occur anymore in our improved expression: the effective couplings $\tilde{\alpha}_S$ and $\tilde{\alpha_S^2}$ do not diverge for small values of the argument and actually remain acceptably small.
Our treatment of the running coupling constant resembles the exponentiation of the $\pi^2$ terms in the $K$-factor of Drell-Yan coefficient function [@prima]. It is well known that, going from the space-like kinematics of Deep-Inelastic-Scattering ($q^2<0$) to the time-like kinematics of Drell-Yan ($q^2>0$), double logarithms in the vertex correction generate a large constant term proportional to $\pi^2$, because: $$\log^2(-q^2-i0) \,\rightarrow\, \log^2(q^2) - \pi^2 + \cdots.$$ Since double logs exponentiate, it has been suggested that also the $\pi^2$ terms exponentiate: $$e^{ -c\,\alpha_S\,\log^2 (- q^2 - i 0) } \, \rightarrow \,
e^{ c\,\alpha_S\,\pi^2 } \, e^{ -c\,\alpha_S\,\log^2 (q^2) },$$ where $c$ is a positive constant [^7]. It has been however established that the exponentiation of $\pi^2$ is not exact, being violated at two loops by terms of the form $\zeta(2)\log^2(-q^2-i0)$, where $\zeta$ is the Rieman zeta function ($\zeta(2)=\pi^2/6$) [@prima2]. Analogously, in our case, we cannot expect to have a complete resummation; we resum, however, a set of constants which are certainly there, and this seems to be a reasonable approximation.
The functions $g_1$ and $g_2$
=============================
Let us begin this section by defining the logarithmic accuracy in our model. We have powers of the coupling at the hard scale, $$\alpha\equiv\alpha_S(Q^2),$$ multiplied by powers of the infrared logarithm, $L=\log N$, i.e. monomials of the form $$\alpha^n ~ L^k.$$ Within the usual resummation scheme,
- by leading logarithms ($LL$) we mean all the terms in the exponent of the form factor of the form $$\alpha^n ~ L^{n+1} ~~~~~~~~~~~~~~ {\rm with}~~~n = 0, 1,2,\cdots\infty ;$$
- by next-to-leading logarithms ($NL$) we mean all the terms of the form $$\alpha^n ~ L^n ~~~~~~~~~~~~~~~ {\rm with}~~~n = 0, 1,2,\cdots\infty.$$
The coupling $\alpha_S(k_t^2)$ at one-loop, when expanded in powers of $\alpha$, produces a series of leading logarithms: $$\label{expand1}
\alpha_S(k_t^2) = \alpha - \beta_0\alpha^2\log\frac{k_t^2}{Q^2}
+ \beta_0^2\alpha^3\log^2\frac{k_t^2}{Q^2}
+ \cdots$$ The powers of $\log k_t^2/Q^2$ give indeed powers of $L=\log N$ after the integration over $k_t$ and $z$ in eq. (\[generale\]).
According to eq. (\[newcoup\]), the relation between the standard and the effective coupling reads [^8]: $$\tilde{\alpha}_S(k_t^2) =
\frac{1}{\pi\beta_0}\arctan\left[\pi\beta_0\alpha_S(k_t^2)\right]
~~~~~{\rm for}~~~\alpha_S(k_t^2) > 0.$$ Therefore, the expansion of our effective coupling in the parameter $\alpha$ is: $$\label{expand2}
\tilde{\alpha}_S(k_t^2) = \alpha - \beta_0\alpha^2\log\frac{k_t^2}{Q^2}
+ \beta_0^2\alpha^3\log^2\frac{k_t^2}{Q^2}
- \frac{(\beta_0\pi)^2}{3} \alpha^3 + \cdots$$ Comparing eq. (\[expand1\]) with eq. (\[expand2\]), we see that the expansion of the effective coupling involves subleading logarithms compensated by powers of $\beta_0\pi$, coming from the absorptive effects discussed before. The main point is that the $\beta_0\pi$ terms have a fundamental regulating effect over the logarithmic ones; therefore one has to consider them on equal footing in the counting. The hierarchy in our model can therefore by defined as follows:
- by $LL$ we mean all the terms of the form $$\alpha^n ~ (\beta_0 L)^{n+1-k} ~ (\beta_0\pi)^k ~~~{\rm with}~~~ k
= 0, 1,\cdots n+1;$$
- by $NL$ we mean all the terms of the form $$\alpha^n ~ (\beta_0 L)^{n-k} ~ (\beta_0\pi)^k ~~~{\rm with}~~~k =
0, 1,\cdots n.$$
The usual scheme is “minimal” in the sense that it deals only with powers of the logarithms and of the coupling; the related problem is that of the Landau pole discussed above. Instead, our scheme, in order to solve the Landau problem, has to be non-minimal.
The next step is to perform the integrations within the above defined next-to-leading accuracy. In general, we believe that fixed logarithmic accuracy is a consistent approximation in our scheme because series of logarithms of higher order are suppressed by powers of $\tilde{\alpha}_S$, which is always small. The integrations over transverse and longitudinal gluon momenta are easily done within the standard approximation [@cattren]: $z^{N-1}-1\simeq -\theta \left(1-z-1/n\right),\label{ctapprox} $ where $\theta$ is the step function and $n=N e^{\gamma_E}$, with $\gamma_E$ being the Euler constant. This approximation, as shown in our previous paper [@ug3], misses for instance terms of the form: $ \pi^2 \alpha^n L^{n-1}$, which are leading according to our power counting. However, in our model, we are interested in picking up only those terms proportional to $\beta_0\pi$, which come from the analytic continuation of the coupling from space-like to time-like region. These are the terms coming from the effective coupling, having the regulating effect on the Landau pole.
In order to obtain the functions $g_1$ and $g_2$, we express the logarithm of the hard scale $\log
Q^2/\Lambda^2$, coming from the integrations, in terms of the coupling evaluated at a general renormalization scale $\mu$ by means of the formulas: $$\log \frac{Q^2}{\Lambda^2} = \log\frac{\mu^2}{\Lambda^2} - \log \frac{\mu^2}{Q^2}.$$ and $$\log \frac{\mu^2}{\Lambda^2} =\frac{1}{\beta _{0}\alpha _S(\mu^2) }
+\frac{\beta_1}{\beta_0^2} \log \left(\beta_0\, \alpha _S(\mu^2)
\right).
\label{logsrep}$$ We finally Taylor expand the exponent on the r.h.s of eq. (\[generale\_new\]) in powers of $\gamma_E$, $\beta_1$ and $\log\mu^2/Q^2$ up to first order included. Our result for the leading function $g_1$ reads: $$\begin{aligned}
\label{g1new}
& & g_1(\lambda;r) = \frac{\mathrm{A_1}}{4\,\beta_0\, \lambda}
\biggl\{ -\left( 1 - 2\,\lambda \right) \,
\log [(1 - 2\,\lambda)^2+r^2]
+ 2\, \left( 1 - \lambda \right)
\,\log [(1 - \lambda)^2+r^2]- \log[1+r^2]+
\\
& & + \left. \frac{1}{r}
\left[ -\pi\,\lambda^2 + (1-r^2)\,\arctan \frac{1}{r}+
\left[ \left( 1 - 2\lambda \right)^2-r^2 \right]
\arctan \frac{1-2\lambda}{r} -
2 \left[\left( 1 - \lambda \right)^2-r^2\right]
\arctan \frac{1-\lambda}{r}
\right]
\right\},
\nonumber\end{aligned}$$
The subleading function $g_2$ reads: $$\begin{aligned}
\label{g2new}
g_2(\lambda;r) &=& \frac{\mathrm{D_1}}{ 4 \beta_0}
\left\{
\log [(1 - 2\,\lambda)^2+r^2]-\log [1+r^2]
+ \frac{2}{r}
\left[
- \pi\,\lambda+ \arctan \frac{1}{r} - ( 1 - 2\lambda ) \arctan
\frac{1- 2 \lambda}{r}\right] \right\}+
\nonumber \\
&+& \frac{\mathrm{B_1}}{2\,\beta_0}
\left\{
\log [(1 - \lambda)^2+r^2]-\log [1+r^2]
+ \frac{1}{r} \left[-\pi\,\lambda+
2\,\arctan\frac{1}{r} -2\, ( 1 -
\lambda ) \arctan \frac{1-\lambda}{r}\right] \right\} + \nonumber
\\
&+& \frac{\mathrm{A_2}} {4\, \beta_0^2} \biggl\{ \log[(1- 2
\lambda)^2 + r^2] -2 \log[(1- \lambda)^2 + r^2] + \log[1+ r^2]+
\nonumber \\
&+ & \left. \frac{2}{r} \left[- ( 1- 2 \lambda) \arctan\frac{1-2
\lambda}{r}+ 2 ( 1- \lambda) \arctan\frac{1-\lambda}{r} -
\arctan\frac{1}{r}\right]
\right\} + \nonumber \\
&+&
\frac{\mathrm{A_1 \,\beta_1}}{ 16\, \beta_0^3} \biggl\{
-\log^2[(1- 2 \lambda)^2 + r^2] + 2 \log^2[(1- \lambda)^2 +
r^2]-\log^2[1+ r^2] +
\nonumber \\
&+& \left. \frac{2\,\pi}{r} ( 1-2 \lambda) \left[-1+\frac{2}{\pi}
\arctan\frac{1-2\lambda}{r}\right] \log[(1-2\lambda)^2+r^2] +
\nonumber \right. \\
&+& \left.
\frac{4\,\pi}{r} ( 1- \lambda) \left[1-\frac{2}{\pi}
\arctan\frac{1-\lambda}{r}\right] \log[(1-\lambda)^2+r^2]+
\frac{2\,\pi}{r} \left[-1+\frac{2}{\pi} \arctan\frac{1}{r}\right]
\log[1+r^2] +
\nonumber \right. \\
&+& \left. 4 \arctan \frac{1-2\lambda}{r}\left[-\pi+
\arctan\frac{1-2\lambda}{r}\right]+ 8 \arctan
\frac{1-\lambda}{r}\left[\pi- \arctan\frac{1-\lambda}{r}\right]+
\right. \nonumber \\
&+& \left. 4 \arctan \frac{1}{r}\left[-\pi+
\arctan\frac{1}{r}\right] \right\}
+ \nonumber \\
&+& \frac{\mathrm{A_1}}{ 4\, \beta_0}\,\log\frac{\mu^2}{Q^2}
\biggl\{ \log[(1- 2 \lambda)^2 + r^2] -2 \log[(1- \lambda)^2 +
r^2] + \log[1+ r^2]+
\nonumber \\
&+& \left. \frac{2}{r} \left[- ( 1- 2 \lambda) \arctan\frac{1-2
\lambda}{r}+ 2 ( 1- \lambda) \arctan\frac{1-\lambda}{r} -
\arctan\frac{1}{r}\right]
\right\} + \nonumber \\
& + &
\frac{\mathrm{A_1\, \gamma_E}}{ 2\, \beta_0}
\biggl\{ \log[(1- 2 \lambda)^2 + r^2] - \log[(1- \lambda)^2 +
r^2] + \nonumber \\ &+ & \left. \frac{2}{r} \left[- ( 1- 2
\lambda) \arctan\frac{1-2 \lambda}{r}+
( 1- \lambda) \arctan\frac{1-\lambda}{r}
-\frac{\pi \,\lambda}{2} \right] \right\}.\end{aligned}$$ We have defined — note the general renormalization scale $\mu$ in the coupling: $$\label{lambdardef} \lambda~\equiv~\beta _0\, \alpha _S (\mu^2) \,
L ~~~~~\mathrm{and}~~~~~ r~\equiv~\pi~\beta _{0} \, \alpha
_S(\mu^2).$$ Note that $\lambda$ and $r$ are obtained from each other by interchanging $L$ with $\pi$. All this is in line with the logarithmic accuracy defined above.
A few remarks are in order. The first one is that, unlike the standard case, $g_1$ and $g_2$ do not depend only on $\lambda$ but also on $r$, i.e. on $\alpha_S$ alone. Our $g_1$ and $g_2$ are then “non-minimal”, as a consequence of the inclusion of absorptive constants in the coupling. The parameter $r\neq 0$ acts as a regulator of Landau-pole singularities: $g_1$ and $g_2$ are regular for any value of $\lambda$. As expected, in the limit $r\rightarrow 0$ we recover the standard $g_1$ and $g_2$ [@cattren]: $$\label{g1old}
g_1(\lambda;r=0) = \frac{\mathrm{A_1}}{2\,\beta_0\, \lambda}
\left\{ -\left( 1 - 2\,\lambda \right) \,
\log [1 - 2\,\lambda]
+ 2\, \left( 1 - \lambda \right)
\,\log [1 - \lambda] \right\}$$ and $$\begin{aligned}
g_2(\lambda;r=0)&=&
\frac{\mathrm{D_1}\,\log (1 - 2\,\lambda )}
{2\,{{\beta }_0}} +
\frac{\mathrm{B_1}\,\log (1 - \lambda )}
{{{\beta }_0}}
+\frac{\mathrm{A_2}\,
\left[ \log (1 - 2\,\lambda ) -
2\,\log (1 - \lambda ) \right] }{2\,
{{{\beta }_0}}^2} +
\\ \nonumber
&-&\frac{\mathrm{A_1}\, {{\beta }_1}\,
\left[ 2\,\log (1 - 2\,\lambda ) +
{\log^2 (1 - 2\,\lambda )} -
4\,\log (1 - \lambda ) -
2\,{\log^2 (1 - \lambda )} \right] \,
}{4\,{{{\beta }_0}}^3} +
\\ \nonumber
&+&\frac{\mathrm{A_1}\, \log (\mu^2/Q^2) \,
\left[ \log (1 - 2\,\lambda ) -
2\,\log (1 - \lambda ) \right] \,}{2\,{{\beta }_0}}
+ \frac{\mathrm{A_1}\,{{\gamma }_E}\,
\left[ \log (1 - 2\,\lambda ) -
\log (1 - \lambda ) \right] }
{{{\beta }_0}}.\end{aligned}$$ If we expand our $g_1$ and $g_2$ for small $r$, we find that the leading corrections are of order $r^2 \lambda^n$: this means corrections of next-to-next-to-leading order (NNLO) $\alpha_S^{n+1} L^n$ in the standard counting, which had been overlooked in our previous evaluation of the $g_3$ [@ug3]. Our $g_1$ and $g_2$ contain some terms from the standard $g_3$, $g_4$, $g_5$, $\ldots$ or from the coefficient function $C(\alpha_S)$ multiplying the form factor. It is remarkable that eqs. (\[g1new\]) and (\[g2new\]) involve only powers of the $\mathrm{log}$ and $\mathrm{arctan}$ functions.
=0.40=0.25
=0.40=0.25
In fig. 2 we have plotted the form factor $f_N(\alpha_S)$ as a function of $N$ for the decay of a beauty quark, i.e. for $Q=m_b$[^9], for which $\alpha_S=0.21$ and $n_F=3$. The dashed line is the plot of the standard form factor, i.e. of the same form factor in the limit $r \rightarrow 0$. We have plotted moments up to $N=20$, because above this point the standard form factor, as already discussed, becomes singular. Our form factor is reasonably close to the standard one up to $N\sim 10$, the latter being steeper for larger $N$. For $N=20$ there is a difference of a factor 2 circa. Fig. 3 shows similar plots for the top case, i.e. for $Q=m_t$ for which $\alpha_S=0.11$ and $n_F=5$. Due to the increased hard scale by more than one order of magnitude:
- differences between the two models are much smaller;
- the peak above one, related to the subleading, single-logarithmic effects, is barely visible, while it is rather pronounced in the [*beauty*]{} case.
The scale dependence is shown in fig. 4, where we have plotted $f_N(\alpha_S)$ for the [*beauty*]{} case for $\mu^2=Q^2$, $\mu^2=Q^2/4$ and $\mu^2=4Q^2$. The scale ambiguity is smaller than the difference between the models shown in fig. 2.
=0.40=0.25
Let us now discuss the extension of our model to NNLO, i.e. the computation of the function $g_3$. It is clear that one has to make similar guesses to the ones needed at NLO to evaluate the terms proportional to $A_2$. Apart from that, the computation, although technically rather cumbersome, does not seem to present any specific difficulty. In general, our fixed logarithmic accuracy for the exponent in the form factor, $$\Sigma = L ~ g_1(\lambda;r) + g_2(\lambda;r)
+ \alpha ~ g_3(\lambda;r) + \alpha^2 g_4(\lambda;r) + \cdots ,$$ is an expansion with better convergence properties than the usual one, because the effective coupling becomes at the most of order one. In other words, our expansion is certainly better than an expansion in $1/(\beta_0\pi)^n$, which is already convergent as $1/(\beta_0\pi) < 1$. We expect the extension of our model to NNLO to be relevant also for beauty physics. In a previous work [@ug3] we have found instead that usual NNLO effects could not be included in the case of beauty decay. In fact, the hard scale $Q = m_b$ was not large enough to avoid effects of the divergence of the perturbative series even at relatively low values of $x$. Such divergence was related to the integration over the Landau pole and is therefore absent in the present scheme.
Form Factors in $x$ space
=========================
Up to now we have considered form factors in $N$-space. The $N$-moments are indeed physical quantities, but in practice a measure of the moments for large $N$ is difficult. Therefore, let us transform back to $x$-space. The inverse transform of $f_N(\alpha_S)$ is defined as: $$\label{xspace}
f(x;\alpha_S) = \frac{1}{2 \pi i } \int_{C-i \infty}^{C+i
\infty}dN \,x^{-N} f_N(\alpha_S),$$ where $C$ is a constant such that the integration contour lies to the right of all the singularities of $f_N$. A standard computation gives to next-to-leading $\log(1-x)$ accuracy [@thrust]: $$\label{finx} f(x;\alpha_S) = -\frac{d}{dx} \left\{ \theta(1-x-0) \frac{ \exp
\left[\,l\, g_1(\tau;r)+g_2(\tau;r) \right] } {
\Gamma\left[1-\frac{d}{d\tau}
\left(\tau g_1(\tau;r) \right)\right] } \right\} ,$$ where we have defined: $$\tau~\equiv~\beta_0~\alpha_S(\mu^2)~l~~~~~\mathrm{and}~~~~~l~\equiv~-\log(1-x).$$ In eq. (\[finx\]) we have neglected terms $O(1-x)$, which are small in the threshold region. The form factor $f(x;\alpha_S)$ in eq. (\[finx\]) is plotted in fig. 5 for the [*beauty*]{} case. Also shown is the standard form factor, i.e. the same formula for $r=0$. Our form factor is formally well defined up to $x=1$, while the standard one presents a singularity at $x_{crit}\sim
1-\Lambda/Q$, so it is plotted only below this point. There are substantial differences between the models: our distribution is broader, the peak is smaller and occurs at much larger $x$ than in the standard case. The reason is the following. Our model contains an effective coupling which approaches a constant value for $N\rightarrow\infty$ or, equivalently, $x\rightarrow 1$ (see eq. (\[zerolimit\])). Therefore, for $x\sim 1$ our model resembles a frozen coupling scheme, while the standard form factor contain a coupling divergent at the non-zero momentum $\Lambda$ — the well-known Landau pole. In fig. 6 we plot $f(x;\alpha_S)$ for the [*top*]{} case in both models; as expected, differences are much smaller with respect to the [*beauty*]{} case. The dependence of our $f(x;\alpha_S)$ from the renormalization scale $\mu$ is shown in fig. 7 for the [*beauty*]{} case and turns out to be moderate.
=0.40=0.25
Phenomenology
=============
Let us now discuss the phenomenological relevance of our model. A natural application concerns beauty physics, i.e. a case with a moderately large hard scale — when the hard scale is very large, as in top decays, there are basically no differences between our model and the standard one. In the beauty case, large non-perturbative corrections are expected. The main problem, in our opinion, is that of combining perturbative and non-perturbative effects in the most efficient way. In many phenomenological studies, one assumes a shape for the non-perturbative effects and convolutes it with a frozen-coupling distribution. Since $\alpha_S(m_b)\sim0.21$ (that is, rather small) and the coupling does not increase at small $k_t$, the perturbative effects turn out to be extremely small. Since the theoretical prediction is a convolution of perturbative and non-perturbative effects, it is completely dominated by the assumed form of the latter. The standard resummation scheme — running coupling — includes much more perturbative dynamics, but it is technically more difficult to implement in a phenomenological analysis and it has been barely used. That is because the perturbative form factors in $x$-space are generally computed with a numerical contour integration, by using the so-called minimal prescription [@minimalpre]. This is a consistent prescription based again on “minimality”: the correct logarithms of $1-x$ are included and the distribution is formally well defined up to $x = 1$, but no physically motivated effects are included, only logarithms. Our approach includes instead a (perturbative) physical effect, namely parton decay, giving rise to analytic distributions well-defined in the whole $x$-space. One can then easily convolve our spectra with an assumed form of non-perturbative effects: $$\begin{aligned}
f_{ex}(x) &=& \int_0^1 \int_0^1 dx' dx''
\delta(x-x'x'') f_{np}(x';c_k) f(x'';\alpha_S)
\nonumber\\
&=&
\int_x^1 \frac{dx'}{x'} f_{np}(x';c_k) f(x/x';\alpha_S),\end{aligned}$$ where $f_{np}$ is a non-perturbative function and $f_{ex}$ is the spetrum to be compared with the data. $c_k$ denotes schematically non-perturbative parameters entering $f_{np}$.
It is well known that non-perturbative effects broaden the peaks of perturbative spectra and shift them to smaller $x$’s. They have, as intuitively expected, a smearing effect, representing the final stage of parton evolution, not described by perturbation theory. We have not done yet any detailed phenomenological analysis, but the following points can already be made. Our model gives a broader spectrum compared to the usual one, while it gives a peak at larger values of $x$. The peak we find corresponds indeed to a jet invariant mass below 1 GeV. We then expect that our model has to be supplemented with non-perturbative effects shifting the peak towards smaller $x$.
Conclusions
===========
Our main result is the improved threshold resummation formula given by eq. (\[improved\]). The latter involves — in place of the usual running coupling possessing the Landau pole — an effective coupling which is not singular in the infrared region and remains rather small in all the integration domain. It approaches the standard coupling for large transverse momenta, while it deviates from the latter at a scale of order $$k_t~\sim~\Lambda e^{\pi/2}\sim~1~{\rm GeV}.$$ The effective coupling, unlike the usual one, includes absorptive effects related to the decay of gluon jets. The physical explanation of the absence of the Landau pole in our resummation formula is the following. The inclusive gluon branching, inducing the running coupling in resummation formula, is described by taking into account absorptive effects in addition to the usual dispersive ones. The gluon cascade is then treated, more realistically, as an intrinsically unstable process and there is not enough resolution time to see the effects of the Landau pole.
By performing the integrations over longitudinal and transverse momenta with next-to-leading accuracy, we have obtained the functions $g_1$ and $g_2$ in the exponent of the form factor given in eqs. (\[g1new\]) and (\[g2new\]) respectively. These functions define next-to-leading order resummed threshold distributions which are well defined in the whole moment or physical space. There are large differences between our next-to-leading order form factor and the standard one for beauty decays. We believe that our model can be used to describe the variety of spectra in inclusive or rare $b$ decays. Natural extensions of our approach involve resummation of shape variables or the inclusion of mass effects in semi-leptonic $b\rightarrow c$ transitions.
0.5truecm
**Acknowledgments**
One of us (U.A.) wishes to thank B. Ermolaev for discussions.
[99]{}
G. Parisi and R. Petronzio, Nucl. Phys. B 154, 427 (1979); G. Curci and M. Greco, Phys. Lett. B92, 175 (1980).
J. Kodaira and L. Trentadue, SLAC-PUB-2934 (1982); Phys. Lett. B 112, 66 (1982); S. Catani, E. D’Emilio and L. Trentadue, Phys. Lett. B 211, 335 (1988).
G. Sterman, Nucl. Phys. B 281, 310 (1987).
S. Catani and L. Trentadue, Nucl. Phys. B 327, 323 (1989).
S. Catani and L. Trentadue, Nucl. Phys. B 353, 183 (1991).
B. Ermolaev, M. Greco, S. Troyan, Phys. Lett. B 522, 57 (2001).
M. Pennington, R. Roberts and G. Ross, Nucl. Phys. B 242, 69 (1984);
S. Catani, L. Trentadue, G. Turnock and B. Webber, Nucl. Phys. B 407, 3 (1993).
G. Altarelli, N. Cabibbo, G. Corbó, L. Maiani and G. Martinelli, Nucl. Phys. B 208, 365 (1982); G. Korchemsky and G. Sterman, Phys. Lett. B 340, 96 (1994); R. Akhoury and I. Rothstein, Phys. Rev. D 54, 2349 (1996); M. Cacciari, G. Corcella and A. Mitov, JHEP 0212, 015 (2002).
U. Aglietti and G. Ricciardi, Phys. Rev. D 66, 074003 (2002);
E. Gardi, Nucl. Phys. B 622, 365 (2002).
S. Catani, M. Mangano, P. Nason and L. Trentadue, Nucl. Phys. B 478, 273 (1996).
See for instance, [*Methods of Quantum Field Theory in Statistical Phyisics*]{}, by A. Abrikosov, L. Gorkov and I. Dzyaloshinski, Dover Publications, 1975.
Y. Dokshitzer, D. Dyakonov and S. Troyan, Phys. Rept. 58, 269 (1980).
D. Amati, A. Bassetto, M. Ciafaloni, G. Marchesini and G. Veneziano, Nucl. Phys. B 173, 429 (1980).
G. Parisi, Phys. Lett. B 90, 295 (1980) ; J. Kripfganz, Nucl. Phys. B 177, 509 (1980);
W. Van Neerven, Nucl. Phys. B 268, 453 (1986).
U. Aglietti, Nucl. Phys. B 610, 293 (2001).
[^1]: e-mail address: Ugo.Aglietti@roma1.infn.it
[^2]: e-mail address: Giulia.Ricciardi@na.infn.it
[^3]: The value of $A_2$ is given in the $\overline{MS}$ scheme for the coupling constant.
[^4]: The integral on the discontinuity of the gluon propagator is transformed into the integral over a circle of radius $k_t^2$ [*plus*]{} the residue of the pole in $s=-\Lambda^2$. In our pragmatic approach, we have just omitted the latter.
[^5]: We define the $\arctan$ function as the one being continuous in zero and discontinuous at infinity, with image in the interval $(-\pi/2,+\pi/2)$.
[^6]: With this choice there is only one effective coupling, while with our choice there are two effective couplings.
[^7]: In our work we are dealing with a [*geometric*]{} series (cfr. eq. (\[geometric\])), while in the $K$-factor an [*exponential*]{} series is resummed.
[^8]: One has to use the relation $\arctan(1/x) = \pi/2-\arctan(x)$, which is valid for $x\geq0$ with our definition of the $\arctan$ function.
[^9]: In general, for heavy flavor decays $Q=2E_X$, where $E_X$ is the energy of the hadronic final state. One can set $2E_X=m_b$ in $b\rightarrow s\gamma$, while this is not true in semi-leptonic $b\rightarrow u$ decays [@tripledis].
|
---
abstract: 'We investigate experimentally the dynamics of a non-spherical levitated nanoparticle in vacuum. In addition to translation and rotation motion, we observe the light torque-induced precession and nutation of the trapped particle. We provide a theoretical model, which we numerically simulate and from which we derive approximate expressions for the motional frequencies. Both, the simulation and approximate expressions, we find in good agreement with experiments. We measure a torque of $1.9 \pm 0.5 \times 10^{-23}$ Nm at $1 \times 10^{-1}$ mbar, with an estimated torque sensitivity of $3.6 \pm 1.1 \times 10^{-31}$ Nm/$\sqrt{\text{Hz}}$ at $1 \times 10^{-7}$ mbar.'
author:
- Muddassar Rashid
- Marko Toroš
- Ashley Setter
- Hendrik Ulbricht
- Muddassar Rashid
- Marko Toroš
- Ashley Setter
- Hendrik Ulbricht
bibliography:
- 'rotation.bib'
title:
- Precession Motion in Levitated Optomechanics
- Supplementary Material for Precession Motion in Levitated Optomechanics
---
[*Introduction–*]{} A typically levitated optomechanical setup comprises of a particle, which is trapped using a tightly focussed laser beam. The trapped particle is best described as a harmonic oscillator and can exhibit a rich variety of dynamics. For example, the dynamics can be under-damped or over-damped depending on the number of collisions controlled by the background gas pressure. The dynamics can be linear or can show strong non-linearities depending on the oscillation amplitude, which is controlled by temporal and spatial modulated external electric, magnetic, and light forces: the dynamics can be driven or cooled. This tunability of the dynamics has enabled various studies of the rich physics of this harmonic oscillator, among them Brownian motion [@Li2010a], nonlinear dynamics [@Gieseler2013; @Fonseca2016], test of fluctuation theorems and non-equilibrium physics [@Gieseler2014; @Gieseler2014b; @Hoang2018], and thermodynamics in the single particle regime [@Millen; @Rondin2017; @Aranas2017]. The unique properties of the underlying dynamics and the ability to control levitated optomechanics demonstrate its immense potential for sensing [@Ranjit2016; @Hebestreit2017; @Hempston2017], as well as, to address fundamental questions in physics [@Romero-Isart2011; @Bassi2013; @bateman2014].
The centre-of-mass (c.o.m.) translation motion of the trapped nanoparticle has been studied in quite some detail already. Optical feedback, cavity-assisted schemes and electric forces have been used to control the translation and have been ultimately used to cool the motion to millikelvin temperatures [@Li2011a; @Kiesel2013; @Vovrosh2016; @Setter2017] and below [@vijay2016], already close to the ground state of the harmonic oscillator.
In addition, to exhibiting translation motion, trapped particles can also show rotation [@Arita2013; @Kuhn2015; @Rahman2017] and libration [@Hoang2016] motion. Different light-matter physics has been used to drive and control the rotation of levitated particles, such as the polarizability-anisotropy in silicon rods coupling to the polarization of light [@Kuhn2016]. Rotational frequencies of up to some GHz [@Reimann2018; @Ahn2018] have been observed, only limited by the centrifugal damage threshold of the rotating particle.
Variation of the linear to circular polarization of light gives a handle to switch between rotation and libration. Libration has been demonstrated experimentally with trapped nanodiamonds [@Hoang2016], silicon rods [@Kuhn2016] and dumbbells [@Ahn2018]. Like translation, libration motion is described by a harmonic oscillator model and is therefore a candidate to apply similar optical techniques for cooling with reasonable promise to reach a quantum ground state, making libration a stark contender in the race towards the quantum regime. First proposals discuss the usefulness of libration to generate macroscopic quantum states such as angular superpositions [@Carlesso2017; @Ma2017]. Additionally, both rotation and libration motion promise unprecedented high levels of sensitivity [@Hoang2016; @Kuhn2017; @Stickler2018] for detection of weak forces such as gravity [@Carlesso2017; @Carlesso2017a] and dispersive forces [@Manjavacas2017].
In this letter, we report on the observation of light-induced precession motion of a non-spherical silica particle compound. We give a theoretical description of the system and numerically simulate the model. We identify the mechanical frequencies in the experimental spectrum: in particular, translation, rotation (spin), precession, and nutation motion. We investigate the precession by variation of background gas pressure and of the power of the trapping laser in agreement with the theoretical model. We discuss the possibilities for torque sensing applications.
{width="1\linewidth"}
[*Theoretical model–*]{} We consider an anisotropic polarizable particle, which is optically trapped by an elliptically polarized Gaussian laser beam. Part of the scattered photons are collected and directed, using optical elements, towards a single photodetector. The scattered light is mixed with a local oscillator to obtain the direct homodyne photo-current $I_{\text{exp}}$, see Fig. \[fig:theory\](a) [@Rashid2017]. This experimental situation has been analyzed theoretically using a quantum model [@Toros2018t], as well as numerically, using an approximate classical model [@Toros2018s]. We now summarize the dynamics referring to the latter, where we assume that we are at relatively high pressure, such that we can neglect photon-recoil heating terms.
The dynamics can be described in a twelve dimensional phase space of the following classical variables: center of mass position $\boldsymbol{r}=(x,y,z)^{\top}$, center of mass conjugate momentum $\boldsymbol{p}=(p_{x},p_{y},p_{z})^{\top}$, angle $\boldsymbol{\phi}=(\alpha,\beta,\gamma)^{\top}$, and angle conjugate momentum $\boldsymbol{\pi}=(\pi_{\alpha},\pi_{\beta},\pi_{\gamma})^{\top}$, where the angles are defined in the Euler $z$-$y'$-$z''$ convention (see Fig. \[fig:theory\](b)). In particular, the dynamics is given by (in Itô form): $$\begin{aligned}
{1}
d\boldsymbol{r}= & -\boldsymbol{\partial_{p}}H_{\text{free}}dt\label{eq:dr}\\
d\boldsymbol{p}= & -\boldsymbol{\partial_{r}}H_{\text{grad}}dt+d\boldsymbol{p}_{\text{scatt}}+d\boldsymbol{\boldsymbol{p}}_{\text{coll}}\label{eq:dp}\\
d\boldsymbol{\boldsymbol{\phi}}= & \boldsymbol{\partial_{\pi}}(H_{\text{free}}+H_{\text{grad}})dt\label{eq:dphi}\\
d\boldsymbol{\boldsymbol{\pi}}= & -\boldsymbol{\partial_{\phi}}(H_{\text{free}}+H_{\text{grad}})dt+d\boldsymbol{\boldsymbol{\pi}}_{\text{scatt}}+d\boldsymbol{\boldsymbol{\pi}}_{\text{coll}},\label{eq:dpi}\end{aligned}$$ where $H_{\text{free}}$ and $H_{\text{grad}}$ denote the free Hamiltonian and the gradient potential, respectively, $d\boldsymbol{p}_{\text{scatt}}$, $d\boldsymbol{\boldsymbol{\pi}}_{\text{scatt}}$ denote the non-conservative terms induced by photon scattering, and $d\boldsymbol{p}_{\text{coll}}$, $d\boldsymbol{\boldsymbol{\pi}}_{\text{coll}}$ the non-conservative terms, which arise from gas collisions. Specifically, $d\boldsymbol{p}_{\text{scatt}}$ corresponds to the radiation pressure scattering force, which displaces the particle along the positive $z$ direction, and $d\boldsymbol{p}_{\text{coll}}$ denotes the terms, which tend to thermalize the center of mass motion to the temperature $T$ of the gas of particles. Similarly, $d\boldsymbol{\boldsymbol{\pi}}_{\text{scatt}}$ denotes the terms, which quantify the transfer of angular momentum from the photons to the particle, i.e. the driving terms, and $d\boldsymbol{\pi}_{\text{coll}}$ denotes the terms that tend to thermalize rotations to the temperature of the gas, i.e. the friction and diffusive terms (see [@supplement]).
We consider a specific experimental situation, where we illustrate the physical content of Eqs. (\[eq:dr\])-(\[eq:dpi\]), and we obtain approximate expression for the dominant mechanical frequencies (further details can be found elsewhere [@supplement]). Specifically, we consider the experimental situation that produces the power spectral density in Fig. \[fig:psd\], where the rotational frequencies are significantly higher or lower than the translational ones. To obtain the dominant mechanical frequencies, we can in first approximation treat translation and rotation as decoupled motion.
We start by looking at translational degrees of freedom. We suppose that $\vert\frac{\mathbf{r}}{\lambda}\vert\ll1$, where $\lambda$ is the laser wavelength, which limits translations to harmonic oscillations. In particular, the frequencies for the $x$, $y$, $z$ motion are given by: $$\begin{aligned}
{2}
\omega_{x}^{2}=\frac{2Pa_{1}\chi_{0}}{c\sigma_{L}w_{0}^{2}\rho}, & \quad\omega_{y}^{2}=\frac{2Pa_{2}\chi_{0}}{c\sigma_{L}w_{0}^{2}\rho},\quad & \omega_{z}^{2}=\frac{2P\chi_{0}}{c\sigma_{L}\rho z_{R}^{2}},\label{eq:translations}\end{aligned}$$ respectively, where $P$ is the laser power, $\sigma_{L}=\pi w_0^2$ is the effective laser beam cross section area, $w_0$ is the mean beam waist radius, $a_{1}$,$a_{2}$ quantify the asymmetry of the beam along the $x$, $y$ directions, respectively, $z_{R}$ is the Rayleigh length, $\rho$ is the particle density, $\chi_{0} = \frac{1}{3}\sum_{i=1}^3 \chi_i$ is an effective susceptibility of the particle , and $c$ is the speed of light. These frequencies are obtained directly from $H_{\text{grad}}$ by expanding to order $\mathcal{O}((\frac{\mathbf{r}}{\lambda})^{2})$.
The rotational frequencies arise from (i) the transfer of angular momentum during photon scattering, and from (ii) the gradient torque. On the one hand, the scattering torque drives the system into a fast spinning motion, while, on the other hand, the gradient torque would like to align the system with the polarization of the incoming beam in such a way to minimize the electric dipole potential energy, resulting in nutation and precession. Before deriving the rotational frequencies mathematically we now first give an intuitive picture of the two mechanisms.
The mechanism (i) can be understood in terms of the angular momentum carried by the incoming light beam (in a particle picture one can think of an individual photon carrying a small amount of angular momentum, e.g. $\hbar$ for circular polarization). During scattering the angular momentum is transferred to the nanoparticle, where the amount that is transferred depends on the susceptibility anisotropy and orientation of the nanoparticle. As a consequence, the particle starts to spin, until an asymptotic rotational frequency is reached, which is constrained by friction due to gas collisions. In particular, we consider the experimental situation, where the photon scattering gives rise to high spinning frequencies $\omega_\alpha^{\text{(spin)}}$ and $\omega_\gamma^{\text{(spin)}}$ about the $z$ and $z''$ axis, respectively.
Besides the dominant spinning motions there are also two additional secondary motions, which arise as a consequence of the mechanism (ii). In a nutshell, the gradient torque would like to align the nanoparticle in such a way to minimize the electric dipole potential energy, i.e. $\beta_0=\frac{\pi}{2}$, but once the nanoparticle starts to spin, i.e. it acquires a large angular momenta along the $z$ and $z''$ axis, it is unable to fully align, but rather settles around an equilibrium position $\beta_0\neq \frac{\pi}{2}$, which can be readily understood in terms of angular momenta addition. Any small perturbation, e.g. gas collisions, will make the $\beta$ angle oscillate around the $\beta_0$ angle, which results in libration (nutation) motion with frequency $\omega_\beta^{\text{(nutation)}}$. In addition, the coupling between $\beta$ and $\alpha$ also creates a second frequency for the $\alpha$ motion, which we denote by $\omega_\alpha^{\text{(precession)}}$: this motion can be visualized as a slow precession of the $z''$ axis about the $z$ axis (see Fig \[fig:theory\] (b)). We now derive the rotational frequencies.
The spinning frequencies can be obtained from Eq. , by setting $d\boldsymbol{\boldsymbol{\pi}}=0$, $\beta=\beta_{0}$, and neglecting conservative and stochastic terms, i.e. we consider only the driving term due to photon scattering and the friction term due to gas collisions. We find the asymptotic angle conjugate momentum $\boldsymbol{\boldsymbol{\pi}}^{\text{(spin)}}=-\frac{1}{2\Gamma_{c}}\boldsymbol{N}_{s}$, where $\Gamma_{c}$ is the collisional damping rate, and $\boldsymbol{N}_{s}=({N}_{\alpha},{N}_{\beta},{N}_{\gamma})^\top$ is the photon scattering torque. We then immediately find the spinning frequencies: $$\begin{aligned}
{1}
(\omega_{\alpha}^{\text{(spin)}},0,\omega_{\gamma}^{\text{(spin)}})^{\top} & =\mathbb{E}[Y]\boldsymbol{\pi}^{\text{(spin)}},\label{eq:spin}\end{aligned}$$ where $\mathbb{E}[\,\cdot\,]$ denotes the time-average over fast oscillating terms, $Y=Y(I)$ is the matrix that maps $\boldsymbol{\pi}$ to $\dot{\boldsymbol{\phi}}$, and $I$ is the moment of inertia tensor in the body-frame (see [@supplement]). The explicit expressions for $\omega_{\alpha}^{\text{(spin)}}$ and $\omega_{\gamma}^{\text{(spin)}}$ are given in Eqs. (B3) and (B5), respectively.
{width="100.00000%"}
We now consider the oscillations of $\beta$ about the equilibrium point $\beta_{0}$, which we denote by $\delta\beta_{0}$. This oscillatory, libration motion is induced by the (ii) conservative terms, as well as by the fast spinning motion of $\alpha$ and $\gamma$. In particular, after writing the Hamiltonian eqs. of motion for $\delta\beta_{0}$, using Eqs. and , performing the time average $\mathbb{E}[\,\cdot\,]$, and keeping only the dominant terms, we eventually obtain: $$\omega_{\beta}^{\text{nutation}}=\frac{1}{2}\frac{I_{1}+I_{2}}{I_{1}I_{2}}\csc ^2(\beta_0)\pi_{\alpha}^{\text{(spin)}},\label{eq:nutation}$$ where $I_{1}$ and $I_{2}$ denote the moment of inertia along the $x''$ and $y''$ principal axis. The explicit expressions for $\omega_{\beta}^{\text{(nutation)}}$ and $\beta_0$ are given in Eqs. (B5) and (B6), respectively.
The $\delta\beta_0$ oscillations also perturb the $\alpha$ motion: we denote the perturbation to the spinning motion by $\delta\alpha$, i.e. $\alpha(t)=\omega_\alpha^{\text{(spin)}} t+\delta\alpha$. In particular, from Eq. , performing time-average $\mathbb{E}[\,\cdot\,]$, and using Eq. , we eventually obtain $
\dot{\delta\alpha}=2\omega_\beta^{\text{(nutation)}}\text{cot}(\beta_0)\delta
\beta$, where $\delta
\beta=\mathcal{B}\text{cos}(\omega_\beta^{\text{(nutation)}}t)$, and $\mathcal{B}$ denotes the amplitude of $\delta \beta$ oscillations. However, in a typical detection we do not measure directly $\alpha$, but normally its sine or cosine value. We consider here $\text{sin}(\alpha)$, which we Taylor expand to order $\mathcal{O}(\delta\alpha)$, i.e. $\text{sin}(\alpha)=\text{sin}(\omega_\alpha^{\text{(spin)}}
t)+\text{cos}(\omega_\alpha^{\text{(spin)}} t)\delta\alpha$. In particular, from the last term, using trigonometric identities, and the expressions for $\delta\alpha$ and $\delta\beta_0$, we obtain a term proportional to $\text{sin}(\omega_{\alpha}^{\text{(precession)}}t)$, where $$\omega_{\alpha}^{\text{(precession)}}=\omega_\alpha^{\text{(spin)}}-\omega_\beta^{\text{(nutation)}}. \label{eq:precession}$$ The $\alpha$ degree of freedom has thus two distinct motions: a fast spinning motion with the frequency given in Eq. (\[eq:spin\]) and a slow precession motion with the frequency given in Eq. (\[eq:precession\]). This precession motion can be seen as a consequence of the $\beta$ motion, which perturbs back the $\alpha$ motion.
From Eqs. (\[eq:spin\]) and (\[eq:nutation\]), noting that $\boldsymbol{\pi}^\text{(spin)} \propto
\frac{\boldsymbol{N}_{s}}{\Gamma_c} \propto \frac{P}{p}$, we find that $\omega_\alpha^{\text{(spin)}}$ and $\omega_\beta^{\text{(nutation)}}$ scale linearly with the laser power $P$, and are inversely proportional to the gas pressure $p$, where we have assumed that the equilibrium position $\beta_0$ does not change significantly near an initially chosen power $P_0$ and pressure $p_0$, $\boldsymbol{N}_{s}$ is the torque due to photon scattering, and $\Gamma_c$ is the damping rate due to gas collisions. On the other hand, the precession frequency $\omega_{\alpha}^{\text{(precession)}}$ given in Eq. (\[eq:precession\]) can scale differently depending on the values of $\partial_P \Delta \omega\vert_{P_0}$ and $\partial_p \Delta \omega\vert_{p_0}$, where $\Delta
\omega=\omega_\alpha^{\text{(spin)}}-\omega_\beta^{\text{(nutation)}}$.
We have confirmed the validity of the obtained approximate formulae in Eqs. (\[eq:translations\])-(\[eq:precession\]) by numerically simulating Eqs. (\[eq:dr\])-(\[eq:dpi\]). More details of the simulations will be discussed elsewhere [@Toros2018s].
[*Experiments–*]{} The optical trap is shown in Fig. \[fig:theory\](a). In the experiments presented we use initially individual silica nanoparticles dispersed in water. We observe ageing of the solution with the result of clustering of the nanospheres into compounds of two to three nanospheres, a few weeks after preparation, as shown in Fig. \[fig:theory\](c). The aged solution is delivered to the trap using a nebuliser. The optical scattering force, limits the maximum particle mean radius that can be optically trapped to about 150 nm to 200 nm.
[*Results–*]{} The power spectral density (PSD) shown in Fig. \[fig:psd\](a) is generated from the time trace recorded as the photodetector signal, $I_{\text{exp}}$, over a time interval of one second. The PSD shows a rich spectrum of frequencies, which respond differently for changing laser power and background gas pressure.
Translation motion is observed at frequencies, $\omega_{x}$ = $2\pi \times$ 196 kHz, $\omega_{y}$ = $2\pi \times$ 246 kHz, $\omega_{z}$ = $2\pi \times$ 124 kHz in Fig. \[fig:psd\](a). We do not observe pressure dependency of the translation frequencies (see Fig. \[fig:psd\](b)), but find that they scale proportionately to the square root of laser power, $P$, in agreement with Eq.(\[eq:translations\]), see Fig. \[fig:psd\](c). The $x$ and $y$ peaks are separated because we use elliptically polarized light, which after it is reflected from the paraboloidal mirror, generates an asymmetric optical trap. The polarization of light was kept constant during the course of the experiments in this letter.
From the experimental data, we find the fundamental frequencies for the rotational motions, $\omega_{\gamma}^{\text{spin}}$ and $\omega_{\alpha}^{\text{spin}}$ to be $2\pi \times 1.9$ MHz and $2\pi \times 3.8$ MHz, respectively, which are perfectly reproduced using Eqs. (\[eq:spin\]), (B3) and (B5). The rotational frequency $\omega_{\gamma}^{(\text{spin})}$ changes with the damping, $\Gamma_c$, which depends linearly on the gas pressure, $p$, i.e. $\omega_{\gamma}^{\text{(spin)}} \propto \frac{1}{\Gamma_c} \propto
\frac{1}{p} $. This is a clear signature of rotation motion, as shown in Fig. \[fig:psd\](b). We also observe a dependency of the frequency on the laser power $P$, as shown in Fig \[fig:psd\](d) in agreement with the dependency on the photon scattering torque, $\boldsymbol{N}_s \propto P$. Zooming to the fundamental frequency $\omega_{\gamma}^{(\text{spin})}$ reveals sidebands, see Fig. \[fig:psd\](e), which are the addition and subtraction of the three translational frequencies. Using the numerical simulation we identify another mode in $\alpha$ rotation, with frequency $\omega_{\alpha}^{'} = 2 \pi \times 393$ kHz: this gives rise to the sideband in $\omega_{\gamma}^{(\text{spin})}$ (see [@supplement]). In the presented data set, we only resolve one of the $\omega_{\alpha}^{'}$ sideband peaks. The light-matter interactions introduce couplings between translation and rotation, which explain the observed sidebands and higher harmonics in agreement with numerical simulation, see Eqs. (A2), (A5)-(A7) in [@supplement] for further details. We also observe $\omega_{\alpha}^{(\text{spin})}$ to scale linearly with power and inversely with pressure.
Further to observing translation and rotation peaks, we observe a frequency at $2\pi \times 5.4$ kHz, as shown in Fig. \[fig:psd\](a). This frequency is well-isolated and characterised by its dependency on laser power, $P$ and gas pressure, $p$. We associate this low frequency with the gradient torque-induced precession, $\omega_{\alpha}^{\text{(precession)}}$. From Eq. (\[eq:precession\]), Taylor expanding about the initial pressure $p_0$, we find that the dominant term is linear in $p$ (the constant terms cancel), i.e. $\omega_{\alpha}^{\text{(precession)}} =
\partial_p(\Delta\omega)\vert_{p_0} p$. On the other hand, we find a weak dependence on the laser power $P$, i.e. $\omega_{\alpha}^{\text{(precession)}} =\Delta\omega\vert_{P_0}$, where $P_0$ is the initial laser power. We verify these frequency dependencies on gas pressure and laser power in Figs. \[fig:psd\](b) and \[fig:psd\](d), respectively. We exclude $\omega_{\alpha}^{\text{(precession)}}$ to be caused by nonlinear translational motion, as the translation frequencies do not change with pressure. This is evident from Fig. \[fig:psd\](b) where $\omega_{z}$ is constant. Thus, we conclude, that the observed frequency and its behaviour is signature of precession motion as described by Eq. (\[eq:precession\]).
The precession motion arises due to the fast spinning $\alpha$ degree of freedom, as well as, the nutation motion of $\beta$. The observation of precession is thus an indirect indication of nutation motion. We associate the time trace of the $\beta$ motion to libration motion, which is linearly dependant on power and inverse proportional to gas pressure, as a result of the numerical simulation. The $\beta$ libration is due to the coupling between $\alpha$ and $\beta$ motion.
[*Discussion–*]{} From the theoretical analysis, the precession motion arises due to an optical-torque acting upon the trapped particle. Torque can also be generated by an external force, which opens the way for sensitive detection of forces by precession. In particular, by combining Eqs. (\[eq:spin\]), (\[eq:nutation\]), and (\[eq:precession\]), we get an expression for the $\alpha$-component of the photon scattering torque: $$N_\alpha= \Gamma_c \sin^2(\beta_0) \left[\frac{2 I_1 I_2 (I_1 + I_2)}{I_1^2 + I_2^2} \right]
\omega_{\alpha}^{\text{(precession)}},
\label{nalpha}$$ where we have kept only the dominant terms, and denote the term in the square brackets by $\mathcal{J}$, and name it the effective moment of inertia. Using the experimentally measured $\omega_{\alpha}^{\text{(precession)}}$, and estimating $\mathcal{J}$, we achieve a measured torque of $N_\alpha= 1.9 \pm 0.5 \times 10^{-23}\text{Nm}$ at $1 \times 10^{-1}$ mbar, in comparison to the measurement of nanoscale torque sensors reported down to $10^{-20}$ Nm [@kim2013nanoscale] and to estimates of $10^{-22}$ Nm [@Kuhn2017] for silicon nanorods.
We now consider an additional small external torque acting on $\alpha$, which we denote by $\delta N_{\text{ext}}$. We suppose that Eq. remains valid, when we formally make the replacement $N_\alpha\rightarrow N_\alpha+\delta N_{\text{ext}}$, and we denote the corresponding change in the precession frequency by $\delta \omega_{\alpha}$. Furthermore, assuming that the equilibrium value of $\beta_0$ remains largely unaffected, we obtain the torque sensitivity $\delta N_{\text{ext}}=\Gamma_c\text{sin}(\beta_0)^2\mathcal{J}\delta
\omega_{\alpha}$. Using experimental parameters we estimate a torque sensitivity of $3.6 \pm 1.1 \times 10^{-31}$ Nm$/\sqrt{\text{Hz}}$ at $1 \times 10^{-7}$ mbar, limited only by photon shot noise (see supplementary D [@supplement]) in comparison to the torque sensitivity using libration motion, which is at $1 \times 10^{-29}$ Nm$/\sqrt{\text{Hz}}$ at $10^{-9}$ mbar [@Hoang2016].
[*Conclusions–*]{} We have observed precession motion in levitated optomechanics and inferred the presence of nutation motion. We present a theory of this motion and show that it arises from the equations of motion for a rotating object experiencing scattering and gradient forces and torques. Additionally, we characterise the rich spectrum detected with translation, rotation and higher harmonics. We further show a measured torque of $ 10^{-23}$ Nm at $10^{-1}$ mbar, and predict the ability to reach sensitivities down to $10^{-31}$ Nm$/\sqrt{\text{Hz}}$ at $10^{-7}$ mbar.
Thus, precession motion is a degree of freedom that could be utilised for torque sensing with the sensitivities to resolve single electron [@Rugar2004] and even nuclear spins [@Hoang2016] at low pressure. This work paves the way for gyroscope applications, as shown in [@nagornykh2017optical]. The precession motion can also be used for dynamical model selection to distinguish between quantum and classical evolution due to the inherent nonlinearities in rotation motion [@ralph2017dynamical], if sufficient coherence can be prepared. Additionally, this work can be used to reconstruct the shape and effective moment of inertia, $\mathcal{J}$, from the knowledge of the full spectrum of freedom containing all motional degrees of the trapped particle.
*Acknowledgments* We thank C. Timberlake and G. Winstone for discussion and M. Rademacher for assistance with the TEM image. We thank for funding the Leverhulme Trust [\[]{}RPG-2016-046[\]]{} and the EU Horizon 2020 research and innovation programme under grant agreement No 766900 \[TEQ\]. A.S. is supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under Centre for Doctoral Training Grant No. EP/L015382/1. All data supporting this study are openly available from the University of Southampton repository at https://doi.org/10.5258/SOTON/D0523"
Dynamics
=========
In this supplementary section we list the terms obtained in [@Toros2018t; @Toros2018s]. We start by specifying the conservative terms. The free Hamiltonian is given by:
$$\begin{aligned}
{1}
H_{\text{free}}=\frac{p_{x}{}^{2}+p_{y}{}^{2}+p_{z}{}^{2}}{2M}+\bigg( & \frac{\csc^{2}(\beta)(\cos(\gamma)(\pi_{\alpha}-\pi_{\gamma}\cos(\beta))-\text{\ensuremath{\pi_{\beta}}}\sin(\beta)\sin(\gamma))^{2}}{2I_{1}}\nonumber \\
& +\frac{\csc^{2}(\beta)(\sin(\gamma)(\text{\ensuremath{\pi_{\alpha}}}-\pi_{\gamma}\cos(\beta))+\pi_{\beta}\sin(\beta)\cos(\gamma))^{2}}{2I_{2}}+\frac{\pi_{\gamma}{}^{2}}{2I_{3}}\bigg),\label{eq:Hfree}\end{aligned}$$
where $M$ is the mass of the nanoparticle, and $I=\text{diag}(I_{1},I_{2},I_{3})$ is the moment of inertia tensor in the body frame.
The gradient potential is given by
$$\begin{aligned}
{1}
H_{\text{grad}}=-\frac{VP}{c\sigma_{L}}\vert u(\boldsymbol{r})\vert^{2} & \bigg(a^{2}\big(\text{\ensuremath{\chi_{1}}}(\cos(\alpha)\cos(\beta)\cos(\gamma)-\sin(\alpha)\sin(\gamma))^{2}\nonumber \\
& \qquad\qquad+\text{\ensuremath{\chi}}_{2}(\cos(\alpha)\cos(\beta)\sin(\gamma)+\sin(\alpha)\cos(\gamma))^{2}+\text{\ensuremath{\chi}}_{3}\cos^{2}(\alpha)\sin^{2}(\beta)\big)\nonumber \\
& \quad+b^{2}\bigg(\text{\ensuremath{\chi}}_{1}(\sin(\alpha)\cos(\beta)\cos(\gamma)+\cos(\alpha)\sin(\gamma))^{2}\nonumber \\
& \qquad\qquad+\text{\ensuremath{\chi}}_{2}(\cos(\alpha)\cos(\gamma)-\sin(\alpha)\cos(\beta)\sin(\gamma))^{2}+\text{\ensuremath{\chi}}_{3}\sin^{2}(\alpha)\sin^{2}(\beta)\big)\bigg).\label{eq:Hgradient}\end{aligned}$$
where $u$ is a modified Gaussian mode function:
$$u(\boldsymbol{r})=\frac{w_{0}}{w(z)}\text{exp}\left(-\frac{a_{1}x^{2}+a_{2}y^{2}}{w(z)^{2}}\right)e^{-ikz},\label{eq:umode}$$
$w_{0}$ is an effective beam waist, $\sigma_{L}=\pi w_{0}^{2}$, $a_{1}$, $a_{2}$ denote the asymmetry along the $x$, $y$ axis, respectively, $a_{1}a_{2}=1$, $w(z)=w_{0}\sqrt{(1+\frac{z^{2}}{z_{R}^2})}$, $z_{R}$ is the Rayleigh range, $k=\frac{2\pi}{\lambda}$, $\lambda$ is the laser wavelength, $P$ is the laser power, $c$ is the speed of light, and $\chi=\text{diag}(\chi_{1},\chi_{2},\chi_{3})$ is the susceptibility tensor in the body-frame.
We now specify the non-conservative terms. We first discuss the deterministic terms related to photon scattering, namely $d\boldsymbol{\boldsymbol{p}}_{\text{scatt}}=(0,0,dp_{z}^{\text{(ds)}})^\top$ and $d\boldsymbol{\boldsymbol{\pi}}_{\text{scatt}}=(d\pi_{\alpha}^{\text{(ds)}},d\pi_{\beta}^{\text{(ds)}},d\pi_{\gamma}^{\text{(ds)}})^\top$. In particular, we have:
$$\begin{aligned}
{1}
dp_{z}^{\text{(ds)}} & =\frac{16\pi\hbar\Gamma_{s}}{3}\frac{2\pi}{\lambda}\vert u(\boldsymbol{r})\vert^{2}dt,\label{eq:sforce}\\
d\pi_{\alpha}^{\text{(ds)}} & =\frac{4\pi b_{x}b_{y}\hbar\Gamma_{s}}{3}\vert u(\boldsymbol{r})\vert^{2}\bigg(-2\sin^{2}(\beta)\cos(2\gamma)(\text{\ensuremath{\chi_{1}}}-\text{\ensuremath{\chi}}_{2})(\text{\ensuremath{\chi}}_{1}+\text{\ensuremath{\chi}}_{2}-2\text{\ensuremath{\chi}}_{3})\nonumber \\
& \qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,\,+\cos(2\beta)\left(\text{\ensuremath{\chi}}_{1}^{2}+2\text{\ensuremath{\chi}}_{3}(\text{\ensuremath{\chi}}_{1}+\text{\ensuremath{\chi}}_{2})-4\text{\ensuremath{\chi}}_{1}\text{\ensuremath{\chi}}_{2}+\text{\ensuremath{\chi}}_{2}^{2}-2\text{\ensuremath{\chi}}_{3}^{2}\right)\nonumber \\
& \qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,\,+3\text{\ensuremath{\chi}}_{1}^{2}-2\text{\ensuremath{\chi}}_{3}(\text{\ensuremath{\chi}}_{1}+\text{\ensuremath{\chi}}_{2})-4\text{\ensuremath{\chi}}_{1}\text{\ensuremath{\chi}}_{2}+3\text{\ensuremath{\chi}}_{2}^{2}+2\text{\ensuremath{\chi}}_{3}^{2}\bigg)dt,\label{eq:ds1}\\
d\pi_{\beta}^{\text{(ds)}} & =\frac{16\pi b_{x}b_{y}\hbar\Gamma_{s}}{3}\vert u(\boldsymbol{r})\vert^{2}\sin(\beta)\sin(\gamma)\cos(\gamma)(\text{\ensuremath{\chi}}_{1}-\text{\ensuremath{\chi}}_{2})(\text{\ensuremath{\chi}}_{1}+\text{\ensuremath{\chi}}_{2}-2\text{\ensuremath{\chi}}_{3})dt,\label{eq:ds2}\\
d\pi_{\gamma}^{\text{(ds)}} & =\frac{16\pi b_{x}b_{y}\hbar\Gamma_{s}}{3}\vert u(\boldsymbol{r})\vert^{2}\cos(\beta)(\text{\ensuremath{\chi}}_{1}-\text{\ensuremath{\chi}}_{2})^{2}dt,\label{eq:ds3}\end{aligned}$$
where $\Gamma_{s}=\frac{\tilde{\sigma}_{R}}{\sigma_{L}}\frac{P}{\hbar\omega_{L}}$ is the scattering rate, $\omega_{L}=\frac{2\pi c}{\lambda}$, $\tilde{\sigma}_{R}=\frac{\pi^{2}V_{0}^{2}}{\lambda^{4}}$ is an effective scattering cross-section, $b_{x}$ and $b_{y}$ are the components of the Gaussian beam polarization vector $\boldsymbol{\boldsymbol{\epsilon}}_{d}=(b_{x},ib_{y},0)^{\top}$, and $b_{x}^{2}+b_{y}^{2}=1$.
We now discuss the gas collision terms denoted by $d\boldsymbol{\boldsymbol{p}}_{\text{coll}}=d\boldsymbol{p}^{\text{(dc)}}+d\boldsymbol{p}^{\text{(sc)}}$ and $d\boldsymbol{\boldsymbol{\pi}}_{\text{coll}}=d\boldsymbol{\boldsymbol{\pi}}^{\text{(dc)}}+d\boldsymbol{\boldsymbol{\pi}}^{\text{(sc)}}$. The terms denoted by the superscripts (ds) and (sc) corresponds to non-conservative deterministic and stochastic terms, respectively. In particular, we have $$\begin{aligned}
{1}
d\boldsymbol{p}^{\text{(dc)}} & =-2\Gamma_{C}\boldsymbol{p}dt,\label{eq:Pd}\\
d\boldsymbol{\boldsymbol{\pi}}^{\text{(dc)}} & =-2\Gamma_{C}\boldsymbol{\boldsymbol{\pi}}dt,\label{eq:Pid}\\
d\boldsymbol{p}_{k}^{\text{(sc)}} & =\sqrt{4mk_{B}T\Gamma_{c}}dV_{k},\label{eq:Pnc}\\
d\boldsymbol{\boldsymbol{\pi}}_{k}^{\text{(sc)}} &
=\sum_{\zeta,j=1}^{3}\sqrt{4k_{B}mT\tilde{D}_{\zeta}\Gamma_{c}}(\partial_{\mathbf{\mathbf{\phi}}_{k}}R)_{j,\zeta}dZ_{\zeta,j},\label{eq:Pinc}\end{aligned}$$ where $\Gamma_{c}=\frac{\pi p_{g}r_{g}^{2}}{\sqrt{8m_{g}k_{B}T}}$ is a characteristic collision rate, $p_{g}$ is the gas pressure, $r_{g}$ and $m_{g}$ are the radius and mass of a gas particle, respectively, $k_{B}$ is Boltzmann’s constant, $T$ is the gas temperature, and $\tilde{D}_{\zeta}=\frac{1}{2}(\text{tr}(I)-I_{\zeta})$. $V_{k}$ and $Z_{\zeta,j}$ are zero mean independent Wiener processes.
System frequencies
==================
Here we list the formulae, which have been used in the main text, to obtain the dominant frequencies of the system. The photon scattering torque is given by
$$\boldsymbol{N}_{s}=\mathbb{E}\left[\left(\frac{d\pi_{\alpha}^{\text{(ds)}}}{dt},\frac{d\pi_{\beta}^{\text{(ds)}}}{dt},\frac{d\pi_{\gamma}^{\text{(ds)}}}{dt}\right)^{\top}\right],$$
where $d\pi_{\alpha}^{\text{(ds)}}$,$d\pi_{\beta}^{\text{(ds)}}$, and $d\pi_{\gamma}^{\text{(ds)}}$ are given in Eqs. (\[eq:ds1\]), (\[eq:ds2\]), and (\[eq:ds3\]), respectively, and $\mathbb{E}$ denotes the time average over fast oscillating terms.
The conversion matrix $Y$ from the conjugate angle momenta $\boldsymbol{\boldsymbol{\pi}} $ to the time-derivative of the angle vector $\dot{\boldsymbol{\boldsymbol{\phi}}}$ is defined as $$Y=\left(\left(\begin{array}{ccc}
-\sin(\beta)\cos(\gamma) & \sin(\gamma) & 0\\
\sin(\beta)\sin(\gamma) & \cos(\gamma) & 0\\
\cos(\beta) & 0 & 1
\end{array}\right)^{\top}\left(\begin{array}{ccc}
I_{1} & 0 & 0\\
0 & I_{2} & 0\\
0 & 0 & I_{3}
\end{array}\right)\left(\begin{array}{ccc}
-\sin(\beta)\cos(\gamma) & \sin(\gamma) & 0\\
\sin(\beta)\sin(\gamma) & \cos(\gamma) & 0\\
\cos(\beta) & 0 & 1
\end{array}\right)\right)^{-1}.$$ The dominant frequencies for the spin motion are given by $$\begin{aligned}
{1}
\omega_{\alpha}^{\text{(spin)}}= & \frac{2b_{x}b_{y}\pi\hbar}{\frac{3}{4}\left(I_{1}^{2}+2I_{1}I_{2}+I_{2}^{2}\right)I_{3}}\frac{\text{\ensuremath{\Gamma_{s}}}}{\Gamma_{c}}\csc^{2}(\text{\ensuremath{\beta_{0}}})\bigg[\frac{1}{2}(I_{1}+I_{2})I_{1}(\text{\ensuremath{\chi_{1}}}-\text{\ensuremath{\chi_{2}}})^{2}\cos^{2}(\text{\ensuremath{\beta_{0}}})-(\frac{1}{2}I_{1}I_{3}+\frac{1}{2}I_{2}I_{3})\nonumber \\
& \left(3\text{\ensuremath{\chi_{1}^{2}}}-4\text{\ensuremath{\chi_{2}}}\text{\ensuremath{\chi_{1}}}+3\text{\ensuremath{\chi_{2}^{2}}}+2\text{\ensuremath{\chi_{3}^{2}}}-2(\text{\ensuremath{\chi_{1}}}+\text{\ensuremath{\chi_{2}}})\text{\ensuremath{\chi_{3}}}+\left(\text{\ensuremath{\chi_{1}^{2}}}-4\text{\ensuremath{\chi_{2}}}\text{\ensuremath{\chi_{1}}}+\text{\ensuremath{\chi_{2}^{2}}}-2\text{\ensuremath{\chi_{3}^{2}}}+2(\text{\ensuremath{\chi_{1}}}+\text{\ensuremath{\chi_{2}}})\text{\ensuremath{\chi_{3}}}\right)\cos(2\text{\ensuremath{\beta_{0}}})\right)\bigg], \label{alphaspin}\end{aligned}$$ and $$\begin{aligned}
{1}
\omega_{\gamma}^{\text{(spin)}}= & \frac{2b_{x}b_{y}\pi\hbar}{\frac{3}{4}\left(I_{1}^{2}+2I_{1}I_{2}+I_{2}^{2}\right)I_{3}}\frac{\text{\ensuremath{\Gamma_{s}}}}{\ensuremath{\Gamma}_{c}}\cot(\text{\ensuremath{\beta}}_{0})\csc(\text{\ensuremath{\beta_{0}}})\nonumber \\
& \bigg[-\frac{1}{2}\left((I_{1}+I_{2})I_{3}\cos^{2}(\text{\ensuremath{\beta_{0}}})+\left(\frac{1}{2}I_{1}^{2}+I_{2}I_{1}+\frac{1}{2}I_{2}^{2}\right)\sin^{2}(\text{\ensuremath{\beta_{0}}})\right)(\text{\ensuremath{\chi_{1}}}-\text{\ensuremath{\chi_{2}}})^{2}\bigg(-(-\frac{1}{2}I_{1}-\frac{1}{2}I_{2})I_{3}\nonumber \\
& \left(3\text{\ensuremath{\chi_{1}^{2}}}-4\text{\ensuremath{\chi_{2}}}\text{\ensuremath{\chi_{1}}}+3\text{\ensuremath{\chi_{2}^{2}}}+2\text{\ensuremath{\chi_{3}^{2}}}-2(\text{\ensuremath{\chi_{1}}}+\text{\ensuremath{\chi_{2}}})\text{\ensuremath{\chi_{3}}}+\left(\text{\ensuremath{\chi_{1}^{2}}}-4\text{\ensuremath{\chi_{2}}}\text{\ensuremath{\chi_{1}}}+\text{\ensuremath{\chi_{2}^{2}}}-2\text{\ensuremath{\chi_{3}^{2}}}+2(\text{\ensuremath{\chi_{1}}}+\text{\ensuremath{\chi_{2}}})\text{\ensuremath{\chi_{3}}}\right)\cos(2\text{\ensuremath{\beta_{0}}})\right)\bigg)\bigg],\end{aligned}$$ while the nutation frequency is given by $$\begin{aligned}
{1}
\omega_{\beta}^{\text{nutation}} = \frac{1}{2}\frac{I_{1}+I_{2}}{I_{1}I_{2}}\csc ^2(\beta_0)
\frac{2 \pi b_x b_y \hbar }{3 }\frac{\Gamma_s}{\Gamma_c}
\bigg[& \cos (2 \beta_0 )
\big( \chi_1^2+2 \chi_3
(\chi_1+\chi_2)-4
\chi_1 \chi_2+
\chi_2^2-2 \chi_3^2\big) \nonumber \\
&+3 \chi_1^2-2 \chi_3
(\chi_1+\chi_2)-4 \chi_1
\chi_2+3 \chi_2^2
+2 \chi_3^2\bigg]. \label{betanutation}\end{aligned}$$ The equilibrium position of the $\beta$ angle is given approximately by: $$\beta_0=\sin ^{-1}\left( \sqrt[4]{
\frac{(I_1+I_2)\pi c w_0^2 \pi_\alpha^2}
{I_1 I_2 P V (2 \chi_3-\chi_1-\chi_2)}}\right).\label{beta0}$$ The fact that $\beta_0$ depends on $\pi_\alpha$ is a consequence of the coupling in Eqs. and .
To estimate the dominant frequencies using Eqs. - we have numerically simulated the system and extracted the fitted parameters (see supplementary material \[sec:SimSpectrum\]). In particular, we obtain from the simulation the following values $\omega_{\alpha}^{\text{(spin)}}=2 \pi \times 3.919\text{MHz}$, $\omega_{\gamma}^{\text{(spin)}}= 2 \pi \times 1.957 \text{MHz}$, $\omega_{\beta}^{\text{(nutation)}}=2 \pi \times 3.924 \text{MHz}$, and $\omega_{\alpha}^{\text{(precession)}}=2 \pi \times 5.5 \text{kHz}$ in perfect agreement with experimental data. Using the simulation parameters, we also find good agreement within order of magnitude with the approximate expressions in Eqs. - . Thus, Eqs. - , can be used for estimating the initial simulation parameters for the fitting algorithm.
.
Numerical Simulation {#sec:SimSpectrum}
====================
In this section we further discuss the different contributions to the simulated spectrum shown in Fig. 2(a). The solution to the twelve coupled SDEs, give information of $x(t)$, $y(t)$, $z(t)$, $\alpha(t)$, $\beta(t)$ and $\gamma(t)$. From these we can extract the frequency spectrum, as shown in Fig. \[fig:SimSpectrum\]. The spectrum for each degree of freedom demonstrates the source of the numerous frequencies observed in Fig. 2(a).
![\[fig:SimSpectrum\] **Simulated Spectrum:** The power spectral density of the six different degrees of motion, where $\omega_{\gamma}^{\text{(spin)}}$ (in green), $\omega_{\beta}^{\text{(nutation)}}$ (in magenta), $\omega_{\alpha}^{\text{(spin)}}$ (in blue) are the rotation motions; $\omega_{x}$ (in grey), $\omega_{y}$ (in brown), and $\omega_{z}$ (in dark green) are the translation motions.](simulatedspectrum.pdf){width="1\linewidth"}
*$\alpha$-rotation* (blue line in Fig. \[fig:SimSpectrum\]): Starting from the right to left, the $\alpha$ rotation contains frequencies relating the main rotation peak, $\omega_{\alpha}^{\text{(spin)}}$ and its higher harmonics. In addition to this there are sidebands to $\omega_{\alpha}^{\text{(spin)}}$ which relate to another mode of motion designated as $\omega_{\alpha}^{'}$. Farther to the left, of the spectrum we observe this additional mode, $\omega_{\alpha}^{'}$ and its second harmonic. The far left of the $\alpha$ spectrum show the precession motion.
*$\beta$-nutation* (magenta line in Fig. \[fig:SimSpectrum\]): The spectrum for $\beta$ motion shows the central nutation frequency $\omega_{\beta}^{\text{(nutation)}}$ and its higher harmonics. The sidebands refer to $\omega_{\alpha}^{'}$ and $2\omega_{\alpha}^{'}$. Both these frequencies also appear in the actual spectrum as well.
*$\gamma$-rotation* (green line in Fig. \[fig:SimSpectrum\]): The spectrum for $\gamma$ rotation shows $\omega_{\alpha}^{\text{(spin)}}$ and its higher harmonics. The sidebands relate to $\omega_{\alpha}^{'}$ and come about due to the coupling of $\gamma$-rotation with $\alpha$-rotation. In addition to the rotation degrees of freedom, Fig. \[fig:SimSpectrum\] also shows the translation motions, $\omega_{x}$, $\omega_{y}$ and $\omega_{z}$.
Recoil heating
==============
We can estimate the recoil heating rates for translational and rotational degrees of freedom due to gas collisions and photon scattering using the Table. \[table1\]. These expressions can be derived heuristically by considering the amount of linear and angular momentum carried by a single gas particle or photon. Specifically, the amount of linear and angular momentum carried by a gas particle can be estimated as $\sqrt{2m_g k_b T}$ and $\sqrt{2m_g k_b T R^2}$, respectively, where $m_g$ is the mass of a gas particle and $R$ is an effective radius of the nanoparticle. To obtain the net effect on the nanoparticle we have to take into account the gas collisions scattering cross section: we formally replace $m_g$ with the mass of the nanoparticle $M$ (we do not change the expressions in $\Gamma_c$, which is associated with a single atom of the nanoparticle). We can make a similar argument for photons: the amount of linear and angular momentum carried by a photon particle is $\hbar \frac{2\pi}{\lambda}$ and $\hbar$, respectively. However, the photon scattering cross section is already included in $\Gamma_s$, and thus we immediately obtain the linear and angular momentum fluctuations per unit time in Table. \[table1\]. A more rigorous derivation, based on the quantum model [@Toros2018t], will be given elsewhere [@Toros2018s].
To get a rough numerical estimate we have considered the moment of inertia tensor of a sphere and a susceptibility tensor that in the body frame has the diagonal elements close to unity.
gas collisions photon scattering
-------------- ------------------------------------- ------------------------------------------------------------
translations $ 4k_{B}TM\Gamma_{c}$ $\Gamma_{s}\hbar^{2}\left(\frac{2\pi}{\lambda}\right)^{2}$
rotations $\frac{4}{5}k_{B}TMR^{2}\Gamma_{c}$ $\Gamma_{s}\hbar^{2}$
: The expressions denote estimates for the variance of momentum (angular momentum) fluctuations per unit time for translations (rotations), induced by gas collisions and photon scattering. Note that this expressions are good estimates for a system that is not highly anisotropic, while for a highly anisotropic objects, the moment of inertia and electric susceptibility tensors have to be taken into account [@Toros2018s]. $M$ and $R$ denote an effective radius and mass of the nanoparticle, $\Gamma_{s}$ and $\Gamma_{c}$ denote photon scattering and gas collision rate, respectively, $T$ is the temperature of the gas, and $\lambda$ is the laser wavelength (see supplementary material A).[]{data-label="table1"}
![Comparison of recoil heating from gas collisions and from photon scattering. We plot the ratio of the estimates in each row from Table. \[table1\]. Values bigger (smaller) than 1 mean that the recoil heating from gas collisions (photon scattering) is dominant. \[fig:recoil\_heating\]](recoil){width="50.00000%"}
Using the formulae in Table. \[table1\] we compare the strength of the heating mechanisms in Fig. \[fig:recoil\_heating\]. This analysis also reproduces the transition to the photon recoil heating regime for translations is in accordance with Fig. 3 from [@vijay2016].
|
---
abstract: 'Renewable-energy-based grids development needs new methods to maintain the balance between the load and generation using the efficient energy storages models. Most of the available energy storages models do not take into account such important features as the nonlinear dependence of efficiency on lifetime and changes in capacity over time horizon, the distribution of load between several independent storages. In order to solve these problems the Volterra integral dynamical models are employed. Such models allow to determine the alternating power function for given/forecasted load and generation datasets. In order to efficiently solve this problem, the load forecasting models were proposed using deep learning and support vector regression models. Forecasting models use various features including average daily temperature, load values with time shift and moving averages. Effectiveness of the proposed energy balancing method using the state-of-the-art forecasting models is demonstrated on the real datasets of Germany’s electric grid.'
author:
- 'Denis Sidorov, Qing Tao , Ildar Muftahov, Aleksei Zhukov, Dmitriy Karamov , Aliona Dreglea and Fang Liu'
title: 'Energy balancing using charge/discharge storages control and load forecasts in a renewable-energy-based grids'
---
Introduction
============
Energy industry is on the verge of new changes due to the various renewable energy sources employment. Relations liberalisation among electricity generators, suppliers and consumers poses new fundamental mathematical problems in the various fields of the modern power engineering. The high proportion of various renewable energy sources increases the power generation variability, disrupting the optimal mode of operation of traditional power systems. The studies of various energy storages usage, such as pumped power plants [@punys2013assessment], compressed air storages [@karellas2014comparison], rechargeable batteries [@dunn2011electrical], and others, are especially important in such challenging conditions of power systems upgrade. Extensive research studies have been conducted on the modeling and optimization of electric energy storages, see e.g. [@di2014modelling; @makarov2012sizing].
R. Dufo-Lopez, J. L. Bernal-Agustin, D. Tsuanyo, E. Dursun [@dufo2014comparison; @tsuanyo2015modeling; @dursun2012comparative] used the method of chronological modeling of the operation of electrochemical energy storages for the determination of the state of charge, voltage, resistance, current charge/discharge optimal behaviour. These studies contribute to a better understanding of the current state of research of power systems using the storages, their technical characteristics, functional limitations and design capabilities of systems using renewable energy sources. A lot of attention is paid to the analysis of performance, technical characteristics and cost of using different storages including the batteries. It is to be noted that most of the works employs the linear models of the storages. Such approaches do not take into account the nonlinear processes of reduction of the available capacity of the energy storages over time and other important features like the state of charge (SOC).
This paper employs a new models based on Volterra integral equations [@sidorov2015integral], which take into account such principal characteristics of storages as capacity, efficiency, number of cycles and discharge / charge rate, load distribution between the available storages. Proposed integral dynamical model relies on electric load forecast, generation from the renewable energy sources and traditional generation. This article proposes load forecasting models using the state-of-the-art machine learning methods.
Inaccurate forecasts reduce the quality of grids’ management: forecast errors lead to the need to use an expensive emergency power plants or to purchase a missing capacity from neighboring manufacturers at higher prices; an overestimated prediction leads to an increase in the costs of maintaining the excess reserve capacity.
There are many forecasting methods available in the literature, most of which are based on traditional methods, not considering the latest advances in machine learning and data analysis. This paper proposes an approach based on data analysis techniques using the following machine learning methods: support vector regression [@drucker1997support] (Support Vector Machine based Regression, SVR), deep learning based on recurrent neural networks (Long Short-Term Memory, LSTM; Gated Recurrent Units, GRU) [@lstm; @gru], random forest [@breiman2001random; @zhukov2016random] and gradient boosting [@friedman2002stochastic].
Features engineering is the principal part of the forecasts models construction. It is important to correctly identify and formalize the factors influencing the forecast. They can be divided into socio-economic and meteorological, which in turn can be cyclical (for example, day of the week), natural (reflecting the natural activity of the technological or natural environment, for example, the heating season or the atmospheric pressure) or random or sudden changes in weather conditions.
All these factors serve as the input for the forecast. In order to represent such information it in the most convenient form for the model, the transformations like the principal component analysis or the Hilbert–Huang transform can be used.
Forecasting mathematical models can be classified into the following two groups:
1. based on traditional statistical and probabilistic methods for analyzing time series;
2. the artificial intelligence based methods.
The first group of methods includes traditional statistical methods for time series analysis, such as multiparametric regression (hereinafter, linear model, LM), exponential smoothing, autoregression models, moving average, and their modifications, such as ARMA, ARIMA. Such models enjoy good accuracy if the input features do not correlate with each other, and have linear dependence on the target variable. Therefore, they often fail to make an sufficiently accurate forecast for data with complex load dependencies on meteorological and socio-economic factors. In addition, without additional filtering, such methods are unstable with respect to outliers and data errors. The probabilistic forecasting methods include the statistical gradient method, Bayes models and other. In order to obtain continuous prediction, filters can be used: for predicting stationary processes, the Wiener-Hopf filter is used; for non-stationary processes, a Kalman filter is used. These methods are usually jointly employed with the regression to improve the prediction result and to make it more resistant to incorrect data.
The second approach, which is most often used, uses traditional and the state-of-the-art models of machine learning, optimization methods and aggregation of expert knowledge. Among the models of machine learning, the most commonly used are both the well-known feedforward artificial neural networks (ANN), random forest (RF), gradient boosting decision trees (GBDTs), support vector regression (SVR), and the deep ANNs. To solve the problems of selecting their parameters, various optimization methods can be used including the genetic algorithms.
The effectiveness of the artificial intelligence (AI) methods can be explained not only by their ability to approximate the complex hidden dependencies, but AI methods also allow to integrate the various techniques into a single model. For example, expert knowledge can be added to the model as an additional feature. The significance of variables assessment helps to dynamically find out the most important features in specific conditions for the forecast of specific parameters. More details on the different approaches to load forecasting can be found in review [@kuster2017electrical].
The remainder of this paper is organized as follows. Sec. 2 focuses on the mathematical dynamical model of the storages in general settings. Experimental studies are fulfilled in Sec. 3, where first the dataset is described, then results of load forecasting are given in Sec. 3.1 and results concerning the application of the Volterra model for charge/discharge storage control are provided and discussed in Sec. 3.2. Concluding remarks are given in Sec.4.
Storage Dynamical Model
=======================
In order to efficiently model the storages operation, it is proposed [@volt] to use the following nonlinear integral model with constraints $$\label{eq1}
\left\{ \begin{array}{ll}
\mbox{$\int\limits_{0}^{t} K(t,s,x(s)) \; ds = f(t), \,\,\, 0 \leq s\leq t \leq T,\,\, f(0)=0$},\\
%\mbox{$\smash{\displaystyle\max_{i=1,N}}{\;\left(\sum\limits_{j=1}^{i}{x_j(t)}\right) \leqslant v_{max}}$}\\
\mbox{${v(t) = \int\limits_{0}^{t}{x(s) ds}, \; \smash{\displaystyle\max_{t \in [0, T]}}{\;v(t)} \leq v_{max}}$},\\
%\mbox{$E_{min}(t) \leqslant \sum\limits_{i=1}^{N-1}\sum\limits_{j=1}^{N}{x_j(t)}\leqslant E_{max}(t)$}\\
\mbox{$E_{min}(t) \leq \int\limits_{0}^{t}{v(s) ds}\leq E_{max}(t)$},\\
\mbox{$0 < \alpha_1(t)<\alpha_2(t)<$ $\dots$ $< \alpha_{n-1}(t)<t,$}\\
\end{array} \right.$$ where the kernel is represented as follows $$\label{eq2}
K(t,s,x(s)) = \left\{ \begin{array}{ll}
\mbox{$K_1(t,s)G_1(s,{x(s)}), \,\, t,s \in m_1,$} \\
\mbox{\,\, \dots \,\, \dots \dots \dots } \\
\mbox{$K_n(t,s)G_n(s,{x(s)}), \,\, t,s \in m_n.$} \\
\end{array} \right.$$ Here\
$m_i = \{ t, s\,\, \bigl |\,\, \alpha_{i-1}(t) < s < \alpha_i(t) \};$\
${ \alpha_0(t)=0,\,\, \alpha_n(t)=t,\, i=\overline{1,n.}};$\
$\alpha_i(t),$ $f(t) \in \mathcal{C}_{[0,T]}^1,$\
$K_i(t,s) \in \mathcal{C}_{[0,T]}^1$ for $t,s \in \overline{m_i};$\
$ K_n(t,t) \neq 0$, $K_i(t,s), \, G_i(s,x(s)) \in \mathcal{C}_{[0,T]};$\
$\alpha_i(0)=0,$ $0 < \alpha_1(t)<\alpha_2(t)<$ $\dots$ $< \alpha_{n-1}(t)<t;$\
$\alpha_1(t), \dots , \alpha_{n-1}(t) $ increase for small $\tau,$ $0\leq t \leq \tau;$\
$0< \alpha_1^{\prime}(0) \leq$ $\dots$ $\leq \alpha_{n-1}^{\prime}(0)<1. $
The theory of such integral models with piecewise continuous kernels was first proposed in [@sidorov2011volterra] and developed in [@sidorov2015integral; @volt; @igu18]. The special linear discrete case of the model of grid-connected storage was employed by R. Dufo-López, see [@dl] and other publications.
Here $ f(t) $ is the load imbalance $$\label{eq3}
f(t)=f_{RES}(t) +f_{gen} - f_{load}(t),$$ where $ f_ {gen} (t) $ is the generation of traditional energy sources, $ f_ {RES} $ is the generation of renewable energy sources and $ f_ {load} (t) $ is the predicted load of consumers.
The functions $ \alpha_i (t) $ show the load distribution between $ n $ drives, and $ K_i (t, s) G_i (s, x (s)) $ - the efficiency of each drive, changing under the influence of two factors - the lifetime $ K_i (t, s) $ and the intensity of current use of time $ G_i (s, x (s)) $, depending on the alternating function $ x (s) $. In this case, the proposed mathematical model allows us to take into account the nonlinear nature of changes in efficiency depending on the service life and/or the behavior of $ x(s) $. In the equation (\[eq1\]), the alternating function of changing the power $ x(s) $ is the desired one. It allows for known $ v_{max} $ (maximum speed of the charge):
1. to determine $ E(t) $ is the storage state of charge under the constraints $ E_ {min} (t) \leq E (t) \leq E_ {max} (t) $ depending on the type of storage;
2. to determine the minimum total capacity of the storage to cover the load shortage of consumers;
3. to calculate the number of cycles based on behavious of function $ E(t) $ function;
4. to predict the lifetime of the storage.
The problem of solution to the equation (\[eq1\]) with respect to $x(t)$ is the typical inverse problem [@sizbook]. The author’s numerical method from [@muftahov2015perturbation] is employed.
The advantages of the nonlinear mathematical model presented above are as follows:
1. [definition of operating parameters of storages when various renewable energy sources and storages are used jointly;]{}
2. [consideration of such characteristics of storages operation as power, charge/discharge rate, maximum number of work cycles, SOC limit;]{}
3. [minor impact on the computational complexity of the algorithm when using a large number of storages ($ n> 10 $);]{}
4. [accounting for the nonlinear nature of efficiency changes;]{}
5. [the ability to flexibly customize the time distribution functions of the load between the drives.]{}
It is to be noted that for efficient application of the proposed Volterra model it is necessary to construct the accurate forecasts of the generation from RES $ f_ {RES} $ (see e.g. [@fang] for the short term wind power forecasting models) and consumer loads $ f_{load} (t) $. This problem is attacked using the state-of-the-art machine learning methods. The next section of the article is focused on the forecasting models and verification of the Voltera model on the real datasets.
Experimental Results
====================
For testing, the Germany’s electrical grid dataset was chosen, since this grid is rather large and has many renewable energy sources and electrical energy storage for leveling the daily heterogeneity of the electrical load graph. At the moment, about 6.7 GW of pumped storage power plants (PSPP) are installed in Germany[^1].
To test the proposed approaches, the publicly available data on the German power grid load, provided by ENTSO-E, for the period from the beginning of 2006 to the end of 2013 was used. It should be noted a significant percentage of renewable energy, which for 2007 was 13.6 and to date, according to the statistical service of the European Union[^2] almost tripled. We also used data on the generation of electricity from various sources located in Germany[^3].
However, in the analyzed dataset, the influence of the nonstationary nature of wind turbines is insignificant; this was revealed from the results of a statistical Dickey-Fuller’s test for the stationarity. Thus, this series can be considered as stationary. However, it cannot be guaranteed that this situation will remain the same in the future. There are no strict requirements for the period of training the models, then its updating can occur periodically. In the case of non-stationary data, online models can be used, such as OzaBag [@oza2005online] and PDSRF [@zhukov2016random].
Most of the load (47%) according to German Association of Energy and Water Industries (BDEW) is related to the industrial sector (unlike other countries with a large percentage of wind turbines, where population form most of the load), which may be the cause of the most ordered nature of the electric load. The German population form 26% of the load, the service sector is 25%, transport is about 2%.
These loads were supplemented by average daily temperature data obtained by the European Climate Assessment & Dataset from the weather stations in Hamburg, Munich, Stuttgart, Bochum, as well as an indicator of working days and holidays in Germany. Also, the following indicators are used for the forecast: current load, day of week, time of day, load a day ago, load value an hour ago, load value a week ago, average load for yesterday, minimum load for yesterday, and exponential moving averages with periods 12, 24, 48, 168 hours.
![Generation and actual load: from the German grid dataset.[]{data-label="fig:GenForecastLoadGermany"}](veq_loadlevl_de_load-eps-converted-to.pdf){width="1.0\linewidth"}
The dataset consists of 69713 examples, including data collected from 2006-01-08 to 2013-12-30. Of these, 60953 were used for training and testing predictive models (using block cross-validation), the remaining 8760 for validation.
Electric load prediction
------------------------
Six popular models were chosen for prediction: support vector machine, LSTM, GRU, RF, GBDT and multiparametric linear regression. Model’s parameters are included in the Tab.1.
As can be seen from Table 1, all the tested machine learning methods under consideration have similar errors. Three following metrics were selected: $ RMSE ~ = ~ \sqrt {\frac{1}{n}\sum_{t = 1}^{n}(x_t-\bar{x_t})^ 2}$ is the root mean square error, mean absolute error $ MAE =\frac{1}{n}\sum_{t = 1}^{n}|x_t-\bar{x_t}|$ and the average absolute error in percent $MAPE=\frac{1}{n}\sum_ {t = 1}^{n}\frac {| x_t- \bar{x_t}|}{\bar{x_t}}*100\% $. Here $\bar{x_t}$ is the real target value, and $x_t$ is the predicted value.
[l|l|l|l|l]{} Method & Notes & RMSE & MAE & MAPE %\
SVR & RBF kernel & 2472.93 & 1551.38 & 3.41\
$\,$ & $C = 32$ & $\,$ & $\,$\
$\,$ & $\gamma=0.05515674$ & $\,$ & $\,$\
LSTM & lstm\_1(300) & 2134.19 & 1236.25 & 2.74\
$\,$ & lstm\_2(300) & &\
$\,$ & Dense\_1(16) & &\
GRU & gru\_1(300) & 2114.17 & 1207.96 & 2.68\
$\,$ & gru\_2(300) & & &\
$\,$ & Dense\_1(16) & & &\
RF & [mtry: 4]{} &2145.59 & 1667.25& 2.77\
GB & [interaction.depth: 9]{} &2144.89 & 1293.78 & 2.89\
$\,$ & [shrinkage: 0.1]{} & & &\
$\,$ & [n.minobsinnode: 10]{} & & &\
LM & & 4774.54 & 3735.50 & 8.07\
Also of interest are errors by days of the week and time of day. It should be noted that all models based on machine learning show similar errors, the largest for Monday, Thursday and the time interval from 7 to 8 hours. This data can be used to further analyze the nature of the load.
It is also necessary to mention the limitations of the proposed predictive model, which include the fact that it does not take into account the structure of the load. For example, since in the shown example most of the industry is occupied, then by including the parameters of the work of large enterprises, it is possible to improve the quality of the forecast. Another factor in the improvement of the model is the consideration of market conditions as additional input parameters.
Experiments with integral model of energy storage
-------------------------------------------------
Proposed dynamical model can use nonlinear dependence on time and changes in the power of storage operation in terms of the storage efficiency. The process of determining the efficiency is not a trivial task, since It consists of many factors. For sake of simplicity, in this paper, the constant efficiency of 92% is used.
Application of model (\[eq1\]) is shown in Fig.\[fig:BatteryGenGermany\]. It demonstrates the alternating power function based on the actual load and its different forecasts shown in Fig.\[fig:GenForecastLoadGermany\]. Here the positive values correspond to the process of the storages charging, and negative values corresponds to the generation to cover the load imbalance. It can be noted that the value of the desired function $x(s)$ heavily depends on the forecast accuracy.
![Generation and load forecasting on the data of German grid (deep learning and SVR).[]{data-label="fig:GenForecastLoadGermany"}](veq_loadlevl_de_prog_china-eps-converted-to.pdf){width="1.0\linewidth"}
![Calculated alternating power functions based on fact and forecasted loads using deep learning and SVR.[]{data-label="fig:BatteryGenGermany"}](veq_loadlevl_de_res_china-eps-converted-to.pdf){width="1.0\linewidth"}
![Generation and load forecasting on the data of German grid (RF, GB and LM).[]{data-label="fig:GenForecastLoadGermany2"}](veq_loadlevl_de_prog_irk-eps-converted-to.pdf){width="1.0\linewidth"}
![Calculated alternating power functions based on fact and forecasted loads using RF, GB and LM.[]{data-label="fig:BatteryGenGermany2"}](veq_loadlevl_de_res_irk-eps-converted-to.pdf){width="1.0\linewidth"}
Fig. \[fig:BatteryGenGermany\] shows the alternating power functions (APF). As you can see, even a small deviation of the forecast from the actual load gives large differences in the operation of the drive.
As shown by the calculation results (Fig. \[fig:BatteryGenGermany\]), to cover the imbalance between generation and consumption with storage devices, a minimum of 12 GW of total storage capacity is required.
Conclusion
==========
This article proposes a new mathematical model of the storages control which allows to take into account the dynamics of efficiency with a nonlinear dependence on the time of use of the storages and APF. The efficient forecasting models of the electric load for the day ahead are employed, which are based on the deep learning models, random forest, gradient boosting decision trees, support vector machine based regression and multiparametric regression. The load forecasting is used as an input parameter for the energy storage model. The proposed models are tested on real dataset of German grid. The load forecasting models’ absolute errors can be as small as 2.68%. Comparison of various models shows that the best results were achieved using random forest and GRU models. It is to be noted that accuracy of APF is mostly depend on accuracy of the forecast. In our experiments the best accuracy of APF was achieved based on the GRU forecast with MAE = 1272.17. According to the results of calculations with this forecast’s accuracy, a full coverage of the imbalance between generation and consumption of energy with storages requires a minimum of 12 GW of total storage capacity.
Our next work will be focused on employment of the self-regularization property of considered class of the Volterra models [@igu16] and Lavrentiev regularization method [@sizbook; @sidorov2015integral] in order to cope with inaccurate forecasts of both electric loads and generation.
[0]{}
H. Ali Mohd, B. Wu, R.A. Dougal, An overview of SMES applications in power and energy systems. *IEEE Transactions on Sustainable Energy*. 2010: vol. 1, no. 1. 38–47. https://doi.org/10.1109/tste2010.2044901.
L. Breiman, Random forests. *Machine learning*. 2001: vol. 45, no. 1. pp. 5–32.
M.L. Di Silvestre, E.R. Sanseverino, Modelling energy storage systems using Fourier analysis: An application for smart grids optimal management. *Applied Soft Computing*. 201: vol. 14. 469–481. https://doi.org/10.1016/j.asoc.2013.08.018
R. Dufo-L[ó]{}pez , J.M. Lujano-Rojas, A.J. Bernal, Comparison of different lead–acid battery lifetime prediction models for use in simulation of standalone photovoltaic systems. *Applied Energy*. 2014: vol. 115. 242–253. https://doi.org/10.1016j.apenergy.2013.11.021
B. Dunn, H. Kamath, J.-M. Tarascon, Electrical energy storage for the grid: a battery of choices. *Science*. 2011: vol. 334, no. 6058. 928–935. https://doi.org/10.1126/science.1212741
E. Dursun, O. Kilic, Comparative evaluation of different power management strategies of a stand-alone PV/Wind/PEMFC hybrid power system. *International Journal of Electrical Power & Energy Systems*. 2012: vol. 34, no. 1. pp. 81–89. https://doi.org/10.1016/j.ijepes.2011.08.025
H. Drucker, C.J. Burges, L. Kaufman et al., Support vector regression machines.*Advances in neural information processing systems*. 1997: 155–161.
S. Hochreiter, J. Schmidhuber, Long short-term memory. *Neural Computation*. 1997: 9 (8), 1735–1780. https://doi.org/10.1162/neco.1997.9.8.1735.
K. Cho, B. van Merrienboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, Y. Bengio. Learning phrase representations using RNN encoder-decoder for statistical machine translation". arXiv:1406.1078, 2014.
D. Sidorov, A. Zhukov, A. Foley, A. Tynda, I. Muftahov, D. Panasetsky and Y. Li, Volterra models in load leveling problem. *E3S Web Conf.* International Conference Green Energy and Smart Grids (GESG 2018), 2018: vol. 69, 2018. https://doi.org/10.1051/e3sconf/20186901015
D. Sidorov, A. Zhukov and I. Muftahov, Volterra equation based models for energy storage usage based on load forecast in EPS with renewable generation. *The Bulletin of Irkutsk State University. Ser. Mathematics.* 2018: Vol. 26, pp. 76–90. https://doi.org/10.26516/1977-7670.2018.26.76
J.H. Friedman, Stochastic gradient boosting.*Computational Statistics & Data Analysis*. 2002: vol. 38, no. 4. 367–378. https://doi.org/10.1016/s0167-9473(01)00065-2
S. Karellas, N. Tzouganatos, Comparison of the performance of compressedair and hydrogen energy storage systems: Karpathos island case study. *Renewable and Sustainable Energy Reviews*. 2014: vol. 29. 865–882. https://doi.org/10.1016/j.rser.2013.07.019
C. Kuster,Y. Rezgui,M. Mourshed, Electrical load forecasting models: A critical systematic review. *Sustainable Cities and Society*. 2017: vol. 35. pp. 257–270. https://doi.org/10.1016/j.scs.2017.08.009
Y.V. Makarov, P. Du, M.C.W. Kintner-Meyer et al., Sizing energy storage to accommodate high penetration of variable energy resources. *IEEE Transactions on sustainable Energy*. 2012: vol. 3, no. 1. 34–40. https://doi.org/10.1109/tste.2011.2164101
Y. P. Petrov, V. S. Sizikov, *Well-posed, Ill-posed, and Intermediate Problems with Applications.* Inverse and Ill-Posed Problems Series. Berlin: DeGruyter, 2011.
I. Muftahov, A. Tynda, D. Sidorov, Numeric solution of Volterra integral equations of the first kind with discontinuous kernels. *Journal of Computational and Applied Mathematics*. 2017: vol. 313. 113–128. https://doi.org/10.1016/j.cam.2016.09.003
J.R. Noriega, O.D. Iyore, C. Budime et al., Characterization system for research on energy storage capacitors. *Review of Scientific Instruments*. 2013: vol. 84, no. 5. 055109. https://doi.org/10.1063/1.4804165
N. C. Oza, Online bagging and boosting. *2005 IEEE International Conference on Systems, Man and Cybernetics*. 2005: vol. 3. 2340–2345. https://doi.org/10.1109/icsmc.2005.1571498
P. Punys, R. Baublys, E. Kasiulis et al., Assessment of renewable electricity generation by pumped storage power plants in EU Member States. *Renewable and Sustainable Energy Reviews*. 2013: vol. 26. 190–200. https://doi.org/10.1016/j.rser.2013.05.072
R. Sebasti[á]{}n, R.P. Alzola, Flywheel energy storage systems: Review and simulation for an isolated wind power system.*Renewable and Sustainable Energy Reviews*. 2012: vol. 16, no. 9. 6803–6813. https://doi.org/10.1016/j.rser.2012.08.008
D.N. Sidorov, On parametric families of solutions of Volterra equations of the first kind with piecewise smooth kernel. *Differential Equations*. 2013: vol. 49, no. 2. 210–216. https://doi.org/10.1134/S0012266113020079
D.N. Sidorov, Integral dynamical models: singularities, signals and control. *World Scientific*. 2015: 300.
R. Dufo-López, Optimisation of size and control of grid-connected storage under real time electricity pricing conditions. *Applied Energy.* 2015: vol. 140, 395–408.
D. Tsuanyo,Y. Azoumah, D. Aussel, P. Neveu, Modeling and optimization of batteryless hybrid PV (photovoltaic) / Diesel systems for off-grid applications. *Energy*. 2015: vol. 86. 152–163. https://doi.org/10.1016/j.energy.2015.03.128
A.V. Zhukov, D.N. Sidorov, A.M. Foley, Random forest based approach for concept drift handling. *International Conference on Analysis of Images, Social Networks and Texts.* Springer. 2016: 69–77. https://doi.org/10.1007/978-3-319-52920-2\_7
F. Liu, R. Li, Y. Li, Y. et al. Short-term wind power forecasting based on TS fuzzy model. *2016 IEEE PES Asia-Pacifc Power and Energy Engineering Conference*: 414–418. https://doi.org/10.1109/APPEEC.2016.7779537
I.R. Muftahov, D.N. Sidorov, N.A. Sidorov, Lavrentiev regularization of integral equations of the first kind in the space of continuous functions. *The Bulletin of Irkutsk State University. Ser. Mathematics.* 2016: Vol. 15, pp. 62–77.
[^1]: https://www.dena.de/en/topics-projects/energy-systems/flexibility-and-storage/pumped-storage/
[^2]: http://appsso.eurostat.ec.europa.eu/nui/submitViewTableAction.do
[^3]: https://www.energy-charts.de/power.htm
|
---
abstract: 'Standard models of accretion discs study the transport of mass on a viscous timescale but do not consider the transport of magnetic flux. The evolution of a large-scale poloidal magnetic field is however an important problem because of its role in the launching of jets and winds and in determining the intensity of turbulence. As a consequence, the transport of poloidal magnetic flux should be considered on an equal basis to the transport of mass. In this paper, we develop a formalism to study such a transport of mass and magnetic flux in a thin accretion disc. The governing equations are derived by performing an asymptotic expansion in the limit of a thin disc, in the regime where the magnetic field is dominated by its vertical component. Turbulent viscosity and resistivity are included, with an arbitrary vertical profile that can be adjusted to mimic the vertical structure of the turbulence. At a given radius and time, the rates of transport of mass and magnetic flux are determined by a one-dimensional problem in the vertical direction, in which the radial gradients of various quantities appear as source terms. We solve this problem to obtain the transport rates and the vertical structure of the disc. The present paper is then restricted to the idealised case of uniform diffusion coefficients, while a companion paper will study more realistic vertical profiles of these coefficients. We show the advection of weak magnetic fields to be significantly faster than the advection of mass, contrary to what a crude vertical averaging might suggest. This results from the larger radial velocities away from the mid-plane, which barely affect the mass accretion owing to the low density in these regions but do affect the advection of magnetic flux. Possible consequences of this larger accretion velocity include a potentially interesting time-dependence with the magnetic flux distribution evolving faster than the mass distribution. If the disc is not too thin, this fast advection may also partially solve the long-standing problem of too efficient diffusion of an inclined magnetic field.'
author:
- |
Jérôme Guilet and Gordon I. Ogilvie\
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences,\
Wilberforce Road, Cambridge CB3 0WA
bibliography:
- 'discs.bib'
title: 'Transport of magnetic flux and the vertical structure of accretion discs: I. Uniform diffusion coefficients'
---
\[firstpage\]
accretion, accretion discs – magnetic fields – MHD – ISM: jets and outflows – galaxies: jets.
Introduction
============
The presence of a large-scale magnetic field in an accretion disc has several potentially important consequences. It may play an essential role in the acceleration and collimation of jets and winds from the disc [@blandford82] and in harnessing the rotational energy of a central black hole [@blandford77]. It also strongly affects the intensity of magnetohydrodynamic (MHD) turbulence and the resulting transport of angular momentum within the disc [@balbus98].
The origin (and existence) of such a large-scale magnetic field is, however, debated from a theoretical perspective. There are in principle two possibilities: either the field is created in situ by a dynamo process, or it is brought in by the gas that is being accreted. It is still unclear whether a dynamo process can create a significant magnetic field that would be coherent over a scale comparable to the radius. Indeed the MHD turbulence, presumed to be at the origin of the dynamo, most likely has a radial correlation length of the order of the vertical thickness of the disc, which is usually much smaller the radius (see the discussion in @spruit10).
In the second possibility, an initially weak field present in the gas supplied to the disc could be strongly amplified as it is transported inwards and the magnetic flux accumulates in the central part of the disc. The transport of magnetic flux is due to two processes: the inward advection by the accretion flow, and the diffusion due to a turbulent resistivity which tends to counteract the advection and therefore transport the magnetic flux outwards. The scenario therefore requires that the field diffuse outwards at a slower rate than it is advected for the total transport to be directed inwards (at least initially). Which of the advection or the diffusion is faster is not obvious a priori, as both are due to the same phenomenon: the turbulence which is causing an effective viscosity (enabling advection) and an effective resistivity (enabling the magnetic field to diffuse). The first theoretical studies of the evolution of the magnetic field in the presence of advection and diffusion have found that the diffusion is much faster than the advection, if the disc is thin and the field lines bend significantly across it [@lubow94a; @heyvaerts96]. This can be understood by the following argument. If the angular momentum transport is due to a turbulent viscosity $\nu$, the advection speed is approximately $$v_{\rm adv} \sim {\frac}{3}{2}\frac{\nu}{r}.
\label{eq:vadv}$$ The diffusion of the magnetic flux is predominantly due to the bending of the magnetic field lines across the disc, which is associated with an azimuthal electric current $J_\phi \sim B_r/(\mu_0H)$, where $H$ is the scale-height of the disc. The action of the turbulent magnetic diffusivity $\eta$ on this current induces a diffusion speed $$v_{\rm diff} \sim \frac{\eta}{H}\frac{B_r}{B_z}.
\label{eq:vdiff}$$ Equating the advection and diffusion speeds, one finds the maximum inclination of the magnetic field lines that can be induced by the advection: $$\frac{B_r}{B_z} \sim \frac{3}{2} {\mathcal{P}}\frac{H}{r},
\label{eq:vadv_diff}$$ where ${\mathcal{P}}\equiv \nu/\eta$ is the turbulent magnetic Prandtl number, which is usually expected to be of order unity. The bending is thus very small for a thin disc, unless the magnetic Prandtl number is unexpectedly large (${\mathcal{P}}\sim r/H$).
This result is rather disappointing as it is problematic for models of jets and winds. Indeed, the acceleration of a jet or wind by the magneto-centrifugal mechanism requires the field lines to bend by more than $30$ degrees from the vertical direction [@blandford82]. Furthermore, outward bending of the field lines is a natural consequence of the accumulation of magnetic flux in the central part of the disc. The result of @lubow94a thus suggests that the inward advection of magnetic flux cannot significantly amplify the magnetic field, and raises doubts on the very existence of a significant large-scale magnetic field.
Various potential solutions to this magnetic diffusion problem have been considered:
- A high effective magnetic Prandtl number associated with the turbulence could solve the problem. However, theoretical expectations [@parker71; @pouquet76] as well as shearing-box simulations of MHD turbulence suggest that the turbulent magnetic Prandtl number is of order unity [@lesur09; @guan09; @fromang09]. Note however that the main contribution to the diffusion (the vertical diffusion of a radial field) has not been measured so far. It is also possible that the effects of turbulence are more complicated than is described by effective diffusion coefficients.
- Breaking of the axial symmetry: @spruit05 proposed a scenario allowing a fast advection of the magnetic flux while preventing its diffusion. In this scenario a weak magnetic field concentrates into strongly magnetised small patches, owing to the tendency of turbulent flows to expel magnetic flux (e.g. @weiss66). The diffusion is slowed down because the strong magnetic field prevents turbulence inside the patch. On the other hand, rapid advection occurs as the patch loses angular momentum through a magneto-centrifugal wind. This interesting idea needs however much more work to be confirmed.
- Vertical structure: the analysis by [@lubow94a] uses a vertical averaging of the disc, which implicitly assumes that the magnetic flux is advected at the same speed as the mass. However, [@ogilvie01] have questioned the validity of this averaging procedure and suggested that the vertical dependence of the radial velocity, the density and the resistivity could have an influence. One objective of this article is to investigate further this question. @bisnovatyi-kogan07 [@bisnovatyi-kogan12], @rothstein08, and @lovelace09 also considered the vertical structure of the disc, studying the effect of a non-turbulent layer at the surface. They suggested that most of the bending of the magnetic field lines would take place in the highly conducting non-turbulent layer. The low resistivity of the layer would thus drastically reduce the diffusion of the magnetic field. A proper self-consistent calculation of this effect is still missing, however, because the solutions of @lovelace09 and @bisnovatyi-kogan12 stop at the base of the conducting surface layer. The formalism developed in this article will be used in a companion paper to test this scenario.
The main purpose of this article and the companion one (Guilet & Ogilvie in preparation, hereafter called paper II) is to clarify how the vertical structure of the disc affects the transport of magnetic flux. We are particularly interested in the poloidal magnetic flux threading the disc, as this quantity satisfies a conservation equation and can be modified only by advective or diffusive transport, possibly enhanced by turbulence. Since the poloidal magnetic field strongly affects the intensity of turbulence in an accretion disc and is an essential component in magneto-centrifugal jet launching, the transport of poloidal flux needs to be considered on an equal basis to the transport of mass in an accretion disc.
Although future work may involve direct numerical simulations of turbulent discs, in our present analysis the turbulence is simply modelled by an effective viscosity and an effective resistivity. The formalism described in Sections \[sec:asymptotic\] and Section \[sec:local\] allows any vertical profile of these coefficients, which will be used in paper II to study the effect of the variation of the turbulent intensity with distance from the mid-plane. The rest of this article however focuses on the idealised case of uniform diffusion coefficients. Under these assumptions and in the limit of a thin accretion disc, we self-consistently solve for the vertical profile of the mean velocity and magnetic field. We focus in particular on the regime in which the magnetic field is dominated by its vertical component. This is done partly to simplify the analysis and take advantage of a linear system of equations, and partly in order to avoid the complications of magneto-centrifugal jet launching from the disc. A separate paper will address the launching process [@ogilvie12]. The plan of this paper is as follows. In Section \[sec:asymptotic\], we use the basic equations of MHD, together with an asymptotic expansion in the limit of a thin disc, to derive the equations describing the evolution of the mass and magnetic flux distributions on a viscous or resistive timescale. In Section \[sec:local\], we focus on the quasi-local problem describing the vertical structure of the thin disc and enabling the calculation of the transport rates. This problem is solved for the special case of uniform diffusion coefficients in Section \[sec:cstalpha\]. The results are discussed and summarised in Section \[sec:discussion\].
An asymptotic expansion within a thin disc {#sec:asymptotic}
==========================================
We start from the equations of MHD, in the form $${\frac}{{\partial}\rho}{{\partial}t}+{{\bmath\nabla}}\cdot(\rho{{\bmath v}})=0,$$ $$\begin{aligned}
\lefteqn{\rho\left({\frac}{{\partial}{{\bmath v}}}{{\partial}t}+{{\bmath v}}\cdot{{\bmath\nabla}}{{\bmath v}}\right)=-\rho{{\bmath\nabla}}\Phi-{{\bmath\nabla}}p}&\nonumber\\
&&+{\frac}{1}{\mu_0}({{\bmath\nabla}}\times{{\bmath B}})\times{{\bmath B}}+{{\bmath\nabla}}\cdot{\mathbf{T}},\end{aligned}$$ $${\frac}{{\partial}{{\bmath B}}}{{\partial}t}={{\bmath\nabla}}\times({{\bmath v}}\times{{\bmath B}}-\eta{{\bmath\nabla}}\times{{\bmath B}}),$$ $${{\bmath\nabla}}\cdot{{\bmath B}}=0,$$ $${\mathbf{T}}=\rho\nu\left[{{\bmath\nabla}}{{\bmath v}}+({{\bmath\nabla}}{{\bmath v}})^{\mathrm{T}}-{\frac}{2}{3}({{\bmath\nabla}}\cdot{{\bmath v}})\,{\mathbf{I}}\right],$$ where $\rho$ is the density, ${{\bmath v}}$ the velocity, $\Phi$ the gravitational potential, $p$ the pressure, ${{\bmath B}}$ the magnetic field, ${\mathbf{T}}$ the viscous stress, $\eta$ the magnetic diffusivity, $\nu$ the kinematic viscosity and ${\mathbf{I}}$ the unit tensor of second rank. (A bulk viscosity could be included, but would have no effect on this problem.) We neglect self-gravity and do not consider a thermal energy equation at this stage.
We consider an axisymmetric solution of these equations in cylindrical polar coordinates $(r,\phi,z)$. The poloidal part of the magnetic field can be described in the usual way by a poloidal magnetic flux function $\psi(r,z,t)$. Thus $${{\bmath B}}={{\bmath\nabla}}\psi\times{{\bmath\nabla}}\phi+B_\phi\,{{\bmath e}}_\phi=-{\frac}{1}{r}{{\bmath e}}_\phi\times{{\bmath\nabla}}\psi+B_\phi\,{{\bmath e}}_\phi.$$ This representation satisfies the constraint ${{\bmath\nabla}}\cdot{{\bmath B}}=0$ automatically. Poloidal magnetic field lines correspond to curves $\psi={\mathrm{constant}}$ in the $(r,z)$ plane, and (up to an additive constant) $2\pi\psi$ is the magnetic flux contained within a circular loop at position $(r,z)$. The poloidal part of the induction equation can then be integrated to give $${\frac}{{\partial}\psi}{{\partial}t}+{{\bmath v}}\cdot{{\bmath\nabla}}\psi=\eta r^2{{\bmath\nabla}}\cdot\left({\frac}{1}{r^2}{{\bmath\nabla}}\psi\right),
\label{dpsidt}$$ which shows that the poloidal magnetic flux satisfies a conservation law and is affected by advection and diffusion.
We are interested in solving these equations both inside and outside a thin accretion disc that lies close to the plane $z=0$. In the absence of magnetic fields, there is a well understood ordering scheme for thin discs. If the characteristic angular semithickness of the disc is $H/r=O(\epsilon)$, where $0<\epsilon\ll1$ is a small dimensionless parameter, and the Shakura–Sunyaev viscosity parameter is $\alpha$, then the ratio of the sound speed to the orbital velocity is $O(\epsilon)$, and the ratio of the radial velocity to the orbital velocity is $O(\alpha\epsilon^2)$. The fractional deviation of the azimuthal velocity from the Keplerian value is $O(\epsilon^2)$, and so on. Furthermore, the disc evolves on the viscous timescale, which is longer than the orbital timescale by $O(\alpha^{-1}\epsilon^{-2})$. Although this is rarely done, the equations governing thin accretion discs can be obtained by a formal asymptotic expansion of the basic equations of fluid dynamics.
For magnetised discs there is more than one possible ordering scheme, depending on the strength (and orientation) of the magnetic field [@ogilvie97]. We consider here a situation in which the disc has comparable values of $\nu$ and $\eta$, with (formally) $\alpha=O(1)$. The magnetic field is dominated by the vertical component, which has a magnetic pressure comparable to the gas pressure. This assumption allows us to avoid a consideration of magnetocentrifugal jet launching.
We resolve the internal structure of the thin disc and its slow evolution in time through the introduction of a rescaled vertical coordinate $$\zeta=\epsilon^{-1}z$$ and a time variable $$\tau=\epsilon^2t.$$ We then propose the following expansion of the fluid variables. (The scheme for indexing the variables is debatable, but the choice made here is convenient for the present purposes.) $$\rho=\rho_0(r,\zeta,\tau)+\epsilon^2\rho_2(r,\zeta,\tau)+O(\epsilon^4),$$ $$p=\epsilon^2\left[p_0(r,\zeta,\tau)+\epsilon^2p_2(r,\zeta,\tau)+O(\epsilon^4)\right],$$ $$\Phi=\Phi_0(r)+\epsilon^2\Phi_2(r){{\textstyle\frac{1}{2}}}\zeta^2+O(\epsilon^4),$$ $$v_r=\epsilon^2v_{r2}(r,\zeta,\tau)+\epsilon^4v_{r4}(r,\zeta,\tau)+O(\epsilon^6),$$ $$v_\phi=r\Omega_0(r)+\epsilon^2v_{\phi2}(r,\zeta,\tau)+\epsilon^4v_{\phi4}(r,\zeta,\tau)+O(\epsilon^6),$$ $$v_z=\epsilon\left[\epsilon^2v_{z2}(r,\zeta,\tau)+\epsilon^4v_{z4}(r,\zeta,\tau)+O(\epsilon^6)\right],$$ $$\psi=\epsilon\left[\psi_0(r,\tau)+\epsilon^2\psi_2(r,\zeta,\tau)+O(\epsilon^4)\right],$$ $$B_r=\epsilon\left[\epsilon B_{r1}(r,\zeta,\tau)+\epsilon^3B_{r3}(r,\zeta,\tau)+O(\epsilon^5)\right],$$ $$B_\phi=\epsilon\left[\epsilon B_{\phi1}(r,\zeta,\tau)+\epsilon^3B_{\phi3}(r,\zeta,\tau)+O(\epsilon^5)\right],$$ $$B_z=\epsilon\left[B_{z0}(r,\tau)+\epsilon^2B_{z2}(r,\zeta,\tau)+O(\epsilon^4)\right],$$ $$\nu=\epsilon^2\left[\nu_0(r,\zeta,\tau)+O(\epsilon^2)\right],$$ $$\eta=\epsilon^2\left[\eta_0(r,\zeta,\tau)+O(\epsilon^2)\right].$$ We make the following observations. The overall scaling of the density is irrelevant provided that self-gravity is neglected; however, the relative scaling of the pressure and density is significant and implies that the sound speed is $O(\epsilon)$ (in comparison to the orbital velocity). The expansion of $\Phi$ is the Taylor expansion of a smooth, symmetric external potential about the midplane $z=0$; in the case of a point-mass potential $\Phi=-GM(r^2+z^2)^{-1/2}$, we have $\Phi_0=-GM/r$ and $\Phi_2=GM/r^3$. The scaling of ${{\bmath B}}$ is such that, as mentioned above, the magnetic field is dominated by the vertical component, which has a magnetic pressure comparable to the gas pressure. The expansion of $\psi$ is of the form required to produce this. The scalings of $\nu$ and $\eta$ correspond to having (formally) $\alpha=O(1)$.
Substituting these expansions into the basic equations and comparing the coefficients of powers of $\epsilon$, we obtain a succession of equations. The radial component of the equation of motion at leading order yields $$-\rho_0r\Omega_0^2=-\rho_0{\partial}_r\Phi_0,$$ which implies $r\Omega_0^2={\partial}_r\Phi_0$, i.e. that $\Omega_0(r)$ is the angular velocity of a circular particle orbit of radius $r$ in the midplane. In the case of a point-mass potential, we obtain the Keplerian value $\Omega_0=(GM/r^3)^{1/2}$.
The vertical component of the equation of motion at leading order yields $$0=-\rho_0\Phi_2\zeta-{\partial}_\zeta p_0,$$ which is the usual condition of hydrostatic equilibrium in which the vertical pressure gradient balances the vertical gravitational force. Under the present scalings, the Lorentz force does not affect this balance at leading order. The solution of this equation depends on additional assumptions regarding the thermal physics of the disc. In the simplest situation of a vertically isothermal disc in which $p_0=c_{{\mathrm{s}}0}^2\rho_0$, where the isothermal sound speed $c_{{\mathrm{s}}0}(r,\tau)$ does not depend on $\zeta$, we obtain the familiar solution $$\rho_0={\frac}{\Sigma_0}{(2\pi)^{1/2}H_0}\exp\left(-{\frac}{\zeta^2}{2H_0^2}\right),
\label{density}$$ where $H_0(r,\tau)=c_{{\mathrm{s}}0}/\Phi_2^{1/2}$ is the isothermal scaleheight and $\Sigma_0(r,\tau)$ is the surface density.
Next, the horizontal components of the equation of motion and of the induction equation yield the four equations $$\begin{aligned}
\lefteqn{-2\rho_0\Omega_0v_{\phi2}=-\rho_0{\partial}_r\Phi_2{{\textstyle\frac{1}{2}}}\zeta^2-{\partial}_r\left(p_0+{\frac}{B_{z0}^2}{2\mu_0}\right)}&\nonumber\\
&&+{\frac}{B_{z0}}{\mu_0}{\partial}_\zeta B_{r1}+{\partial}_\zeta(\rho_0\nu_0{\partial}_\zeta v_{r2}), \label{motion_r}\end{aligned}$$ $$\begin{aligned}
\lefteqn{\rho_0v_{r2}{\frac}{1}{r}{\partial}_r(r^2\Omega_0)={\frac}{B_{z0}}{\mu_0}{\partial}_\zeta B_{\phi1}+{\frac}{1}{r^2}{\partial}_r(\rho_0\nu_0r^3{\partial}_r\Omega_0)}&\nonumber\\
&&+{\partial}_\zeta(\rho_0\nu_0{\partial}_\zeta v_{\phi2}), \label{motion_phi}\end{aligned}$$ $$0=B_{z0}{\partial}_\zeta v_{r2}+{\partial}_\zeta[\eta_0({\partial}_\zeta B_{r1}-{\partial}_rB_{z0})],
\label{induction_br}$$ $$0=B_{r1}r{\partial}_r\Omega_0+B_{z0}{\partial}_\zeta v_{\phi2}+{\partial}_\zeta(\eta_0{\partial}_\zeta B_{\phi1}).
\label{induction_bphi}$$ These can be regarded as four *linear* equations for the unknowns $v_{r2}$, $v_{\phi2}$, $B_{r1}$ and $B_{\phi1}$. They are linear because of the assumption of small deviations from orbital motion and a vertical magnetic field, implicit in the asymptotic expansion. Although these four quantities depend on $r$, $\zeta$ and $\tau$, only derivatives with respect to $\zeta$ appear in these equations and they can therefore be regarded an eighth-order system of ordinary differential equations (ODEs) in $\zeta$ at each $r$ and $\tau$. This situation arises because the vertical dependence is assumed to be more rapid than the radial dependence in a thin disc, and the time-dependence is assumed to be slow. The equations are inhomogeneous and are driven by various source terms: the vertical dependence of the radial gravitational force, the radial gradient of total pressure, the azimuthal viscous force resulting from the orbital motion, etc.
The expected symmetry of the solutions is that $v_{r2}$ and $v_{\phi2}$ are even functions of $\zeta$ while $B_{r1}$ and $B_{\phi1}$ are odd. The behaviour of the solutions at large $|\zeta|$ can be illustrated by considering the case of a vertically isothermal disc in which $\nu_0$ and $\eta_0$ are bounded as $|\zeta|\to\infty$ (e.g. if they are independent of $\zeta$). In this case the solution is of the form $$v_{r2}\sim C_1+E_1(\zeta), \label{ur_infinity}$$ $$v_{\phi2}\sim C_2{{\textstyle\frac{1}{2}}}\zeta^2\pm C_3\zeta+C_4+E_2(\zeta), \label{uphi_infinity}$$ $$B_{r1}\sim C_5\zeta\pm C_6+E_3(\zeta), \label{Br_infinity}$$ $$B_{\phi1}\sim \pm C_7 + E_4(\zeta) \label{Bphi_infinity}$$ as $\zeta\to\pm\infty$, where the $C_i$ are constants and the $E_i$ are (like the density and pressure) exponentially small; here the parametric dependence of the solution on $r$ and $\tau$ has been suppressed. In order to satisfy the governing equations, the values of the constants are constrained by $$C_2B_{z0}=-C_5r{\partial}_r\Omega_0,$$ $$C_3B_{z0}=-C_6r{\partial}_r\Omega_0,$$ $$C_5={\partial}_rB_{z0};$$ while $C_6$ and $C_7$ have the nature of input parameters, $C_1$ and $C_4$ have the nature of output parameters. The $C_5$ term is required in order to produce a force-free field in the low-density exterior at large $|\zeta|$. (In fact it is a current-free or potential field, because an axisymmetric poloidal force-free field must be current-free.) The $C_6$ term represents, in some sense, the inclination of the poloidal field at the upper surface of the disc, and we call $C_6=B_{r{\mathrm{s}}}$ accordingly. More precisely, it is the radial component of the magnetic field obtained by extrapolating the parabolic form of the field lines in the force-free region above the disc down to the midplane $\zeta=0$. The $C_7$ term represents the azimuthal component of the magnetic field at the upper surface and we call it $B_{\phi{\mathrm{s}}}$. If we are considering discs without outflows (because of the condition $|B_{r{\mathrm{s}}}|\ll|B_z|$) then we should set $B_{\phi{\mathrm{s}}}=0$ in order that the magnetic stress $B_\phi B_z/\mu_0$ vanish at large $|\zeta|$. However, we retain this term as a convenient way to model the effect of angular momentum removal by an outflow without the complications of solving for the acceleration of the flow. Angular momentum is removed by specifying a negative value for $B_{\phi{\mathrm{s}}}B_z$.
In terms of the flux function $\psi$, we also have the relations $$B_{z0}={\frac}{1}{r}{\partial}_r\psi_0,$$ $$B_{r1}=-{\frac}{1}{r}{\partial}_\zeta\psi_2.$$ Neither $\psi_0$ nor $B_{z0}$ depends on $\zeta$, so $B_{z0}$ can be regarded as an input parameter in the above system of ODEs.
Furthermore, the integrated form of the poloidal part of the induction equation (equation \[dpsidt\]) yields $${\partial}_\tau\psi_0+v_{r2}{\partial}_r\psi_0=\eta_0\left[r{\partial}_r\left({\frac}{1}{r}{\partial}_r\psi_0\right)+{\partial}_\zeta^2\psi_2\right].
\label{psidot}$$ Note that $-(1/r){\partial}_\zeta$ of equation (\[psidot\]) produces equation (\[induction\_br\]). Once we have solved the system of ODEs, equation (\[psidot\]) can be evaluated at any value of $\zeta$ (or indeed averaged in any convenient way with respect to $\zeta$) to discover the value of ${\partial}_\tau\psi_0$. This gives the rate of poloidal magnetic flux transport at the given values of $r$ and $\tau$, which we can associate with an effective transport velocity $v_\psi$ through ${\partial}_\tau\psi_0+v_\psi{\partial}_r\psi_0=0$, which implies ${\partial}_\tau B_{z0}+(1/r){\partial}_r(rv_\psi B_{z0})=0$. In fact, if we evaluate equation (\[psidot\]) in the limit $\zeta\to\infty$, we find ${\partial}_\tau\psi_0=-C_1rB_{z0}$ and therefore $v_\psi=C_1$. Since the electric current vanishes in this limit, the radial velocity corresponds to the speed at which the magnetic field is transported.
In addition, the equation of mass conservation yields $${\partial}_\tau\rho_0+{\frac}{1}{r}{\partial}_r(r\rho_0v_{r2})+{\partial}_\zeta(\rho_0v_{z2})=0.
\label{drhodt}$$ We define the surface density at leading order, $$\Sigma_0(r,\tau)=\int_{-\infty}^\infty\rho_0(r,\zeta,\tau)\,{\mathrm{d}}\zeta.$$ Then (assuming no vertical mass loss from the disc) $${\partial}_\tau\Sigma_0+{\frac}{1}{r}{\partial}_r\left(r\int_{-\infty}^\infty\rho_0v_{r2}\,{\mathrm{d}}\zeta\right)=0.$$ Let $$m_0(r,\tau)=\int_0^r\Sigma_0(r',\tau)\,2\pi r'\,{\mathrm{d}}r'$$ be the mass contained within radius $r$ at time $\tau$; then ${\partial}_\tau
m_0+v_m{\partial}_rm_0=0$, where $$v_m={\frac}{1}{\Sigma_0}\int_{-\infty}^\infty\rho_0v_{r2}\,{\mathrm{d}}\zeta$$ is the effective transport velocity for mass, which can be determined from the solution of the system of ODEs.
In summary, then, the input parameters at given values of $r$ and $\tau$ are $\Omega_0$, $\Phi_2$, $c_{{\mathrm{s}}0}$, $\Sigma_0$, $\nu_0$, $\eta_0$, $B_{z0}$, $B_{r{\mathrm{s}}}$, $B_{\phi{\mathrm{s}}}$, ${\partial}_r\Omega_0$, ${\partial}_r^2\Omega_0$, ${\partial}_rc_{{\mathrm{s}}0}$, ${\partial}_r\Sigma_0$, ${\partial}_r\nu_0$, ${\partial}_r\eta_0$ and ${\partial}_rB_{z0}$. The output parameters of greatest interest are $v_m$ and $v_\psi$. The problem can be simplified, and made dimensionless, by assuming a Keplerian disc ($\Omega_0\propto r^{-3/2}$ and $\Phi_2=\Omega_0^2$) and alpha prescriptions of the form $\nu_0=\alpha
c_{{\mathrm{s}}0}^2/\Omega_0$, $\eta_0=\nu_0/{\mathcal{P}}$ with $\alpha$ and ${\mathcal{P}}$ being constants.
In this analysis we have simplified the problem by assuming that the magnetic field is dominated by its vertical component within the disc. This field needs to be matched to a force-free (in fact, current-free) field outside the disc, which involves the solution of a global problem [e.g. @ogilvie97]. The nature of this problem is that the poloidal flux distribution on the disc determines the inclination of the poloidal field at each radial location, and in general this angle is not small. An alternative ordering scheme is possible, in which all three components of the magnetic field are $O(\epsilon)$. In such a scheme, the inclination of the poloidal field could be large and magneto-centrifugal jet launching would be possible [@ogilvie01]. The pressure of the horizontal magnetic field would also contribute to the vertical force balance and the problem would become more nonlinear. (In fact, it is easy to add this effect by hand to the present system of equations, but we will not do so here.) However, the transport of magnetic flux by diffusion would be formally faster by a factor $O(\epsilon^{-1})$ than that by advection, if ${\mathcal{P}}=O(1)$; this inequality is merely an expression of the result of @lubow94a. Therefore no ordering scheme is fully satisfactory. We adopt the present scheme because of the transparency of its derivation, the linearity of the resulting equations, and because it avoids the complications of jet launching, while missing only the effect of the magnetic compression of the disc.
A quasi-local problem {#sec:local}
=====================
Equations (\[motion\_r\])–(\[induction\_bphi\]) define a quasi-local problem, as only vertical derivatives of the unknowns appear, while radial derivatives appear only as source terms. The solution of this local problem provides the radial transport velocities of mass and poloidal magnetic flux (at a given time and radius), which are necessary to determine the global evolution of the disc on a viscous/resistive timescale. The local problem on the other hand is effectively stationary (no time derivatives appear), because stationarity is assumed on the much shorter dynamical timescale. In this article and paper II, we solve this local problem and leave for future work the study of the global evolution of a disc. In this section we rewrite in a dimensionless form the equations governing the local problem. We discuss the dependence on the different parameters, and describe the numerical method used to solve the equations.
Differential system {#sec:system}
-------------------
We choose as independent variables $\rho_0$, $v_{r2}$, $v_{\phi1}$, $B_{r1}$ and $B_{\phi1}$. These variables are non-dimensionalised by defining $$\begin{aligned}
\tilde{\rho} &\equiv & \frac{\rho_0 H}{\Sigma}, \\
u_r &\equiv & \frac{r}{H}\frac{v_{r2}}{c_{\mathrm{s}}}, \\
u_\phi &\equiv & \frac{r}{H}\frac{v_{\phi2}}{c_{\mathrm{s}}}, \\
b_r &\equiv & \frac{r}{H}\frac{B_{r1}}{B_z}, \\
b_\phi &\equiv & \frac{r}{H}\frac{B_{\phi1}}{B_z}, \end{aligned}$$ where we have dropped the subscript $0$ on quantities such as $H$, $c_{\mathrm{s}}$, $B_z$ and $\Sigma$. A dimensionless vertical spatial coordinate is also defined as: $$\zeta \equiv z/H.$$ Note that this variable is different from $\zeta$ used in Section \[sec:asymptotic\], which was a rescaled but dimensional variable. In the rest of the paper when we use the symbol $\zeta$, we refer to the dimensionless variable.
We assume a point-mass potential and therefore circular Keplerian orbital motion at leading order. We use the standard alpha prescription for the viscosity, $\nu = \alpha c_{\mathrm{s}}^2/\Omega = \alpha
c_{\mathrm{s}}H$, allowing for a vertical (but not radial) dependence of the $\alpha$ parameter through $\alpha = \alpha_0g(\zeta)$. The resistivity $\eta$ is then related to the viscosity through the magnetic Prandtl number ${\mathcal{P}}\equiv \nu/\eta $, which can also have a vertical dependence. An isothermal equation of state is assumed for simplicity. Equation (\[density\]) gives the corresponding density profile, which can be written in dimensionless form as $$\tilde{\rho} = \frac{1}{\sqrt{2\pi}}\,\mathrm{e}^{-\zeta^2/2}.$$
The differential system given by equations (\[motion\_r\])–(\[induction\_bphi\]) is rewritten as $$\begin{aligned}
-\frac{1}{\tilde\rho}\lefteqn{{\partial}_\zeta\left(\tilde{\rho}\alpha{\partial}_\zeta u_r \right) - 2u_\phi - \frac{1}{\beta_0\tilde\rho} {\partial}_\zeta b_r = \frac{3}{2} + D_H - D_{\nu\Sigma}}&\nonumber\\
&& + \left(\frac{3}{2}-D_H\right)\zeta^2 - \frac{D_B}{\beta_0\tilde\rho}, \label{eq:motion_r}\end{aligned}$$ $$\begin{aligned}
-\frac{1}{\tilde\rho}\lefteqn{{\partial}_\zeta\left(\tilde{\rho}\alpha{\partial}_\zeta u_\phi \right) + \frac{1}{2} u_r - \frac{1}{\beta_0\tilde\rho}{\partial}_\zeta b_\phi = \frac{3}{2}\alpha \bigg\lbrack - \frac{1}{2} }&\nonumber\\
&& + D_H\left(1 + \frac{{{\rm d}}\ln\alpha}{{{\rm d}}\ln\zeta} \right)- D_{\nu\Sigma} - D_H\zeta^2 \bigg\rbrack, \label{eq:motion_phi}\end{aligned}$$ $$-{\partial}_\zeta\left( \frac{\alpha}{{\mathcal{P}}}{\partial}_\zeta b_r \right) - {\partial}_\zeta
u_r = -D_B {\partial}_\zeta\left(\frac{\alpha}{{\mathcal{P}}}\right), \label{eq:induction_r}$$ $$-{\partial}_\zeta\left(\frac{\alpha}{{\mathcal{P}}}{\partial}_\zeta b_\phi\right) - {\partial}_\zeta u_\phi + \frac{3}{2}b_r = 0,\label{eq:induction_phi}$$ where we have defined the following dimensionless parameters: $$\begin{aligned}
\beta_0 &\equiv& \frac{\mu_0}{B_z^2}\frac{\Sigma c_s^2}{H}, \\
D_H &\equiv&\frac{{\partial}\ln H}{{\partial}\ln r}, \\
D_{\nu\Sigma} &\equiv& 2D_H - \frac{3}{2} + \frac{{\partial}\ln\Sigma}{{\partial}\ln r}, \\
D_B &\equiv & \frac{{\partial}\ln B_z}{{\partial}\ln r}.\end{aligned}$$ Here $\beta_0$ corresponds roughly to the midplane value of the plasma $\beta$ parameter (the ratio of the thermal pressure to the magnetic pressure); more precisely, the two are related by $\beta(\zeta=0) = \sqrt{2/\pi}\,\beta_0 $. The parameter $D_{\nu\Sigma}$ equals ${\partial}\ln(\nu\Sigma)/{\partial}\ln r$, given that $\nu=\alpha H^2\Omega$ and $\alpha$ does not depend on $r$.
Boundary conditions
-------------------
From equations (\[ur\_infinity\])–(\[Bphi\_infinity\]), one can deduce that the following quantities vanish exponentially fast as $\zeta \rightarrow \pm \infty$: $$\rho u_r \to0, \label{eq:boundary_ur} \\$$ $$\rho u_\phi \to0, \label{eq:boundary_uphi}$$ $$b_r-(D_B\zeta\pm b_{r{\mathrm{s}}})\to0, \label{eq:boundary_br}$$ $$b_\phi - (\pm b_{\phi{\mathrm{s}}}) \to 0. \label{eq:boundary_bphi} \\$$ Let us recall that these conditions are motivated by the absence of outflow, owing to the low inclination of the field lines. As a consequence the magnetic field at infinity is force-free. The inclusion of a non-vanishing azimuthal component of the magnetic field is not self-consistent but allows us, if we wish, to mimic the effect of angular momentum removal by an outflow. These boundary conditions are homogeneous, except for the linear source terms proportional to $D_B$, $b_{r{\mathrm{s}}}$, and $b_{\phi{\mathrm{s}}}$ in equations (\[eq:boundary\_br\]) and (\[eq:boundary\_bphi\]).
As the differential equations, the source terms and boundary conditions have reflectional symmetry about the midplane, the stationary profile we seek shares the same symmetry (implying a symmetric horizontal velocity and an antisymmetric horizontal magnetic field). As a consequence, the following conditions apply at the midplane $\zeta=0$: $${\partial}_\zeta u_r=0, \label{eq:midplane_ur}$$ $${\partial}_\zeta u_\phi=0, \label{eq:midplane_uphi}$$ $$b_r=0, \label{eq:midplane_br}$$ $$b_\phi=0. \label{eq:midplane_bphi}$$ Reflectional symmetry also implies that the solution need be computed only in the half-space $\zeta>0$.
Transport velocities of mass and magnetic flux
----------------------------------------------
The transport rates of mass and magnetic flux can be obtained from the solution of the above equations and expressed in terms of a transport velocity in the following way. Multiplying the azimuthal component (\[eq:motion\_phi\]) of the equation of motion by $2\tilde\rho$, integrating over all $\zeta$ and applying the boundary conditions, we obtain $$\int_{-\infty}^\infty\tilde\rho u_r\,{\mathrm{d}}\zeta=-{\frac}{3}{2}(1+2D_{\nu\Sigma})\int_{-\infty}^\infty\tilde\rho\alpha\,{\mathrm{d}}\zeta + \frac{4}{\beta_0}b_{\phi{\mathrm{s}}}.$$ In the case of uniform $\alpha$ this gives the transport velocity $$u_m=-{\frac}{3}{2}\alpha(1+2D_{\nu\Sigma}) + \frac{4}{\beta_0}b_{\phi{\mathrm{s}}},
\label{um}$$ where $$u_m\equiv{\frac}{r}{H}{\frac}{v_m}{c_{\mathrm{s}}}=\int_{-\infty}^\infty\tilde\rho u_r\,{\mathrm{d}}\zeta$$ is the dimensionless mass transport velocity. More generally, $\alpha$ should be replaced by $\alpha_0$ in equation (\[um\]) and a factor of $$\int_{-\infty}^\infty{\mathrm{e}}^{-\zeta^2/2}g(\zeta)\,{\mathrm{d}}\zeta\Bigg/\int_{-\infty}^\infty{\mathrm{e}}^{-\zeta^2/2}\,{\mathrm{d}}\zeta$$ should be included. For a vanishing $b_{\phi{\mathrm{s}}}$ these results are consistent with the familiar expression from the standard theory of a Keplerian accretion disc, $$\bar v_r=-{\frac}{3}{r^{1/2}\Sigma}{\partial}_r(r^{1/2}\bar\nu\Sigma),$$ where the overbar refers to a density-weighted vertical average.
The integrated version of the radial component (\[eq:induction\_r\]) of the induction equation is $$u_\psi = u_r + \frac{\alpha}{{\mathcal{P}}}({\partial}_\zeta b_r-D_B) = \mathrm{const},
\label{upsi}$$ where $$u_\psi\equiv{\frac}{r}{H}{\frac}{v_\psi}{c_{\mathrm{s}}}$$ is the dimensionless magnetic flux transport velocity. It can be evaluated using equation (\[upsi\]) at any convenient hight.
Relation to previous works
--------------------------
@ogilvie01 solved for the vertical structure of a magnetised accretion disc, allowing for the possibility of magneto-centrifugal jet launching [see also @ogilvie97; @ogilvie98a]. The equations they solved are similar to ours, but with the following differences. The thermal structure of the disc was determined by a balance between viscous and resistive heating and radiative cooling, rather than being assumed to be isothermal. This solution was matched to an isothermal atmosphere with a force-free magnetic field. The radial gradients of thermal and magnetic pressure were neglected, as was the vertical transport of momentum by viscosity. The pressure of the horizontal magnetic field was taken into account in the vertical force balance.
@lovelace09 and @bisnovatyi-kogan12 also performed a similar calculation of the vertical structure of a magnetised accretion disc. The differences between their approach and ours are the following. The equations in the limit of a thin disc were derived more systematically in this paper, and describe more precisely the vertical structure: contrary to @lovelace09 we take into account the vertical dependence of the pressure gradient and viscous stress. We also did not discard the radial gradient of vertical magnetic field, whose contribution to the magnetic flux transport appears at the same order in the expansion as the advection by an accretion flow. Furthermore, we solve a more general problem, as we do not assume that the disc is stationary on a viscous/resistive timescale but only on a dynamical timescale, and we consider a much wider range of $\beta$. The vertical boundary conditions also differ substantially. We calculate the vertical profiles up to the region that is magnetically dominated, where one can impose the magnetic field to be that dictated by the exterior solution. On the other hand, @lovelace09 and @bisnovatyi-kogan12 do not solve the transition between the interior of the disc and the magnetically dominated region; instead they apply special boundary conditions at the surface of the disc. The validity of these boundary conditions may be questioned in light of the results of Section \[sec:cstalpha\], since we find a significant jump in $b_r$ and $v_\phi$ at the transition to the force-free regime, while their boundary conditions preclude such jumps (note however that their diffusion coefficients vanish in the surface layer, which is not considered in this article but delayed to paper II). Another difference is the choice of $\alpha$: considering the transition to the magnetically dominated regime forces us to determine $\alpha$ through the marginal stability hypothesis in order to avoid unphysical effects (see Section \[sec:marginal\]). Finally, they use a disc model where the density is independent of height inside the disc, while we use an isothermal model where the density is a Gaussian function of height. Allowing a variation of the density is essential in our result that the advection speed of mass and magnetic flux can differ dramatically (see Section \[sec:cstalpha\]).
Relation to the shearing sheet
------------------------------
Both our equations and those of @ogilvie01 are related to the local approximation (shearing sheet or shearing box), commonly used in the study of accretion discs. Starting with the equations of MHD in this approximation and assuming a solution that depends only on the vertical coordinate $z$, we would obtain equations equivalent to ours except that the source terms would be absent and the pressure of the horizontal magnetic field would be taken into account in the vertical force balance.
Relation to the magnetorotational instability
---------------------------------------------
The four equations (\[eq:motion\_r\])–(\[eq:induction\_phi\]) governing the horizontal components of the velocity and magnetic field are closely related to those appearing in the analysis of the magnetorotational instability [MRI; @balbus98]. Let us write the equations in the symbolic form $$\mathbf{L}\bmath{X}=\bmath{F},
\label{lx}$$ where $\bmath{X}=[u_r\;u_\phi\;b_r\;b_\phi]^\mathrm{T}$ is a vector of unknowns, $\mathbf{L}$ is the linear operator that generates the left-hand sides of the equations, and $\bmath{F}$ represents the source terms on the right-hand sides. To analyse the MRI of an isothermal disc with a vertical magnetic field, we would instead solve the eigenvalue problem $$\mathbf{L}\bmath{X}=\lambda\bmath{X},
\label{lx_mri}$$ with homogeneous boundary conditions, corresponding to solutions of the linearized equations proportional to $\mathrm{e}^{\lambda t}$ and with no horizontal spatial dependence. The MRI would correspond to a solution with $\mathrm{Re}(\lambda)>0$. The close connection between the MRI and the equilibrium of magnetised discs has been noted before [@ogilvie98b; @ogilvie01].
Marginal stability hypothesis {#sec:marginal}
-----------------------------
The value of $\alpha$ (or $\alpha_0$ if a vertical dependence is permitted) can be estimated by supposing that the MRI is made marginally stable by the effect of the viscosity and resistivity [@ogilvie01]. This approach has the desirable property of ensuring that the obtained stationary solution is not unstable (at least to modes with no horizontal dependence, which are invariably the most dangerous). @ogilvie01 showed that this assumption avoids unphysical effects like multiple bending of the field lines (see also Appendix A).
Stability depends on the strength of the magnetic field (through $\beta_0$) and on the viscosity and resistivity (through $\alpha$ and ${\mathcal{P}}$). As expected, the most difficult mode to stabilise is the largest-scale channel mode, which is antisymmetric about the midplane [@ogilvie01], and this mode has $\lambda=0$ at marginal stability. [^1] For weak fields ($\beta_0 \ga 10^4$) the marginal mode is localised around $\zeta =\pm\zeta_B$ (Figure \[fig:channel\]), where $$\zeta_B = \sqrt{\ln\left(\frac{2}{\pi}\beta_0^2\right)}
\label{eq:zetab}$$ is the height at which the magnetic pressure equals the thermal pressure.
![The value of the $\alpha$ parameter determined by the marginal stability hypothesis as a function of the magnetic field strength. The calculation is done with uniform diffusion coefficients. The full line illustrates the value of $\alpha$ obtained with the magnetic Prandtl number mostly used in this paper, ${\mathcal{P}}=1$. The dotted line shows the $\alpha$ values obtained without viscosity (${\mathcal{P}}=0$), and the dashed line the corresponding analytical estimate for large $\beta_0$ of Appendix A (equation \[eq:alpha\_marginal\]). []{data-label="fig:alpha"}](figures/figure2.ps){width="\columnwidth"}
While physically motivated, this way of determining the effective turbulent diffusion coefficients should be taken with a grain of salt. Particularly worrying is the tendency of this method to produce large values of $\alpha$. Indeed, in the case of uniform diffusion coefficients, $\alpha$ is of order unity over a large range of magnetic field strength (Figure \[fig:alpha\]). Such a high value is most probably not realistic for weak magnetic fields ($\beta_0 \gg
1$), although we note that direct numerical simulations of discs with vertical gravity and a net vertical magnetic flux are computationally difficult and few results are available, perhaps precisely because the turbulence is very intense when $\beta_0$ is not very large. The somewhat surprisingly slow decline of $\alpha$ at large $\beta_0$ is due to the fact that its value is determined mostly by the region where the channel mode is localised, which is rather strongly magnetised ($\beta\sim1$) even when the midplane is very weakly magnetised. Obviously, assuming the same value of $\alpha$ in these regions with vastly different magnetisation is unrealistic. The use of a vertical profile for $\alpha$, as is done in paper II, alleviates this issue by reducing the midplane value ($\alpha(\zeta=0)
\sim 0.1$ is more realistic although still probably too high for very weak fields), while keeping similar values far from the midplane.
Fortunately (given the uncertainty on the value of $\alpha$), the balance between the advection and diffusion of the magnetic field is only weakly dependent on $\alpha$, when the angular transport enabling advection is caused by turbulence. Indeed, both advection and diffusion processes are (at least approximately) proportional to $\alpha$, through respectively the viscosity and the resistivity.
Parameters and form of the solution {#sec:parameters}
-----------------------------------
The free parameters of the dimensionless problem are the magnetic Prandtl number ${\mathcal{P}}$ and the magnetisation parameter $\beta_0$, as well as the source terms $D_H$, $D_{\nu\Sigma}$, $D_B$, $b_{r{\mathrm{s}}}$ and $b_{\phi{\mathrm{s}}}$.
An important consequence of the linearity of the equations is that the solution depends linearly on the source terms, appearing either on the right-hand side of the system of equations or as a non-vanishing boundary condition at infinity ($b_{r{\mathrm{s}}}$ and $b_{\phi{\mathrm{s}}}$). Thus, the general solution is a linear combination of the solution vectors corresponding to each source term: $$\begin{aligned}
\lefteqn{\bmath{X} = \bmath{X}_\mathrm{K} + \bmath{X}_{DH}D_H + \bmath{X}_{D\nu\Sigma}D_{\nu\Sigma} + \bmath{X}_{DB}D_B}&\nonumber\\
&& + \bmath{X}_{br{\mathrm{s}}}b_{r{\mathrm{s}}} + \bmath{X}_{b\phi{\mathrm{s}}}b_{\phi{\mathrm{s}}},\end{aligned}$$ where $\bmath{X}_{DH}$ is the solution vector corresponding to the source term proportional to $D_H$ and so on. $\bmath{X}_\mathrm{K}$ is the solution vector corresponding to the source terms $\bmath{F}$ that are independent of $D_H$, $D_{\nu\Sigma}$, $D_B$, $b_{r{\mathrm{s}}}$, and $b_{\phi{\mathrm{s}}}$; these terms contain the radial derivative of the leading-order angular velocity which is assumed to be Keplerian (hence the notation), as well as a term describing the vertical dependence of the gravitational potential and other geometrical terms coming from differentiating factors depending on $r$ in the viscous term and the radial pressure gradient. Furthermore, we define $$\bmath{X}_\mathrm{hyd} = \bmath{X}_\mathrm{K} + \bmath{X}_{DH},$$ being the solution corresponding to the hydrodynamic source terms with the standard parameters $D_H=1$ (relevant to a disc with a constant aspect ratio $H/r$) and $D_{\nu\Sigma}=0$ (relevant to a steady accretion disc far from the inner boundary). For these parameters, we thus have $$\bmath{X} = \bmath{X}_\mathrm{hyd} + \bmath{X}_{DB}D_B + \bmath{X}_{br{\mathrm{s}}}b_{r{\mathrm{s}}} + \bmath{X}_{b\phi{\mathrm{s}}}b_{\phi{\mathrm{s}}}.$$
The linearity of the problem makes the exploration of the parameter space and the physical understanding of the solutions much easier. Indeed, given the marginal stability hypothesis, the solution depends in a nonlinear way only on the two parameters $\beta_0$ and ${\mathcal{P}}$. For each pair of values of these two parameters, one needs to compute the six solution vectors $\bmath{X}_\mathrm{K}$, $\bmath{X}_{DH}$, $
\bmath{X}_{D\nu\Sigma}$, $\bmath{X}_{DB}$, $\bmath{X}_{br{\mathrm{s}}}$ and $\bmath{X}_{b\phi{\mathrm{s}}}$, each of which can be represented by plotting the profiles of $u_r$, $u_\phi$, $b_r$ and $b_\phi$. The general solution is then just a linear combination of these solution vectors with the appropriate coefficients.
Note that another input to the model is the vertical profile of the diffusion coefficients, the shape of which can be freely imposed (although its normalisation is determined by the marginal stability hypothesis). In Section \[sec:cstalpha\], we study in detail the simplest case of a uniform resistivity and viscosity. The effect of the vertical structure of the diffusion coefficients will be considered in paper II.
Method of numerical solution
----------------------------
We solve the problem described in the previous subsections with the use of two different methods: the shooting method, and a (more successful) spectral method using a decomposition on a basis of Whittaker cardinal functions, which are well suited to problems on an infinite interval [e.g. @boyd01; @latter10]. In the shooting method, we shoot from the midplane by guessing the values of $u_r$, $u_\phi$, ${\partial}_\zeta
b_r$, and ${\partial}_\zeta b_\phi$ there (equations \[eq:midplane\_ur\]–\[eq:midplane\_bphi\]). The differential system given by equations (\[eq:motion\_r\])–(\[eq:induction\_phi\]) is then used to evolve this solution up to a height $\zeta_{\rm num}$, where the boundary conditions at infinity are applied. Newton iteration is used to converge to the desired solution. $\zeta_{\rm num}$ should be chosen large enough to lie in the force-free regime, in which case the solution is independent of $\zeta_{\rm num}$. We found that choosing $\zeta_{\rm num}$ typically one or two scale-heights above $\zeta_B$ was sufficient for this purpose. Similarly, the ‘grid’ of the spectral method should extend up to several scale-heights above $\zeta_B$ to obtain converged results. The decomposition on Whittaker cardinal functions implicitly imposes the condition that the variables tend to zero exponentially fast at infinity. For this reason, in the numerical calculation using the spectral method we replace $b_r$ and $b_\phi$ by the following variables: $$\tilde{b}_r \equiv b_r - D_B\zeta - b_{r{\mathrm{s}}}\tanh(\zeta^3),$$ $$\tilde{b}_\phi \equiv b_\phi - b_{\phi{\mathrm{s}}}\tanh(\zeta^3),$$ which should vanish exponentially fast at infinity according to the boundary conditions stated in equations (\[eq:boundary\_br\])–(\[eq:boundary\_bphi\]). The differential system had to be changed accordingly. We obtained very good agreement between the two methods (Figure \[fig:simu\]).
We also compared the results (with no source terms except $b_{r{\mathrm{s}}}$) with one-dimensional time-dependent direct numerical simulations of a stratified shearing box in which the value of $B_r$ is imposed at the vertical boundaries. The simulations were performed with the code RAMSES [@teyssier02; @fromang06]. The comparison allows us to evaluate the limitation due to our assumption that $|B_r| \ll |B_z|$ (Figure \[fig:simu\]). Fortunately even when the field is significantly inclined ($B_r =0.5 B_z$, just below the inclination threshold above which a magnetocentrifugal jet is launched), the velocity and magnetic field profiles remain very close to those obtained under the assumption of a small inclination. This is observed to be true as long as $\beta_0$ is rather large. For $\beta_0$ values of order unity magnetic compression (neglected here) would be expected to play a significant role if the inclination is large.
The results described in the remainder of this paper were obtained with the spectral method, which gave a good convergence on a broader parameter space than the shooting method.
The case of uniform diffusion coefficients {#sec:cstalpha}
==========================================
The formalism developed in Sections \[sec:asymptotic\] and \[sec:local\] allows in principle for a vertical dependence of the diffusion coefficients. However, in the remainder of this paper we focus on the case of uniform diffusion coefficients. This simple, but probably unrealistic, model is a first step and has the advantage of allowing for some analytical treatment. More general models involving non-uniform diffusion coefficients are studied in paper II. In the remainder of this section, unless otherwise noted, the numerical calculations use our fiducial parameters ${\mathcal{P}}=1$ and $\beta_0=10^4$.
Approximate analytical model {#sec:anal}
----------------------------
Before discussing the numerical solutions of our problem, we develop an approximate analytical model that is helpful in their interpretation. The magnetic field and velocity profiles can be easily understood in the two limiting cases of weak (passive) magnetic field ($\beta \gg 1$), and very strong (force-free) magnetic field ($\beta \ll 1$). In the case where the magnetic pressure is small compared to the midplane pressure ($\beta_0 \gg 1$), both regimes appear: close to the midplane the field is passive, while at $\zeta \rightarrow \infty$ the field is force-free. The transition between the two regimes takes place around $\zeta_B$ where the magnetic pressure is comparable to the thermal pressure ($\zeta_B$ is given by equation (\[eq:zetab\])). In the following two subsections we describe the two different regimes. In a third subsection we construct a simple model that connects the two regions to obtain an analytical estimate of the profiles, and in particular of the transport velocity of the magnetic flux.
### Passive magnetic field {#sec:passive_field}
In the limit of very large $\beta$, the Lorentz force is negligible and the velocity profile is unaffected by the magnetic field. If the kinematic viscosity is uniform, the purely hydrodynamic velocity profiles can be computed analytically and are parabolic: $$\begin{aligned}
u_r &=& u_{r0} + u_{r2}\zeta^2, \label{eq:ur_hydro} \\
u_\phi &=& u_{\phi0} + u_{\phi2}\zeta^2, \label{eq:uphi_hydro} \end{aligned}$$ with the following coefficients: $$\begin{aligned}
u_{r0} &=& \alpha\left(-\frac{9}{2} + 5D_H - 3D_{\nu\Sigma}\right) + \frac{4\alpha^3}{1+4\alpha^2}\left(3-5D_H\right), \nonumber\\ \\
u_{r2} &=& \frac{\alpha}{1+4\alpha^2}\left(3-5D_H\right), \\
u_{\phi0} &=& -{\frac}{1}{2}\left(D_H - D_{\nu\Sigma} + \frac{3}{2}\right) - \frac{\alpha^2}{1+4\alpha^2}\left(3-5D_H\right), \nonumber\\ \\
u_{\phi2} &=& {\frac}{1}{2}\left(D_H - \frac{3}{2}\right) + \frac{\alpha^2}{1+4\alpha^2}\left(3-5D_H\right). \label{eq:uphi2}\end{aligned}$$
The magnetic field profiles correspond then to a situation where the stretching of the magnetic field lines by the (imposed) velocity field is compensated by the diffusion following equations (\[eq:induction\_r\])–(\[eq:induction\_phi\]). These can be solved if some boundary conditions can be applied, which is not obvious as the region where the magnetic field is passive does not extend to infinity (see Section \[sec:anal\_model\] for a simple model). After two successive integrations, the radial induction equation gives $$b_r(\zeta) = b_{r1}\zeta -\frac{{\mathcal{P}}}{\alpha}\frac{u_{r2}}{3}\zeta^3 , \label{eq:br_passive}$$ where $b_{r1}$ is some unknown constant to be determined by the boundary conditions, and we used $b_r(\zeta=0) = 0$ to remove one integration constant. Similarly, the azimuthal induction equation gives $$b_\phi(\zeta) = b_{\phi1}\zeta + \frac{{\mathcal{P}}}{\alpha}\left\lbrack \left(\frac{b_{r1}}{4}- \frac{u_{\phi2}}{3}\right)\zeta^3 -
\frac{{\mathcal{P}}}{\alpha}\frac{u_{r2}}{40}\zeta^5 \right\rbrack, \label{eq:bphi_passive}$$ where $b_{\phi1}$ is to be determined by boundary conditions.
### Force-free magnetic field {#sec:force_free}
When $\beta \ll 1$, nothing can compensate the Lorentz force, so the magnetic field has to be force-free: the current is parallel to the magnetic field lines. With our assumption that the field is almost vertical, this means that the radial and azimuthal currents vanish: $$\begin{aligned}
{\partial}_\zeta b_\phi &=& 0, \\
{\partial}_\zeta b_r - D_B &=& 0.\end{aligned}$$ Applying the boundary conditions at infinity, we find that they are satisfied anywhere in the force-free region: $$\begin{aligned}
b_\phi &=& \pm b_{\phi{\mathrm{s}}}, \\
b_r &=& \pm b_{r{\mathrm{s}}} + D_B\zeta,\end{aligned}$$ where $\pm$ stands for $\mathrm{sgn}(\zeta)$.
Because the current vanishes, the magnetic field cannot diffuse: the velocity is dictated by the fact that the fluid is frozen into the magnetic field lines. In particular, isorotation is enforced along the magnetic field lines, determining the azimuthal velocity to be $$u_\phi = \frac{3}{2}\left(b_{r{\mathrm{s}}}|\zeta| + \frac{D_B}{2}\zeta^2 \right) + u_{\phi0}^{\prime},$$ where $u_{\phi0}^{\prime}$ is again a constant to be determined by boundary conditions. The radial velocity is constant and equals the speed $u_\psi$ at which the magnetic field is being advected or diffused.
### Two-zone model {#sec:anal_model}
In this subsection, we build an approximate model of the vertical profiles of velocity and magnetic field by assuming that for $\zeta < \zeta_B$ they behave as described in Section \[sec:passive\_field\] (passive field) and for $\zeta >
\zeta_B$ they behave as in Section \[sec:force\_free\] (force-free field). By doing so we neglect the thickness of the transition where $\beta$ is of order unity. To build the model, we need to connect the two regions of passive field ($\zeta < \zeta_B$) and force-free field ($\zeta > \zeta_B $), thus determining the proper boundary conditions at $\zeta = \zeta_B$ (the conditions at $\zeta=-\zeta_B$ then being given simply by reflectional symmetry). Four conditions are needed to constrain the four unknowns $b_{r1}$, $b_{\phi1}$, $u_{\phi0}^\prime$ and $u_\psi$.
Two boundary conditions can be obtained from the analysis of the induction equation. The radial component states that the azimuthal electric field is independent of the height $\zeta$ (equation \[upsi\]). Using equations (\[eq:ur\_hydro\]) and (\[eq:br\_passive\]) at $\zeta = 0$, we obtain a first condition: $$u_{\psi} = \frac{\alpha}{{\mathcal{P}}}\left( b_{r1} - D_B\right) + u_{r0}. \label{eq:const_ur}$$
The azimuthal component of the induction equation is more complicated because $b_r$ acts as a source term in this equation. By integrating between two heights $\zeta_1$ and $\zeta_2$, we get the relation $$\left\lbrack \frac{\alpha}{{\mathcal{P}}}{\partial}_\zeta b_\phi + u_\phi \right\rbrack_{\zeta_1}^{\zeta_2}
= \frac{3}{2}\int_{\zeta_1}^{\zeta_2}b_r \, {{\rm d}}\zeta.$$ We can use this relation between $\zeta_B^-$ and $\zeta_B^+$ to connect the passive-field region and the force-free region. As mentioned above, we neglect the width of the intermediate region, so that the right-hand side of the equation vanishes. Then $$u_\phi(\zeta_B^+) - u_\phi(\zeta_B^-) = \frac{\alpha}{{\mathcal{P}}}{\partial}_\zeta
b_\phi(\zeta_B^-), \label{eq:const_uphi}$$ where we have used the force-free condition ${\partial}_\zeta b_\phi(\zeta_B^+)=0$. This boundary condition corresponds to a continuous radial electric field across $\zeta_B$.
Finally, two more conditions are needed in order to constrain the model. These should come from the two components of the equation of motion. Integrating this equation across the intermediate region is not straightforward as it contains terms proportional to $\tilde\rho\beta_0$, which vary under our simple model from infinity in the passive-field region to zero in the force-free region. To build a simple model, we first make the simplest assumption that the magnetic field does not vary significantly at the transition between the two regions: $$\begin{aligned}
b_r (\zeta_B^-)&=& b_r (\zeta_B^+)= b_{r{\mathrm{s}}} + D_B \zeta_B \label{eq:const_br} \\
b_\phi (\zeta_B^-)&=& b_\phi (\zeta_B^+)= b_{\phi{\mathrm{s}}} \label{eq:const_bphi}\end{aligned}$$ Although not well motivated theoretically, this assumption might seem intuitively consistent with the assumption of an infinitely thin transition. As shown in the next subsections, it is rather well verified for the azimuthal field but not so well for the radial field. Indeed, the profile of the radial component of the magnetic field can show a significant variation around $\zeta_B$, which is not reproduced by the above description. The comparison with numerical calculations described in the next few subsections, however shows that this model is a useful approximate description, which reproduces most relevant effects. A better description of the boundary conditions is developed in Appendix A for the special case of vanishing viscosity, and heuristically generalised in Appendix B to non-vanishing viscosity in order to obtain more precise analytical expressions in a two-zone model.
Magnetic field diffusion ($b_{r{\mathrm{s}}}$ and $D_B$) {#sec:diffusion}
--------------------------------------------------------
We recall that our problem is linear and that the full solution of the problem consists of a superposition of the response of the system to various source terms (see Section \[sec:parameters\]). We consider first the source terms $b_{r{\mathrm{s}}}$ and $D_B$, which cause the magnetic field to bend and diffuse through the disc. The profiles computed numerically (including only the source terms $b_{r{\mathrm{s}}}$ and $D_B$) are illustrated in Figures \[fig:cstalpha\_brs0\] and \[fig:cstalpha\_DB0\] for three different strengths of the magnetic field. The quantities that are plotted are the dimensionless magnetic field and velocity variables as defined in Section \[sec:system\]. Note that in each case ($b_{r{\mathrm{s}}}=1$ and $D_B=1$) the poloidal magnetic field bends outwards.
The transition between the passive-field region close to the midplane and the force-free region at $\zeta>\zeta_B$ is clearly visible for the weak fields ($\beta_0 =10^4$ and $10^8$) as the height above which the velocity departs from zero and the magnetic field tends to the values dictated by the boundary conditions at infinity. Note that in the case of $\beta_0=10$ the magnetic field is sufficiently strong so that it does not really behave as a passive field at the midplane. As expected from Section \[sec:passive\_field\], in the passive-field region, the velocity vanishes while the radial component of the magnetic field has a linear profile and the azimuthal component looks like a third-order polynomial. In the force-free region, the azimuthal magnetic field vanishes and the radial component is either a constant ($b_{r{\mathrm{s}}}$, Figure \[fig:cstalpha\_brs0\]) or linear ($D_B\zeta$, Figure \[fig:cstalpha\_DB0\]) as required by the boundary conditions at infinity. Also as expected, the radial velocity is constant and equals the transport velocity of the magnetic flux, while the azimuthal velocity is either linearly increasing or parabolic.
Let us now use the two-zone model of Section \[sec:anal\] to derive approximate vertical profiles. As already mentioned, when only these source terms are included, the velocity vanishes in the passive-field region. The radial magnetic field profile in this region is then linear and can be obtained using equation (\[eq:const\_br\]): $$b_r = b_{r1}\zeta = \left(\frac{b_{r{\mathrm{s}}}}{\zeta_B} + D_B \right)\zeta.$$ Using equation (\[eq:const\_bphi\]) (with $b_{\phi{\mathrm{s}}}=0$), one can obtain the azimuthal magnetic field profile arising from the stretching of this radial field: $$b_\phi = \frac{1}{4}\frac{{\mathcal{P}}}{\alpha}\left(\frac{b_{r{\mathrm{s}}}}{\zeta_B} + D_B \right)\zeta(\zeta^2-\zeta_B^2).$$ The transport velocity of the magnetic flux is then obtained from equation (\[eq:const\_ur\]): $$u_{\psi} = \frac{\alpha}{{\mathcal{P}}} \frac{b_{r{\mathrm{s}}}}{\zeta_B},
\label{eq:uflux_mag}$$ and the azimuthal velocity profile in the force-free region is obtained using equation (\[eq:const\_uphi\]): $$u_\phi = b_{r{\mathrm{s}}}\left(\frac{3}{2}|\zeta| - \zeta_B \right) + \frac{D_B}{4}\left(3\zeta^2-\zeta_B^2 \right).$$
At the transition between the two regions, the radial and azimuthal components of the velocity show a jump qualitatively consistent with the boundary conditions of the approximate model (equations \[eq:const\_ur\]–\[eq:const\_uphi\]). As assumed in the model, the azimuthal component of the magnetic field goes smoothly to zero at $\zeta_B$; however, the radial component has a behaviour that is not described by the simple boundary conditions of Section \[sec:anal\]. Instead of smoothly increasing to its final value at infinity, the radial magnetic field reaches a maximum and then decreases to its value at infinity. This can be qualitatively understood with the following argument. At $\zeta\sim\zeta_B$, the azimuthal velocity has to increase and take positive (i.e. super-Keplerian) values, due to the requirement that the radial electric current be continuous (equation \[eq:const\_uphi\]). This requires a radial Lorentz force directed inward, which corresponds to a radial magnetic field decreasing with $\zeta$. Given that in the passive-field region the radial component of the magnetic field is increasing, it has to show a maximum at the transition around $\zeta\sim\zeta_B$. A quantitative description of this effect is given in Appendix A and B.
To go further in the comparison between the approximate analytical model and the numerically computed profiles, we remark that the dependence on the magnetic Prandtl number and the magnetic field strength can be scaled out of the analytical expressions in the following way (here for the source term $b_{r{\mathrm{s}}}$ only): $$\begin{aligned}
b_r(\zeta<\zeta_B) &=& b_{r{\mathrm{s}}}\frac{\zeta}{\zeta_B}, \label{eq:scaled_br} \\
\frac{\alpha}{{\mathcal{P}}\zeta_B^2} b_\phi(\zeta < \zeta_B) &=&
\frac{b_{r{\mathrm{s}}}}{4}\frac{\zeta}{\zeta_B}\left(\frac{\zeta^2}{\zeta_B^2}-1
\right), \label{eq:scaled_bphi} \\
\frac{{\mathcal{P}}\zeta_B}{\alpha}u_r(\zeta> \zeta_B) &=& b_{r{\mathrm{s}}}, \label{eq:scaled_ur} \\
\frac{1}{\zeta_B}u_\phi(\zeta>\zeta_B) &=&
b_{r{\mathrm{s}}}\left(\frac{3}{2}\frac{|\zeta|}{\zeta_B}-1 \right). \label{eq:scaled_uphi}\end{aligned}$$ In Figures \[fig:cstalpha\_brs\_scaled\] and \[fig:cstalpha\_brs\_scaled\_pm\] we show the scaled profiles (i.e. of $b_r$, $\alpha b_\phi/({\mathcal{P}}\zeta_B^2)$, ${\mathcal{P}}\zeta_B u_r/\alpha$, and $u_\phi/\zeta_B$) as a function of $\zeta/\zeta_B$ for different values of the magnetic field strength (Figure \[fig:cstalpha\_brs\_scaled\]) and of the magnetic Prandtl number (Figure \[fig:cstalpha\_brs\_scaled\_pm\]). The model suggests that all these curves should lie on top of each other, and coincide with the dashed curves that represent the analytical profiles given by equations (\[eq:scaled\_br\])–(\[eq:scaled\_uphi\]). The different profiles do indeed lie very close to each other (except maybe for the $\beta_0=100$ case, where the passive-field hypothesis at the midplane is less well verified) showing that the dependences on the magnetic field strength and the magnetic Prandtl number have been properly scaled out.
One exception to this agreement, however, is the bump in radial velocity close to $\zeta=\zeta_B$, which is more and more pronounced as $\beta_0$ increases. The two-zone model fails to reproduce this feature as it takes place in the intermediate region where the magnetic field is neither negligible nor dominant. Furthermore, note that the agreement between the numerical profiles and the analytical ones is not perfect. All the differences can actually be traced back to the profile of radial magnetic field, which goes through a maximum at $\zeta\sim\zeta_B$ as discussed earlier. The presence of the maximum leads to a larger radial field at $\zeta=\zeta_B^-$ than assumed in the model. The bump in the radial velocity is due to the presence of the maximum in the radial magnetic field, together with the requirement that the azimuthal electric field is uniform. The improved boundary conditions described in the Appendix B yield a much better agreement between the numerical profiles and analytical ones (shown with dot-dashed lines in Figures \[fig:cstalpha\_DB0\], \[fig:cstalpha\_brs\_scaled\] and \[fig:cstalpha\_brs\_scaled\_pm\]).
Figure \[fig:cstalpha\_uflux\_beta\] shows the dependence of the transport velocity of the magnetic flux on the strength of the vertical field (upper left and right panels for $b_{r{\mathrm{s}}}$ and $D_B$ respectively). The prediction of the analytical model agrees well with the numerical calculation for $\beta_0 > 10^3-10^4$, especially when using the improved boundary conditions of Appendix B. We recall that for a thin disc and a significant inclination of the surface magnetic field, the dominant contribution to the diffusion is that of the radial field $u_{br{\mathrm{s}}}$ (which is larger than $u_{DB}$ by a factor $\sim r/H$). It is interesting to note that this diffusion process is less efficient for weak magnetic fields. Indeed for the parameters explored here, the diffusion is up to a factor of 4 slower than the crude estimate stated in the introduction (equation (\[eq:vdiff\]), which corresponds to $u_{br{\mathrm{s}}}=1$ in Figure \[fig:cstalpha\_uflux\_beta\]). This slower diffusion is a consequence of the fact that field lines bend on a length-scale $\zeta_B$, which is larger for weak fields.
Transport due to angular momentum loss in an outflow ($b_{\phi{\mathrm{s}}}$) {#sec:outflow}
-----------------------------------------------------------------------------
Applying the two-zone model with the simple boundary conditions of Section \[sec:anal\] predicts that no advection is induced by the source term $b_{\phi{\mathrm{s}}}$. Indeed, in the passive-field region the velocity vanishes, as does the radial magnetic field. Alone, the azimuthal magnetic field has a linear dependence on $\zeta$: $$b_\phi = b_{\phi{\mathrm{s}}}\frac{\zeta}{\zeta_B}
\label{eq:bphis_bphi}$$ The transport of magnetic flux vanishes: $u_\psi=0$, while the azimuthal velocity jumps around $\zeta_B$ to its final value: $$u_\phi = \frac{\alpha}{{\mathcal{P}}}\frac{b_{\phi{\mathrm{s}}}}{\zeta_B}
\label{eq:bphis_uphi}$$
This is not the whole story however, because two effects have not been taken into account. First, the jump in radial magnetic field around $\zeta_B$ (Figure \[fig:cstalpha\_bphis\]) induces some transport of magnetic flux. This is described quantitatively in Appendix B, and the analytical prediction for the transport velocity of magnetic flux is compared to numerical results in Figure \[fig:cstalpha\_uflux\_beta\]. The agreement is very good for rather weak fields ($\beta_0>10^2$). For larger values of the magnetic field, another effect neglected in the two-zone model becomes important. Indeed, the magnetic force cannot be neglected anymore and the torque exerted by the azimuthal field induces a non-negligible radial velocity, which contributes to the transport of magnetic flux. For $\beta_0<100$, the transport velocity of magnetic flux is then very close to that of mass (given by equation (\[um\])).
As might have been expected, the efficiency of the surface azimuthal magnetic field to transport mass and magnetic flux strongly increases as the magnetic field strength is increased (Figure \[fig:cstalpha\_uflux\_beta\]). It is interesting to note however that the transport velocity of magnetic flux is much larger than that of mass, when the magnetic field is rather weak.
Advection due to the turbulent viscosity {#sec:hydro}
----------------------------------------
All the remaining source terms ($D_{\nu\Sigma}$, $D_H$ and those which solution vector is $\mathbf{X}_K$) are hydrodynamical in nature and describe the advection process due the turbulent viscosity. First, we note that the source term $D_{\nu\Sigma}$ admits an exact solution with uniform radial and azimuthal velocities and vanishing radial and azimuthal magnetic fields: $$\begin{aligned}
b_r &=&0, \\
b_\phi &=&0, \\
u_r &=&-3\alpha D_{\nu\Sigma}, \\
u_\phi &=& \frac{D_{\nu\Sigma}}{2}.\end{aligned}$$
We now focus on the other hydrodynamical source terms, and their solution in the approximate analytical model. In the passive-field region the velocity profiles are the parabolae given by equations (\[eq:ur\_hydro\])–(\[eq:uphi2\]) (note that the parabolic shape is clear in Figure \[fig:cstalpha\_hydro0\]). The magnetic field profiles result from the stretching of the magnetic field lines by the parabolic velocity profile, and from the condition that both radial and azimuthal components vanish at $\zeta=\zeta_B$ (for the simple boundary conditions of Section \[sec:anal\]): $$b_r = \frac{{\mathcal{P}}}{\alpha}\frac{u_{r2}}{3}\zeta(\zeta_B^2-\zeta^2),$$ $$b_\phi = \frac{{\mathcal{P}}}{\alpha}(\zeta_B^2-\zeta^2)\left\lbrack\frac{u_{\phi2}}{3}\zeta +
\frac{u_{r2}}{120}\frac{{\mathcal{P}}}{\alpha}\zeta(3\zeta^2-7\zeta_B^2) \right\rbrack.$$ Then the velocities in the force-free region and, most importantly, the advection speed of the magnetic flux are found using equations (\[eq:const\_ur\]) and (\[eq:const\_uphi\]): $$u_{\psi} = u_{r0} + \frac{u_{r2}}{3}\zeta_B^2, \label{eq:uflux_hydro}$$ $$u_\phi(\zeta>\zeta_B) = u_{\phi0} + \frac{u_{\phi2}}{3}\zeta_B^2 + \frac{u_{r2}}{15}\frac{{\mathcal{P}}}{\alpha}\zeta_B^4.$$ These profiles are compared with the numerical solution in Figure \[fig:cstalpha\_hydro0\] for the case $\beta_0=10^8$ (the analytical profile is the dashed line). The agreement is acceptable though the magnetic profiles are slightly underestimated. Again, a quantitatively better agreement is found when using the improved boundary conditions of Appendix B (dot-dashed lines).
Rewriting the transport velocity of the magnetic flux as a function of the source terms, one finds $$\frac{u_{\psi}}{\alpha} = -\frac{9}{2} + 5D_H - 3D_{\nu\Sigma} +
\frac{3-5D_H}{4\alpha^2+1}\left(4\alpha^2 + \frac{\zeta_B^2}{3}
\right). \label{eq:uflux_hydro_bis}$$ This shows that the inward transport is faster for weak fields (i.e. large $\zeta_B$) if $D_H>0.6$. It is for example the case for our fiducial values of $D_H=1$ and $D_{\nu\Sigma}=0$: $$u_{\psi,\mathrm{hyd}} = \alpha\left\lbrack \frac{1}{2} - \frac{2}{4\alpha^2+1}\left(4\alpha^2+\frac{\zeta_B^2}{3} \right) \right\rbrack.
\label{eq:uflux_hyd}$$ The lower left panel of Figure \[fig:cstalpha\_uflux\_beta\] compares the transport velocity of the magnetic flux $u_\mathrm{hyd}$ (full black line) with equation (\[eq:uflux\_hyd\]) (dashed line) and the transport velocity of mass (red line). Interestingly, the transport velocities of mass and magnetic flux coincide for strong fields $\beta_0<10$, but for weak fields the advection of magnetic flux is much faster than that of mass (by a factor up to 10 for the present parameters). This behaviour is correctly reproduced by the approximate analytical model, and its interpretation will be discussed in the following subsection.
Advection/diffusion speed as a vertical average {#sec:uflux_average}
-----------------------------------------------
The results of Sections \[sec:diffusion\], \[sec:outflow\] and \[sec:hydro\] show that the transport of magnetic flux is significantly affected by the vertical structure, in the sense that it differs from the simple vertical average used by many previous works such as @lubow94a. This can be understood by noting that the transport velocity of magnetic flux, contrary to that of mass, is not given by a density weighted average. Below, we repeat with our notation the argument given by @ogilvie01 on the correct way of averaging the equations in order to obtain the effective transport velocities of mass and magnetic flux. We then use it to interpret our results.
Dividing equation (\[upsi\]) by the dimensionless resistivity $\alpha/{\mathcal{P}}$, and integrating between $\zeta=0$ and $\zeta$, we obtain: $$u_{\psi} = \bar{\eta}\left(\frac{b_r}{\zeta} - D_B \right) + \bar{u}_r,
\label{eq:upsi_average}$$ where $\bar{\eta}$ is an average dimensionless resistivity defined in the following way (it is actually the inverse of the height-averaged conductivity): $$\bar{\eta} = \frac{\zeta}{\int_0^\zeta \frac{{\mathcal{P}}}{\alpha}\, {{\rm d}}\zeta^{\prime}}, \label{eq:eta_average}$$ and $\bar{u}_r$ is the average of the radial velocity weighted by the conductivity ($1/\eta$): $$\bar{u}_r = \frac{1}{\int_0^\zeta \frac{{\mathcal{P}}}{\alpha}\, {{\rm d}}\zeta^{\prime}}
\int_0^\zeta \frac{{\mathcal{P}}}{\alpha} u_r\, {{\rm d}}\zeta^{\prime}. \label{eq:vadv_average}$$
This averaging procedure shows that the contribution of advection to the velocity at which the magnetic flux is transported is the vertically averaged radial velocity weighted by the electrical conductivity. This average may be very different from the density-weighted average $u_m$ that describes mass transport.
Note that equation (\[eq:upsi\_average\]) is valid for any height $\zeta$ up to which the averaging procedure is performed, and which can be chosen in the most convenient way. Integrating up to an infinite height may not be very useful as the force-free regime would dominate the average and one would simply recover $v_\psi=C_1$, as was already found in Section \[sec:asymptotic\]. Instead, the two-zone model developed in Section \[sec:anal\] suggests that performing the average up to the height $\zeta_B$ is most useful. By doing so, one can indeed recover the results for $ u_{\psi}$ of Sections \[sec:diffusion\] and \[sec:hydro\] in a simpler way: $$u_{\psi} = \frac{\alpha}{{\mathcal{P}}} \left(\frac{b_r(\zeta_B^-)}{\zeta_B} - D_B \right) + \bar{u}_r,
\label{eq:uflux_average_zetaB}$$ where $\bar{u}_r$ is the average advection velocity as defined by equation (\[eq:vadv\_average\]), performing the averaging procedure between $\zeta=0$ and $\zeta=\zeta_B^-$, and we used $\bar{\eta}=\alpha/{\mathcal{P}}$ since the diffusion coefficients are uniform. Following the two-zone modelling of Section \[sec:anal\], the average advection velocity can be computed using the purely hydrodynamical velocity profile (because the field is assumed to be passive for $\zeta<\zeta_B^-$): $$\bar{u}_r = \frac{1}{\zeta_B}\int_0^{\zeta_B} (u_{r0}+u_{r2}\zeta^2) \, {{\rm d}}\zeta = u_{r0} + \frac{u_{r2}}{3}\zeta_B^2.$$ $\bar{u}_r$ gives the value of the advection speed due to the hydrodynamic source terms, as described by equation (\[eq:uflux\_hydro\]). The value of $b_r$ at $\zeta_B^-$ then has to be prescribed by the boundary conditions. Using that of Section \[sec:anal\] gives $b_r(\zeta_B^-) = b_{r{\mathrm{s}}}+D_B\zeta_B$, and one recovers equation (\[eq:uflux\_mag\]). Using the boundary conditions of Appendix B is also possible in principle but it requires first solving for the full vertical profiles.
The above argument allows for a deeper interpretation of the higher advection speed of weak fields, which was observed in the previous section. Indeed, the mass transport velocity $u_m$ corresponds to a density-weighted average and is therefore dominated by the velocity close to the midplane, where the density is large. By contrast, the magnetic flux transport velocity $u_\psi$ is a conductivity-weighted average, which for the case of uniform diffusion coefficients is equivalent to a height-average. As a consequence the advection of a weak field is significantly affected by the low-density region away from the midplane. This low-density region at high $\zeta$ can have a much faster inward radial velocity than in the midplane, if $u_{r2}<0$ as is readily found for standard parameters. Such parabolic velocity profiles, where the radial velocity becomes more and more negative with the distance from the midplane, are well known solutions of viscous hydrodynamical disc models [e.g. @takeuchi02].
Inclination of the field lines {#sec:inclination}
------------------------------
![Radial inclination of the field lines at the surface of the disc obtained in a stationary situation. The upper panel shows the inclination ($B_{r{\mathrm{s}}}/B_z$) in units of $H/r$ such that the diffusion compensates the advection due to the turbulent viscosity (‘hydrodynamic’ source terms). The lower panel shows the ratio $-b_{r{\mathrm{s}}}/b_{\phi{\mathrm{s}}}$ such that diffusion compensates advection due to an outflow. In both panels, the full black line corresponds to the numerical results, while dashed and dash-dotted lines correspond to the analytical model respectively with the simple continuous magnetic field boundary conditions (Section \[sec:anal\]) and with the more precise boundary conditions of Appendix B. Note that in the lower panel the dashed line is not shown as it would equal 0.[]{data-label="fig:inclination"}](figures/figure12.ps){width="\columnwidth"}
As outlined in the introduction, one of the most important diagnostics of the relative efficiency of the magnetic flux advection and diffusion processes is the inclination of the field lines at the surface of the disc that can be achieved in a stationary situation. In such a stationary situation the diffusion velocity due to the inclination, $u_{br{\mathrm{s}}}b_{r{\mathrm{s}}}$ (recall that $b_{r{\mathrm{s}}}=r/H\times B_{r{\mathrm{s}}}/B_z$), compensates the advection velocity $u_\mathrm{hyd}$ (as an example, let us focus on the case $D_H=1$, $D_{\nu\Sigma}=0$). Figure \[fig:cstalpha\_stat\] illustrates the vertical profiles obtained in such a stationary situation. The inclination is then: $$\frac{B_{r{\mathrm{s}}}}{B_z} = -\frac{H}{r}\frac{u_\mathrm{hyd}}{u_{br{\mathrm{s}}}}.$$ In the limit of an infinitely thin disc, and if $u_\mathrm{hyd}$ and $u_{br{\mathrm{s}}}$ are of order unity, one recovers the classical result that the inclination is vanishingly small. The inclination of the field lines can be estimated with the approximate analytical model (here with the simple continuous magnetic field boundary condition of Section \[sec:anal\]): $$\frac{B_{rs}}{B_z} = \frac{H}{r}{\mathcal{P}}\zeta_B\left\lbrack -\frac{1}{2} +
\frac{2}{4\alpha^2+1}\left(4\alpha^2+\frac{\zeta_B^2}{3} \right) \right\rbrack.$$ In Figure \[fig:inclination\], this estimate (represented by dashed lines) is compared to the numerical result. The more precise estimate represented in dash-dotted line is given in Appendix B. As expected, the formula shows that the bending does not depend much on the value of $\alpha$ (at least if $\alpha^2\ll1$) but rather on the magnetic Prandtl number ${\mathcal{P}}$. In particular one recovers the result that a significant inclination can be achieved if ${\mathcal{P}}\sim r/H$ (but this is not expected to be true). The formula and associated figure however suggest a new way to achieve a large inclination: to be in the weak field regime where $\beta_0$ and $\zeta_B$ are large. Indeed, a significant inclination is possible for weak fields if the disc is not too thin. For example, if $H/r=0.1$ a field of $\beta_0\sim10^3$ can be significantly bent ($B_r\sim B_z$). This is due both to a reduced diffusion and a faster advection speed as compared to the rough estimates quoted in the introduction (as discussed in previous subsections).
Similarly the ratio of the radial and azimuthal components of the surface magnetic field can be estimated in a stationary situation where diffusion is compensated by the advection due to an outflow: $$\frac{B_{r{\mathrm{s}}}}{B_{\phi{\mathrm{s}}}} = - \frac{u_{b\phi{\mathrm{s}}}}{u_{br{\mathrm{s}}}}.
\label{eq:inc_bphis}$$ (Recall that $B_{\phi{\mathrm{s}}}$ is negative so that the outflow removes angular momentum from the disc). This ratio is represented in the lower panel of Figure \[fig:inclination\] and compared with the estimate of the analytical model given in Appendix B. As might be expected, the advection due the outflow is more efficient at bending the magnetic field for strong magnetic fields.
In order to relate the above ratio to a radial inclination of the field with respect to the vertical, one would need to know the value of the azimuthal magnetic field at the surface, which determines the efficiency with which the outflow removes angular momentum. This would require a study of the launching and acceleration of the outflow, which is beyond the scope of this paper. We therefore keep it as a free parameter and obtain the following expression for the inclination: $$\frac{B_{r{\mathrm{s}}}}{B_z} = -\frac{B_{\phi{\mathrm{s}}}}{B_{z}} \frac{u_{b\phi{\mathrm{s}}}}{u_{br{\mathrm{s}}}}.$$ For a given value of $B_{\phi{\mathrm{s}}}/B_{z}$, Figure \[fig:inclination\] suggests that only magnetic fields above a certain strength can sustain a significant inclination of the field lines, if the advection is due to an outflow. For $B_{\phi{\mathrm{s}}}/B_{z} = -1$, the threshold is $\beta_0\sim10$, while for $B_{\phi{\mathrm{s}}}/B_{z} = -10$ it reaches $\beta_0\sim10^3$. Although magneto-centrifugal acceleration is usually thought to produce an outflow if the magnetic field is strong ($\beta_0$ of order or slightly above unity), we note that outflows with weaker fields may be driven by discs with superheated surface layers. Indeed, @murphy10 have recently reported the production of a magneto-centrifugal outflow from a weakly magnetised disc ($\beta_0 \sim 100-1000$). This opens the possibility that an outflow from a weakly magnetised disc could significantly affect the transport of magnetic flux.
A scenario to obtain strong inclined magnetic fields {#sec:scenario}
----------------------------------------------------
![Upper panel: maximum strength of the magnetic field (i.e. minimum $\beta_0$) for which an inclination of $30\degree$ of the surface magnetic field can be sustained, if the advection is due to the turbulent viscosity. This maximum strength is plotted as a function of the aspect ratio of the disc $H/r$. Bottom panel: minimum strength of the magnetic field (i.e. maximum $\beta_0$) for which an inclination of $30\degree$ of the surface magnetic field can be sustained, if the magnetic flux advection is due to an outflow. This minimum strength is shown as a function of the azimuthal magnetic field component at the surface of the disc, parametrising the efficiency at which the outflow extracts angular momentum from the disc.[]{data-label="fig:beta_stationary"}](figures/figure13.ps){width="\columnwidth"}
![Aspect ratio above which a disc can accumulate magnetic flux and create strong magnetic fields in the scenario outlined in Section \[sec:scenario\]. This aspect ratio depends strongly on the efficiency at which an outflow removes angular momentum, parameterised by $-B_{\phi {\mathrm{s}}}/B_z$.[]{data-label="fig:aspect_ratio"}](figures/figure14.ps){width="\columnwidth"}
The above discussion suggests a scenario by which strong and inclined magnetic fields could be obtained in a moderately thin disc through the inward advection of an initially weak magnetic field. In such a disc, the advection of a weak field by the accretion flow due to the turbulent viscosity is possible up to a certain strength, which depends strongly on the aspect ratio of the disc (Figure \[fig:beta\_stationary\], upper panel). Unless the disc is extremely thick, this strength is however rather weak: for example $\beta_0\sim10^3$ is reached for an already rather thick disc with $H/r > 0.07$. Creating and sustaining a stronger inclined field might then be possible if the advection of the magnetic flux is primarily due to an outflow[^2], which is possible above a certain strength of the magnetic field. The minimum strength allowing the outflow to compensate diffusion is shown as a function of $B_{\phi{\mathrm{s}}}/B_z$ (parameterising the ability of an outflow to remove angular momentum from the disc) in Figure \[fig:beta\_stationary\] (lower panel).
A condition for this scenario to work is that the maximum strength for advection through turbulence should be larger than the minimum strength for advection through an outflow. This requirement is fulfilled above a certain aspect ratio, shown as a function of $B_{\phi{\mathrm{s}}}/B_z$ in Figure \[fig:aspect\_ratio\]. This suggests that for reasonable values of the surface azimuthal field, the disc needs to be rather thick to enable the production of a strong magnetic field: $H/r > 0.2$ for $B_{\phi{\mathrm{s}}}/B_{z} = -1$, or $H/r>0.07$ for $B_{\phi{\mathrm{s}}}/B_{z} = -10$.
Obviously, these numerical values should be taken with a grain of salt for several reasons. The effect of an outflow has been taken into account in a quite approximate way. A proper description of this scenario would need to include the launching and acceleration process of the jet, which should determine self consistently the value of the azimuthal component of the magnetic field at the surface of the disc. Furthermore a variation of the diffusion coefficients in the vertical direction might be expected to change quantitatively the result, as will be studied in paper II. The turbulence may also not have such a simple effect as isotropic diffusion coefficients ; for example, the vertical diffusion may be expected to be less efficient than the radial one due to smaller vertical velocities in MRI turbulence. As a consequence, the numerical values for the minimum aspect ratio should not be considered as a strong constraint, and it might turn out that this scenario is effective for somewhat lower aspect ratio.
Conclusion and discussion {#sec:discussion}
=========================
We have developed a formalism enabling us to compute self-consistently the radial transport of poloidal magnetic flux and mass in a thin accretion disc, where the effect of turbulence is modelled by an effective viscosity and resistivity. For this purpose, we performed a systematic asymptotic expansion in the limit of a thin disc. We assumed that the magnetic field is only weakly inclined with respect to the vertical direction, in order to avoid the complication of modelling a magneto-centrifugal outflow, as well as for self-consistency in the expansion. Owing to this assumption, the solution (and thus the transport velocity of the magnetic flux) depends linearly on a number of source terms. These source terms are the radial gradients of various quantities (surface density, aspect ratio, magnetic field strength), and non-vanishing values of the radial and azimuthal magnetic field at the surface of the disc. Each of them describes a different physical process which can then be studied and understood independently from the others: the diffusion due the bending of the field lines across the disc ($b_{r{\mathrm{s}}}$), the diffusion due the radial gradient of its intensity ($D_B$), the advection due to angular momentum loss in an outflow ($b_{\phi{\mathrm{s}}}$), and that due the turbulent viscosity (‘hydrodynamic’ source terms).
An important aspect of this calculation is that it takes into account the vertical structure of the disc. The vertical profiles of velocity and magnetic field are indeed explicitly computed. The formalism also allows for a vertical dependence of the diffusion coefficients. In this paper, we have however only considered the idealised case of uniform coefficients, leaving the question of their vertical structure for paper II. Even in this simplest case, we found the vertical structure to have an important impact on the transport of magnetic flux. Indeed, the simple vertical averaging usually performed gives results that can be in error by a factor of 10 or more, because of its improper treatment of the vertical structure. We have developed an approximate analytical model which is able to describe this behaviour, and provides a physical explanation.
Two main effects of the vertical structure on the transport of magnetic flux have been uncovered by this study. Both of these effects are important when the poloidal magnetic field is rather weak in the sense that the magnetic pressure is small compared to the thermal pressure in the midplane. First, we showed that the diffusion of weak fields is less efficient because the bending of the field lines takes place on a larger vertical scale. Second, the advection of magnetic flux can be significantly faster than that of mass, owing to fast radial velocities in the low-density regions away from the midplane.
Taken together, these two effects enable a more efficient inward transport of magnetic flux, which partly alleviates the long-standing problem of too fast outward diffusion of an inclined magnetic field. Indeed, they suggest that in a moderately thin disc, magnetic fields below a certain strength can be advected inwards without significant outward diffusion. This critical strength depends steeply on the aspect ratio of the disc, however, and is probably rather weak for reasonably thin discs. We propose a scenario in which the advection due to the presence of an outflow could then do the rest of the job, and provide a means to obtain yet stronger magnetic fields. A condition for a successful advection is that the aspect ratio of the disc be larger than a critical value, which is dependent on the efficiency of the outflow to extract angular momentum from the disc. Confirming or invalidating this scenario will require a better understanding of the magnetic flux advection induced by an outflow with a weak magnetic field.
The faster advection of weak magnetic fields could also have interesting consequences on the time-dependence of magnetised accretion discs. Indeed, the magnetic flux distribution would have the counterintuitive ability to evolve significantly faster than the mass distribution. These two potential consequences should be studied in a global time-dependent disc model, where the inclination at the disc surface is computed self-consistently as a result of the distribution of magnetic flux. The time-evolution of the mass and magnetic flux could then be described using the transport rates computed with the formalism described in this paper (with the equations given in Section \[sec:asymptotic\]). This question is left for future work.
One limitation of our analysis is that it is restricted to small values of the inclination of the magnetic field with respect to the vertical direction. By comparing with direct numerical simulations of a stratified shearing box, we found that the small-inclination approximation gives good results even at significant inclination as long as an outflow is not launched ($i<30\degree$) and the field is not very strong ($\beta_0 \sim 10^2-10^4$). However, it is expected to break down at stronger fields because magnetic compression would become important, or at still larger inclination, in which case the launching process of an outflow should be included.
A related worry could come from the large values of $b_\phi$ obtained in some cases for weak magnetic fields (e.g. Figure \[fig:cstalpha\_hydro0\]). If the aspect ratio is only moderately small (e.g. $H/r \sim 10^{-2}-10^{-1}$) then the azimuthal component of the magnetic field may be larger than the vertical one. In addition to contradicting the initial assumption (in particular the neglect of compression), one might worry about the stability of such a solution.
We have underlined in this paper the importance of the vertical profile of radial velocity. [@Fromang11] have studied this profile by means of global MHD simulations and found that it differed from the prediction of viscous alpha models. This comes from the different vertical profile of the turbulent stress, compared to a viscous model with a viscosity independent of height (as was assumed in this paper). This shows the limitation of the idealised model studied in this paper. In paper II, we will consider a vertical profile of the viscosity in order to obtain a more realistic vertical profile of the stress. This study confirms the robustness of our main result: that the magnetic flux is advected faster than mass when the field is weak.
Both the faster advection and the reduced diffusion of weak magnetic fields rely on the dynamics of the low-density region away from the midplane. Turbulent MHD simulations both of stratified shearing boxes and global models show that this region tends to be strongly magnetised [@miller00; @fromang06b], and it is often called a ‘corona’. (Note that ‘strongly magnetised’ here refers to the fluctuating component of the magnetic field which has a pressure comparable to the thermal pressure, but this does not mean that the large-scale poloidal magnetic field is strong.) This article thus suggests that the dynamics of the corona and its connection with the transport of magnetic flux should be studied in more detail. It is unclear whether this corona should be modelled with an effective viscosity and resistivity like the main part of the disc, since some authors refer to this corona as non-turbulent due to its strong magnetisation. The vertical extension of the corona would also be of some interest for the diffusion process. Indeed, a more extended corona (because it is hot or because of magnetic support) could help reduce even more the diffusion of weak fields by increasing the length-scale on which the field lines are bent.
Finally we should stress that all this analysis relies on the assumption that the turbulence can be modelled by an effective resistivity and viscosity. It is not clear that this assumption is valid, especially for diffusive processes acting on a scale comparable to the scale-height. Indeed, in that case there might not be a real scale separation since the correlation length of the turbulence could be of the same order of magnitude. In principle then, the right method would be to perform turbulent numerical simulations that self-consistently describe the effect of turbulence. MHD simulations of a stratified disc containing a significant net vertical flux have however proved problematic so far. To avoid the further complication and the necessarily low resolution of global simulations, it would be useful to study the advection and diffusion of the magnetic flux in a local model. The classical stratified shearing box model would need some modification in order to take into account the relevant source terms. The inclusion of the non-vanishing horizontal component of the magnetic field at the surface of the disc could be achieved simply by changing the vertical boundary conditions, as has been done in this article for 1D non-turbulent simulations. A boundary condition that imposes the magnetic field may be problematic in the presence of turbulence however. The inclusion of other source terms describing the advection due to the effect of turbulence is much less obvious, but may deserve to be further investigated.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by STFC. We thank Henk Spruit, Henrik Latter, Clément Baruteau, and Sébastien Fromang for useful discussions. We also thank Sébastien Fromang for providing us with the stratified shearing-box implementation of RAMSES.
Appendix A : a description of the intermediate region in the case of zero viscosity {#sec:appendix_A .unnumbered}
===================================================================================
In this appendix we provide an analytical description of the intermediate region around $\zeta_B$ connecting the passive field and the force free field regimes. For this description to be tractable we assume this intermediate region to be thin, and we consider only the case of vanishing magnetic Prandtl number (i.e. zero viscosity $\alpha$, but finite resistivity $\alpha/{\mathcal{P}}$). We concentrate on the source terms $D_B$ and $b_{rs}$ in order to explain the bump observed in the profile of the radial component of the magnetic field (the hydrodynamical source terms with vanishing viscosity would not be very meaningful anyway).
The density profile around $\zeta_B$ is approximated by an exponential: $$\rho \simeq \frac{1}{2\beta_0}e^{-\zeta_Bx},$$ where we defined $x\equiv\zeta-\zeta_B$. This expansion of the density profile is valid if $x\ll \zeta_B$. Note that the intermediate region where $\beta \sim 1$ has a typical thickness of $x \sim 1/\zeta_B$, therefore the assumption of a thin transition is valid if $\zeta_B^2 \gg 1$. The differential system is then rewritten in the following way: $$\begin{aligned}
u_\phi + e^{\zeta_Bx}\left({\partial}_xb_r - D_B\right) & = & 0, \label{eq:app_uphi} \\
u_r - 4 e^{\zeta_Bx}{\partial}_xb_\phi & = & 0, \label{eq:app_ur} \\
{\partial}_x\left[\frac{\alpha}{{\mathcal{P}}}({\partial}_xb_r - D_B) + u_r \right] & = & 0, \label{eq:app_br} \\
{\partial}_x\left[\frac{\alpha}{{\mathcal{P}}}{\partial}_xb_\phi + u_\phi \right] & = & \frac{3}{2}b_r. \label{eq:app_bphi}\end{aligned}$$ The term on the right hand side of Equation (\[eq:app\_bphi\]) is furthermore neglected due to our assumption that the transition region is thin. Equations (\[eq:app\_br\])-(\[eq:app\_bphi\]) then simply state that the azimuthal and radial electric field are constant, and their value can be estimated at $\zeta_Bx \rightarrow\pm\infty$ (i.e. at $\zeta=\zeta_B^\pm$): this is consistent with the boundary conditions given by equations (\[eq:const\_ur\]) and (\[eq:const\_uphi\]). Thus $$\begin{aligned}
u_r &=& u_\psi -\frac{\alpha}{{\mathcal{P}}}({\partial}_xb_r-D_B), \\
u_\phi &=& \Delta u_\phi -\frac{\alpha}{{\mathcal{P}}}{\partial}_xb_\phi,\end{aligned}$$ where $u_\psi = u_r(\zeta_B^+)= \alpha/{\mathcal{P}}({\partial}_\zeta b_r(\zeta_B^-)-D_B)$ and $\Delta u_\phi = u_\phi(\zeta_B^+) = \alpha/{\mathcal{P}}{\partial}_\zeta b_\phi(\zeta_B^-)$. Introducing these relations into equations (\[eq:app\_uphi\])-(\[eq:app\_ur\]), one obtains the system of two equations: $$\begin{aligned}
\frac{\alpha}{{\mathcal{P}}}{\partial}_xb_\phi - e^{\zeta_Bx}({\partial}_x b_r-D_B) &=& \Delta u_\phi, \\
\frac{\alpha}{{\mathcal{P}}}({\partial}_xb_r-D_B) + 4 e^{\zeta_Bx} {\partial}_xb_\phi &=& u_\psi, \end{aligned}$$ which can be solved to obtain explicit functions for ${\partial}_x b_r$ and ${\partial}_x b_\phi$: $$\begin{aligned}
{\partial}_x b_r &=& D_B + \frac{\frac{\alpha}{{\mathcal{P}}}u_\psi - 4 e^{\zeta_Bx}\Delta u_\phi}{\frac{\alpha^2}{{\mathcal{P}}^2} + 4e^{2\zeta_Bx}}, \\
{\partial}_x b_\phi &=& \frac{\frac{\alpha}{{\mathcal{P}}}\Delta u_\phi + e^{\zeta_Bx} u_\psi}{\frac{\alpha^2}{{\mathcal{P}}^2} + 4e^{2\zeta_Bx}}.\end{aligned}$$ Integrating over $x$, one can compute the jump in $b_r$ across the intermediate region: $$b_r(x) - b_r(-x) \simeq - \frac{{\mathcal{P}}\pi}{\alpha}\frac{\Delta u_\phi}{\zeta_B} + \left[\zeta_Bx - \ln\left(\frac{2{\mathcal{P}}}{\alpha}\right)\right]\frac{{\mathcal{P}}}{\alpha}\frac{u_\psi}{\zeta_B} + 2D_Bx,$$ if $e^{\zeta_Bx} \gg 1$. The two terms proportional to $x$ represent the smooth variation in the passive field regime (from $-x$ to 0) and force free field regime (from 0 to $x$), and do not correspond to a jump in the transition region, thus $$\Delta b_r \equiv b_r(\zeta_B^+) - b_r(\zeta_B^-) = - \frac{{\mathcal{P}}\pi}{\alpha}\frac{\Delta u_\phi}{\zeta_B} - \ln\left(\frac{2{\mathcal{P}}}{\alpha}\right)\frac{{\mathcal{P}}}{\alpha}\frac{u_\psi}{\zeta_B}.$$
Similarly, one obtains the jump in $b_\phi$: $$\Delta b_\phi \equiv b_\phi(\zeta_B^+) - b_\phi(\zeta_B^-) = \frac{{\mathcal{P}}\pi}{4\alpha}\frac{u_\psi}{\zeta_B} - \ln\left(\frac{2{\mathcal{P}}}{\alpha}\right)\frac{{\mathcal{P}}}{\alpha}\frac{\Delta u_\phi}{\zeta_B}$$
These new boundary conditions can then be injected into the two-zone model. It turns out that, in the limit $\zeta_B^2\gg1$, only one term is significant, and the jump conditions can be simplified to $$\begin{aligned}
\Delta b_r &=& - \frac{{\mathcal{P}}\pi}{\alpha}\frac{\Delta u_\phi}{\zeta_B}, \label{eq:jump_br} \\
\Delta b_\phi &=& 0. \label{eq:jump_bphi} \end{aligned}$$ This comes from the fact that $b_r/b_\phi = O(1/\zeta_B^2)$. The two-zone model then gives the following slope of the profile of the radial component of the magnetic field: $$b_{r1} = \frac{1}{1-\frac{{\mathcal{P}}\pi}{2\alpha}}\left(\frac{b_{rs}}{\zeta_B} + D_B \right). \label{eq:app_br1}$$
Using the same two-zone description for the marginal antisymmetric MRI mode, one can compute the value of the magnetic diffusivity under the marginal stability hypothesis. For this purpose, one assumes the mode to be antisymmetric about the midplane (contrary to the calculation in Section \[sec:anal\]). We obtain $$\frac{\alpha}{{\mathcal{P}}} = \frac{3\pi}{2}, \label{eq:alpha_marginal}$$ which is in good agreement with the numerical calculations at large $\beta_0$ and without viscosity (respectively dashed and dotted line in Figure \[fig:alpha\]). Using this value of $\alpha/{\mathcal{P}}$ in equation (\[eq:app\_br1\]), one finally obtains the slope of the radial magnetic field profile: $$b_{r1} = \frac{3}{2}\left(\frac{b_{rs}}{\zeta_B} + D_B \right),$$ which reproduces the larger slope observed in numerical profiles in Section \[sec:diffusion\]. Note that equation (\[eq:app\_br1\]) indicates that relaxing the marginal stability hypothesis would change the slope of the radial magnetic field profile to a different factor than $3/2$. It even shows that if $\alpha/{\mathcal{P}}< \pi/2$ the slope and hence the transport rate would change sign ! This is not believed to be physical, and is probably an artefact of considering a stationary flow that is unstable to the MRI. This assertion is supported by the fact that the value $\alpha/{\mathcal{P}}=\pi/2$ corresponds to the largest-scale MRI mode *symmetric* about the midplane being marginally stable (this can be obtained using the two-zone model). At still lower values of $\alpha$, the numerical profiles show multiple bending of the field lines, which is also an artefact due to the fact that the flow is unstable to the MRI [@ogilvie01].
Appendix B : two-zone model with more precise boundary conditions {#sec:appendix_B .unnumbered}
=================================================================
Here we assume that the boundary conditions found in Appendix A can be generalised to non-vanishing magnetic Prandtl number in the following way: $$\begin{aligned}
\Delta b_r &=& - \frac{2}{3}\frac{\Delta u_\phi}{\zeta_B}, \label{eq:jump_br_heuristic} \\
\Delta b_\phi &=& 0.\end{aligned}$$ These boundary conditions are the same as equations (\[eq:jump\_br\]) and (\[eq:jump\_bphi\]) with the value of $\alpha/{\mathcal{P}}$ given by the marginal stability analysis (equation \[eq:alpha\_marginal\]). This heuristic generalisation is motivated by the numerical results, which show that the jump in radial magnetic field is independent of the magnetic Prandtl number (see Figure \[fig:cstalpha\_brs\_scaled\_pm\]). This feature is well reproduced by the above boundary conditions. Another way of looking at this generalisation is to assume a more general form inspired by equations (\[eq:jump\_br\]) and (\[eq:jump\_bphi\]): $$\begin{aligned}
\Delta b_r &=& - \frac{{\mathcal{P}}A}{\alpha}\frac{\Delta u_\phi}{\zeta_B}, \label{eq:jump_br_A}\\
\Delta b_\phi &=& 0,\end{aligned}$$ where $A$ is some unspecified numerical factor which may depend on ${\mathcal{P}}$. Using then a two-zone model to describe the marginal MRI mode leads to the following value: $A=2\alpha/(3{\mathcal{P}})$. Introducing this value into equation (\[eq:jump\_br\_A\]) gives equation (\[eq:jump\_br\_heuristic\]).
We now use the new boundary conditions in conjunction with the two-zone model of Section \[sec:anal\] to construct analytical profiles corresponding to the different source terms. The analytical profiles for the source terms $b_{r{\mathrm{s}}}$ and $D_B$ are then $$\begin{aligned}
b_r(\zeta<\zeta_B) &=& \frac{3}{2}\left(\frac{b_{r{\mathrm{s}}}}{\zeta_B} + D_B \right)\zeta, \\
b_\phi(\zeta<\zeta_B) &=& \frac{3}{8}\frac{{\mathcal{P}}}{\alpha}\left(\frac{b_{r{\mathrm{s}}}}{\zeta_B} +
D_B \right)\zeta(\zeta^2-\zeta_B^2), \\
u_{\psi} &=& \frac{\alpha}{{\mathcal{P}}}\left( \frac{3}{2}
\frac{b_{r{\mathrm{s}}}}{\zeta_B}+ \frac{1}{2}D_B \right) ,
\label{eq:uflux_mag_bis} \\
u_\phi(\zeta>\zeta_B) &=& \frac{3}{4}b_{r{\mathrm{s}}}\left(2|\zeta| - \zeta_B \right) +
\frac{3D_B}{4}\zeta^2.\end{aligned}$$
Those for the source term $b_{\phi{\mathrm{s}}}$ are $$\begin{aligned}
b_r(\zeta<\zeta_B) &=& \frac{\alpha}{{\mathcal{P}}}\frac{b_{\phi{\mathrm{s}}}}{\zeta_B^3}, \\
b_\phi(\zeta<\zeta_B) &=& \frac{b_{\phi{\mathrm{s}}}}{4}\left(3\frac{\zeta}{\zeta_B} + \frac{\zeta^3}{\zeta_B^3}\right), \\
u_{\psi} &=& \frac{\alpha^2}{{\mathcal{P}}^2} \frac{b_{\phi{\mathrm{s}}}}{\zeta_B^2} , \label{eq:uflux_bphis} \\
u_\phi(\zeta>\zeta_B) &=& \frac{3}{2}\frac{\alpha}{{\mathcal{P}}}\frac{b_{\phi{\mathrm{s}}}}{\zeta_B}.\end{aligned}$$
And finally the analytical profiles for the hydrodynamic source terms are the following: $$\begin{aligned}
b_r &=& \frac{{\mathcal{P}}}{\alpha}\frac{u_{r2}}{30}\zeta(12\zeta_B^2-10\zeta^2) - \frac{2}{3}u_{\phi2}\zeta, \\
b_\phi &=& \frac{{\mathcal{P}}}{\alpha}(\zeta_B^2-\zeta^2)\left\lbrack\frac{2}{3}u_{\phi2}\zeta +
\frac{u_{r2}}{120}\frac{{\mathcal{P}}}{\alpha}\zeta(3\zeta^2-9\zeta_B^2) \right\rbrack. \\
u_{\psi} &=& u_{r0} + \frac{12}{30}u_{r2}\zeta_B^2 - \frac{2}{3}\frac{\alpha}{{\mathcal{P}}}u_{\phi2}, \label{eq:uflux_hydro} \\
u_\phi(\zeta>\zeta_B) &=& u_{\phi0} + \frac{{\mathcal{P}}}{\alpha}\frac{u_{r2}}{10}\zeta_B^4.\end{aligned}$$
From these, one can deduce the inclination of the field lines at the surface of the disc, when the diffusion is compensated by advection due to effective viscosity (here for $D_H=1$, and $D_{\nu\Sigma}=0$): $$\frac{B_r}{B_z} = \frac{2}{3} \frac{H}{r}{\mathcal{P}}\zeta_B\left\lbrack -\frac{1}{2}-\frac{1}{6{\mathcal{P}}} + \frac{2}{4\alpha^2+1}\left((4-\frac{2}{3{\mathcal{P}}})\alpha^2+\frac{12\zeta_B^2}{30} \right) \right\rbrack,$$ as well as the ratio of the radial to azimuthal components of the surface magnetic field, when the diffusion is compensated by the transport due to an outflow: $$\frac{B_{r{\mathrm{s}}}}{B_{\phi{\mathrm{s}}}} = - \frac{u_{b\phi{\mathrm{s}}}}{u_{br{\mathrm{s}}}} = -\frac{2}{3}\frac{\alpha}{{\mathcal{P}}}\frac{1}{\zeta_B^2}.
\label{eq:inc_bphis}$$
\[lastpage\]
[^1]: Although this means that the linear operator $\mathbf{L}$ is singular and possesses an antisymmetric null eigenfunction, equation (\[lx\]) still has a unique solution that is symmetric about the midplane, because the operator is invertible within the symmetric subspace.
[^2]: Note that this does not mean that the angular momentum transport and hence the advection of mass is primarily due to the outflow, since the advection speed of mass and magnetic flux can be very different.
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